Subgroups — C₆₇×#34993 #30714244

Order 17152.

2866 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#543×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#534×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#2850	C₂	1	4	yes	maximal, complemented
4	C₆₇×#2639	C₂	1	4	yes	maximal, complemented
5	C₆₇×#2488	C₂	1	8	yes	maximal, complemented
6	C₆₇×#2472	C₂	1	8	yes	maximal, complemented
7	C₆₇×#2308	C₂	1	2	yes	maximal, complemented
8	C₂×#386×C₆₇	C₂	1	2	yes	maximal, complemented
9	C₂×C₄×C₂²⋊C₄×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂³×D₄×C₆₇	C₂²	1	2	yes	—
11	C₂⁴×C₄×C₆₇	C₂²	1	1	yes	characteristic
12	#543×C₆₇	—	2	8	no	—
13	#534×C₆₇	—	2	8	no	—
14	C₂×#124×C₆₇	C₂²	1	4	yes	complemented
15	C₂×#123×C₆₇	C₂²	1	4	yes	—
16	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	4	yes	complemented
17	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	8	yes	complemented
18	C₂×#121×C₆₇	C₂²	1	4	yes	—
19	C₂×#120×C₆₇	C₂²	1	4	yes	—
20	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
21	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
22	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
23	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	4	yes	complemented
24	C₂²×C₄²×C₆₇	C₂²	1	2	yes	complemented
25	#400×C₆₇	C₂²	1	8	yes	complemented
26	#397×C₆₇	C₂²	1	8	yes	complemented
27	#394×C₆₇	C₂²	1	8	yes	complemented
28	#393×C₆₇	C₂²	1	8	yes	complemented
29	#391×C₆₇	C₂²	1	4	yes	complemented
30	#390×C₆₇	C₂²	1	4	yes	complemented
31	#388×C₆₇	C₂²	1	8	yes	complemented
32	#388×C₆₇	C₂²	1	8	yes	complemented
33	#386×C₆₇	C₂²	1	8	yes	—
34	#386×C₆₇	C₂²	1	8	yes	—
35	#386×C₆₇	C₂²	1	4	yes	complemented
36	#386×C₆₇	C₂²	1	4	yes	complemented
37	#386×C₆₇	C₂²	1	8	yes	complemented
38	#382×C₆₇	C₂²	1	4	yes	complemented
39	#381×C₆₇	C₂²	1	4	yes	complemented
40	#379×C₆₇	C₂²	1	4	yes	complemented
41	C₄×C₂²⋊C₄×C₆₇	C₂²	1	8	yes	complemented
42	C₂×#95×C₆₇	C₂²	1	2	yes	complemented
43	C₂×#95×C₆₇	C₂²	1	4	yes	complemented
44	C₂⁵×C₆₇	C₂³	1	2	yes	—
45	C₂⁵×C₆₇	C₂³	1	1	yes	characteristic
46	C₂²×Q₈×C₆₇	C₂³	1	4	yes	—
47	C₂²×D₄×C₆₇	—	2	8	no	—
48	C₂²×D₄×C₆₇	—	2	8	no	—
49	C₂²×D₄×C₆₇	—	2	8	no	—
50	C₂²×D₄×C₆₇	C₂³	1	4	yes	—
51	C₂²×D₄×C₆₇	C₂³	1	4	yes	—
52	C₂²×D₄×C₆₇	C₂³	1	4	yes	—
53	C₂³×C₄×C₆₇	—	2	4	no	—
54	C₂³×C₄×C₆₇	—	4	4	no	—
