Subgroups — C₂×C₆₇×#2976 #30739475

Order 17152.

2306 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#525×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#2976	C₂	1	16	yes	maximal, direct product factor
3	C₂×#436×C₆₇	C₂	1	2	yes	maximal, complemented
4	C₂×#435×C₆₇	C₂	1	2	yes	maximal, complemented
5	C₂×#435×C₆₇	C₂	1	2	yes	maximal, complemented
6	C₂×C₈×D₄×C₆₇	C₂	1	2	yes	maximal, complemented
7	C₂×#433×C₆₇	C₂	1	2	yes	maximal, complemented
8	C₂×#408×C₆₇	C₂	1	2	yes	maximal, complemented
9	C₂×#407×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×C₂²⋊C₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂²×#41×C₆₇	C₂²	1	1	yes	characteristic
12	C₂⁴×C₄×C₆₇	C₂²	1	1	yes	characteristic
13	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
14	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
15	C₂²×M₁₆×C₆₇	C₂²	1	2	yes	complemented
16	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
17	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
18	C₂³×C₈×C₆₇	C₂²	1	2	yes	complemented
19	#525×C₆₇	C₄	1	8	yes	—
20	#525×C₆₇	C₂²	1	8	yes	—
21	C₂×#123×C₆₇	C₄	1	2	yes	—
22	C₂×C₂²⋊Q₈×C₆₇	C₄	1	1	yes	characteristic
23	C₂×C₂²⋊Q₈×C₆₇	C₄	1	1	yes	characteristic
24	C₂×#121×C₆₇	C₄	1	1	yes	characteristic
25	C₂×#121×C₆₇	C₄	1	1	yes	characteristic
26	C₂×#120×C₆₇	C₄	1	2	yes	—
27	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
28	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
29	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
30	#436×C₆₇	C₂²	1	16	yes	complemented
31	#435×C₆₇	C₂²	1	16	yes	complemented
32	#435×C₆₇	C₂²	1	16	yes	complemented
33	C₈×D₄×C₆₇	C₂²	1	16	yes	complemented
34	#433×C₆₇	C₂²	1	16	yes	complemented
35	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	complemented
36	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	complemented
37	#408×C₆₇	C₂²	1	16	yes	complemented
38	#407×C₆₇	C₂²	1	8	yes	complemented
39	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
40	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
41	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
42	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
43	C₂×C₂²⋊C₈×C₆₇	C₂²	1	8	yes	complemented
44	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	complemented
45	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	complemented
46	C₂×#97×C₆₇	C₂²	1	2	yes	complemented
47	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	complemented
48	C₂⁵×C₆₇	C₂×C₄	1	1	yes	characteristic
49	C₂×#41×C₆₇	—	4	2	no	—
50	C₂×#41×C₆₇	—	4	2	no	—
51	C₂×#41×C₆₇	C₂×C₄	1	4	yes	—
52	C₂×#41×C₆₇	C₂³	1	4	yes	—
53	C₂²×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
54	C₂²×D₄×C₆₇	C₂×C₄	1	2	yes	—
55	C₂²×D₄×C₆₇	C₂×C₄	1	2	yes	—
56	C₂²×D₄×C₆₇	C₂×C₄	1	2	yes	—
57	C₂²×D₄×C₆₇	C₂×C₄	1	1	yes	characteristic
58	C₂³×C₄×C₆₇	—	4	2	no	—
59	C₂³×C₄×C₆₇	—	4	2	no	—
60	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
61	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
62	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
63	C₂³×C₄×C₆₇	C₂×C₄	1	4	yes	—
64	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
