Subgroups — C₆₇×#59863 #30739114

Order 17152.

1852 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#2808	C₂	1	4	yes	maximal, complemented
2	C₆₇×#2592	C₂	1	4	yes	maximal, complemented
3	C₆₇×#2563	C₂	1	2	yes	maximal, complemented
4	C₆₇×#2548	C₂	1	2	yes	maximal
5	C₆₇×#2547	C₂	1	2	yes	maximal, complemented
6	C₆₇×#2513	C₂	1	4	yes	maximal, complemented
7	C₆₇×#2485	C₂	1	4	yes	maximal, complemented
8	C₆₇×#2355	C₂	1	2	yes	maximal, complemented
9	C₆₇×#2354	C₂	1	2	yes	maximal, complemented
10	C₆₇×#2354	C₂	1	2	yes	maximal, complemented
11	C₄×C₄⋊Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₄×C₄⋊Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
13	C₄×#127×C₆₇	C₂	1	1	yes	maximal, characteristic
14	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
16	C₂×#127×C₆₇	C₂²	1	1	yes	characteristic
17	C₂×#124×C₆₇	C₂²	1	4	yes	—
18	C₂×C₄×Q₈×C₆₇	C₂²	1	2	yes	—
19	C₂×C₄×Q₈×C₆₇	C₂²	1	2	yes	—
20	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
21	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
22	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
23	C₂×#117×C₆₇	C₂²	1	4	yes	—
24	#397×C₆₇	C₂²	1	8	yes	—
25	#397×C₆₇	C₂²	1	8	yes	—
26	#394×C₆₇	C₂²	1	8	yes	—
27	#393×C₆₇	C₂²	1	8	yes	—
28	#391×C₆₇	C₂²	1	4	yes	—
29	#391×C₆₇	C₂²	1	2	yes	—
30	#391×C₆₇	C₂²	1	2	yes	—
31	#390×C₆₇	C₂²	1	2	yes	—
32	#390×C₆₇	C₂²	1	4	yes	—
33	#390×C₆₇	C₂²	1	4	yes	—
34	#390×C₆₇	C₂²	1	4	yes	—
35	#390×C₆₇	C₂²	1	2	yes	—
36	#390×C₆₇	C₂²	1	2	yes	—
37	#389×C₆₇	C₂²	1	2	yes	—
38	#389×C₆₇	C₂²	1	2	yes	—
39	#389×C₆₇	C₂²	1	2	yes	—
40	#389×C₆₇	C₂²	1	2	yes	—
41	#388×C₆₇	C₂²	1	8	yes	—
42	#388×C₆₇	C₂²	1	8	yes	—
43	#388×C₆₇	C₂²	1	8	yes	—
44	#387×C₆₇	C₂²	1	8	yes	—
45	#384×C₆₇	C₂²	1	1	yes	characteristic
46	#384×C₆₇	C₂²	1	1	yes	characteristic
47	#384×C₆₇	C₂²	1	1	yes	characteristic
48	#382×C₆₇	C₂²	1	4	yes	—
49	#380×C₆₇	C₂²	1	4	yes	—
50	#380×C₆₇	C₂²	1	4	yes	—
51	#380×C₆₇	C₂²	1	4	yes	—
52	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
53	C₄×C₄⋊C₄×C₆₇	C₂²	1	4	yes	—
54	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
55	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
56	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
57	C₄×C₄⋊C₄×C₆₇	C₂²	1	4	yes	—
58	C₄³×C₆₇	C₂²	1	1	yes	characteristic
59	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
60	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
61	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
62	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
63	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
64	C₂³×C₄×C₆₇	C₂³	1	4	yes	—
65	C₄⋊Q₈×C₆₇	—	2	4	no	—
66	C₄⋊Q₈×C₆₇	—	2	4	no	—
67	#127×C₆₇	—	2	4	no	—
68	#124×C₆₇	—	4	8	no	—
69	C₄×Q₈×C₆₇	—	2	8	no	—
70	C₄×Q₈×C₆₇	—	4	4	no	—
71	C₄×D₄×C₆₇	—	2	8	no	—
72	C₄×D₄×C₆₇	—	2	8	no	—
73	C₄×D₄×C₆₇	—	4	4	no	—
74	#117×C₆₇	—	4	8	no	—
75	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
76	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
77	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
78	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
79	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
80	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
81	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
82	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
83	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