55	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
56	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
57	C₂³×C₄×C₆₇	C₂³	1	4	yes	—
58	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
59	C₂³×C₄×C₆₇	C₂³	1	4	yes	—
60	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
61	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
62	#124×C₆₇	—	4	8	no	—
63	#123×C₆₇	—	2	16	no	—
64	C₂²⋊Q₈×C₆₇	—	2	16	no	—
65	C₂²⋊Q₈×C₆₇	—	2	32	no	—
66	#121×C₆₇	—	2	16	no	—
67	#120×C₆₇	—	2	16	no	—
68	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
69	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
70	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
71	C₂×C₂²⋊C₄×C₆₇	—	2	8	no	—
72	C₂×C₂²⋊C₄×C₆₇	—	2	4	no	—
73	C₂×C₂²⋊C₄×C₆₇	—	4	8	no	—
74	C₂×C₂²⋊C₄×C₆₇	D₄	1	8	yes	—
75	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
76	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
77	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
78	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
79	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
80	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
81	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
82	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
83	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
84	C₂×C₄²×C₆₇	—	4	4	no	—
85	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
86	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
87	#95×C₆₇	C₂³	1	8	yes	—
88	#95×C₆₇	C₂³	1	8	yes	—
89	#95×C₆₇	C₂³	1	8	yes	—
90	#95×C₆₇	C₂³	1	8	yes	—
91	#95×C₆₇	C₂³	1	8	yes	—
92	C₂⁴×C₆₇	—	2	4	no	—
93	C₂⁴×C₆₇	—	2	4	no	—
94	C₂⁴×C₆₇	—	2	4	no	—
95	C₂⁴×C₆₇	—	4	2	no	—
96	C₂⁴×C₆₇	—	4	2	no	—
97	C₂⁴×C₆₇	—	4	4	no	—
98	C₂⁴×C₆₇	—	4	4	no	—
99	C₂⁴×C₆₇	C₂×D₄	1	4	yes	—
100	C₂⁴×C₆₇	#41	1	4	yes	—
101	C₂⁴×C₆₇	#41	1	4	yes	—
102	C₂⁴×C₆₇	C₂⁴	1	4	yes	—
103	C₂⁴×C₆₇	C₂⁴	1	2	yes	—
104	C₂⁴×C₆₇	C₂⁴	1	1	yes	characteristic
105	C₂×Q₈×C₆₇	—	2	8	no	—
106	C₂×Q₈×C₆₇	—	4	4	no	—
107	C₂×Q₈×C₆₇	—	4	4	no	—
108	C₂×D₄×C₆₇	—	2	8	no	—
109	C₂×D₄×C₆₇	—	2	8	no	—
110	C₂×D₄×C₆₇	—	2	8	no	—
111	C₂×D₄×C₆₇	—	2	8	no	—
112	C₂×D₄×C₆₇	—	2	8	no	—
113	C₂×D₄×C₆₇	—	2	8	no	—
114	C₂×D₄×C₆₇	—	2	8	no	—
115	C₂×D₄×C₆₇	—	4	4	no	—
116	C₂×D₄×C₆₇	—	4	4	no	—
117	C₂×D₄×C₆₇	—	4	32	no	—
118	C₂²×C₄×C₆₇	—	2	2	no	—