65	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
66	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
67	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
68	C₂³×C₄×C₆₇	C₂³	1	4	yes	—
69	C₂×M₁₆×C₆₇	—	4	2	no	—
70	C₂×M₁₆×C₆₇	—	4	2	no	—
71	C₂×M₁₆×C₆₇	—	4	4	no	—
72	C₂×M₁₆×C₆₇	C₂³	1	4	yes	—
73	C₂×M₁₆×C₆₇	C₂³	1	4	yes	—
74	C₂×M₁₆×C₆₇	C₂³	1	8	yes	complemented
75	C₂²×C₈×C₆₇	—	4	2	no	—
76	C₂²×C₈×C₆₇	—	4	2	no	—
77	C₂²×C₈×C₆₇	—	4	4	no	—
78	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
79	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
80	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
81	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
82	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
83	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
84	C₂²×C₈×C₆₇	C₂³	1	8	yes	complemented
85	#123×C₆₇	C₂×C₄	1	16	yes	—
86	C₂²⋊Q₈×C₆₇	C₂×C₄	1	8	yes	—
87	C₂²⋊Q₈×C₆₇	C₂×C₄	1	8	yes	—
88	#121×C₆₇	C₂×C₄	1	8	yes	—
89	#121×C₆₇	C₂×C₄	1	8	yes	—
90	#120×C₆₇	C₂×C₄	1	16	yes	—
91	C₄×D₄×C₆₇	C₂×C₄	1	8	yes	—
92	C₄×D₄×C₆₇	C₂×C₄	1	8	yes	—
93	C₄×D₄×C₆₇	C₂³	1	8	yes	—
94	C₄×D₄×C₆₇	C₂³	1	8	yes	—
95	#117×C₆₇	C₂×C₄	1	4	yes	—
96	#117×C₆₇	C₂³	1	4	yes	—
97	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
98	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
99	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
100	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
101	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
102	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
103	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
104	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
105	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
106	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
107	C₄⋊C₈×C₆₇	C₂³	1	8	yes	complemented
108	C₄⋊C₈×C₆₇	C₂³	1	8	yes	complemented
109	C₂²⋊C₈×C₆₇	C₂³	1	8	yes	—
110	C₂²⋊C₈×C₆₇	C₂³	1	8	yes	—
111	C₂²⋊C₈×C₆₇	C₂³	1	8	yes	—
112	C₂²⋊C₈×C₆₇	C₂³	1	8	yes	—
113	C₂²⋊C₈×C₆₇	C₂³	1	8	yes	complemented
114	C₂²⋊C₈×C₆₇	C₂³	1	8	yes	complemented
115	#97×C₆₇	C₂³	1	8	yes	complemented
116	C₄×C₈×C₆₇	C₂³	1	8	yes	complemented
117	C₂⁴×C₆₇	—	2	4	no	—
118	C₂⁴×C₆₇	—	4	2	no	—
119	C₂⁴×C₆₇	—	4	2	no	—
120	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
121	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
122	C₂⁴×C₆₇	C₂²×C₄	1	2	yes	—
123	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
124	C₂⁴×C₆₇	C₂²×C₄	1	4	yes	—
125	#41×C₆₇	—	4	8	no	—
126	#41×C₆₇	—	4	4	no	—
127	#41×C₆₇	—	4	4	no	—
128	C₂×Q₈×C₆₇	—	4	2	no	—
129	C₂×Q₈×C₆₇	C₂²×C₄	1	4	yes	—
130	C₂×D₄×C₆₇	—	2	8	no	—
131	C₂×D₄×C₆₇	—	4	4	no	—
132	C₂×D₄×C₆₇	—	4	2	no	—
133	C₂×D₄×C₆₇	—	4	4	no	—
134	C₂×D₄×C₆₇	C₂²×C₄	1	8	yes	—
135	C₂×D₄×C₆₇	C₂²×C₄	1	8	yes	—
136	C₂×D₄×C₆₇	C₂²×C₄	1	8	yes	—
137	C₂×D₄×C₆₇	C₂²×C₄	1	4	yes	—
138	C₂²×C₄×C₆₇	—	2	2	no	—
139	C₂²×C₄×C₆₇	—	2	2	no	—
140	C₂²×C₄×C₆₇	—	2	4	no	—
141	C₂²×C₄×C₆₇	—	2	2	no	—
142	C₂²×C₄×C₆₇	—	2	2	no	—
143	C₂²×C₄×C₆₇	—	2	4	no	—
144	C₂²×C₄×C₆₇	—	2	2	no	—
145	C₂²×C₄×C₆₇	—	2	2	no	—
146	C₂²×C₄×C₆₇	—	4	8	no	—
