Subgroups — C₂×C₆₇×#2993 #30739511

Order 17152.

1926 conjugacy classes of subgroups.

The subgroup lattice has 518 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#529×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#2993	C₂	1	16	yes	maximal, direct product factor
3	C₂×#446×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₈×Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#436×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#436×C₆₇	C₂	1	2	yes	maximal, complemented
7	C₂×#435×C₆₇	C₂	1	2	yes	maximal, complemented
8	C₂×C₈×D₄×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#433×C₆₇	C₂	1	2	yes	maximal, complemented
10	C₂×#431×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×#424×C₆₇	C₂	1	2	yes	maximal, complemented
12	C₂×#423×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂²×M₁₆×C₆₇	C₂²	1	2	yes	—
14	C₂²×M₁₆×C₆₇	C₂²	1	2	yes	—
15	C₂³×C₈×C₆₇	C₂²	1	2	yes	—
16	#529×C₆₇	C₄	1	8	yes	—
17	#529×C₆₇	C₂²	1	8	yes	—
18	C₂×#126×C₆₇	C₄	1	2	yes	—
19	C₂×#125×C₆₇	C₄	1	1	yes	characteristic
20	C₂×#124×C₆₇	C₄	1	1	yes	characteristic
21	C₂×#123×C₆₇	C₄	1	2	yes	—
22	C₂×C₂²⋊Q₈×C₆₇	C₄	1	1	yes	characteristic
23	C₂×#121×C₆₇	C₄	1	1	yes	characteristic
24	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
25	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
26	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
27	C₂×#117×C₆₇	C₂²	1	2	yes	—
28	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
29	#446×C₆₇	C₂²	1	8	yes	complemented
30	C₈×Q₈×C₆₇	C₂²	1	8	yes	complemented
31	#436×C₆₇	C₂²	1	8	yes	complemented
32	#436×C₆₇	C₂²	1	16	yes	complemented
33	#435×C₆₇	C₂²	1	16	yes	complemented
34	C₈×D₄×C₆₇	C₂²	1	8	yes	complemented
35	#433×C₆₇	C₂²	1	16	yes	complemented
36	#431×C₆₇	C₂²	1	8	yes	complemented
37	#424×C₆₇	C₂²	1	16	yes	complemented
38	#423×C₆₇	C₂²	1	8	yes	complemented
39	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
40	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
41	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
42	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
43	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
44	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
45	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
46	C₂×C₂²⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
47	C₂×C₂²⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
48	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
49	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
50	C₂×#97×C₆₇	C₂²	1	2	yes	—
51	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
52	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
53	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
54	C₂²×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
55	C₂²×D₄×C₆₇	C₂×C₄	1	1	yes	characteristic
56	C₂²×D₄×C₆₇	C₂×C₄	1	2	yes	—
57	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
58	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
59	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
60	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
61	C₂×M₁₆×C₆₇	—	4	4	no	—
62	C₂×M₁₆×C₆₇	—	4	4	no	—
63	C₂×M₁₆×C₆₇	C₂³	1	8	yes	—
64	C₂×M₁₆×C₆₇	C₂³	1	8	yes	—
65	C₂²×C₈×C₆₇	—	4	4	no	—