84	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
85	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
86	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
87	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
88	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
89	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
90	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
91	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
92	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
93	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
94	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
95	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
96	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
97	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
98	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
99	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
100	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
101	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
102	C₂×C₄²×C₆₇	C₂³	1	8	yes	—
103	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
104	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
105	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
106	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
107	#95×C₆₇	C₂³	1	8	yes	—
108	#95×C₆₇	C₂³	1	8	yes	—
109	#95×C₆₇	C₂³	1	4	yes	—
110	#95×C₆₇	C₂³	1	4	yes	—
111	#95×C₆₇	C₂³	1	4	yes	—
112	#95×C₆₇	C₂³	1	4	yes	—
113	#95×C₆₇	C₂³	1	8	yes	—
114	C₂⁴×C₆₇	C₂⁴	1	4	yes	—
115	C₂×Q₈×C₆₇	—	2	4	no	—
116	C₂×Q₈×C₆₇	—	2	4	no	—
117	C₂×Q₈×C₆₇	—	2	4	no	—
118	C₂×Q₈×C₆₇	—	2	4	no	—
119	C₂×Q₈×C₆₇	—	4	2	no	—
120	C₂×Q₈×C₆₇	—	4	2	no	—
121	C₂×D₄×C₆₇	—	2	4	no	—
122	C₂×D₄×C₆₇	—	2	4	no	—
123	C₂×D₄×C₆₇	—	2	4	no	—
124	C₂×D₄×C₆₇	—	2	4	no	—
125	C₂×D₄×C₆₇	—	2	4	no	—
126	C₂×D₄×C₆₇	—	2	4	no	—
127	C₂×D₄×C₆₇	—	2	4	no	—
128	C₂×D₄×C₆₇	—	4	2	no	—
129	C₂×D₄×C₆₇	—	4	2	no	—
130	C₂²×C₄×C₆₇	—	2	8	no	—
131	C₂²×C₄×C₆₇	—	2	8	no	—
132	C₂²×C₄×C₆₇	—	2	8	no	—
133	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
134	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
135	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
136	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
137	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
138	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
139	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
140	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
141	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
142	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
143	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
144	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
145	C₄⋊C₄×C₆₇	—	2	4	no	—
146	C₄⋊C₄×C₆₇	—	2	4	no	—
147	C₄⋊C₄×C₆₇	—	2	4	no	—
148	C₄⋊C₄×C₆₇	—	2	8	no	—
149	C₄⋊C₄×C₆₇	—	2	4	no	—
150	C₄⋊C₄×C₆₇	—	2	4	no	—
151	C₄⋊C₄×C₆₇	—	2	4	no	—
152	C₄⋊C₄×C₆₇	—	2	4	no	—
153	C₄⋊C₄×C₆₇	—	2	4	no	—
154	C₄⋊C₄×C₆₇	—	2	4	no	—
155	C₄⋊C₄×C₆₇	—	2	4	no	—
156	C₄⋊C₄×C₆₇	—	4	2	no	—
157	C₄⋊C₄×C₆₇	—	4	2	no	—
158	C₄⋊C₄×C₆₇	—	4	2	no	—
159	C₄⋊C₄×C₆₇	—	4	4	no	—
160	C₂²⋊C₄×C₆₇	—	2	8	no	—
161	C₂²⋊C₄×C₆₇	—	2	8	no	—
162	C₂²⋊C₄×C₆₇	—	4	8	no	—
163	C₂²⋊C₄×C₆₇	—	4	8	no	—
164	C₂²⋊C₄×C₆₇	—	4	4	no	—
165	C₂²⋊C₄×C₆₇	—	4	8	no	—
166	C₂²⋊C₄×C₆₇	—	4	8	no	—
167	C₄²×C₆₇	—	2	8	no	—
168	C₄²×C₆₇	—	2	2	no	—
169	C₄²×C₆₇	—	2	2	no	—