147	C₂²×C₄×C₆₇	—	4	4	no	—
148	C₂²×C₄×C₆₇	—	4	4	no	—
149	C₂²×C₄×C₆₇	—	4	2	no	—
150	C₂²×C₄×C₆₇	—	4	2	no	—
151	C₂²×C₄×C₆₇	—	4	2	no	—
152	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
153	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
154	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
155	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
156	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
157	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
158	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
159	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
160	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
161	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
162	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
163	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
164	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
165	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
166	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
167	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
168	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
169	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
170	M₁₆×C₆₇	—	4	4	no	—
171	M₁₆×C₆₇	—	4	4	no	—
172	M₁₆×C₆₇	—	4	8	no	—
173	C₂×C₈×C₆₇	—	2	4	no	—
174	C₂×C₈×C₆₇	—	2	4	no	—
175	C₂×C₈×C₆₇	—	2	4	no	—
176	C₂×C₈×C₆₇	—	2	4	no	—
177	C₂×C₈×C₆₇	—	4	4	no	—
178	C₂×C₈×C₆₇	—	4	4	no	—
179	C₂×C₈×C₆₇	—	4	8	no	—
180	C₂×C₈×C₆₇	C₂⁴	1	4	yes	—
181	C₂×C₈×C₆₇	C₂⁴	1	4	yes	—
182	C₂×C₈×C₆₇	C₂⁴	1	4	yes	—
183	C₂×C₈×C₆₇	C₂⁴	1	4	yes	—
184	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
185	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
186	C₄⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
187	C₄⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
188	C₂²⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
189	C₂²⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
190	C₂²⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
191	C₂²⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
192	C₂²⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
193	C₄²×C₆₇	C₂²×C₄	1	4	yes	—
194	C₄²×C₆₇	C₂⁴	1	4	yes	—
195	C₂³×C₆₇	—	2	2	no	—
196	C₂³×C₆₇	—	2	2	no	—
197	C₂³×C₆₇	—	2	2	no	—
198	C₂³×C₆₇	—	2	2	no	—
199	C₂³×C₆₇	—	2	4	no	—
200	C₂³×C₆₇	—	2	4	no	—
201	C₂³×C₆₇	—	2	8	no	—
202	C₂³×C₆₇	—	2	2	no	—
203	C₂³×C₆₇	—	2	2	no	—
204	C₂³×C₆₇	—	2	2	no	—
205	C₂³×C₆₇	—	2	4	no	—
206	C₂³×C₆₇	—	2	2	no	—
207	C₂³×C₆₇	—	4	4	no	—
208	C₂³×C₆₇	—	4	4	no	—
209	C₂³×C₆₇	—	4	16	no	—
210	C₂³×C₆₇	#131	1	4	yes	—
211	C₂³×C₆₇	C₂³×C₄	1	2	yes	—
212	C₂³×C₆₇	C₂³×C₄	1	2	yes	—
213	C₂³×C₆₇	C₂³×C₄	1	4	yes	—
214	C₂³×C₆₇	C₂³×C₄	1	2	yes	—
215	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
216	Q₈×C₆₇	—	4	4	no	—
217	D₄×C₆₇	—	2	16	no	—
218	D₄×C₆₇	—	4	8	no	—
219	D₄×C₆₇	—	4	4	no	—
220	D₄×C₆₇	—	4	8	no	—
221	C₂×C₄×C₆₇	—	2	2	no	—
222	C₂×C₄×C₆₇	—	2	2	no	—
223	C₂×C₄×C₆₇	—	2	2	no	—
224	C₂×C₄×C₆₇	—	2	4	no	—
225	C₂×C₄×C₆₇	—	2	4	no	—
226	C₂×C₄×C₆₇	—	2	4	no	—
227	C₂×C₄×C₆₇	—	2	4	no	—
228	C₂×C₄×C₆₇	—	2	4	no	—
229	C₂×C₄×C₆₇	—	2	4	no	—
