Subgroups — C₂×C₆₇×#2974 #30739473

Order 17152.

2654 conjugacy classes of subgroups.

The subgroup lattice has 494 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×D₄×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#2974	C₂	1	16	yes	maximal, direct product factor
3	C₂×#436×C₆₇	C₂	1	2	yes	maximal, complemented
4	C₂×#435×C₆₇	C₂	1	4	yes	maximal, complemented
5	C₂×C₈×D₄×C₆₇	C₂	1	2	yes	maximal, complemented
6	C₂×#432×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#431×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#407×C₆₇	C₂	1	2	yes	maximal, complemented
9	C₂²×C₂²⋊C₈×C₆₇	C₂	1	2	yes	maximal, complemented
10	C₂³×D₄×C₆₇	C₄	1	1	yes	characteristic
11	C₂⁴×C₄×C₆₇	C₂²	1	2	yes	—
12	C₂²×M₁₆×C₆₇	C₂²	1	4	yes	complemented
13	C₂³×C₈×C₆₇	C₂²	1	4	yes	complemented
14	C₂×C₄×D₄×C₆₇	C₄	1	2	yes	—
15	C₂×C₄×D₄×C₆₇	C₄	1	8	yes	—
16	C₂×C₄×D₄×C₆₇	C₄	1	1	yes	characteristic
17	C₂×C₄×D₄×C₆₇	C₄	1	1	yes	characteristic
18	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
19	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
20	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
21	C₂×C₄×D₄×C₆₇	C₂²	1	8	yes	—
22	C₂²×C₄⋊C₄×C₆₇	C₄	1	1	yes	characteristic
23	C₂²×C₂²⋊C₄×C₆₇	C₄	1	2	yes	—
24	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
25	#436×C₆₇	C₂²	1	16	yes	complemented
26	#435×C₆₇	C₂²	1	32	yes	complemented
27	C₈×D₄×C₆₇	C₂²	1	16	yes	complemented
28	#432×C₆₇	C₂²	1	8	yes	complemented
29	#431×C₆₇	C₂²	1	8	yes	complemented
30	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	complemented
31	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	complemented
32	#407×C₆₇	C₂²	1	16	yes	complemented
33	C₂×C₂²⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
34	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
35	C₂×C₂²⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
36	C₂×C₂²⋊C₈×C₆₇	C₂²	1	4	yes	complemented
37	C₂×C₂²⋊C₈×C₆₇	C₂²	1	4	yes	complemented
38	C₂×C₂²⋊C₈×C₆₇	C₂²	1	16	yes	complemented
39	C₂×#97×C₆₇	C₂²	1	2	yes	complemented
40	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	complemented
41	C₂⁵×C₆₇	C₂×C₄	1	2	yes	—
42	C₂²×D₄×C₆₇	—	2	8	no	—
43	C₂²×D₄×C₆₇	C₂×C₄	1	2	yes	—
44	C₂²×D₄×C₆₇	C₂×C₄	1	8	yes	—
45	C₂²×D₄×C₆₇	C₂×C₄	1	1	yes	characteristic
46	C₂²×D₄×C₆₇	C₂×C₄	1	1	yes	characteristic
47	C₂³×C₄×C₆₇	—	4	8	no	—
48	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
49	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
50	C₂³×C₄×C₆₇	C₂×C₄	1	2	yes	—
51	C₂³×C₄×C₆₇	C₂×C₄	1	8	yes	—
52	C₂³×C₄×C₆₇	C₂×C₄	1	2	yes	—
53	C₂³×C₄×C₆₇	C₂×C₄	1	2	yes	—
54	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
55	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