170	C₄²×C₆₇	—	2	8	no	—
171	C₄²×C₆₇	—	2	2	no	—
172	C₄²×C₆₇	—	4	1	no	—
173	C₄²×C₆₇	—	4	8	no	—
174	C₄²×C₆₇	#41	1	4	yes	—
175	C₄²×C₆₇	#41	1	4	yes	—
176	C₄²×C₆₇	#41	1	4	yes	—
177	C₂³×C₆₇	—	2	4	no	—
178	C₂³×C₆₇	—	2	4	no	—
179	C₂³×C₆₇	—	2	4	no	—
180	C₂³×C₆₇	—	2	4	no	—
181	C₂³×C₆₇	—	2	4	no	—
182	C₂³×C₆₇	—	2	4	no	—
183	C₂³×C₆₇	—	2	4	no	—
184	C₂³×C₆₇	C₂⁵	1	1	yes	characteristic
185	Q₈×C₆₇	—	4	8	no	—
186	Q₈×C₆₇	—	4	8	no	—
187	D₄×C₆₇	—	4	8	no	—
188	D₄×C₆₇	—	4	8	no	—
189	D₄×C₆₇	—	4	8	no	—
190	C₂×C₄×C₆₇	—	2	1	no	—
191	C₂×C₄×C₆₇	—	2	1	no	—
192	C₂×C₄×C₆₇	—	2	1	no	—
193	C₂×C₄×C₆₇	—	2	1	no	—
194	C₂×C₄×C₆₇	—	2	1	no	—
195	C₂×C₄×C₆₇	—	2	1	no	—
196	C₂×C₄×C₆₇	—	2	4	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	4	no	—
201	C₂×C₄×C₆₇	—	2	4	no	—
202	C₂×C₄×C₆₇	—	2	1	no	—
203	C₂×C₄×C₆₇	—	2	4	no	—
204	C₂×C₄×C₆₇	—	2	4	no	—
205	C₂×C₄×C₆₇	—	2	1	no	—
206	C₂×C₄×C₆₇	—	2	4	no	—
207	C₂×C₄×C₆₇	—	2	4	no	—
208	C₂×C₄×C₆₇	—	2	4	no	—
209	C₂×C₄×C₆₇	—	2	4	no	—
210	C₂×C₄×C₆₇	—	2	1	no	—
211	C₂×C₄×C₆₇	—	2	4	no	—
212	C₂×C₄×C₆₇	—	2	4	no	—
213	C₂×C₄×C₆₇	—	2	4	no	—
214	C₂×C₄×C₆₇	—	4	16	no	—
215	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
216	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
217	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
218	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
219	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
220	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
221	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
222	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
223	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
224	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
225	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
226	C₂×C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
227	C₂²×C₆₇	—	4	4	no	—
228	C₂²×C₆₇	—	4	4	no	—
229	C₂²×C₆₇	—	4	4	no	—
230	C₂²×C₆₇	—	4	4	no	—
231	C₂²×C₆₇	—	4	4	no	—
232	C₂²×C₆₇	—	4	4	no	—
233	C₂²×C₆₇	—	4	4	no	—
234	C₂²×C₆₇	C₂²×#41	1	1	yes	characteristic
235	C₂²×C₆₇	C₂²×#41	1	1	yes	characteristic
236	C₂²×C₆₇	C₂²×#41	1	1	yes	characteristic
237	C₂²×C₆₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
238	C₂²×C₆₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
239	C₂²×C₆₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
240	C₂²×C₆₇	#585	1	1	yes	characteristic
241	C₄×C₆₇	—	2	2	no	—
242	C₄×C₆₇	—	2	2	no	—
243	C₄×C₆₇	—	2	2	no	—
244	C₄×C₆₇	—	4	4	no	—
245	C₄×C₆₇	—	4	4	no	—
246	C₄×C₆₇	—	4	4	no	—
247	C₄×C₆₇	—	4	4	no	—
248	C₄×C₆₇	—	4	1	no	—
249	C₄×C₆₇	—	4	4	no	—
250	C₄×C₆₇	#555	1	4	yes	—
251	C₄×C₆₇	#556	1	4	yes	—
252	C₄×C₆₇	#559	1	4	yes	—
253	#59863	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
254	C₂×C₆₇	—	8	4	no	—
255	C₂×C₆₇	C₂×#555	1	1	yes	characteristic
256	C₂×C₆₇	C₂×#556	1	1	yes	characteristic
257	C₂×C₆₇	C₂×#559	1	1	yes	characteristic
258	C₂×C₆₇	#3506	1	1	yes	characteristic
259	C₂×C₆₇	#3506	1	1	yes	characteristic
260	C₂×C₆₇	#3507	1	1	yes	characteristic
261	C₂×C₆₇	#3548	1	1	yes	characteristic
262	#2808	C₂×C₆₇	1	4	yes	complemented
263	#2592	C₂×C₆₇	1	4	yes	complemented
264	#2563	C₂×C₆₇	1	2	yes	complemented
265	#2548	C₂×C₆₇	1	2	yes	—
266	#2547	C₂×C₆₇	1	2	yes	complemented
267	#2513	C₂×C₆₇	1	4	yes	complemented
268	#2485	C₂×C₆₇	1	4	yes	complemented
269	#2355	C₂×C₆₇	1	2	yes	complemented
270	#2354	C₂×C₆₇	1	2	yes	complemented