230	C₂×C₄×C₆₇	—	2	8	no	—
231	C₂×C₄×C₆₇	—	2	2	no	—
232	C₂×C₄×C₆₇	—	2	2	no	—
233	C₂×C₄×C₆₇	—	2	4	no	—
234	C₂×C₄×C₆₇	—	2	2	no	—
235	C₂×C₄×C₆₇	—	2	2	no	—
236	C₂×C₄×C₆₇	—	2	4	no	—
237	C₂×C₄×C₆₇	—	2	2	no	—
238	C₂×C₄×C₆₇	—	2	2	no	—
239	C₂×C₄×C₆₇	—	4	4	no	—
240	C₂×C₄×C₆₇	—	4	4	no	—
241	C₂×C₄×C₆₇	—	4	4	no	—
242	C₂×C₄×C₆₇	C₂³×C₄	1	4	yes	—
243	C₂×C₄×C₆₇	C₂³×C₄	1	4	yes	—
244	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
245	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
246	C₂×C₄×C₆₇	C₂³×C₄	1	4	yes	—
247	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
248	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
249	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
250	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
251	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
252	C₈×C₆₇	—	2	4	no	—
253	C₈×C₆₇	—	2	4	no	—
254	C₈×C₆₇	—	2	4	no	—
255	C₈×C₆₇	—	2	4	no	—
256	C₂²×C₆₇	—	2	2	no	—
257	C₂²×C₆₇	—	2	8	no	—
258	C₂²×C₆₇	—	2	2	no	—
259	C₂²×C₆₇	—	2	2	no	—
260	C₂²×C₆₇	—	2	2	no	—
261	C₂²×C₆₇	—	2	4	no	—
262	C₂²×C₆₇	—	2	4	no	—
263	C₂²×C₆₇	—	2	2	no	—
264	C₂²×C₆₇	—	2	2	no	—
265	C₂²×C₆₇	—	2	2	no	—
266	C₂²×C₆₇	—	2	4	no	—
267	C₂²×C₆₇	—	2	2	no	—
268	C₂²×C₆₇	—	4	16	no	—
269	C₂²×C₆₇	—	4	8	no	—
270	C₂²×C₆₇	—	4	4	no	—
271	C₂²×C₆₇	—	4	4	no	—
272	C₂²×C₆₇	#518	1	2	yes	—
273	C₂²×C₆₇	C₂×#131	1	4	yes	—
274	C₂²×C₆₇	C₂×#131	1	2	yes	—
275	C₂²×C₆₇	#568	1	2	yes	—
276	C₂²×C₆₇	C₂⁴×C₄	1	1	yes	characteristic
277	C₄×C₆₇	—	2	2	no	—
278	C₄×C₆₇	—	2	2	no	—
279	C₄×C₆₇	—	2	2	no	—
280	C₄×C₆₇	—	2	4	no	—
281	C₄×C₆₇	—	2	4	no	—
282	C₄×C₆₇	—	2	4	no	—
283	C₄×C₆₇	#518	1	2	yes	—
284	C₄×C₆₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
285	C₄×C₆₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
286	C₂×#2976	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
287	C₂×C₆₇	—	2	4	no	—
288	C₂×C₆₇	—	4	4	no	—
289	C₂×C₆₇	—	4	2	no	—
290	C₂×C₆₇	—	4	2	no	—
291	C₂×C₆₇	#2976	1	4	yes	direct product factor
292	C₂×C₆₇	C₂×#518	1	1	yes	characteristic
293	C₂×C₆₇	C₂²×#131	1	1	yes	characteristic
294	C₂×C₆₇	C₂×#568	1	1	yes	characteristic
295	C₂×#525	C₂×C₆₇	1	1	yes	characteristic
296	#2976	C₂×C₆₇	1	16	yes	direct product factor
297	C₂×#436	C₂×C₆₇	1	2	yes	complemented
298	C₂×#435	C₂×C₆₇	1	2	yes	complemented
299	C₂×#435	C₂×C₆₇	1	2	yes	complemented
300	C₂×C₈×D₄	C₂×C₆₇	1	2	yes	complemented
301	C₂×#433	C₂×C₆₇	1	2	yes	complemented
302	C₂×#408	C₂×C₆₇	1	2	yes	complemented
303	C₂×#407	C₂×C₆₇	1	1	yes	characteristic, complemented
304	C₂²×C₂²⋊C₈	C₂×C₆₇	1	1	yes	characteristic, complemented
305	C₆₇	C₂×#2976	1	1	yes	characteristic, direct product factor, Sylow
306	C₂²×#41	C₂²×C₆₇	1	1	yes	characteristic
307	C₂⁴×C₄	C₂²×C₆₇	1	1	yes	characteristic
308	C₂²×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
309	C₂²×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
310	C₂²×M₁₆	C₂²×C₆₇	1	2	yes	complemented
311	C₂³×C₈	C₂²×C₆₇	1	1	yes	characteristic
312	C₂³×C₈	C₂²×C₆₇	1	1	yes	characteristic
313	C₂³×C₈	C₂²×C₆₇	1	2	yes	complemented
314	#525	C₄×C₆₇	1	8	yes	—
315	#525	C₂²×C₆₇	1	8	yes	—
316	C₂×#123	C₄×C₆₇	1	2	yes	—