119	C₂²×C₄×C₆₇	—	2	4	no	—
120	C₂²×C₄×C₆₇	—	2	4	no	—
121	C₂²×C₄×C₆₇	—	2	4	no	—
122	C₂²×C₄×C₆₇	—	2	8	no	—
123	C₂²×C₄×C₆₇	—	2	2	no	—
124	C₂²×C₄×C₆₇	—	2	4	no	—
125	C₂²×C₄×C₆₇	—	2	4	no	—
126	C₂²×C₄×C₆₇	—	2	8	no	—
127	C₂²×C₄×C₆₇	—	2	8	no	—
128	C₂²×C₄×C₆₇	—	4	2	no	—
129	C₂²×C₄×C₆₇	—	4	2	no	—
130	C₂²×C₄×C₆₇	—	4	4	no	—
131	C₂²×C₄×C₆₇	—	4	2	no	—
132	C₂²×C₄×C₆₇	—	4	2	no	—
133	C₂²×C₄×C₆₇	—	4	2	no	—
134	C₂²×C₄×C₆₇	—	4	4	no	—
135	C₂²×C₄×C₆₇	—	4	4	no	—
136	C₂²×C₄×C₆₇	—	4	16	no	—
137	C₂²×C₄×C₆₇	C₂×D₄	1	8	yes	—
138	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
139	C₂²×C₄×C₆₇	C₂⁴	1	8	yes	—
140	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
141	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
142	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
143	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
144	C₄⋊C₄×C₆₇	—	2	8	no	—
145	C₄⋊C₄×C₆₇	—	2	8	no	—
146	C₄⋊C₄×C₆₇	—	2	16	no	—
147	C₂²⋊C₄×C₆₇	—	2	8	no	—
148	C₂²⋊C₄×C₆₇	—	2	8	no	—
149	C₂²⋊C₄×C₆₇	—	2	8	no	—
150	C₂²⋊C₄×C₆₇	—	2	16	no	—
151	C₂²⋊C₄×C₆₇	—	2	16	no	—
152	C₂²⋊C₄×C₆₇	—	4	8	no	—
153	C₂²⋊C₄×C₆₇	—	4	8	no	—
154	C₂²⋊C₄×C₆₇	—	4	8	no	—
155	C₂²⋊C₄×C₆₇	C₂×D₄	1	16	yes	—
156	C₄²×C₆₇	—	2	8	no	—
157	C₄²×C₆₇	—	4	4	no	—
158	C₂³×C₆₇	—	2	2	no	—
159	C₂³×C₆₇	—	2	4	no	—
160	C₂³×C₆₇	—	2	2	no	—
161	C₂³×C₆₇	—	2	4	no	—
162	C₂³×C₆₇	—	2	4	no	—
163	C₂³×C₆₇	—	2	4	no	—
164	C₂³×C₆₇	—	2	4	no	—
165	C₂³×C₆₇	—	2	8	no	—
166	C₂³×C₆₇	—	2	8	no	—
167	C₂³×C₆₇	—	4	4	no	—
168	C₂³×C₆₇	—	4	8	no	—
169	C₂³×C₆₇	—	4	8	no	—
170	C₂³×C₆₇	—	4	4	no	—
171	C₂³×C₆₇	—	4	16	no	—
172	C₂³×C₆₇	—	4	8	no	—
173	C₂³×C₆₇	—	8	4	no	—
174	C₂³×C₆₇	—	8	4	no	—
175	C₂³×C₆₇	—	8	8	no	—
176	C₂³×C₆₇	#124	1	16	yes	complemented
177	C₂³×C₆₇	C₂²×D₄	1	2	yes	—
178	C₂³×C₆₇	C₂²×D₄	1	4	yes	—
179	C₂³×C₆₇	C₂×#41	1	4	yes	—
180	C₂³×C₆₇	C₂×#41	1	4	yes	—
181	C₂³×C₆₇	C₂×#41	1	2	yes	—
182	C₂³×C₆₇	C₂×#41	1	2	yes	—
183	C₂³×C₆₇	C₂⁵	1	1	yes	characteristic
184	Q₈×C₆₇	—	8	8	no	—
185	D₄×C₆₇	—	4	16	no	—
186	D₄×C₆₇	—	4	16	no	—
187	D₄×C₆₇	—	8	8	no	—
188	C₂×C₄×C₆₇	—	2	8	no	—
189	C₂×C₄×C₆₇	—	2	2	no	—
190	C₂×C₄×C₆₇	—	2	4	no	—
191	C₂×C₄×C₆₇	—	2	2	no	—
192	C₂×C₄×C₆₇	—	2	4	no	—
193	C₂×C₄×C₆₇	—	2	8	no	—
194	C₂×C₄×C₆₇	—	2	4	no	—
195	C₂×C₄×C₆₇	—	2	4	no	—
196	C₂×C₄×C₆₇	—	2	2	no	—
197	C₂×C₄×C₆₇	—	2	4	no	—
198	C₂×C₄×C₆₇	—	2	4	no	—