56	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
57	C₂³×C₄×C₆₇	C₂³	1	8	yes	—
58	C₂×M₁₆×C₆₇	—	4	8	no	—
59	C₂×M₁₆×C₆₇	C₂³	1	16	yes	complemented
60	C₂²×C₈×C₆₇	—	4	8	no	—
61	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
62	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
63	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
64	C₂²×C₈×C₆₇	C₂³	1	16	yes	complemented
65	C₄×D₄×C₆₇	C₂×C₄	1	8	yes	—
66	C₄×D₄×C₆₇	C₂×C₄	1	4	yes	—
67	C₄×D₄×C₆₇	C₂×C₄	1	4	yes	—
68	C₄×D₄×C₆₇	C₂×C₄	1	16	yes	—
69	C₄×D₄×C₆₇	C₂×C₄	1	8	yes	—
70	C₄×D₄×C₆₇	C₂×C₄	1	8	yes	—
71	C₄×D₄×C₆₇	C₂³	1	4	yes	—
72	C₄×D₄×C₆₇	C₂³	1	8	yes	—
73	C₄×D₄×C₆₇	C₂³	1	4	yes	—
74	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
75	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	8	yes	—
76	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
77	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
78	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
79	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
80	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
81	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
82	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	16	yes	—
83	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
84	C₂×C₄²×C₆₇	C₂×C₄	1	4	yes	—
85	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
86	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
87	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
88	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
89	C₄⋊C₈×C₆₇	C₂³	1	8	yes	complemented
90	C₄⋊C₈×C₆₇	C₂³	1	8	yes	complemented
91	C₂²⋊C₈×C₆₇	C₂³	1	4	yes	—
92	C₂²⋊C₈×C₆₇	C₂³	1	8	yes	—
93	C₂²⋊C₈×C₆₇	C₂³	1	4	yes	—
94	C₂²⋊C₈×C₆₇	C₂³	1	16	yes	complemented
95	C₂²⋊C₈×C₆₇	C₂³	1	16	yes	complemented
96	#97×C₆₇	C₂³	1	8	yes	complemented
97	C₄×C₈×C₆₇	C₂³	1	8	yes	complemented
98	C₂⁴×C₆₇	—	2	8	no	—
99	C₂⁴×C₆₇	—	2	8	no	—
100	C₂⁴×C₆₇	—	4	4	no	—
101	C₂⁴×C₆₇	C₂²×C₄	1	2	yes	—
102	C₂⁴×C₆₇	C₂²×C₄	1	8	yes	—
103	C₂⁴×C₆₇	C₂²×C₄	1	2	yes	—
104	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
105	C₂×D₄×C₆₇	—	2	8	no	—
106	C₂×D₄×C₆₇	—	2	32	no	—
107	C₂×D₄×C₆₇	—	2	4	no	—
108	C₂×D₄×C₆₇	M₁₆	1	8	yes	complemented
109	C₂×D₄×C₆₇	C₂²×C₄	1	8	yes	—
110	C₂×D₄×C₆₇	C₂²×C₄	1	4	yes	—
111	C₂×D₄×C₆₇	C₂²×C₄	1	4	yes	—
112	C₂²×C₄×C₆₇	—	2	4	no	—
113	C₂²×C₄×C₆₇	—	2	4	no	—
114	C₂²×C₄×C₆₇	—	2	8	no	—
115	C₂²×C₄×C₆₇	—	2	8	no	—
116	C₂²×C₄×C₆₇	—	2	4	no	—
117	C₂²×C₄×C₆₇	—	4	16	no	—
118	C₂²×C₄×C₆₇	—	4	16	no	—
119	C₂²×C₄×C₆₇	—	4	4	no	—
120	C₂²×C₄×C₆₇	—	4	4	no	—
121	C₂²×C₄×C₆₇	—	4	4	no	—
122	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
123	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
124	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