271	#2354	C₂×C₆₇	1	2	yes	complemented
272	C₄×C₄⋊Q₈	C₂×C₆₇	1	1	yes	characteristic, complemented
273	C₄×C₄⋊Q₈	C₂×C₆₇	1	1	yes	characteristic, complemented
274	C₄×#127	C₂×C₆₇	1	1	yes	characteristic
275	C₆₇	#59863	1	1	yes	characteristic, direct product factor, Sylow
276	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
277	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
278	C₂×#127	C₂²×C₆₇	1	1	yes	characteristic
279	C₂×#124	C₂²×C₆₇	1	4	yes	—
280	C₂×C₄×Q₈	C₂²×C₆₇	1	2	yes	—
281	C₂×C₄×Q₈	C₂²×C₆₇	1	2	yes	—
282	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
283	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
284	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
285	C₂×#117	C₂²×C₆₇	1	4	yes	—
286	#397	C₂²×C₆₇	1	8	yes	—
287	#397	C₂²×C₆₇	1	8	yes	—
288	#394	C₂²×C₆₇	1	8	yes	—
289	#393	C₂²×C₆₇	1	8	yes	—
290	#391	C₂²×C₆₇	1	2	yes	—
291	#391	C₂²×C₆₇	1	2	yes	—
292	#391	C₂²×C₆₇	1	4	yes	—
293	#390	C₂²×C₆₇	1	2	yes	—
294	#390	C₂²×C₆₇	1	2	yes	—
295	#390	C₂²×C₆₇	1	2	yes	—
296	#390	C₂²×C₆₇	1	4	yes	—
297	#390	C₂²×C₆₇	1	4	yes	—
298	#390	C₂²×C₆₇	1	4	yes	—
299	#389	C₂²×C₆₇	1	2	yes	—
300	#389	C₂²×C₆₇	1	2	yes	—
301	#389	C₂²×C₆₇	1	2	yes	—
302	#389	C₂²×C₆₇	1	2	yes	—
303	#388	C₂²×C₆₇	1	8	yes	—
304	#388	C₂²×C₆₇	1	8	yes	—
305	#388	C₂²×C₆₇	1	8	yes	—
306	#387	C₂²×C₆₇	1	8	yes	—
307	#384	C₂²×C₆₇	1	1	yes	characteristic
308	#384	C₂²×C₆₇	1	1	yes	characteristic
309	#384	C₂²×C₆₇	1	1	yes	characteristic
310	#382	C₂²×C₆₇	1	4	yes	—
311	#380	C₂²×C₆₇	1	4	yes	—
312	#380	C₂²×C₆₇	1	4	yes	—
313	#380	C₂²×C₆₇	1	4	yes	—
314	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
315	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
316	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
317	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
318	C₄×C₄⋊C₄	C₂²×C₆₇	1	4	yes	—
319	C₄×C₄⋊C₄	C₂²×C₆₇	1	4	yes	—
320	C₄³	C₂²×C₆₇	1	1	yes	characteristic
321	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
322	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
323	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
324	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
325	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
326	C₂³×C₄	C₂³×C₆₇	1	4	yes	—
327	C₄⋊Q₈	—	2	4	no	—
328	C₄⋊Q₈	—	2	4	no	—
329	#127	—	2	4	no	—
330	#124	—	4	8	no	—
331	C₄×Q₈	—	2	8	no	—
332	C₄×Q₈	—	4	4	no	—
333	C₄×D₄	—	2	8	no	—
334	C₄×D₄	—	2	8	no	—
335	C₄×D₄	—	4	4	no	—
336	#117	—	4	8	no	—
337	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
338	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
339	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
340	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
341	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
342	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
343	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
344	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
345	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
346	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
347	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
348	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
349	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
350	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
351	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