317	C₂×C₂²⋊Q₈	C₄×C₆₇	1	1	yes	characteristic
318	C₂×C₂²⋊Q₈	C₄×C₆₇	1	1	yes	characteristic
319	C₂×#121	C₄×C₆₇	1	1	yes	characteristic
320	C₂×#121	C₄×C₆₇	1	1	yes	characteristic
321	C₂×#120	C₄×C₆₇	1	2	yes	—
322	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
323	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
324	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
325	#436	C₂²×C₆₇	1	16	yes	complemented
326	#435	C₂²×C₆₇	1	16	yes	complemented
327	#435	C₂²×C₆₇	1	16	yes	complemented
328	C₈×D₄	C₂²×C₆₇	1	16	yes	complemented
329	#433	C₂²×C₆₇	1	16	yes	complemented
330	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	complemented
331	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	complemented
332	#408	C₂²×C₆₇	1	16	yes	complemented
333	#407	C₂²×C₆₇	1	8	yes	complemented
334	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
335	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
336	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
337	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
338	C₂×C₂²⋊C₈	C₂²×C₆₇	1	8	yes	complemented
339	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	complemented
340	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	complemented
341	C₂×#97	C₂²×C₆₇	1	2	yes	complemented
342	C₂×C₄×C₈	C₂²×C₆₇	1	2	yes	complemented
343	C₂⁵	C₂×C₄×C₆₇	1	1	yes	characteristic
344	C₂×#41	—	4	2	no	—
345	C₂×#41	—	4	2	no	—
346	C₂×#41	C₂×C₄×C₆₇	1	4	yes	—
347	C₂×#41	C₂³×C₆₇	1	4	yes	—
348	C₂²×Q₈	C₂×C₄×C₆₇	1	1	yes	characteristic
349	C₂²×D₄	C₂×C₄×C₆₇	1	2	yes	—
350	C₂²×D₄	C₂×C₄×C₆₇	1	2	yes	—
351	C₂²×D₄	C₂×C₄×C₆₇	1	2	yes	—
352	C₂²×D₄	C₂×C₄×C₆₇	1	1	yes	characteristic
353	C₂³×C₄	—	4	2	no	—
354	C₂³×C₄	—	4	2	no	—
355	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
356	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
357	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
358	C₂³×C₄	C₂×C₄×C₆₇	1	4	yes	—
359	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
360	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
361	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
362	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
363	C₂³×C₄	C₂³×C₆₇	1	4	yes	—
364	C₂×M₁₆	—	4	2	no	—
365	C₂×M₁₆	—	4	2	no	—
366	C₂×M₁₆	—	4	4	no	—
367	C₂×M₁₆	C₂³×C₆₇	1	4	yes	—
368	C₂×M₁₆	C₂³×C₆₇	1	4	yes	—
369	C₂×M₁₆	C₂³×C₆₇	1	8	yes	complemented
370	C₂²×C₈	—	4	2	no	—
371	C₂²×C₈	—	4	2	no	—
372	C₂²×C₈	—	4	4	no	—
373	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
374	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
375	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
376	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
377	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
378	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
379	C₂²×C₈	C₂³×C₆₇	1	8	yes	complemented
380	#123	C₂×C₄×C₆₇	1	16	yes	—
381	C₂²⋊Q₈	C₂×C₄×C₆₇	1	8	yes	—
382	C₂²⋊Q₈	C₂×C₄×C₆₇	1	8	yes	—
383	#121	C₂×C₄×C₆₇	1	8	yes	—
384	#121	C₂×C₄×C₆₇	1	8	yes	—
385	#120	C₂×C₄×C₆₇	1	16	yes	—
386	C₄×D₄	C₂×C₄×C₆₇	1	8	yes	—
387	C₄×D₄	C₂×C₄×C₆₇	1	8	yes	—
388	C₄×D₄	C₂³×C₆₇	1	8	yes	—
