Subgroups — C₂×C₆₇×#2592 #30714152

Order 17152.

2918 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#124×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#2592	C₂	1	16	yes	maximal, direct product factor
4	C₂×#397×C₆₇	C₂	1	2	yes	maximal, complemented
5	C₂×#393×C₆₇	C₂	1	2	yes	maximal, complemented
6	C₂×#391×C₆₇	C₂	1	2	yes	maximal, complemented
7	C₂×#390×C₆₇	C₂	1	2	yes	maximal
8	C₂×#388×C₆₇	C₂	1	4	yes	maximal, complemented
9	C₂×C₄×C₄⋊C₄×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂³×Q₈×C₆₇	C₂²	1	1	yes	characteristic
11	C₂³×Q₈×C₆₇	C₂²	1	1	yes	characteristic
12	C₂³×D₄×C₆₇	C₂²	1	1	yes	characteristic
13	C₂×#124×C₆₇	—	2	8	no	—
14	C₂×#124×C₆₇	C₂²	1	8	yes	—
15	C₂×C₄×Q₈×C₆₇	—	2	8	no	—
16	C₂×C₄×Q₈×C₆₇	C₂²	1	8	yes	complemented
17	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
18	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
19	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
20	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
21	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	4	yes	—
22	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
23	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	4	yes	—
24	C₂²×C₄²×C₆₇	C₂²	1	2	yes	—
25	C₂²×C₄²×C₆₇	C₂²	1	2	yes	—
26	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
27	#397×C₆₇	C₂²	1	16	yes	complemented
28	#393×C₆₇	C₂²	1	16	yes	complemented
29	#391×C₆₇	C₂²	1	16	yes	complemented
30	#390×C₆₇	C₂²	1	16	yes	—
31	#388×C₆₇	C₂²	1	32	yes	complemented
32	C₄×C₄⋊C₄×C₆₇	C₂²	1	8	yes	complemented
33	C₂×#95×C₆₇	C₂²	1	2	yes	—
34	C₂×#95×C₆₇	C₂²	1	4	yes	—
35	C₂×#95×C₆₇	C₂²	1	4	yes	—
36	C₂⁵×C₆₇	C₂³	1	2	yes	—
37	C₂²×Q₈×C₆₇	—	2	4	no	—
38	C₂²×Q₈×C₆₇	—	2	4	no	—
39	C₂²×Q₈×C₆₇	—	2	8	no	—
40	C₂²×Q₈×C₆₇	—	2	8	no	—
41	C₂²×Q₈×C₆₇	C₂³	1	4	yes	—
42	C₂²×Q₈×C₆₇	C₂³	1	4	yes	—
43	C₂²×D₄×C₆₇	—	2	4	no	—
44	C₂²×D₄×C₆₇	—	2	8	no	—
45	C₂²×D₄×C₆₇	C₂³	1	4	yes	—
46	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
47	C₂³×C₄×C₆₇	C₂³	1	4	yes	—
48	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
49	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
50	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
51	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
52	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
53	#124×C₆₇	—	2	32	no	—
54	C₄×Q₈×C₆₇	—	2	32	no	—
55	C₂×C₄⋊C₄×C₆₇	—	2	8	no	—
56	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
57	C₂×C₄⋊C₄×C₆₇	—	2	8	no	—
58	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
59	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