352	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
353	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
354	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
355	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
356	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
357	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
358	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
359	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
360	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
361	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
362	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
363	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
364	C₂×C₄²	C₂³×C₆₇	1	8	yes	—
365	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
366	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
367	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
368	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
369	#95	C₂³×C₆₇	1	8	yes	—
370	#95	C₂³×C₆₇	1	8	yes	—
371	#95	C₂³×C₆₇	1	4	yes	—
372	#95	C₂³×C₆₇	1	4	yes	—
373	#95	C₂³×C₆₇	1	4	yes	—
374	#95	C₂³×C₆₇	1	4	yes	—
375	#95	C₂³×C₆₇	1	8	yes	—
376	C₂⁴	C₂⁴×C₆₇	1	4	yes	—
377	C₂×Q₈	—	2	4	no	—
378	C₂×Q₈	—	2	4	no	—
379	C₂×Q₈	—	2	4	no	—
380	C₂×Q₈	—	2	4	no	—
381	C₂×Q₈	—	4	2	no	—
382	C₂×Q₈	—	4	2	no	—
383	C₂×D₄	—	2	4	no	—
384	C₂×D₄	—	2	4	no	—
385	C₂×D₄	—	2	4	no	—
386	C₂×D₄	—	2	4	no	—
387	C₂×D₄	—	2	4	no	—
388	C₂×D₄	—	2	4	no	—
389	C₂×D₄	—	2	4	no	—
390	C₂×D₄	—	4	2	no	—
391	C₂×D₄	—	4	2	no	—
392	C₂²×C₄	—	2	8	no	—
393	C₂²×C₄	—	2	8	no	—
394	C₂²×C₄	—	2	8	no	—
395	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
396	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
397	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
398	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
399	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
400	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
401	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
402	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
403	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
404	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
405	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
406	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
407	C₄⋊C₄	—	2	4	no	—
408	C₄⋊C₄	—	2	4	no	—
409	C₄⋊C₄	—	2	4	no	—
410	C₄⋊C₄	—	2	8	no	—
411	C₄⋊C₄	—	2	4	no	—
412	C₄⋊C₄	—	2	4	no	—
413	C₄⋊C₄	—	2	4	no	—
414	C₄⋊C₄	—	2	4	no	—
415	C₄⋊C₄	—	2	4	no	—
416	C₄⋊C₄	—	2	4	no	—
417	C₄⋊C₄	—	2	4	no	—
418	C₄⋊C₄	—	4	2	no	—
419	C₄⋊C₄	—	4	2	no	—
420	C₄⋊C₄	—	4	4	no	—
421	C₄⋊C₄	—	4	2	no	—
422	C₂²⋊C₄	—	2	8	no	—
423	C₂²⋊C₄	—	2	8	no	—
424	C₂²⋊C₄	—	4	8	no	—
425	C₂²⋊C₄	—	4	4	no	—
426	C₂²⋊C₄	—	4	8	no	—
427	C₂²⋊C₄	—	4	8	no	—
428	C₂²⋊C₄	—	4	8	no	—
429	C₄²	—	2	8	no	—
430	C₄²	—	2	2	no	—
431	C₄²	—	2	2	no	—
432	C₄²	—	2	8	no	—
433	C₄²	—	2	2	no	—
434	C₄²	—	4	8	no	—
435	C₄²	—	4	1	no	—
436	C₄²	#41×C₆₇	1	4	yes	—
437	C₄²	#41×C₆₇	1	4	yes	—
438	C₄²	#41×C₆₇	1	4	yes	—
439	C₂³	—	2	4	no	—
440	C₂³	—	2	4	no	—
441	C₂³	—	2	4	no	—
442	C₂³	—	2	4	no	—
443	C₂³	—	2	4	no	—
444	C₂³	—	2	4	no	—
445	C₂³	—	2	4	no	—
446	C₂³	C₂⁵×C₆₇	1	1	yes	characteristic
447	Q₈	—	4	8	no	—
448	Q₈	—	4	8	no	—
449	D₄	—	4	8	no	—
450	D₄	—	4	8	no	—
451	D₄	—	4	8	no	—
452	C₂×C₄	—	2	1	no	—
453	C₂×C₄	—	2	1	no	—
454	C₂×C₄	—	2	1	no	—
455	C₂×C₄	—	2	1	no	—