125	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
126	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
127	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
128	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
129	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
130	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
131	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
132	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
133	C₂²×C₄×C₆₇	C₂²×C₄	1	8	yes	—
134	C₂²×C₄×C₆₇	C₂²×C₄	1	8	yes	—
135	C₂²×C₄×C₆₇	C₂²×C₄	1	8	yes	—
136	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
137	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
138	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
139	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
140	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
141	M₁₆×C₆₇	—	4	16	no	—
142	C₂×C₈×C₆₇	—	2	4	no	—
143	C₂×C₈×C₆₇	—	2	8	no	—
144	C₂×C₈×C₆₇	—	2	4	no	—
145	C₂×C₈×C₆₇	—	4	16	no	—
146	C₂×C₈×C₆₇	C₂⁴	1	4	yes	—
147	C₂×C₈×C₆₇	C₂⁴	1	8	yes	—
148	C₂×C₈×C₆₇	C₂⁴	1	4	yes	—
149	C₄⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
150	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
151	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
152	C₂²⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
153	C₂²⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
154	C₂²⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
155	C₂²⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
156	C₄²×C₆₇	C₂²×C₄	1	4	yes	—
157	C₄²×C₆₇	C₂²×C₄	1	4	yes	—
158	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
159	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
160	C₄²×C₆₇	C₂⁴	1	2	yes	—
161	C₄²×C₆₇	C₂⁴	1	2	yes	—
162	C₂³×C₆₇	—	2	2	no	—
163	C₂³×C₆₇	—	2	4	no	—
164	C₂³×C₆₇	—	2	2	no	—
165	C₂³×C₆₇	—	2	4	no	—
166	C₂³×C₆₇	—	2	4	no	—
167	C₂³×C₆₇	—	2	16	no	—
168	C₂³×C₆₇	—	2	4	no	—
169	C₂³×C₆₇	—	2	16	no	—
170	C₂³×C₆₇	—	2	4	no	—
171	C₂³×C₆₇	—	4	8	no	—
172	C₂³×C₆₇	—	4	32	no	—
173	C₂³×C₆₇	C₂×M₁₆	1	8	yes	—
174	C₂³×C₆₇	#131	1	4	yes	—
175	C₂³×C₆₇	C₂³×C₄	1	4	yes	—
176	C₂³×C₆₇	C₂³×C₄	1	4	yes	—
177	C₂³×C₆₇	C₂³×C₄	1	2	yes	—
178	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
179	D₄×C₆₇	—	2	16	no	—
180	D₄×C₆₇	—	2	8	no	—
181	D₄×C₆₇	C₂×M₁₆	1	16	yes	complemented
182	C₂×C₄×C₆₇	—	2	2	no	—
183	C₂×C₄×C₆₇	—	2	4	no	—
184	C₂×C₄×C₆₇	—	2	4	no	—
185	C₂×C₄×C₆₇	—	2	2	no	—
186	C₂×C₄×C₆₇	—	2	8	no	—
187	C₂×C₄×C₆₇	—	2	2	no	—
188	C₂×C₄×C₆₇	—	2	2	no	—
189	C₂×C₄×C₆₇	—	2	4	no	—
190	C₂×C₄×C₆₇	—	2	8	no	—
191	C₂×C₄×C₆₇	—	2	8	no	—
192	C₂×C₄×C₆₇	—	2	8	no	—
193	C₂×C₄×C₆₇	—	2	8	no	—
194	C₂×C₄×C₆₇	—	2	8	no	—
195	C₂×C₄×C₆₇	—	2	4	no	—