60	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
61	C₂×C₂²⋊C₄×C₆₇	—	2	8	no	—
62	C₂×C₂²⋊C₄×C₆₇	—	2	8	no	—
63	C₂×C₂²⋊C₄×C₆₇	—	2	16	no	—
64	C₂×C₂²⋊C₄×C₆₇	—	4	8	no	—
65	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
66	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
67	C₂×C₂²⋊C₄×C₆₇	C₂³	1	16	yes	—
68	C₂×C₂²⋊C₄×C₆₇	C₂³	1	16	yes	—
69	C₂×C₄²×C₆₇	—	2	8	no	—
70	C₂×C₄²×C₆₇	—	4	4	no	—
71	C₂×C₄²×C₆₇	D₄	1	8	yes	—
72	C₂×C₄²×C₆₇	C₂³	1	8	yes	—
73	C₂×C₄²×C₆₇	C₂³	1	8	yes	—
74	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
75	#95×C₆₇	C₂³	1	8	yes	—
76	#95×C₆₇	C₂³	1	16	yes	—
77	#95×C₆₇	C₂³	1	16	yes	—
78	C₂⁴×C₆₇	—	2	4	no	—
79	C₂⁴×C₆₇	—	2	8	no	—
80	C₂⁴×C₆₇	—	2	8	no	—
81	C₂⁴×C₆₇	—	2	4	no	—
82	C₂⁴×C₆₇	—	2	4	no	—
83	C₂⁴×C₆₇	C₂⁴	1	4	yes	—
84	C₂⁴×C₆₇	C₂⁴	1	1	yes	characteristic
85	C₂×Q₈×C₆₇	—	2	32	no	—
86	C₂×Q₈×C₆₇	—	2	8	no	—
87	C₂×Q₈×C₆₇	—	2	8	no	—
88	C₂×Q₈×C₆₇	—	2	16	no	—
89	C₂×Q₈×C₆₇	—	2	16	no	—
90	C₂×Q₈×C₆₇	—	4	16	no	—
91	C₂×D₄×C₆₇	—	2	8	no	—
92	C₂×D₄×C₆₇	—	2	16	no	—
93	C₂×D₄×C₆₇	—	4	16	no	—
94	C₂²×C₄×C₆₇	—	2	4	no	—
95	C₂²×C₄×C₆₇	—	2	4	no	—
96	C₂²×C₄×C₆₇	—	2	4	no	—
97	C₂²×C₄×C₆₇	—	2	2	no	—
98	C₂²×C₄×C₆₇	—	2	2	no	—
99	C₂²×C₄×C₆₇	—	2	4	no	—
100	C₂²×C₄×C₆₇	—	2	4	no	—
101	C₂²×C₄×C₆₇	—	2	8	no	—
102	C₂²×C₄×C₆₇	—	2	8	no	—
103	C₂²×C₄×C₆₇	—	2	8	no	—
104	C₂²×C₄×C₆₇	—	2	8	no	—
105	C₂²×C₄×C₆₇	—	2	4	no	—
106	C₂²×C₄×C₆₇	—	2	4	no	—
107	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
108	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
109	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
110	C₂²×C₄×C₆₇	#41	1	8	yes	—
111	C₂²×C₄×C₆₇	#41	1	8	yes	—
112	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
113	C₂²×C₄×C₆₇	C₂⁴	1	8	yes	—
114	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
115	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
116	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
117	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
118	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
119	C₄⋊C₄×C₆₇	—	2	16	no	—
120	C₄⋊C₄×C₆₇	—	2	8	no	—
121	C₄⋊C₄×C₆₇	—	2	16	no	—
122	C₂²⋊C₄×C₆₇	—	2	16	no	—
123	C₂²⋊C₄×C₆₇	—	2	16	no	—
124	C₂²⋊C₄×C₆₇	—	2	32	no	—
125	C₂²⋊C₄×C₆₇	—	4	16	no	—
126	C₄²×C₆₇	—	2	16	no	—
127	C₄²×C₆₇	—	4	8	no	—
128	C₄²×C₆₇	C₂×D₄	1	16	yes	—
129	C₂³×C₆₇	—	2	4	no	—
130	C₂³×C₆₇	—	2	8	no	—
131	C₂³×C₆₇	—	2	8	no	—
132	C₂³×C₆₇	—	2	4	no	—
133	C₂³×C₆₇	—	2	4	no	—
134	C₂³×C₆₇	—	4	8	no	—
135	C₂³×C₆₇	—	4	8	no	—
136	C₂³×C₆₇	—	4	8	no	—
137	C₂³×C₆₇	—	4	16	no	—
138	C₂³×C₆₇	—	4	16	no	—