389	C₄×D₄	C₂³×C₆₇	1	8	yes	—
390	#117	C₂×C₄×C₆₇	1	4	yes	—
391	#117	C₂³×C₆₇	1	4	yes	—
392	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
393	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
394	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
395	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
396	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
397	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
398	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
399	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
400	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
401	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
402	C₄⋊C₈	C₂³×C₆₇	1	8	yes	complemented
403	C₄⋊C₈	C₂³×C₆₇	1	8	yes	complemented
404	C₂²⋊C₈	C₂³×C₆₇	1	8	yes	—
405	C₂²⋊C₈	C₂³×C₆₇	1	8	yes	—
406	C₂²⋊C₈	C₂³×C₆₇	1	8	yes	—
407	C₂²⋊C₈	C₂³×C₆₇	1	8	yes	—
408	C₂²⋊C₈	C₂³×C₆₇	1	8	yes	complemented
409	C₂²⋊C₈	C₂³×C₆₇	1	8	yes	complemented
410	#97	C₂³×C₆₇	1	8	yes	complemented
411	C₄×C₈	C₂³×C₆₇	1	8	yes	complemented
412	C₂⁴	—	2	4	no	—
413	C₂⁴	—	4	2	no	—
414	C₂⁴	—	4	2	no	—
415	C₂⁴	C₂²×C₄×C₆₇	1	1	yes	characteristic
416	C₂⁴	C₂²×C₄×C₆₇	1	1	yes	characteristic
417	C₂⁴	C₂²×C₄×C₆₇	1	2	yes	—
418	C₂⁴	C₂²×C₄×C₆₇	1	1	yes	characteristic
419	C₂⁴	C₂²×C₄×C₆₇	1	4	yes	—
420	#41	—	4	8	no	—
421	#41	—	4	4	no	—
422	#41	—	4	4	no	—
423	C₂×Q₈	—	4	2	no	—
424	C₂×Q₈	C₂²×C₄×C₆₇	1	4	yes	—
425	C₂×D₄	—	2	8	no	—
426	C₂×D₄	—	4	2	no	—
427	C₂×D₄	—	4	4	no	—
428	C₂×D₄	—	4	4	no	—
429	C₂×D₄	C₂²×C₄×C₆₇	1	8	yes	—
430	C₂×D₄	C₂²×C₄×C₆₇	1	8	yes	—
431	C₂×D₄	C₂²×C₄×C₆₇	1	8	yes	—
432	C₂×D₄	C₂²×C₄×C₆₇	1	4	yes	—
433	C₂²×C₄	—	2	2	no	—
434	C₂²×C₄	—	2	2	no	—
435	C₂²×C₄	—	2	4	no	—
436	C₂²×C₄	—	2	2	no	—
437	C₂²×C₄	—	2	2	no	—
438	C₂²×C₄	—	2	4	no	—
439	C₂²×C₄	—	2	2	no	—
440	C₂²×C₄	—	2	2	no	—
441	C₂²×C₄	—	4	8	no	—
442	C₂²×C₄	—	4	4	no	—
443	C₂²×C₄	—	4	4	no	—
444	C₂²×C₄	—	4	2	no	—
445	C₂²×C₄	—	4	2	no	—
446	C₂²×C₄	—	4	2	no	—
447	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
448	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
449	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
450	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
451	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
452	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
453	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
454	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
455	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
456	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
457	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
458	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
459	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
460	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
461	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
462	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
463	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
464	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
465	M₁₆	—	4	4	no	—
466	M₁₆	—	4	4	no	—
467	M₁₆	—	4	8	no	—
468	C₂×C₈	—	2	4	no	—