66	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
67	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
68	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
69	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
70	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
71	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
72	C₂²×C₈×C₆₇	C₂³	1	8	yes	—
73	#126×C₆₇	C₂×C₄	1	16	yes	—
74	#125×C₆₇	C₂×C₄	1	8	yes	—
75	#124×C₆₇	C₂×C₄	1	8	yes	—
76	#123×C₆₇	C₂×C₄	1	16	yes	—
77	C₂²⋊Q₈×C₆₇	C₂×C₄	1	8	yes	—
78	#121×C₆₇	C₂×C₄	1	8	yes	—
79	C₄×Q₈×C₆₇	C₂×C₄	1	4	yes	—
80	C₄×Q₈×C₆₇	C₂³	1	4	yes	—
81	C₄×D₄×C₆₇	C₂×C₄	1	4	yes	—
82	C₄×D₄×C₆₇	C₂×C₄	1	8	yes	—
83	C₄×D₄×C₆₇	C₂³	1	4	yes	—
84	C₄×D₄×C₆₇	C₂³	1	8	yes	—
85	#117×C₆₇	C₂×C₄	1	8	yes	—
86	#117×C₆₇	C₂³	1	8	yes	—
87	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
88	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
89	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
90	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
91	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
92	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
93	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
94	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
95	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
96	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
97	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
98	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
99	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
100	C₂×C₄²×C₆₇	C₂×C₄	1	4	yes	—
101	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
102	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
103	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
104	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
105	C₄⋊C₈×C₆₇	C₂³	1	8	yes	—
106	C₄⋊C₈×C₆₇	C₂³	1	4	yes	—
107	C₄⋊C₈×C₆₇	C₂³	1	4	yes	—
108	C₄⋊C₈×C₆₇	C₂³	1	4	yes	—
109	C₄⋊C₈×C₆₇	C₂³	1	8	yes	—
110	C₄⋊C₈×C₆₇	C₂³	1	8	yes	—
111	C₄⋊C₈×C₆₇	C₂³	1	4	yes	—
112	C₂²⋊C₈×C₆₇	C₂³	1	4	yes	—
113	C₂²⋊C₈×C₆₇	C₂³	1	4	yes	—
114	C₂²⋊C₈×C₆₇	C₂³	1	8	yes	—
115	C₂²⋊C₈×C₆₇	C₂³	1	8	yes	—
116	#97×C₆₇	C₂³	1	8	yes	—
117	C₄×C₈×C₆₇	C₂³	1	4	yes	—
118	C₄×C₈×C₆₇	C₂³	1	4	yes	—
119	C₄×C₈×C₆₇	C₂³	1	8	yes	—
120	C₂⁴×C₆₇	C₂²×C₄	1	2	yes	—
121	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
122	C₂×Q₈×C₆₇	—	2	4	no	—
123	C₂×Q₈×C₆₇	C₂²×C₄	1	4	yes	—
124	C₂×D₄×C₆₇	—	2	4	no	—
125	C₂×D₄×C₆₇	—	4	4	no	—
126	C₂×D₄×C₆₇	C₂²×C₄	1	4	yes	—
127	C₂×D₄×C₆₇	C₂²×C₄	1	8	yes	—
128	C₂²×C₄×C₆₇	—	2	4	no	—
129	C₂²×C₄×C₆₇	—	2	4	no	—
130	C₂²×C₄×C₆₇	—	2	2	no	—
131	C₂²×C₄×C₆₇	—	2	2	no	—
132	C₂²×C₄×C₆₇	—	2	4	no	—
133	C₂²×C₄×C₆₇	—	4	2	no	—
134	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
135	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
136	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
137	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
138	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
139	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
140	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
141	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
142	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