199	C₂×C₄×C₆₇	—	2	4	no	—
200	C₂×C₄×C₆₇	—	2	8	no	—
201	C₂×C₄×C₆₇	—	2	4	no	—
202	C₂×C₄×C₆₇	—	2	4	no	—
203	C₂×C₄×C₆₇	—	2	8	no	—
204	C₂×C₄×C₆₇	—	4	8	no	—
205	C₂×C₄×C₆₇	—	4	16	no	—
206	C₂×C₄×C₆₇	—	4	8	no	—
207	C₂×C₄×C₆₇	—	4	4	no	—
208	C₂×C₄×C₆₇	—	4	8	no	—
209	C₂×C₄×C₆₇	—	8	8	no	—
210	C₂×C₄×C₆₇	C₂²×D₄	1	8	yes	—
211	C₂×C₄×C₆₇	H(5,𝔽₂)	1	4	yes	—
212	C₂²×C₆₇	—	2	4	no	—
213	C₂²×C₆₇	—	2	8	no	—
214	C₂²×C₆₇	—	2	2	no	—
215	C₂²×C₆₇	—	2	4	no	—
216	C₂²×C₆₇	—	2	4	no	—
217	C₂²×C₆₇	—	2	2	no	—
218	C₂²×C₆₇	—	2	4	no	—
219	C₂²×C₆₇	—	2	8	no	—
220	C₂²×C₆₇	—	4	16	no	—
221	C₂²×C₆₇	—	4	4	no	—
222	C₂²×C₆₇	—	4	8	no	—
223	C₂²×C₆₇	—	4	4	no	—
224	C₂²×C₆₇	—	4	4	no	—
225	C₂²×C₆₇	—	4	8	no	—
226	C₂²×C₆₇	—	8	16	no	—
227	C₂²×C₆₇	C₂×#124	1	8	yes	—
228	C₂²×C₆₇	C₂×#124	1	4	yes	—
229	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
230	C₂²×C₆₇	C₂²×#41	1	1	yes	characteristic
231	C₂²×C₆₇	C₂²×#41	1	1	yes	characteristic
232	C₂²×C₆₇	C₂×H(5,𝔽₂)	1	2	yes	—
233	C₂²×C₆₇	#585	1	2	yes	—
234	C₄×C₆₇	—	2	4	no	—
235	C₄×C₆₇	—	2	8	no	—
236	C₄×C₆₇	—	4	4	no	—
237	C₄×C₆₇	—	4	4	no	—
238	C₄×C₆₇	—	4	2	no	—
239	C₄×C₆₇	—	4	8	no	—
240	#34993	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
241	C₂×C₆₇	—	2	8	no	—
242	C₂×C₆₇	—	2	4	no	—
243	C₂×C₆₇	—	8	4	no	—
244	C₂×C₆₇	C₂²×#124	1	1	yes	characteristic
245	C₂×C₆₇	#3459	1	2	yes	—
246	C₂×C₆₇	#3490	1	1	yes	characteristic
247	C₂×C₆₇	#3491	1	1	yes	characteristic
248	C₂×C₆₇	#3496	1	2	yes	—
249	C₂×#543	C₂×C₆₇	1	1	yes	characteristic, complemented
250	C₂×#534	C₂×C₆₇	1	1	yes	characteristic
251	#2850	C₂×C₆₇	1	4	yes	complemented
252	#2639	C₂×C₆₇	1	4	yes	complemented
253	#2488	C₂×C₆₇	1	8	yes	complemented
254	#2472	C₂×C₆₇	1	8	yes	complemented
255	#2308	C₂×C₆₇	1	2	yes	complemented
256	C₂×#386	C₂×C₆₇	1	2	yes	complemented
257	C₂×C₄×C₂²⋊C₄	C₂×C₆₇	1	1	yes	characteristic, complemented
258	C₆₇	#34993	1	1	yes	characteristic, direct product factor, Sylow
259	C₂³×D₄	C₂²×C₆₇	1	2	yes	—
260	C₂⁴×C₄	C₂²×C₆₇	1	1	yes	characteristic
261	#543	—	2	8	no	—
262	#534	—	2	8	no	—
263	C₂×#124	C₂²×C₆₇	1	4	yes	complemented
264	C₂×#123	C₂²×C₆₇	1	4	yes	—
265	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	4	yes	complemented
266	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	8	yes	complemented
267	C₂×#121	C₂²×C₆₇	1	4	yes	—
268	C₂×#120	C₂²×C₆₇	1	4	yes	—