196	C₂×C₄×C₆₇	—	4	8	no	—
197	C₂×C₄×C₆₇	—	4	8	no	—
198	C₂×C₄×C₆₇	—	4	8	no	—
199	C₂×C₄×C₆₇	C₂×M₁₆	1	4	yes	—
200	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
201	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
202	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
203	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
204	C₂×C₄×C₆₇	C₂³×C₄	1	4	yes	—
205	C₂×C₄×C₆₇	C₂³×C₄	1	4	yes	—
206	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
207	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
208	C₂×C₄×C₆₇	H(5,𝔽₂)	1	4	yes	—
209	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
210	C₈×C₆₇	—	2	4	no	—
211	C₈×C₆₇	—	2	8	no	—
212	C₈×C₆₇	—	2	4	no	—
213	C₂²×C₆₇	—	2	2	no	—
214	C₂²×C₆₇	—	2	4	no	—
215	C₂²×C₆₇	—	2	8	no	—
216	C₂²×C₆₇	—	2	16	no	—
217	C₂²×C₆₇	—	2	2	no	—
218	C₂²×C₆₇	—	2	4	no	—
219	C₂²×C₆₇	—	2	4	no	—
220	C₂²×C₆₇	—	2	4	no	—
221	C₂²×C₆₇	—	2	4	no	—
222	C₂²×C₆₇	—	4	32	no	—
223	C₂²×C₆₇	—	4	8	no	—
224	C₂²×C₆₇	#518	1	2	yes	—
225	C₂²×C₆₇	C₂²×M₁₆	1	8	yes	—
226	C₂²×C₆₇	C₂²×M₁₆	1	2	yes	—
227	C₂²×C₆₇	C₂×#131	1	4	yes	—
228	C₂²×C₆₇	C₂×#131	1	2	yes	—
229	C₂²×C₆₇	C₂⁴×C₄	1	1	yes	characteristic
230	C₄×C₆₇	—	2	2	no	—
231	C₄×C₆₇	—	2	4	no	—
232	C₄×C₆₇	—	2	4	no	—
233	C₄×C₆₇	—	2	2	no	—
234	C₄×C₆₇	—	2	2	no	—
235	C₄×C₆₇	—	2	2	no	—
236	C₄×C₆₇	#518	1	2	yes	—
237	C₄×C₆₇	C₂²×M₁₆	1	4	yes	—
238	C₄×C₆₇	C₂×H(5,𝔽₂)	1	2	yes	—
239	C₂×#2974	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
240	C₂×C₆₇	—	2	4	no	—
241	C₂×C₆₇	—	2	8	no	—
242	C₂×C₆₇	—	4	4	no	—
243	C₂×C₆₇	#2974	1	4	yes	direct product factor
244	C₂×C₆₇	C₂×#518	1	1	yes	characteristic
245	C₂×C₆₇	C₂³×M₁₆	1	1	yes	characteristic
246	C₂×C₆₇	C₂²×#131	1	1	yes	characteristic
247	C₂²×C₄×D₄	C₂×C₆₇	1	1	yes	characteristic
248	#2974	C₂×C₆₇	1	16	yes	direct product factor
249	C₂×#436	C₂×C₆₇	1	2	yes	complemented
250	C₂×#435	C₂×C₆₇	1	4	yes	complemented
251	C₂×C₈×D₄	C₂×C₆₇	1	2	yes	complemented
252	C₂×#432	C₂×C₆₇	1	1	yes	characteristic, complemented
253	C₂×#431	C₂×C₆₇	1	1	yes	characteristic, complemented
254	C₂×#407	C₂×C₆₇	1	2	yes	complemented
255	C₂²×C₂²⋊C₈	C₂×C₆₇	1	2	yes	complemented
256	C₆₇	C₂×#2974	1	1	yes	characteristic, direct product factor, Sylow
257	C₂³×D₄	C₄×C₆₇	1	1	yes	characteristic
258	C₂⁴×C₄	C₂²×C₆₇	1	2	yes	—
259	C₂²×M₁₆	C₂²×C₆₇	1	4	yes	complemented
260	C₂³×C₈	C₂²×C₆₇	1	4	yes	complemented
261	C₂×C₄×D₄	C₄×C₆₇	1	2	yes	—
262	C₂×C₄×D₄	C₄×C₆₇	1	8	yes	—
263	C₂×C₄×D₄	C₄×C₆₇	1	1	yes	characteristic
264	C₂×C₄×D₄	C₄×C₆₇	1	1	yes	characteristic
265	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
266	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
267	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
268	C₂×C₄×D₄	C₂²×C₆₇	1	8	yes	—