143	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
144	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
145	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
146	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
147	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
148	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
149	M₁₆×C₆₇	—	4	8	no	—
150	M₁₆×C₆₇	—	4	8	no	—
151	C₂×C₈×C₆₇	—	2	2	no	—
152	C₂×C₈×C₆₇	—	2	4	no	—
153	C₂×C₈×C₆₇	—	2	4	no	—
154	C₂×C₈×C₆₇	—	2	2	no	—
155	C₂×C₈×C₆₇	—	2	2	no	—
156	C₂×C₈×C₆₇	—	2	2	no	—
157	C₂×C₈×C₆₇	—	4	8	no	—
158	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
159	C₂×C₈×C₆₇	C₂⁴	1	4	yes	—
160	C₂×C₈×C₆₇	C₂⁴	1	4	yes	—
161	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
162	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
163	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
164	C₄⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
165	C₄⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
166	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
167	C₄⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
168	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
169	C₄⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
170	C₂²⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
171	C₂²⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
172	C₂²⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
173	C₂²⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
174	C₂²⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
175	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
176	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
177	C₄²×C₆₇	C₂²×C₄	1	4	yes	—
178	C₄²×C₆₇	C₂²×C₄	1	4	yes	—
179	C₄²×C₆₇	C₂²×C₄	1	4	yes	—
180	C₄²×C₆₇	C₂⁴	1	2	yes	—
181	C₄²×C₆₇	C₂⁴	1	2	yes	—
182	C₄²×C₆₇	C₂⁴	1	4	yes	—
183	C₂³×C₆₇	—	2	4	no	—
184	C₂³×C₆₇	—	2	2	no	—
185	C₂³×C₆₇	—	2	2	no	—
186	C₂³×C₆₇	—	2	4	no	—
187	C₂³×C₆₇	—	2	2	no	—
188	C₂³×C₆₇	—	2	4	no	—
189	C₂³×C₆₇	C₂³×C₄	1	4	yes	—
190	C₂³×C₆₇	C₂³×C₄	1	2	yes	—
191	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
192	Q₈×C₆₇	—	2	8	no	—
193	D₄×C₆₇	—	2	8	no	—
194	D₄×C₆₇	—	4	8	no	—
195	C₂×C₄×C₆₇	—	2	4	no	—
196	C₂×C₄×C₆₇	—	2	2	no	—
197	C₂×C₄×C₆₇	—	2	2	no	—
198	C₂×C₄×C₆₇	—	2	4	no	—
199	C₂×C₄×C₆₇	—	2	2	no	—
200	C₂×C₄×C₆₇	—	2	2	no	—
201	C₂×C₄×C₆₇	—	2	4	no	—
202	C₂×C₄×C₆₇	—	2	8	no	—
203	C₂×C₄×C₆₇	—	2	4	no	—
204	C₂×C₄×C₆₇	—	2	8	no	—
205	C₂×C₄×C₆₇	—	2	4	no	—
206	C₂×C₄×C₆₇	—	2	2	no	—
207	C₂×C₄×C₆₇	—	2	2	no	—
208	C₂×C₄×C₆₇	—	2	4	no	—
209	C₂×C₄×C₆₇	—	4	4	no	—
210	C₂×C₄×C₆₇	#131	1	4	yes	—
211	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
212	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
213	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
214	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
215	C₂×C₄×C₆₇	C₂³×C₄	1	4	yes	—
216	C₂×C₄×C₆₇	C₂³×C₄	1	4	yes	—
217	C₂×C₄×C₆₇	C₂³×C₄	1	4	yes	—
218	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
219	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
220	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
221	C₂×C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
222	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