269	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
270	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
271	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
272	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	4	yes	complemented
273	C₂²×C₄²	C₂²×C₆₇	1	2	yes	complemented
274	#400	C₂²×C₆₇	1	8	yes	complemented
275	#397	C₂²×C₆₇	1	8	yes	complemented
276	#394	C₂²×C₆₇	1	8	yes	complemented
277	#393	C₂²×C₆₇	1	8	yes	complemented
278	#391	C₂²×C₆₇	1	4	yes	complemented
279	#390	C₂²×C₆₇	1	4	yes	complemented
280	#388	C₂²×C₆₇	1	8	yes	complemented
281	#388	C₂²×C₆₇	1	8	yes	complemented
282	#386	C₂²×C₆₇	1	8	yes	—
283	#386	C₂²×C₆₇	1	8	yes	—
284	#386	C₂²×C₆₇	1	8	yes	complemented
285	#386	C₂²×C₆₇	1	4	yes	complemented
286	#386	C₂²×C₆₇	1	4	yes	complemented
287	#382	C₂²×C₆₇	1	4	yes	complemented
288	#381	C₂²×C₆₇	1	4	yes	complemented
289	#379	C₂²×C₆₇	1	4	yes	complemented
290	C₄×C₂²⋊C₄	C₂²×C₆₇	1	8	yes	complemented
291	C₂×#95	C₂²×C₆₇	1	2	yes	complemented
292	C₂×#95	C₂²×C₆₇	1	4	yes	complemented
293	C₂⁵	C₂³×C₆₇	1	2	yes	—
294	C₂⁵	C₂³×C₆₇	1	1	yes	characteristic
295	C₂²×Q₈	C₂³×C₆₇	1	4	yes	—
296	C₂²×D₄	—	2	8	no	—
297	C₂²×D₄	—	2	8	no	—
298	C₂²×D₄	—	2	8	no	—
299	C₂²×D₄	C₂³×C₆₇	1	4	yes	—
300	C₂²×D₄	C₂³×C₆₇	1	4	yes	—
301	C₂²×D₄	C₂³×C₆₇	1	4	yes	—
302	C₂³×C₄	—	2	4	no	—
303	C₂³×C₄	—	4	4	no	—
304	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
305	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
306	C₂³×C₄	C₂³×C₆₇	1	4	yes	—
307	C₂³×C₄	C₂³×C₆₇	1	4	yes	—
308	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
309	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
310	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
311	#124	—	4	8	no	—
312	#123	—	2	16	no	—
313	C₂²⋊Q₈	—	2	16	no	—
314	C₂²⋊Q₈	—	2	32	no	—
315	#121	—	2	16	no	—
316	#120	—	2	16	no	—
317	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
318	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
319	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
320	C₂×C₂²⋊C₄	—	2	8	no	—
321	C₂×C₂²⋊C₄	—	2	4	no	—
322	C₂×C₂²⋊C₄	—	4	8	no	—
323	C₂×C₂²⋊C₄	D₄×C₆₇	1	8	yes	—
324	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
325	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
326	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