469	C₂×C₈	—	2	4	no	—
470	C₂×C₈	—	2	4	no	—
471	C₂×C₈	—	2	4	no	—
472	C₂×C₈	—	4	4	no	—
473	C₂×C₈	—	4	4	no	—
474	C₂×C₈	—	4	8	no	—
475	C₂×C₈	C₂⁴×C₆₇	1	4	yes	—
476	C₂×C₈	C₂⁴×C₆₇	1	4	yes	—
477	C₂×C₈	C₂⁴×C₆₇	1	4	yes	—
478	C₂×C₈	C₂⁴×C₆₇	1	4	yes	—
479	C₄⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
480	C₄⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
481	C₄⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
482	C₄⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
483	C₂²⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
484	C₂²⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
485	C₂²⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
486	C₂²⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
487	C₂²⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
488	C₄²	C₂²×C₄×C₆₇	1	4	yes	—
489	C₄²	C₂⁴×C₆₇	1	4	yes	—
490	C₂³	—	2	2	no	—
491	C₂³	—	2	2	no	—
492	C₂³	—	2	2	no	—
493	C₂³	—	2	2	no	—
494	C₂³	—	2	4	no	—
495	C₂³	—	2	4	no	—
496	C₂³	—	2	8	no	—
497	C₂³	—	2	2	no	—
498	C₂³	—	2	2	no	—
499	C₂³	—	2	2	no	—
500	C₂³	—	2	4	no	—
501	C₂³	—	2	2	no	—
502	C₂³	—	4	4	no	—
503	C₂³	—	4	4	no	—
504	C₂³	—	4	16	no	—
505	C₂³	#131×C₆₇	1	4	yes	—
506	C₂³	C₂³×C₄×C₆₇	1	2	yes	—
507	C₂³	C₂³×C₄×C₆₇	1	2	yes	—
508	C₂³	C₂³×C₄×C₆₇	1	4	yes	—
509	C₂³	C₂³×C₄×C₆₇	1	2	yes	—
510	C₂³	C₂³×C₄×C₆₇	1	1	yes	characteristic
511	Q₈	—	4	4	no	—
512	D₄	—	2	16	no	—
513	D₄	—	4	4	no	—
514	D₄	—	4	8	no	—
515	D₄	—	4	8	no	—
516	C₂×C₄	—	2	2	no	—
517	C₂×C₄	—	2	2	no	—
518	C₂×C₄	—	2	2	no	—
519	C₂×C₄	—	2	4	no	—
520	C₂×C₄	—	2	4	no	—
521	C₂×C₄	—	2	4	no	—
522	C₂×C₄	—	2	4	no	—
523	C₂×C₄	—	2	4	no	—
524	C₂×C₄	—	2	4	no	—
525	C₂×C₄	—	2	8	no	—
526	C₂×C₄	—	2	2	no	—
527	C₂×C₄	—	2	2	no	—
528	C₂×C₄	—	2	4	no	—
529	C₂×C₄	—	2	2	no	—
530	C₂×C₄	—	2	2	no	—
531	C₂×C₄	—	2	4	no	—
532	C₂×C₄	—	2	2	no	—
533	C₂×C₄	—	2	2	no	—
534	C₂×C₄	—	4	4	no	—
535	C₂×C₄	—	4	4	no	—
536	C₂×C₄	—	4	4	no	—
537	C₂×C₄	C₂³×C₄×C₆₇	1	4	yes	—
538	C₂×C₄	C₂³×C₄×C₆₇	1	4	yes	—
539	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
540	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
541	C₂×C₄	C₂³×C₄×C₆₇	1	4	yes	—
542	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
543	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
544	C₂×C₄	H(5,𝔽₂)×C₆₇	1	2	yes	—
545	C₂×C₄	H(5,𝔽₂)×C₆₇	1	2	yes	—
546	C₂×C₄	C₂⁵×C₆₇	1	1	yes	characteristic
547	C₈	—	2	4	no	—
548	C₈	—	2	4	no	—
549	C₈	—	2	4	no	—
550	C₈	—	2	4	no	—
551	C₂²	—	2	2	no	—
552	C₂²	—	2	2	no	—
553	C₂²	—	2	8	no	—
554	C₂²	—	2	4	no	—
555	C₂²	—	2	2	no	—
556	C₂²	—	2	2	no	—
557	C₂²	—	2	2	no	—
558	C₂²	—	2	4	no	—
559	C₂²	—	2	2	no	—
560	C₂²	—	2	2	no	—
561	C₂²	—	2	2	no	—
562	C₂²	—	2	4	no	—
563	C₂²	—	4	16	no	—
564	C₂²	—	4	4	no	—
565	C₂²	—	4	4	no	—
566	C₂²	—	4	8	no	—
567	C₂²	#518×C₆₇	1	2	yes	—
568	C₂²	C₂×#131×C₆₇	1	4	yes	—
569	C₂²	C₂×#131×C₆₇	1	2	yes	—
570	C₂²	#568×C₆₇	1	2	yes	—
571	C₂²	C₂⁴×C₄×C₆₇	1	1	yes	characteristic
572	C₄	—	2	4	no	—
573	C₄	—	2	4	no	—
574	C₄	—	2	2	no	—
575	C₄	—	2	2	no	—
576	C₄	—	2	2	no	—
577	C₄	—	2	4	no	—
578	C₄	#518×C₆₇	1	2	yes	—
579	C₄	C₂×H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