139	C₂³×C₆₇	C₂²×D₄	1	2	yes	—
140	C₂³×C₆₇	C₂×#41	1	4	yes	—
141	C₂³×C₆₇	C₂×#41	1	4	yes	—
142	C₂³×C₆₇	H(5,𝔽₂)	1	2	yes	—
143	C₂³×C₆₇	2₋¹⁺⁴	1	2	yes	—
144	C₂³×C₆₇	C₂⁵	1	1	yes	characteristic
145	Q₈×C₆₇	—	2	32	no	—
146	Q₈×C₆₇	—	4	16	no	—
147	D₄×C₆₇	—	4	16	no	—
148	C₂×C₄×C₆₇	—	2	8	no	—
149	C₂×C₄×C₆₇	—	2	16	no	—
150	C₂×C₄×C₆₇	—	2	8	no	—
151	C₂×C₄×C₆₇	—	2	4	no	—
152	C₂×C₄×C₆₇	—	2	4	no	—
153	C₂×C₄×C₆₇	—	2	4	no	—
154	C₂×C₄×C₆₇	—	2	2	no	—
155	C₂×C₄×C₆₇	—	2	2	no	—
156	C₂×C₄×C₆₇	—	2	4	no	—
157	C₂×C₄×C₆₇	—	2	4	no	—
158	C₂×C₄×C₆₇	—	2	8	no	—
159	C₂×C₄×C₆₇	—	2	8	no	—
160	C₂×C₄×C₆₇	—	2	8	no	—
161	C₂×C₄×C₆₇	—	2	8	no	—
162	C₂×C₄×C₆₇	—	2	4	no	—
163	C₂×C₄×C₆₇	—	2	4	no	—
164	C₂×C₄×C₆₇	—	4	8	no	—
165	C₂×C₄×C₆₇	—	4	8	no	—
166	C₂×C₄×C₆₇	—	4	16	no	—
167	C₂×C₄×C₆₇	#124	1	16	yes	—
168	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
169	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
170	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
171	C₂×C₄×C₆₇	C₂×#41	1	8	yes	—
172	C₂×C₄×C₆₇	C₂×#41	1	8	yes	—
173	C₂²×C₆₇	—	4	4	no	—
174	C₂²×C₆₇	—	4	4	no	—
175	C₂²×C₆₇	—	4	4	no	—
176	C₂²×C₆₇	—	4	8	no	—
177	C₂²×C₆₇	—	4	8	no	—
178	C₂²×C₆₇	—	8	16	no	—
179	C₂²×C₆₇	#525	1	4	yes	—
180	C₂²×C₆₇	C₂×#124	1	4	yes	—
181	C₂²×C₆₇	#537	1	4	yes	—
182	C₂²×C₆₇	#556	1	8	yes	—
183	C₂²×C₆₇	#559	1	8	yes	—
184	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
185	C₂²×C₆₇	C₂²×#41	1	2	yes	—
186	C₂²×C₆₇	C₂²×#41	1	2	yes	—
187	C₂²×C₆₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
188	C₂²×C₆₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
189	C₄×C₆₇	—	2	4	no	—
190	C₄×C₆₇	—	2	8	no	—
191	C₄×C₆₇	—	2	4	no	—
192	C₄×C₆₇	—	4	4	no	—
193	C₄×C₆₇	—	4	4	no	—
194	C₄×C₆₇	—	4	8	no	—
195	C₄×C₆₇	C₂×#124	1	8	yes	—
196	C₂×#2592	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
197	C₂×C₆₇	—	8	4	no	—
198	C₂×C₆₇	#2592	1	8	yes	direct product factor
199	C₂×C₆₇	C₂×#525	1	1	yes	characteristic
200	C₂×C₆₇	C₂²×#124	1	1	yes	characteristic
201	C₂×C₆₇	C₂×#537	1	1	yes	characteristic
202	C₂×C₆₇	C₂×#556	1	2	yes	—
203	C₂×C₆₇	C₂×#559	1	2	yes	—
204	C₂²×#124	C₂×C₆₇	1	1	yes	characteristic
205	C₂²×C₄×Q₈	C₂×C₆₇	1	1	yes	characteristic, complemented
206	#2592	C₂×C₆₇	1	16	yes	direct product factor
207	C₂×#397	C₂×C₆₇	1	2	yes	complemented
208	C₂×#393	C₂×C₆₇	1	2	yes	complemented
209	C₂×#391	C₂×C₆₇	1	2	yes	complemented
210	C₂×#390	C₂×C₆₇	1	2	yes	—
211	C₂×#388	C₂×C₆₇	1	4	yes	complemented
212	C₂×C₄×C₄⋊C₄	C₂×C₆₇	1	1	yes	characteristic, complemented
213	C₆₇	C₂×#2592	1	1	yes	characteristic, direct product factor, Sylow