327	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
328	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
329	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
330	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
331	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
332	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
333	C₂×C₄²	—	4	4	no	—
334	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
335	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
336	#95	C₂³×C₆₇	1	8	yes	—
337	#95	C₂³×C₆₇	1	8	yes	—
338	#95	C₂³×C₆₇	1	8	yes	—
339	#95	C₂³×C₆₇	1	8	yes	—
340	#95	C₂³×C₆₇	1	8	yes	—
341	C₂⁴	—	2	4	no	—
342	C₂⁴	—	2	4	no	—
343	C₂⁴	—	2	4	no	—
344	C₂⁴	—	4	2	no	—
345	C₂⁴	—	4	2	no	—
346	C₂⁴	—	4	4	no	—
347	C₂⁴	—	4	4	no	—
348	C₂⁴	C₂×D₄×C₆₇	1	4	yes	—
349	C₂⁴	#41×C₆₇	1	4	yes	—
350	C₂⁴	#41×C₆₇	1	4	yes	—
351	C₂⁴	C₂⁴×C₆₇	1	4	yes	—
352	C₂⁴	C₂⁴×C₆₇	1	2	yes	—
353	C₂⁴	C₂⁴×C₆₇	1	1	yes	characteristic
354	C₂×Q₈	—	2	8	no	—
355	C₂×Q₈	—	4	4	no	—
356	C₂×Q₈	—	4	4	no	—
357	C₂×D₄	—	2	8	no	—
358	C₂×D₄	—	2	8	no	—
359	C₂×D₄	—	2	8	no	—
360	C₂×D₄	—	2	8	no	—
361	C₂×D₄	—	2	8	no	—
362	C₂×D₄	—	2	8	no	—
363	C₂×D₄	—	2	8	no	—
364	C₂×D₄	—	4	4	no	—
365	C₂×D₄	—	4	4	no	—
366	C₂×D₄	—	4	32	no	—
367	C₂²×C₄	—	2	2	no	—
368	C₂²×C₄	—	2	4	no	—
369	C₂²×C₄	—	2	4	no	—
370	C₂²×C₄	—	2	4	no	—
371	C₂²×C₄	—	2	8	no	—
372	C₂²×C₄	—	2	2	no	—
373	C₂²×C₄	—	2	4	no	—
374	C₂²×C₄	—	2	4	no	—
375	C₂²×C₄	—	2	8	no	—
376	C₂²×C₄	—	2	8	no	—
377	C₂²×C₄	—	4	2	no	—
378	C₂²×C₄	—	4	2	no	—
379	C₂²×C₄	—	4	4	no	—
380	C₂²×C₄	—	4	2	no	—
381	C₂²×C₄	—	4	2	no	—
382	C₂²×C₄	—	4	2	no	—
383	C₂²×C₄	—	4	4	no	—
384	C₂²×C₄	—	4	4	no	—
385	C₂²×C₄	—	4	16	no	—
386	C₂²×C₄	C₂×D₄×C₆₇	1	8	yes	—
387	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
388	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
389	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
390	C₂²×C₄	C₂⁴×C₆₇	1	8	yes	—
391	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
392	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
393	C₄⋊C₄	—	2	8	no	—
394	C₄⋊C₄	—	2	8	no	—
395	C₄⋊C₄	—	2	16	no	—
396	C₂²⋊C₄	—	2	8	no	—
397	C₂²⋊C₄	—	2	8	no	—
398	C₂²⋊C₄	—	2	8	no	—
399	C₂²⋊C₄	—	2	16	no	—
400	C₂²⋊C₄	—	2	16	no	—
401	C₂²⋊C₄	—	4	8	no	—
402	C₂²⋊C₄	—	4	8	no	—
403	C₂²⋊C₄	—	4	8	no	—