269	C₂²×C₄⋊C₄	C₄×C₆₇	1	1	yes	characteristic
270	C₂²×C₂²⋊C₄	C₄×C₆₇	1	2	yes	—
271	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
272	#436	C₂²×C₆₇	1	16	yes	complemented
273	#435	C₂²×C₆₇	1	32	yes	complemented
274	C₈×D₄	C₂²×C₆₇	1	16	yes	complemented
275	#432	C₂²×C₆₇	1	8	yes	complemented
276	#431	C₂²×C₆₇	1	8	yes	complemented
277	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	complemented
278	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	complemented
279	#407	C₂²×C₆₇	1	16	yes	complemented
280	C₂×C₂²⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
281	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
282	C₂×C₂²⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
283	C₂×C₂²⋊C₈	C₂²×C₆₇	1	16	yes	complemented
284	C₂×C₂²⋊C₈	C₂²×C₆₇	1	4	yes	complemented
285	C₂×C₂²⋊C₈	C₂²×C₆₇	1	4	yes	complemented
286	C₂×#97	C₂²×C₆₇	1	2	yes	complemented
287	C₂×C₄×C₈	C₂²×C₆₇	1	2	yes	complemented
288	C₂⁵	C₂×C₄×C₆₇	1	2	yes	—
289	C₂²×D₄	—	2	8	no	—
290	C₂²×D₄	C₂×C₄×C₆₇	1	2	yes	—
291	C₂²×D₄	C₂×C₄×C₆₇	1	8	yes	—
292	C₂²×D₄	C₂×C₄×C₆₇	1	1	yes	characteristic
293	C₂²×D₄	C₂×C₄×C₆₇	1	1	yes	characteristic
294	C₂³×C₄	—	4	8	no	—
295	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
296	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
297	C₂³×C₄	C₂×C₄×C₆₇	1	2	yes	—
298	C₂³×C₄	C₂×C₄×C₆₇	1	8	yes	—
299	C₂³×C₄	C₂×C₄×C₆₇	1	2	yes	—
300	C₂³×C₄	C₂×C₄×C₆₇	1	2	yes	—
301	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
302	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
303	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
304	C₂³×C₄	C₂³×C₆₇	1	8	yes	—
305	C₂×M₁₆	—	4	8	no	—
306	C₂×M₁₆	C₂³×C₆₇	1	16	yes	complemented
307	C₂²×C₈	—	4	8	no	—
308	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
309	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
310	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
311	C₂²×C₈	C₂³×C₆₇	1	16	yes	complemented
312	C₄×D₄	C₂×C₄×C₆₇	1	8	yes	—
313	C₄×D₄	C₂×C₄×C₆₇	1	4	yes	—
314	C₄×D₄	C₂×C₄×C₆₇	1	4	yes	—
315	C₄×D₄	C₂×C₄×C₆₇	1	16	yes	—
316	C₄×D₄	C₂×C₄×C₆₇	1	8	yes	—
317	C₄×D₄	C₂×C₄×C₆₇	1	8	yes	—
318	C₄×D₄	C₂³×C₆₇	1	4	yes	—
319	C₄×D₄	C₂³×C₆₇	1	8	yes	—
320	C₄×D₄	C₂³×C₆₇	1	4	yes	—
321	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
322	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	8	yes	—
323	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
324	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
325	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
326	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
327	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
328	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