214	C₂³×Q₈	C₂²×C₆₇	1	1	yes	characteristic
215	C₂³×Q₈	C₂²×C₆₇	1	1	yes	characteristic
216	C₂³×D₄	C₂²×C₆₇	1	1	yes	characteristic
217	C₂×#124	—	2	8	no	—
218	C₂×#124	C₂²×C₆₇	1	8	yes	—
219	C₂×C₄×Q₈	—	2	8	no	—
220	C₂×C₄×Q₈	C₂²×C₆₇	1	8	yes	complemented
221	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
222	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
223	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
224	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
225	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	4	yes	—
226	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
227	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	4	yes	—
228	C₂²×C₄²	C₂²×C₆₇	1	2	yes	—
229	C₂²×C₄²	C₂²×C₆₇	1	2	yes	—
230	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
231	#397	C₂²×C₆₇	1	16	yes	complemented
232	#393	C₂²×C₆₇	1	16	yes	complemented
233	#391	C₂²×C₆₇	1	16	yes	complemented
234	#390	C₂²×C₆₇	1	16	yes	—
235	#388	C₂²×C₆₇	1	32	yes	complemented
236	C₄×C₄⋊C₄	C₂²×C₆₇	1	8	yes	complemented
237	C₂×#95	C₂²×C₆₇	1	2	yes	—
238	C₂×#95	C₂²×C₆₇	1	4	yes	—
239	C₂×#95	C₂²×C₆₇	1	4	yes	—
240	C₂⁵	C₂³×C₆₇	1	2	yes	—
241	C₂²×Q₈	—	2	4	no	—
242	C₂²×Q₈	—	2	4	no	—
243	C₂²×Q₈	—	2	8	no	—
244	C₂²×Q₈	—	2	8	no	—
245	C₂²×Q₈	C₂³×C₆₇	1	4	yes	—
246	C₂²×Q₈	C₂³×C₆₇	1	4	yes	—
247	C₂²×D₄	—	2	4	no	—
248	C₂²×D₄	—	2	8	no	—
249	C₂²×D₄	C₂³×C₆₇	1	4	yes	—
250	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
251	C₂³×C₄	C₂³×C₆₇	1	4	yes	—
252	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
253	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
254	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
255	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
256	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
257	#124	—	2	32	no	—
258	C₄×Q₈	—	2	32	no	—
259	C₂×C₄⋊C₄	—	2	8	no	—
260	C₂×C₄⋊C₄	—	2	4	no	—
261	C₂×C₄⋊C₄	—	2	8	no	—
262	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
263	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
264	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
265	C₂×C₂²⋊C₄	—	2	8	no	—
266	C₂×C₂²⋊C₄	—	2	8	no	—
267	C₂×C₂²⋊C₄	—	2	16	no	—
268	C₂×C₂²⋊C₄	—	4	8	no	—
269	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
270	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
271	C₂×C₂²⋊C₄	C₂³×C₆₇	1	16	yes	—
272	C₂×C₂²⋊C₄	C₂³×C₆₇	1	16	yes	—
273	C₂×C₄²	—	2	8	no	—
274	C₂×C₄²	—	4	4	no	—
275	C₂×C₄²	D₄×C₆₇	1	8	yes	—
276	C₂×C₄²	C₂³×C₆₇	1	8	yes	—
277	C₂×C₄²	C₂³×C₆₇	1	8	yes	—
278	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