456	C₂×C₄	—	2	1	no	—
457	C₂×C₄	—	2	1	no	—
458	C₂×C₄	—	2	4	no	—
459	C₂×C₄	—	2	4	no	—
460	C₂×C₄	—	2	4	no	—
461	C₂×C₄	—	2	4	no	—
462	C₂×C₄	—	2	4	no	—
463	C₂×C₄	—	2	4	no	—
464	C₂×C₄	—	2	1	no	—
465	C₂×C₄	—	2	4	no	—
466	C₂×C₄	—	2	4	no	—
467	C₂×C₄	—	2	1	no	—
468	C₂×C₄	—	2	4	no	—
469	C₂×C₄	—	2	4	no	—
470	C₂×C₄	—	2	4	no	—
471	C₂×C₄	—	2	4	no	—
472	C₂×C₄	—	2	1	no	—
473	C₂×C₄	—	2	4	no	—
474	C₂×C₄	—	2	4	no	—
475	C₂×C₄	—	2	4	no	—
476	C₂×C₄	—	4	16	no	—
477	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
478	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
479	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
480	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
481	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
482	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
483	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
484	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
485	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
486	C₂×C₄	H(5,𝔽₂)×C₆₇	1	2	yes	—
487	C₂×C₄	H(5,𝔽₂)×C₆₇	1	2	yes	—
488	C₂×C₄	2₋¹⁺⁴×C₆₇	1	2	yes	—
489	C₂²	—	4	4	no	—
490	C₂²	—	4	4	no	—
491	C₂²	—	4	4	no	—
492	C₂²	—	4	4	no	—
493	C₂²	—	4	4	no	—
494	C₂²	—	4	4	no	—
495	C₂²	—	4	4	no	—
496	C₂²	C₂²×#41×C₆₇	1	1	yes	characteristic
497	C₂²	C₂²×#41×C₆₇	1	1	yes	characteristic
498	C₂²	C₂²×#41×C₆₇	1	1	yes	characteristic
499	C₂²	C₂×H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
500	C₂²	C₂×H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
501	C₂²	C₂×2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
502	C₂²	#585×C₆₇	1	1	yes	characteristic
503	C₄	—	2	2	no	—
504	C₄	—	2	2	no	—
505	C₄	—	2	2	no	—
506	C₄	—	4	1	no	—
507	C₄	—	4	4	no	—
508	C₄	—	4	4	no	—
509	C₄	—	4	4	no	—
510	C₄	—	4	4	no	—
511	C₄	—	4	4	no	—
512	C₄	#555×C₆₇	1	4	yes	—
513	C₄	#556×C₆₇	1	4	yes	—
514	C₄	#559×C₆₇	1	4	yes	—
515	C₂	—	8	4	no	—
516	C₂	C₂×#555×C₆₇	1	1	yes	characteristic
517	C₂	C₂×#556×C₆₇	1	1	yes	characteristic
518	C₂	C₂×#559×C₆₇	1	1	yes	characteristic
519	C₂	C₆₇×#3506	1	1	yes	characteristic
520	C₂	C₆₇×#3506	1	1	yes	characteristic
521	C₂	C₆₇×#3507	1	1	yes	characteristic
522	C₂	C₆₇×#3548	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁵×C₆₇
prime radical	C₂×#127×C₆₇	C₂²
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. C₆₇×#2808	C₂	semidirect
2. C₆₇×#2592	C₂	semidirect
3. C₆₇×#2563	C₂	semidirect
5. C₆₇×#2547	C₂	semidirect
6. C₆₇×#2513	C₂	semidirect
7. C₆₇×#2485	C₂	semidirect
8. C₆₇×#2355	C₂	semidirect
9. C₆₇×#2354	C₂	semidirect
10. C₆₇×#2354	C₂	semidirect
11. C₄×C₄⋊Q₈×C₆₇	C₂	semidirect
12. C₄×C₄⋊Q₈×C₆₇	C₂	semidirect
253. #59863	C₆₇	direct (N × H)
262. #2808	C₂×C₆₇	semidirect
263. #2592	C₂×C₆₇	semidirect
264. #2563	C₂×C₆₇	semidirect
266. #2547	C₂×C₆₇	semidirect
267. #2513	C₂×C₆₇	semidirect
268. #2485	C₂×C₆₇	semidirect
269. #2355	C₂×C₆₇	semidirect
270. #2354	C₂×C₆₇	semidirect
271. #2354	C₂×C₆₇	semidirect
272. C₄×C₄⋊Q₈	C₂×C₆₇	semidirect
273. C₄×C₄⋊Q₈	C₂×C₆₇	semidirect
275. C₆₇	#59863	direct (N × H)

Maximal subgroups

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

subgroup	index
C₆₇×#2808	2
C₆₇×#2592	2
C₆₇×#2563	2
C₆₇×#2548	2
C₆₇×#2547	2
C₆₇×#2513	2
C₆₇×#2485	2
C₆₇×#2355	2
C₆₇×#2354	2
C₄×C₄⋊Q₈×C₆₇	2
C₄×#127×C₆₇	2
#59863	67

Subgroups — C67×#59863 #30739114

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#59863 #30739114