Subgroups — C₆₇×#24658 #30703909

Order 17152.

1100 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#459×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#432×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#2856	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄×#121×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#2034	C₂	1	2	yes	maximal, complemented
6	C₆₇×#1999	C₂	1	2	yes	maximal, complemented
7	C₆₇×#1982	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₇×#1972	C₂	1	2	yes	maximal, complemented
9	C₆₇×#1948	C₂	1	1	yes	maximal, characteristic, complemented
10	C₆₇×#1936	C₂	1	2	yes	maximal, complemented
11	C₆₇×#1802	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×D₈×C₆₇	C₂²	1	2	yes	—
13	C₂×#127×C₆₇	C₂²	1	1	yes	characteristic
14	C₂×#121×C₆₇	—	2	1	no	—
15	C₂×#121×C₆₇	—	2	1	no	—
16	C₂×#121×C₆₇	C₂²	1	1	yes	characteristic
17	C₂×#121×C₆₇	C₂²	1	2	yes	—
18	C₂×C₄×D₄×C₆₇	—	2	1	no	—
19	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
20	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
21	#459×C₆₇	—	2	4	no	—
22	#432×C₆₇	—	2	4	no	—
23	C₂×#106×C₆₇	C₂²	1	2	yes	—
24	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
25	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
26	C₂×#102×C₆₇	C₂²	1	2	yes	—
27	C₂×#102×C₆₇	C₂²	1	2	yes	—
28	C₂×#102×C₆₇	C₂²	1	2	yes	—
29	C₂×#102×C₆₇	C₂²	1	2	yes	—
30	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
31	C₂×#97×C₆₇	C₂²	1	2	yes	—
32	#390×C₆₇	—	2	1	no	—
33	#390×C₆₇	—	2	1	no	—
34	#390×C₆₇	C₂²	1	1	yes	characteristic
35	#390×C₆₇	C₂²	1	2	yes	—
36	#389×C₆₇	C₂²	1	1	yes	characteristic
37	#388×C₆₇	—	2	1	no	—
38	#386×C₆₇	—	2	1	no	—
39	#384×C₆₇	C₂²	1	1	yes	characteristic
40	#380×C₆₇	C₂²	1	1	yes	characteristic
41	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
42	C₄×C₂²⋊C₄×C₆₇	—	2	1	no	—
43	#340×C₆₇	C₂²	1	2	yes	—
44	#340×C₆₇	C₂²	1	2	yes	—
45	C₂²×D₄×C₆₇	—	2	1	no	—
46	C₂²×D₄×C₆₇	—	2	2	no	—
47	C₂²×D₄×C₆₇	—	2	1	no	—
48	C₂²×D₄×C₆₇	—	2	1	no	—
49	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
50	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
51	C₂³×C₄×C₆₇	—	2	1	no	—
52	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
53	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
54	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
55	C₂×D₈×C₆₇	—	2	4	no	—
56	C₂×D₈×C₆₇	—	2	4	no	—
57	C₂×D₈×C₆₇	—	4	2	no	—
58	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
59	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
60	#127×C₆₇	—	2	2	no	—
61	#127×C₆₇	—	2	2	no	—
62	#121×C₆₇	—	2	2	no	—
63	#121×C₆₇	—	2	2	no	—
64	#121×C₆₇	—	4	2	no	—
65	#121×C₆₇	—	4	2	no	—
66	#121×C₆₇	—	4	4	no	—
67	#121×C₆₇	—	4	4	no	—
68	C₄×D₄×C₆₇	—	2	2	no	—
69	C₄×D₄×C₆₇	—	2	2	no	—
70	C₄×D₄×C₆₇	—	4	4	no	—
71	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
72	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
73	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
74	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
75	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
76	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
77	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
78	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
79	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
80	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