223	C₈×C₆₇	—	2	2	no	—
224	C₈×C₆₇	—	2	4	no	—
225	C₈×C₆₇	—	2	4	no	—
226	C₈×C₆₇	—	2	2	no	—
227	C₈×C₆₇	—	2	2	no	—
228	C₈×C₆₇	—	2	2	no	—
229	C₂²×C₆₇	—	2	4	no	—
230	C₂²×C₆₇	—	2	2	no	—
231	C₂²×C₆₇	—	2	2	no	—
232	C₂²×C₆₇	—	2	4	no	—
233	C₂²×C₆₇	—	2	2	no	—
234	C₂²×C₆₇	—	2	4	no	—
235	C₂²×C₆₇	—	4	4	no	—
236	C₂²×C₆₇	—	4	8	no	—
237	C₂²×C₆₇	#520	1	2	yes	—
238	C₂²×C₆₇	C₂×#131	1	2	yes	—
239	C₂²×C₆₇	#568	1	2	yes	—
240	C₂²×C₆₇	C₂⁴×C₄	1	1	yes	characteristic
241	C₄×C₆₇	—	2	4	no	—
242	C₄×C₆₇	—	2	2	no	—
243	C₄×C₆₇	—	2	2	no	—
244	C₄×C₆₇	—	2	4	no	—
245	C₄×C₆₇	—	2	2	no	—
246	C₄×C₆₇	—	2	2	no	—
247	C₄×C₆₇	—	2	4	no	—
248	C₄×C₆₇	#520	1	2	yes	—
249	C₄×C₆₇	C₂×#131	1	4	yes	—
250	C₄×C₆₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
251	C₄×C₆₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
252	C₂×#2993	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
253	C₂×C₆₇	—	4	2	no	—
254	C₂×C₆₇	—	4	4	no	—
255	C₂×C₆₇	#2993	1	4	yes	direct product factor
256	C₂×C₆₇	C₂×#520	1	1	yes	characteristic
257	C₂×C₆₇	C₂²×#131	1	1	yes	characteristic
258	C₂×C₆₇	C₂×#568	1	1	yes	characteristic
259	C₂×#529	C₂×C₆₇	1	1	yes	characteristic
260	#2993	C₂×C₆₇	1	16	yes	direct product factor
261	C₂×#446	C₂×C₆₇	1	1	yes	characteristic, complemented
262	C₂×C₈×Q₈	C₂×C₆₇	1	1	yes	characteristic, complemented
263	C₂×#436	C₂×C₆₇	1	1	yes	characteristic, complemented
264	C₂×#436	C₂×C₆₇	1	2	yes	complemented
265	C₂×#435	C₂×C₆₇	1	2	yes	complemented
266	C₂×C₈×D₄	C₂×C₆₇	1	1	yes	characteristic, complemented
267	C₂×#433	C₂×C₆₇	1	2	yes	complemented
268	C₂×#431	C₂×C₆₇	1	1	yes	characteristic, complemented
269	C₂×#424	C₂×C₆₇	1	2	yes	complemented
270	C₂×#423	C₂×C₆₇	1	1	yes	characteristic, complemented
271	C₆₇	C₂×#2993	1	1	yes	characteristic, direct product factor, Sylow
272	C₂²×M₁₆	C₂²×C₆₇	1	2	yes	—
273	C₂²×M₁₆	C₂²×C₆₇	1	2	yes	—
274	C₂³×C₈	C₂²×C₆₇	1	2	yes	—
275	#529	C₄×C₆₇	1	8	yes	—
276	#529	C₂²×C₆₇	1	8	yes	—
277	C₂×#126	C₄×C₆₇	1	2	yes	—
278	C₂×#125	C₄×C₆₇	1	1	yes	characteristic
279	C₂×#124	C₄×C₆₇	1	1	yes	characteristic
280	C₂×#123	C₄×C₆₇	1	2	yes	—
281	C₂×C₂²⋊Q₈	C₄×C₆₇	1	1	yes	characteristic
282	C₂×#121	C₄×C₆₇	1	1	yes	characteristic
283	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
284	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
285	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
286	C₂×#117	C₂²×C₆₇	1	2	yes	—
287	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
288	#446	C₂²×C₆₇	1	8	yes	complemented
289	C₈×Q₈	C₂²×C₆₇	1	8	yes	complemented
290	#436	C₂²×C₆₇	1	8	yes	complemented
291	#436	C₂²×C₆₇	1	16	yes	complemented
292	#435	C₂²×C₆₇	1	16	yes	complemented
293	C₈×D₄	C₂²×C₆₇	1	8	yes	complemented
294	#433	C₂²×C₆₇	1	16	yes	complemented
295	#431	C₂²×C₆₇	1	8	yes	complemented
296	#424	C₂²×C₆₇	1	16	yes	complemented
297	#423	C₂²×C₆₇	1	8	yes	complemented
298	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
299	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
300	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
301	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
302	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
303	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
304	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