580	C₄	C₂×H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
581	C₂	—	2	4	no	—
582	C₂	—	4	2	no	—
583	C₂	—	4	2	no	—
584	C₂	—	4	4	no	—
585	C₂	C₆₇×#2976	1	4	yes	direct product factor
586	C₂	C₂×#518×C₆₇	1	1	yes	characteristic
587	C₂	C₂²×#131×C₆₇	1	1	yes	characteristic
588	C₂	C₂×#568×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₆₇	C₂⁴
commutator	C₂²	C₂⁴×C₄×C₆₇
Frattini, cosocle	C₂×C₄	C₂⁵×C₆₇
socle	C₂³×C₆₇	C₂³×C₄
prime radical	C₂×#525×C₆₇	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₆₇ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. C₆₇×#2976	C₂	direct (N × H)
3. C₂×#436×C₆₇	C₂	semidirect
4. C₂×#435×C₆₇	C₂	semidirect
5. C₂×#435×C₆₇	C₂	semidirect
6. C₂×C₈×D₄×C₆₇	C₂	semidirect
7. C₂×#433×C₆₇	C₂	semidirect
8. C₂×#408×C₆₇	C₂	semidirect
9. C₂×#407×C₆₇	C₂	semidirect
10. C₂²×C₂²⋊C₈×C₆₇	C₂	semidirect
15. C₂²×M₁₆×C₆₇	C₂²	semidirect
18. C₂³×C₈×C₆₇	C₂²	semidirect
30. #436×C₆₇	C₂²	semidirect
31. #435×C₆₇	C₂²	semidirect
32. #435×C₆₇	C₂²	semidirect
33. C₈×D₄×C₆₇	C₂²	semidirect
34. #433×C₆₇	C₂²	semidirect
35. C₂×C₄⋊C₈×C₆₇	C₂²	semidirect
36. C₂×C₄⋊C₈×C₆₇	C₂²	semidirect
37. #408×C₆₇	C₂²	semidirect
38. #407×C₆₇	C₂²	semidirect
43. C₂×C₂²⋊C₈×C₆₇	C₂²	semidirect
44. C₂×C₂²⋊C₈×C₆₇	C₂²	semidirect
45. C₂×C₂²⋊C₈×C₆₇	C₂²	semidirect
46. C₂×#97×C₆₇	C₂²	semidirect
47. C₂×C₄×C₈×C₆₇	C₂²	semidirect
74. C₂×M₁₆×C₆₇	C₂³	semidirect
84. C₂²×C₈×C₆₇	C₂³	semidirect
107. C₄⋊C₈×C₆₇	C₂³	semidirect
108. C₄⋊C₈×C₆₇	C₂³	semidirect
113. C₂²⋊C₈×C₆₇	C₂³	semidirect
114. C₂²⋊C₈×C₆₇	C₂³	semidirect
115. #97×C₆₇	C₂³	semidirect
116. C₄×C₈×C₆₇	C₂³	semidirect
286. C₂×#2976	C₆₇	direct (N × H)
291. C₂×C₆₇	#2976	direct (N × H)
296. #2976	C₂×C₆₇	direct (N × H)
297. C₂×#436	C₂×C₆₇	semidirect
298. C₂×#435	C₂×C₆₇	semidirect
299. C₂×#435	C₂×C₆₇	semidirect
300. C₂×C₈×D₄	C₂×C₆₇	semidirect
301. C₂×#433	C₂×C₆₇	semidirect
302. C₂×#408	C₂×C₆₇	semidirect
303. C₂×#407	C₂×C₆₇	semidirect
304. C₂²×C₂²⋊C₈	C₂×C₆₇	semidirect
305. C₆₇	C₂×#2976	direct (N × H)
310. C₂²×M₁₆	C₂²×C₆₇	semidirect
313. C₂³×C₈	C₂²×C₆₇	semidirect
325. #436	C₂²×C₆₇	semidirect
326. #435	C₂²×C₆₇	semidirect
327. #435	C₂²×C₆₇	semidirect
328. C₈×D₄	C₂²×C₆₇	semidirect
329. #433	C₂²×C₆₇	semidirect
330. C₂×C₄⋊C₈	C₂²×C₆₇	semidirect
331. C₂×C₄⋊C₈	C₂²×C₆₇	semidirect
332. #408	C₂²×C₆₇	semidirect
333. #407	C₂²×C₆₇	semidirect
338. C₂×C₂²⋊C₈	C₂²×C₆₇	semidirect
339. C₂×C₂²⋊C₈	C₂²×C₆₇	semidirect
340. C₂×C₂²⋊C₈	C₂²×C₆₇	semidirect
341. C₂×#97	C₂²×C₆₇	semidirect
342. C₂×C₄×C₈	C₂²×C₆₇	semidirect
369. C₂×M₁₆	C₂³×C₆₇	semidirect
379. C₂²×C₈	C₂³×C₆₇	semidirect
402. C₄⋊C₈	C₂³×C₆₇	semidirect
403. C₄⋊C₈	C₂³×C₆₇	semidirect
408. C₂²⋊C₈	C₂³×C₆₇	semidirect
409. C₂²⋊C₈	C₂³×C₆₇	semidirect
410. #97	C₂³×C₆₇	semidirect
411. C₄×C₈	C₂³×C₆₇	semidirect
585. C₂	C₆₇×#2976	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#525×C₆₇	2
C₆₇×#2976	2
C₂×#436×C₆₇	2
C₂×#435×C₆₇	2
C₂×C₈×D₄×C₆₇	2
C₂×#433×C₆₇	2
C₂×#408×C₆₇	2
C₂×#407×C₆₇	2
C₂²×C₂²⋊C₈×C₆₇	2
C₂×#2976	67

Subgroups — C2×C67×#2976 #30739475

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₇×#2976 #30739475