329	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	16	yes	—
330	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
331	C₂×C₄²	C₂×C₄×C₆₇	1	4	yes	—
332	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
333	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
334	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
335	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
336	C₄⋊C₈	C₂³×C₆₇	1	8	yes	complemented
337	C₄⋊C₈	C₂³×C₆₇	1	8	yes	complemented
338	C₂²⋊C₈	C₂³×C₆₇	1	4	yes	—
339	C₂²⋊C₈	C₂³×C₆₇	1	8	yes	—
340	C₂²⋊C₈	C₂³×C₆₇	1	4	yes	—
341	C₂²⋊C₈	C₂³×C₆₇	1	16	yes	complemented
342	C₂²⋊C₈	C₂³×C₆₇	1	16	yes	complemented
343	#97	C₂³×C₆₇	1	8	yes	complemented
344	C₄×C₈	C₂³×C₆₇	1	8	yes	complemented
345	C₂⁴	—	2	8	no	—
346	C₂⁴	—	2	8	no	—
347	C₂⁴	—	4	4	no	—
348	C₂⁴	C₂²×C₄×C₆₇	1	2	yes	—
349	C₂⁴	C₂²×C₄×C₆₇	1	8	yes	—
350	C₂⁴	C₂²×C₄×C₆₇	1	2	yes	—
351	C₂⁴	C₂²×C₄×C₆₇	1	1	yes	characteristic
352	C₂×D₄	—	2	8	no	—
353	C₂×D₄	—	2	32	no	—
354	C₂×D₄	—	2	4	no	—
355	C₂×D₄	M₁₆×C₆₇	1	8	yes	complemented
356	C₂×D₄	C₂²×C₄×C₆₇	1	8	yes	—
357	C₂×D₄	C₂²×C₄×C₆₇	1	4	yes	—
358	C₂×D₄	C₂²×C₄×C₆₇	1	4	yes	—
359	C₂²×C₄	—	2	4	no	—
360	C₂²×C₄	—	2	4	no	—
361	C₂²×C₄	—	2	8	no	—
362	C₂²×C₄	—	2	8	no	—
363	C₂²×C₄	—	2	4	no	—
364	C₂²×C₄	—	4	16	no	—
365	C₂²×C₄	—	4	16	no	—
366	C₂²×C₄	—	4	4	no	—
367	C₂²×C₄	—	4	4	no	—
368	C₂²×C₄	—	4	4	no	—
369	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
370	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
371	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
372	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
373	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
374	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
375	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
376	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
377	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
378	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
379	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
380	C₂²×C₄	C₂²×C₄×C₆₇	1	8	yes	—
381	C₂²×C₄	C₂²×C₄×C₆₇	1	8	yes	—
382	C₂²×C₄	C₂²×C₄×C₆₇	1	8	yes	—
383	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
384	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
385	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
386	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
387	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
388	M₁₆	—	4	16	no	—
389	C₂×C₈	—	2	4	no	—
390	C₂×C₈	—	2	8	no	—
391	C₂×C₈	—	2	4	no	—
392	C₂×C₈	—	4	16	no	—
393	C₂×C₈	C₂⁴×C₆₇	1	4	yes	—
394	C₂×C₈	C₂⁴×C₆₇	1	4	yes	—
395	C₂×C₈	C₂⁴×C₆₇	1	8	yes	—
396	C₄⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