305	C₂×C₂²⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
306	C₂×C₂²⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
307	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
308	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
309	C₂×#97	C₂²×C₆₇	1	2	yes	—
310	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
311	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
312	C₂×C₄×C₈	C₂²×C₆₇	1	2	yes	—
313	C₂²×Q₈	C₂×C₄×C₆₇	1	1	yes	characteristic
314	C₂²×D₄	C₂×C₄×C₆₇	1	1	yes	characteristic
315	C₂²×D₄	C₂×C₄×C₆₇	1	2	yes	—
316	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
317	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
318	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
319	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
320	C₂×M₁₆	—	4	4	no	—
321	C₂×M₁₆	—	4	4	no	—
322	C₂×M₁₆	C₂³×C₆₇	1	8	yes	—
323	C₂×M₁₆	C₂³×C₆₇	1	8	yes	—
324	C₂²×C₈	—	4	4	no	—
325	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
326	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
327	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
328	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
329	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
330	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
331	C₂²×C₈	C₂³×C₆₇	1	8	yes	—
332	#126	C₂×C₄×C₆₇	1	16	yes	—
333	#125	C₂×C₄×C₆₇	1	8	yes	—
334	#124	C₂×C₄×C₆₇	1	8	yes	—
335	#123	C₂×C₄×C₆₇	1	16	yes	—
336	C₂²⋊Q₈	C₂×C₄×C₆₇	1	8	yes	—
337	#121	C₂×C₄×C₆₇	1	8	yes	—
338	C₄×Q₈	C₂×C₄×C₆₇	1	4	yes	—
339	C₄×Q₈	C₂³×C₆₇	1	4	yes	—
340	C₄×D₄	C₂×C₄×C₆₇	1	4	yes	—
341	C₄×D₄	C₂×C₄×C₆₇	1	8	yes	—
342	C₄×D₄	C₂³×C₆₇	1	4	yes	—
343	C₄×D₄	C₂³×C₆₇	1	8	yes	—
344	#117	C₂×C₄×C₆₇	1	8	yes	—
345	#117	C₂³×C₆₇	1	8	yes	—
346	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
347	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
348	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
349	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
350	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
351	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
352	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
353	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
354	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
355	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
356	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
357	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
358	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
359	C₂×C₄²	C₂×C₄×C₆₇	1	4	yes	—
360	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
361	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
362	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
363	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
364	C₄⋊C₈	C₂³×C₆₇	1	8	yes	—
365	C₄⋊C₈	C₂³×C₆₇	1	4	yes	—
366	C₄⋊C₈	C₂³×C₆₇	1	4	yes	—
367	C₄⋊C₈	C₂³×C₆₇	1	4	yes	—
368	C₄⋊C₈	C₂³×C₆₇	1	8	yes	—
369	C₄⋊C₈	C₂³×C₆₇	1	8	yes	—
370	C₄⋊C₈	C₂³×C₆₇	1	4	yes	—
371	C₂²⋊C₈	C₂³×C₆₇	1	4	yes	—
372	C₂²⋊C₈	C₂³×C₆₇	1	4	yes	—
373	C₂²⋊C₈	C₂³×C₆₇	1	8	yes	—
374	C₂²⋊C₈	C₂³×C₆₇	1	8	yes	—
375	#97	C₂³×C₆₇	1	8	yes	—
376	C₄×C₈	C₂³×C₆₇	1	4	yes	—
377	C₄×C₈	C₂³×C₆₇	1	4	yes	—
378	C₄×C₈	C₂³×C₆₇	1	8	yes	—
379	C₂⁴	C₂²×C₄×C₆₇	1	2	yes	—