404	C₂²⋊C₄	C₂×D₄×C₆₇	1	16	yes	—
405	C₄²	—	2	8	no	—
406	C₄²	—	4	4	no	—
407	C₂³	—	2	2	no	—
408	C₂³	—	2	4	no	—
409	C₂³	—	2	2	no	—
410	C₂³	—	2	4	no	—
411	C₂³	—	2	4	no	—
412	C₂³	—	2	4	no	—
413	C₂³	—	2	4	no	—
414	C₂³	—	2	8	no	—
415	C₂³	—	2	8	no	—
416	C₂³	—	4	4	no	—
417	C₂³	—	4	8	no	—
418	C₂³	—	4	8	no	—
419	C₂³	—	4	4	no	—
420	C₂³	—	4	16	no	—
421	C₂³	—	4	8	no	—
422	C₂³	—	8	4	no	—
423	C₂³	—	8	4	no	—
424	C₂³	—	8	8	no	—
425	C₂³	#124×C₆₇	1	16	yes	complemented
426	C₂³	C₂²×D₄×C₆₇	1	2	yes	—
427	C₂³	C₂²×D₄×C₆₇	1	4	yes	—
428	C₂³	C₂×#41×C₆₇	1	4	yes	—
429	C₂³	C₂×#41×C₆₇	1	4	yes	—
430	C₂³	C₂×#41×C₆₇	1	2	yes	—
431	C₂³	C₂×#41×C₆₇	1	2	yes	—
432	C₂³	C₂⁵×C₆₇	1	1	yes	characteristic
433	Q₈	—	8	8	no	—
434	D₄	—	4	16	no	—
435	D₄	—	4	16	no	—
436	D₄	—	8	8	no	—
437	C₂×C₄	—	2	8	no	—
438	C₂×C₄	—	2	2	no	—
439	C₂×C₄	—	2	4	no	—
440	C₂×C₄	—	2	2	no	—
441	C₂×C₄	—	2	4	no	—
442	C₂×C₄	—	2	8	no	—
443	C₂×C₄	—	2	4	no	—
444	C₂×C₄	—	2	4	no	—
445	C₂×C₄	—	2	2	no	—
446	C₂×C₄	—	2	4	no	—
447	C₂×C₄	—	2	4	no	—
448	C₂×C₄	—	2	4	no	—
449	C₂×C₄	—	2	8	no	—
450	C₂×C₄	—	2	4	no	—
451	C₂×C₄	—	2	4	no	—
452	C₂×C₄	—	2	8	no	—
453	C₂×C₄	—	4	8	no	—
454	C₂×C₄	—	4	16	no	—
455	C₂×C₄	—	4	4	no	—
456	C₂×C₄	—	4	8	no	—
457	C₂×C₄	—	4	8	no	—
458	C₂×C₄	—	8	8	no	—
459	C₂×C₄	C₂²×D₄×C₆₇	1	8	yes	—
460	C₂×C₄	H(5,𝔽₂)×C₆₇	1	4	yes	—
461	C₂²	—	2	4	no	—
462	C₂²	—	2	8	no	—
463	C₂²	—	2	2	no	—
464	C₂²	—	2	4	no	—
465	C₂²	—	2	4	no	—
466	C₂²	—	2	2	no	—
467	C₂²	—	2	4	no	—
468	C₂²	—	2	8	no	—
469	C₂²	—	4	16	no	—
470	C₂²	—	4	4	no	—
471	C₂²	—	4	8	no	—
472	C₂²	—	4	4	no	—
473	C₂²	—	4	4	no	—
474	C₂²	—	4	8	no	—
475	C₂²	—	8	16	no	—
476	C₂²	C₂×#124×C₆₇	1	8	yes	—
477	C₂²	C₂×#124×C₆₇	1	4	yes	—
478	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
479	C₂²	C₂²×#41×C₆₇	1	1	yes	characteristic
480	C₂²	C₂²×#41×C₆₇	1	1	yes	characteristic
481	C₂²	C₂×H(5,𝔽₂)×C₆₇	1	2	yes	—
482	C₂²	#585×C₆₇	1	2	yes	—
483	C₄	—	2	4	no	—
484	C₄	—	2	8	no	—
485	C₄	—	4	4	no	—
486	C₄	—	4	4	no	—
487	C₄	—	4	2	no	—
488	C₄	—	4	8	no	—
489	C₂	—	2	8	no	—
490	C₂	—	2	4	no	—
491	C₂	—	8	4	no	—
492	C₂	C₂²×#124×C₆₇	1	1	yes	characteristic
493	C₂	C₆₇×#3459	1	2	yes	—
494	C₂	C₆₇×#3490	1	1	yes	characteristic
495	C₂	C₆₇×#3491	1	1	yes	characteristic
496	C₂	C₆₇×#3496	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁵×C₆₇