397	C₄⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
398	C₄⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
399	C₂²⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
400	C₂²⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
401	C₂²⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
402	C₂²⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
403	C₄²	C₂²×C₄×C₆₇	1	4	yes	—
404	C₄²	C₂²×C₄×C₆₇	1	4	yes	—
405	C₄²	C₂²×C₄×C₆₇	1	2	yes	—
406	C₄²	C₂²×C₄×C₆₇	1	2	yes	—
407	C₄²	C₂⁴×C₆₇	1	2	yes	—
408	C₄²	C₂⁴×C₆₇	1	2	yes	—
409	C₂³	—	2	2	no	—
410	C₂³	—	2	4	no	—
411	C₂³	—	2	2	no	—
412	C₂³	—	2	4	no	—
413	C₂³	—	2	4	no	—
414	C₂³	—	2	16	no	—
415	C₂³	—	2	4	no	—
416	C₂³	—	2	16	no	—
417	C₂³	—	2	4	no	—
418	C₂³	—	4	8	no	—
419	C₂³	—	4	32	no	—
420	C₂³	C₂×M₁₆×C₆₇	1	8	yes	—
421	C₂³	#131×C₆₇	1	4	yes	—
422	C₂³	C₂³×C₄×C₆₇	1	4	yes	—
423	C₂³	C₂³×C₄×C₆₇	1	4	yes	—
424	C₂³	C₂³×C₄×C₆₇	1	2	yes	—
425	C₂³	C₂³×C₄×C₆₇	1	1	yes	characteristic
426	D₄	—	2	16	no	—
427	D₄	—	2	8	no	—
428	D₄	C₂×M₁₆×C₆₇	1	16	yes	complemented
429	C₂×C₄	—	2	2	no	—
430	C₂×C₄	—	2	4	no	—
431	C₂×C₄	—	2	4	no	—
432	C₂×C₄	—	2	2	no	—
433	C₂×C₄	—	2	8	no	—
434	C₂×C₄	—	2	2	no	—
435	C₂×C₄	—	2	2	no	—
436	C₂×C₄	—	2	4	no	—
437	C₂×C₄	—	2	8	no	—
438	C₂×C₄	—	2	8	no	—
439	C₂×C₄	—	2	8	no	—
440	C₂×C₄	—	2	8	no	—
441	C₂×C₄	—	2	8	no	—
442	C₂×C₄	—	2	4	no	—
443	C₂×C₄	—	4	8	no	—
444	C₂×C₄	—	4	8	no	—
445	C₂×C₄	—	4	8	no	—
446	C₂×C₄	C₂×M₁₆×C₆₇	1	4	yes	—
447	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
448	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
449	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
450	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
451	C₂×C₄	C₂³×C₄×C₆₇	1	4	yes	—
452	C₂×C₄	C₂³×C₄×C₆₇	1	4	yes	—
453	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
454	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
455	C₂×C₄	H(5,𝔽₂)×C₆₇	1	4	yes	—
456	C₂×C₄	C₂⁵×C₆₇	1	1	yes	characteristic
457	C₈	—	2	4	no	—
458	C₈	—	2	8	no	—
459	C₈	—	2	4	no	—
460	C₂²	—	2	2	no	—
461	C₂²	—	2	4	no	—
462	C₂²	—	2	2	no	—
463	C₂²	—	2	4	no	—
464	C₂²	—	2	8	no	—
465	C₂²	—	2	16	no	—
466	C₂²	—	2	4	no	—
467	C₂²	—	2	4	no	—
468	C₂²	—	2	4	no	—
469	C₂²	—	4	32	no	—
470	C₂²	—	4	8	no	—
471	C₂²	#518×C₆₇	1	2	yes	—
472	C₂²	C₂²×M₁₆×C₆₇	1	8	yes	—
473	C₂²	C₂²×M₁₆×C₆₇	1	2	yes	—
474	C₂²	C₂×#131×C₆₇	1	4	yes	—
475	C₂²	C₂×#131×C₆₇	1	2	yes	—
476	C₂²	C₂⁴×C₄×C₆₇	1	1	yes	characteristic
477	C₄	—	2	2	no	—
478	C₄	—	2	2	no	—
479	C₄	—	2	2	no	—
480	C₄	—	2	2	no	—
481	C₄	—	2	4	no	—
482	C₄	—	2	4	no	—
483	C₄	#518×C₆₇	1	2	yes	—
484	C₄	C₂²×M₁₆×C₆₇	1	4	yes	—
485	C₄	C₂×H(5,𝔽₂)×C₆₇	1	2	yes	—
486	C₂	—	2	4	no	—
487	C₂	—	2	8	no	—
488	C₂	—	4	4	no	—
489	C₂	C₆₇×#2974	1	4	yes	direct product factor