279	#95	C₂³×C₆₇	1	8	yes	—
280	#95	C₂³×C₆₇	1	16	yes	—
281	#95	C₂³×C₆₇	1	16	yes	—
282	C₂⁴	—	2	4	no	—
283	C₂⁴	—	2	8	no	—
284	C₂⁴	—	2	8	no	—
285	C₂⁴	—	2	4	no	—
286	C₂⁴	—	2	4	no	—
287	C₂⁴	C₂⁴×C₆₇	1	4	yes	—
288	C₂⁴	C₂⁴×C₆₇	1	1	yes	characteristic
289	C₂×Q₈	—	2	32	no	—
290	C₂×Q₈	—	2	8	no	—
291	C₂×Q₈	—	2	8	no	—
292	C₂×Q₈	—	2	16	no	—
293	C₂×Q₈	—	2	16	no	—
294	C₂×Q₈	—	4	16	no	—
295	C₂×D₄	—	2	8	no	—
296	C₂×D₄	—	2	16	no	—
297	C₂×D₄	—	4	16	no	—
298	C₂²×C₄	—	2	4	no	—
299	C₂²×C₄	—	2	4	no	—
300	C₂²×C₄	—	2	4	no	—
301	C₂²×C₄	—	2	2	no	—
302	C₂²×C₄	—	2	2	no	—
303	C₂²×C₄	—	2	4	no	—
304	C₂²×C₄	—	2	4	no	—
305	C₂²×C₄	—	2	8	no	—
306	C₂²×C₄	—	2	8	no	—
307	C₂²×C₄	—	2	8	no	—
308	C₂²×C₄	—	2	8	no	—
309	C₂²×C₄	—	2	4	no	—
310	C₂²×C₄	—	2	4	no	—
311	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
312	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
313	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
314	C₂²×C₄	#41×C₆₇	1	8	yes	—
315	C₂²×C₄	#41×C₆₇	1	8	yes	—
316	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
317	C₂²×C₄	C₂⁴×C₆₇	1	8	yes	—
318	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
319	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
320	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
321	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
322	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
323	C₄⋊C₄	—	2	16	no	—
324	C₄⋊C₄	—	2	8	no	—
325	C₄⋊C₄	—	2	16	no	—
326	C₂²⋊C₄	—	2	16	no	—
327	C₂²⋊C₄	—	2	16	no	—
328	C₂²⋊C₄	—	2	32	no	—
329	C₂²⋊C₄	—	4	16	no	—
330	C₄²	—	2	16	no	—
331	C₄²	—	4	8	no	—
332	C₄²	C₂×D₄×C₆₇	1	16	yes	—
333	C₂³	—	2	4	no	—
334	C₂³	—	2	8	no	—
335	C₂³	—	2	8	no	—
336	C₂³	—	2	4	no	—
337	C₂³	—	2	4	no	—
338	C₂³	—	4	8	no	—
339	C₂³	—	4	8	no	—
340	C₂³	—	4	8	no	—
341	C₂³	—	4	16	no	—
342	C₂³	—	4	16	no	—
343	C₂³	C₂²×D₄×C₆₇	1	2	yes	—
344	C₂³	C₂×#41×C₆₇	1	4	yes	—
345	C₂³	C₂×#41×C₆₇	1	4	yes	—
346	C₂³	H(5,𝔽₂)×C₆₇	1	2	yes	—
347	C₂³	2₋¹⁺⁴×C₆₇	1	2	yes	—
348	C₂³	C₂⁵×C₆₇	1	1	yes	characteristic
349	Q₈	—	2	32	no	—
350	Q₈	—	4	16	no	—
351	D₄	—	4	16	no	—
352	C₂×C₄	—	2	8	no	—
353	C₂×C₄	—	2	16	no	—
354	C₂×C₄	—	2	8	no	—
355	C₂×C₄	—	2	4	no	—
356	C₂×C₄	—	2	4	no	—
357	C₂×C₄	—	2	4	no	—
358	C₂×C₄	—	2	2	no	—
359	C₂×C₄	—	2	2	no	—
360	C₂×C₄	—	2	4	no	—
361	C₂×C₄	—	2	4	no	—
362	C₂×C₄	—	2	8	no	—
363	C₂×C₄	—	2	8	no	—
364	C₂×C₄	—	2	8	no	—
365	C₂×C₄	—	2	8	no	—
366	C₂×C₄	—	2	4	no	—
367	C₂×C₄	—	2	4	no	—
368	C₂×C₄	—	4	8	no	—
369	C₂×C₄	—	4	8	no	—
370	C₂×C₄	—	4	16	no	—
371	C₂×C₄	#124×C₆₇	1	16	yes	—
372	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