380	C₂⁴	C₂²×C₄×C₆₇	1	1	yes	characteristic
381	C₂×Q₈	—	2	4	no	—
382	C₂×Q₈	C₂²×C₄×C₆₇	1	4	yes	—
383	C₂×D₄	—	2	4	no	—
384	C₂×D₄	—	4	4	no	—
385	C₂×D₄	C₂²×C₄×C₆₇	1	4	yes	—
386	C₂×D₄	C₂²×C₄×C₆₇	1	8	yes	—
387	C₂²×C₄	—	2	4	no	—
388	C₂²×C₄	—	2	4	no	—
389	C₂²×C₄	—	2	2	no	—
390	C₂²×C₄	—	2	4	no	—
391	C₂²×C₄	—	2	2	no	—
392	C₂²×C₄	—	4	2	no	—
393	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
394	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
395	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
396	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
397	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
398	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
399	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
400	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
401	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
402	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
403	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
404	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
405	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
406	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
407	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
408	M₁₆	—	4	8	no	—
409	M₁₆	—	4	8	no	—
410	C₂×C₈	—	2	2	no	—
411	C₂×C₈	—	2	4	no	—
412	C₂×C₈	—	2	4	no	—
413	C₂×C₈	—	2	2	no	—
414	C₂×C₈	—	2	2	no	—
415	C₂×C₈	—	2	2	no	—
416	C₂×C₈	—	4	8	no	—
417	C₂×C₈	C₂⁴×C₆₇	1	2	yes	—
418	C₂×C₈	C₂⁴×C₆₇	1	2	yes	—
419	C₂×C₈	C₂⁴×C₆₇	1	2	yes	—
420	C₂×C₈	C₂⁴×C₆₇	1	2	yes	—
421	C₂×C₈	C₂⁴×C₆₇	1	4	yes	—
422	C₂×C₈	C₂⁴×C₆₇	1	4	yes	—
423	C₄⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
424	C₄⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
425	C₄⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
426	C₄⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
427	C₄⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
428	C₄⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
429	C₂²⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
430	C₂²⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
431	C₂²⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
432	C₂²⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
433	C₂²⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
434	C₄²	C₂²×C₄×C₆₇	1	2	yes	—
435	C₄²	C₂²×C₄×C₆₇	1	2	yes	—
436	C₄²	C₂²×C₄×C₆₇	1	4	yes	—
437	C₄²	C₂²×C₄×C₆₇	1	4	yes	—
438	C₄²	C₂²×C₄×C₆₇	1	4	yes	—
439	C₄²	C₂⁴×C₆₇	1	2	yes	—
440	C₄²	C₂⁴×C₆₇	1	4	yes	—
441	C₄²	C₂⁴×C₆₇	1	2	yes	—
442	C₂³	—	2	4	no	—
443	C₂³	—	2	2	no	—
444	C₂³	—	2	2	no	—
445	C₂³	—	2	4	no	—
446	C₂³	—	2	2	no	—
447	C₂³	—	2	4	no	—
448	C₂³	C₂³×C₄×C₆₇	1	4	yes	—
449	C₂³	C₂³×C₄×C₆₇	1	2	yes	—
450	C₂³	C₂³×C₄×C₆₇	1	1	yes	characteristic
451	Q₈	—	2	8	no	—
452	D₄	—	2	8	no	—
453	D₄	—	4	8	no	—
454	C₂×C₄	—	2	4	no	—
455	C₂×C₄	—	2	2	no	—
456	C₂×C₄	—	2	2	no	—
457	C₂×C₄	—	2	4	no	—
458	C₂×C₄	—	2	2	no	—
459	C₂×C₄	—	2	2	no	—
460	C₂×C₄	—	2	4	no	—
461	C₂×C₄	—	2	8	no	—
462	C₂×C₄	—	2	4	no	—
463	C₂×C₄	—	2	8	no	—
464	C₂×C₄	—	2	4	no	—
465	C₂×C₄	—	2	2	no	—
466	C₂×C₄	—	2	2	no	—
467	C₂×C₄	—	2	4	no	—
468	C₂×C₄	—	4	4	no	—
469	C₂×C₄	#131×C₆₇	1	4	yes	—
470	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