490	C₂	C₂×#518×C₆₇	1	1	yes	characteristic
491	C₂	C₂³×M₁₆×C₆₇	1	1	yes	characteristic
492	C₂	C₂²×#131×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	C₂³×D₄×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×#2974	C₂	direct (N × H)
3. C₂×#436×C₆₇	C₂	semidirect
4. C₂×#435×C₆₇	C₂	semidirect
5. C₂×C₈×D₄×C₆₇	C₂	semidirect
6. C₂×#432×C₆₇	C₂	semidirect
7. C₂×#431×C₆₇	C₂	semidirect
8. C₂×#407×C₆₇	C₂	semidirect
9. C₂²×C₂²⋊C₈×C₆₇	C₂	semidirect
12. C₂²×M₁₆×C₆₇	C₂²	semidirect
13. C₂³×C₈×C₆₇	C₂²	semidirect
25. #436×C₆₇	C₂²	semidirect
26. #435×C₆₇	C₂²	semidirect
27. C₈×D₄×C₆₇	C₂²	semidirect
28. #432×C₆₇	C₂²	semidirect
29. #431×C₆₇	C₂²	semidirect
30. C₂×C₄⋊C₈×C₆₇	C₂²	semidirect
31. C₂×C₄⋊C₈×C₆₇	C₂²	semidirect
32. #407×C₆₇	C₂²	semidirect
36. C₂×C₂²⋊C₈×C₆₇	C₂²	semidirect
37. C₂×C₂²⋊C₈×C₆₇	C₂²	semidirect
38. C₂×C₂²⋊C₈×C₆₇	C₂²	semidirect
39. C₂×#97×C₆₇	C₂²	semidirect
40. C₂×C₄×C₈×C₆₇	C₂²	semidirect
59. C₂×M₁₆×C₆₇	C₂³	semidirect
64. C₂²×C₈×C₆₇	C₂³	semidirect
89. C₄⋊C₈×C₆₇	C₂³	semidirect
90. C₄⋊C₈×C₆₇	C₂³	semidirect
94. C₂²⋊C₈×C₆₇	C₂³	semidirect
95. C₂²⋊C₈×C₆₇	C₂³	semidirect
96. #97×C₆₇	C₂³	semidirect
97. C₄×C₈×C₆₇	C₂³	semidirect
108. C₂×D₄×C₆₇	M₁₆	semidirect
181. D₄×C₆₇	C₂×M₁₆	semidirect
239. C₂×#2974	C₆₇	direct (N × H)
243. C₂×C₆₇	#2974	direct (N × H)
248. #2974	C₂×C₆₇	direct (N × H)
249. C₂×#436	C₂×C₆₇	semidirect
250. C₂×#435	C₂×C₆₇	semidirect
251. C₂×C₈×D₄	C₂×C₆₇	semidirect
252. C₂×#432	C₂×C₆₇	semidirect
253. C₂×#431	C₂×C₆₇	semidirect
254. C₂×#407	C₂×C₆₇	semidirect
255. C₂²×C₂²⋊C₈	C₂×C₆₇	semidirect
256. C₆₇	C₂×#2974	direct (N × H)
259. C₂²×M₁₆	C₂²×C₆₇	semidirect
260. C₂³×C₈	C₂²×C₆₇	semidirect
272. #436	C₂²×C₆₇	semidirect
273. #435	C₂²×C₆₇	semidirect
274. C₈×D₄	C₂²×C₆₇	semidirect
275. #432	C₂²×C₆₇	semidirect
276. #431	C₂²×C₆₇	semidirect
277. C₂×C₄⋊C₈	C₂²×C₆₇	semidirect
278. C₂×C₄⋊C₈	C₂²×C₆₇	semidirect
279. #407	C₂²×C₆₇	semidirect
283. C₂×C₂²⋊C₈	C₂²×C₆₇	semidirect
284. C₂×C₂²⋊C₈	C₂²×C₆₇	semidirect
285. C₂×C₂²⋊C₈	C₂²×C₆₇	semidirect
286. C₂×#97	C₂²×C₆₇	semidirect
287. C₂×C₄×C₈	C₂²×C₆₇	semidirect
306. C₂×M₁₆	C₂³×C₆₇	semidirect
311. C₂²×C₈	C₂³×C₆₇	semidirect
336. C₄⋊C₈	C₂³×C₆₇	semidirect
337. C₄⋊C₈	C₂³×C₆₇	semidirect
341. C₂²⋊C₈	C₂³×C₆₇	semidirect
342. C₂²⋊C₈	C₂³×C₆₇	semidirect
343. #97	C₂³×C₆₇	semidirect
344. C₄×C₈	C₂³×C₆₇	semidirect
355. C₂×D₄	M₁₆×C₆₇	semidirect
428. D₄	C₂×M₁₆×C₆₇	semidirect
489. C₂	C₆₇×#2974	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₄×D₄×C₆₇	2
C₆₇×#2974	2
C₂×#436×C₆₇	2
C₂×#435×C₆₇	2
C₂×C₈×D₄×C₆₇	2
C₂×#432×C₆₇	2
C₂×#431×C₆₇	2
C₂×#407×C₆₇	2
C₂²×C₂²⋊C₈×C₆₇	2
C₂×#2974	67

Subgroups — C2×C67×#2974 #30739473

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₇×#2974 #30739473