373	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
374	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
375	C₂×C₄	C₂×#41×C₆₇	1	8	yes	—
376	C₂×C₄	C₂×#41×C₆₇	1	8	yes	—
377	C₂²	—	4	4	no	—
378	C₂²	—	4	4	no	—
379	C₂²	—	4	4	no	—
380	C₂²	—	4	8	no	—
381	C₂²	—	4	8	no	—
382	C₂²	—	8	16	no	—
383	C₂²	#525×C₆₇	1	4	yes	—
384	C₂²	C₂×#124×C₆₇	1	4	yes	—
385	C₂²	#537×C₆₇	1	4	yes	—
386	C₂²	#556×C₆₇	1	8	yes	—
387	C₂²	#559×C₆₇	1	8	yes	—
388	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
389	C₂²	C₂²×#41×C₆₇	1	2	yes	—
390	C₂²	C₂²×#41×C₆₇	1	2	yes	—
391	C₂²	C₂×H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
392	C₂²	C₂×2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
393	C₄	—	2	4	no	—
394	C₄	—	2	8	no	—
395	C₄	—	2	4	no	—
396	C₄	—	4	4	no	—
397	C₄	—	4	4	no	—
398	C₄	—	4	8	no	—
399	C₄	C₂×#124×C₆₇	1	8	yes	—
400	C₂	—	8	4	no	—
401	C₂	C₆₇×#2592	1	8	yes	direct product factor
402	C₂	C₂×#525×C₆₇	1	1	yes	characteristic
403	C₂	C₂²×#124×C₆₇	1	1	yes	characteristic
404	C₂	C₂×#537×C₆₇	1	1	yes	characteristic
405	C₂	C₂×#556×C₆₇	1	2	yes	—
406	C₂	C₂×#559×C₆₇	1	2	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₆₇	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁵×C₆₇
prime radical	C₂³×D₄×C₆₇	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×C₄×Q₈×C₆₇	C₂	semidirect
3. C₆₇×#2592	C₂	direct (N × H)
4. C₂×#397×C₆₇	C₂	semidirect
5. C₂×#393×C₆₇	C₂	semidirect
6. C₂×#391×C₆₇	C₂	semidirect
8. C₂×#388×C₆₇	C₂	semidirect
9. C₂×C₄×C₄⋊C₄×C₆₇	C₂	semidirect
16. C₂×C₄×Q₈×C₆₇	C₂²	semidirect
27. #397×C₆₇	C₂²	semidirect
28. #393×C₆₇	C₂²	semidirect
29. #391×C₆₇	C₂²	semidirect
31. #388×C₆₇	C₂²	semidirect
32. C₄×C₄⋊C₄×C₆₇	C₂²	semidirect
196. C₂×#2592	C₆₇	direct (N × H)
198. C₂×C₆₇	#2592	direct (N × H)
205. C₂²×C₄×Q₈	C₂×C₆₇	semidirect
206. #2592	C₂×C₆₇	direct (N × H)
207. C₂×#397	C₂×C₆₇	semidirect
208. C₂×#393	C₂×C₆₇	semidirect
209. C₂×#391	C₂×C₆₇	semidirect
211. C₂×#388	C₂×C₆₇	semidirect
212. C₂×C₄×C₄⋊C₄	C₂×C₆₇	semidirect
213. C₆₇	C₂×#2592	direct (N × H)
220. C₂×C₄×Q₈	C₂²×C₆₇	semidirect
231. #397	C₂²×C₆₇	semidirect
232. #393	C₂²×C₆₇	semidirect
233. #391	C₂²×C₆₇	semidirect
235. #388	C₂²×C₆₇	semidirect
236. C₄×C₄⋊C₄	C₂²×C₆₇	semidirect
401. C₂	C₆₇×#2592	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×#124×C₆₇	2
C₂²×C₄×Q₈×C₆₇	2
C₆₇×#2592	2
C₂×#397×C₆₇	2
C₂×#393×C₆₇	2
C₂×#391×C₆₇	2
C₂×#390×C₆₇	2
C₂×#388×C₆₇	2
C₂×C₄×C₄⋊C₄×C₆₇	2
C₂×#2592	67

Subgroups — C2×C67×#2592 #30714152

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₇×#2592 #30714152