471	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
472	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
473	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
474	C₂×C₄	C₂³×C₄×C₆₇	1	4	yes	—
475	C₂×C₄	C₂³×C₄×C₆₇	1	4	yes	—
476	C₂×C₄	C₂³×C₄×C₆₇	1	4	yes	—
477	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
478	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
479	C₂×C₄	H(5,𝔽₂)×C₆₇	1	2	yes	—
480	C₂×C₄	2₋¹⁺⁴×C₆₇	1	2	yes	—
481	C₂×C₄	C₂⁵×C₆₇	1	1	yes	characteristic
482	C₈	—	2	2	no	—
483	C₈	—	2	4	no	—
484	C₈	—	2	2	no	—
485	C₈	—	2	4	no	—
486	C₈	—	2	2	no	—
487	C₈	—	2	2	no	—
488	C₂²	—	2	2	no	—
489	C₂²	—	2	4	no	—
490	C₂²	—	2	2	no	—
491	C₂²	—	2	4	no	—
492	C₂²	—	2	4	no	—
493	C₂²	—	2	2	no	—
494	C₂²	—	4	4	no	—
495	C₂²	—	4	8	no	—
496	C₂²	#520×C₆₇	1	2	yes	—
497	C₂²	C₂×#131×C₆₇	1	2	yes	—
498	C₂²	#568×C₆₇	1	2	yes	—
499	C₂²	C₂⁴×C₄×C₆₇	1	1	yes	characteristic
500	C₄	—	2	2	no	—
501	C₄	—	2	4	no	—
502	C₄	—	2	2	no	—
503	C₄	—	2	2	no	—
504	C₄	—	2	4	no	—
505	C₄	—	2	4	no	—
506	C₄	—	2	2	no	—
507	C₄	#520×C₆₇	1	2	yes	—
508	C₄	C₂×#131×C₆₇	1	4	yes	—
509	C₄	C₂×H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
510	C₄	C₂×2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
511	C₂	—	4	2	no	—
512	C₂	—	4	4	no	—
513	C₂	C₆₇×#2993	1	4	yes	direct product factor
514	C₂	C₂×#520×C₆₇	1	1	yes	characteristic
515	C₂	C₂²×#131×C₆₇	1	1	yes	characteristic
516	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×#121×C₆₇	C₄
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×#2993	C₂	direct (N × H)
3. C₂×#446×C₆₇	C₂	semidirect
4. C₂×C₈×Q₈×C₆₇	C₂	semidirect
5. C₂×#436×C₆₇	C₂	semidirect
6. C₂×#436×C₆₇	C₂	semidirect
7. C₂×#435×C₆₇	C₂	semidirect
8. C₂×C₈×D₄×C₆₇	C₂	semidirect
9. C₂×#433×C₆₇	C₂	semidirect
10. C₂×#431×C₆₇	C₂	semidirect
11. C₂×#424×C₆₇	C₂	semidirect
12. C₂×#423×C₆₇	C₂	semidirect
29. #446×C₆₇	C₂²	semidirect
30. C₈×Q₈×C₆₇	C₂²	semidirect
31. #436×C₆₇	C₂²	semidirect
32. #436×C₆₇	C₂²	semidirect
33. #435×C₆₇	C₂²	semidirect
34. C₈×D₄×C₆₇	C₂²	semidirect
35. #433×C₆₇	C₂²	semidirect
36. #431×C₆₇	C₂²	semidirect
37. #424×C₆₇	C₂²	semidirect
38. #423×C₆₇	C₂²	semidirect
252. C₂×#2993	C₆₇	direct (N × H)
255. C₂×C₆₇	#2993	direct (N × H)
260. #2993	C₂×C₆₇	direct (N × H)
261. C₂×#446	C₂×C₆₇	semidirect
262. C₂×C₈×Q₈	C₂×C₆₇	semidirect
263. C₂×#436	C₂×C₆₇	semidirect
264. C₂×#436	C₂×C₆₇	semidirect
265. C₂×#435	C₂×C₆₇	semidirect
266. C₂×C₈×D₄	C₂×C₆₇	semidirect
267. C₂×#433	C₂×C₆₇	semidirect
268. C₂×#431	C₂×C₆₇	semidirect
269. C₂×#424	C₂×C₆₇	semidirect
270. C₂×#423	C₂×C₆₇	semidirect
271. C₆₇	C₂×#2993	direct (N × H)
288. #446	C₂²×C₆₇	semidirect
289. C₈×Q₈	C₂²×C₆₇	semidirect
290. #436	C₂²×C₆₇	semidirect
291. #436	C₂²×C₆₇	semidirect
292. #435	C₂²×C₆₇	semidirect
293. C₈×D₄	C₂²×C₆₇	semidirect
294. #433	C₂²×C₆₇	semidirect
295. #431	C₂²×C₆₇	semidirect
296. #424	C₂²×C₆₇	semidirect
297. #423	C₂²×C₆₇	semidirect
513. C₂	C₆₇×#2993	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×#529×C₆₇	2
C₆₇×#2993	2
C₂×#446×C₆₇	2
C₂×C₈×Q₈×C₆₇	2
C₂×#436×C₆₇	2
C₂×#435×C₆₇	2
C₂×C₈×D₄×C₆₇	2
C₂×#433×C₆₇	2
C₂×#431×C₆₇	2
C₂×#424×C₆₇	2
C₂×#423×C₆₇	2
C₂×#2993	67

Subgroups — C2×C67×#2993 #30739511

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₇×#2993 #30739511