prime radical	C₂×#534×C₆₇	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#543×C₆₇	C₂	semidirect
3. C₆₇×#2850	C₂	semidirect
4. C₆₇×#2639	C₂	semidirect
5. C₆₇×#2488	C₂	semidirect
6. C₆₇×#2472	C₂	semidirect
7. C₆₇×#2308	C₂	semidirect
8. C₂×#386×C₆₇	C₂	semidirect
9. C₂×C₄×C₂²⋊C₄×C₆₇	C₂	semidirect
14. C₂×#124×C₆₇	C₂²	semidirect
16. C₂×C₂²⋊Q₈×C₆₇	C₂²	semidirect
17. C₂×C₂²⋊Q₈×C₆₇	C₂²	semidirect
23. C₂²×C₂²⋊C₄×C₆₇	C₂²	semidirect
24. C₂²×C₄²×C₆₇	C₂²	semidirect
25. #400×C₆₇	C₂²	semidirect
26. #397×C₆₇	C₂²	semidirect
27. #394×C₆₇	C₂²	semidirect
28. #393×C₆₇	C₂²	semidirect
29. #391×C₆₇	C₂²	semidirect
30. #390×C₆₇	C₂²	semidirect
31. #388×C₆₇	C₂²	semidirect
32. #388×C₆₇	C₂²	semidirect
35. #386×C₆₇	C₂²	semidirect
36. #386×C₆₇	C₂²	semidirect
37. #386×C₆₇	C₂²	semidirect
38. #382×C₆₇	C₂²	semidirect
39. #381×C₆₇	C₂²	semidirect
40. #379×C₆₇	C₂²	semidirect
41. C₄×C₂²⋊C₄×C₆₇	C₂²	semidirect
42. C₂×#95×C₆₇	C₂²	semidirect
43. C₂×#95×C₆₇	C₂²	semidirect
176. C₂³×C₆₇	#124	semidirect
240. #34993	C₆₇	direct (N × H)
249. C₂×#543	C₂×C₆₇	semidirect
251. #2850	C₂×C₆₇	semidirect
252. #2639	C₂×C₆₇	semidirect
253. #2488	C₂×C₆₇	semidirect
254. #2472	C₂×C₆₇	semidirect
255. #2308	C₂×C₆₇	semidirect
256. C₂×#386	C₂×C₆₇	semidirect
257. C₂×C₄×C₂²⋊C₄	C₂×C₆₇	semidirect
258. C₆₇	#34993	direct (N × H)
263. C₂×#124	C₂²×C₆₇	semidirect
265. C₂×C₂²⋊Q₈	C₂²×C₆₇	semidirect
266. C₂×C₂²⋊Q₈	C₂²×C₆₇	semidirect
272. C₂²×C₂²⋊C₄	C₂²×C₆₇	semidirect
273. C₂²×C₄²	C₂²×C₆₇	semidirect
274. #400	C₂²×C₆₇	semidirect
275. #397	C₂²×C₆₇	semidirect
276. #394	C₂²×C₆₇	semidirect
277. #393	C₂²×C₆₇	semidirect
278. #391	C₂²×C₆₇	semidirect
279. #390	C₂²×C₆₇	semidirect
280. #388	C₂²×C₆₇	semidirect
281. #388	C₂²×C₆₇	semidirect
284. #386	C₂²×C₆₇	semidirect
285. #386	C₂²×C₆₇	semidirect
286. #386	C₂²×C₆₇	semidirect
287. #382	C₂²×C₆₇	semidirect
288. #381	C₂²×C₆₇	semidirect
289. #379	C₂²×C₆₇	semidirect
290. C₄×C₂²⋊C₄	C₂²×C₆₇	semidirect
291. C₂×#95	C₂²×C₆₇	semidirect
292. C₂×#95	C₂²×C₆₇	semidirect
425. C₂³	#124×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#543×C₆₇	2
C₂×#534×C₆₇	2
C₆₇×#2850	2
C₆₇×#2639	2
C₆₇×#2488	2
C₆₇×#2472	2
C₆₇×#2308	2
C₂×#386×C₆₇	2
C₂×C₄×C₂²⋊C₄×C₆₇	2
#34993	67

Subgroups — C67×#34993 #30714244

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#34993 #30714244