81	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
82	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
83	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
84	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
85	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
86	C₂×C₄²×C₆₇	—	2	2	no	—
87	C₂×C₄²×C₆₇	—	2	2	no	—
88	C₂×C₄²×C₆₇	—	2	1	no	—
89	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
90	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
91	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
92	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
93	#106×C₆₇	—	4	2	no	—
94	C₄⋊C₈×C₆₇	—	2	2	no	—
95	C₄⋊C₈×C₆₇	D₄	1	4	yes	complemented
96	#102×C₆₇	—	2	4	no	—
97	#102×C₆₇	—	2	4	no	—
98	#102×C₆₇	—	4	2	no	—
99	#102×C₆₇	—	4	2	no	—
100	C₂²⋊C₈×C₆₇	—	2	4	no	—
101	#97×C₆₇	—	2	4	no	—
102	#95×C₆₇	—	2	1	no	—
103	#95×C₆₇	—	2	1	no	—
104	#95×C₆₇	—	2	1	no	—
105	C₂⁴×C₆₇	—	2	1	no	—
106	C₂⁴×C₆₇	—	2	2	no	—
107	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
108	C₂×D₄×C₆₇	—	2	4	no	—
109	C₂×D₄×C₆₇	—	2	1	no	—
110	C₂×D₄×C₆₇	—	2	2	no	—
111	C₂×D₄×C₆₇	—	2	1	no	—
112	C₂×D₄×C₆₇	—	2	1	no	—
113	C₂×D₄×C₆₇	—	2	2	no	—
114	C₂×D₄×C₆₇	—	2	2	no	—
115	C₂×D₄×C₆₇	—	2	2	no	—
116	C₂×D₄×C₆₇	—	2	1	no	—
117	C₂×D₄×C₆₇	—	2	1	no	—
118	C₂×D₄×C₆₇	—	2	2	no	—
119	C₂×D₄×C₆₇	—	2	2	no	—
120	C₂×D₄×C₆₇	—	4	2	no	—
121	C₂×D₄×C₆₇	—	4	2	no	—
122	C₂×D₄×C₆₇	—	4	2	no	—
123	C₂×D₄×C₆₇	—	4	2	no	—
124	C₂×D₄×C₆₇	—	4	4	no	—
125	C₂×D₄×C₆₇	—	4	4	no	—
126	C₂×D₄×C₆₇	—	4	4	no	—
127	C₂×D₄×C₆₇	—	4	2	no	—
128	C₂×D₄×C₆₇	—	4	2	no	—
129	C₂×D₄×C₆₇	—	4	2	no	—
130	C₂×D₄×C₆₇	—	4	2	no	—
131	C₂×D₄×C₆₇	#41	1	2	yes	—
132	C₂²×C₄×C₆₇	—	2	2	no	—
133	C₂²×C₄×C₆₇	—	2	1	no	—
134	C₂²×C₄×C₆₇	—	2	2	no	—
135	C₂²×C₄×C₆₇	—	2	2	no	—
136	C₂²×C₄×C₆₇	—	2	2	no	—
137	C₂²×C₄×C₆₇	—	2	2	no	—
138	C₂²×C₄×C₆₇	—	2	1	no	—
139	C₂²×C₄×C₆₇	—	2	1	no	—
140	C₂²×C₄×C₆₇	—	2	1	no	—
141	C₂²×C₄×C₆₇	—	2	1	no	—
142	C₂²×C₄×C₆₇	—	2	1	no	—
143	C₂²×C₄×C₆₇	—	2	1	no	—
144	C₂²×C₄×C₆₇	—	2	1	no	—
145	C₂²×C₄×C₆₇	—	2	2	no	—
146	C₂²×C₄×C₆₇	—	2	1	no	—
147	C₂²×C₄×C₆₇	—	4	1	no	—
148	C₂²×C₄×C₆₇	—	4	2	no	—
149	C₂²×C₄×C₆₇	—	4	2	no	—
150	C₂²×C₄×C₆₇	—	4	2	no	—
151	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
152	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
153	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
154	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
155	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
156	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
157	D₈×C₆₇	—	4	8	no	—
158	C₂×C₈×C₆₇	—	2	2	no	—
159	C₂×C₈×C₆₇	—	2	2	no	—
160	C₂×C₈×C₆₇	—	2	2	no	—
161	C₂×C₈×C₆₇	—	2	2	no	—
162	C₂×C₈×C₆₇	—	2	2	no	—
163	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
164	C₄⋊C₄×C₆₇	—	2	1	no	—
165	C₄⋊C₄×C₆₇	—	2	1	no	—
166	C₄⋊C₄×C₆₇	—	2	1	no	—
167	C₄⋊C₄×C₆₇	—	2	1	no	—
168	C₄⋊C₄×C₆₇	—	2	2	no	—
169	C₄⋊C₄×C₆₇	—	2	1	no	—
170	C₄⋊C₄×C₆₇	—	2	2	no	—
171	C₄⋊C₄×C₆₇	—	4	2	no	—
172	C₄⋊C₄×C₆₇	—	4	2	no	—
173	C₄⋊C₄×C₆₇	—	4	2	no	—
174	C₄⋊C₄×C₆₇	#41	1	2	yes	—
175	C₂²⋊C₄×C₆₇	—	2	4	no	—
176	C₂²⋊C₄×C₆₇	—	4	2	no	—
177	C₂²⋊C₄×C₆₇	—	4	2	no	—
178	C₂²⋊C₄×C₆₇	—	4	4	no	—
179	C₂²⋊C₄×C₆₇	—	4	4	no	—
180	C₂²⋊C₄×C₆₇	—	4	4	no	—
181	C₂²⋊C₄×C₆₇	—	4	2	no	—
182	C₂²⋊C₄×C₆₇	—	4	2	no	—
183	C₄²×C₆₇	—	2	2	no	—
184	C₄²×C₆₇	—	2	2	no	—
185	C₄²×C₆₇	—	2	1	no	—
186	C₄²×C₆₇	—	2	1	no	—
187	C₄²×C₆₇	—	4	2	no	—
188	C₄²×C₆₇	C₂×D₄	1	2	yes	—
189	C₄²×C₆₇	#41	1	2	yes	—
190	C₂³×C₆₇	—	2	1	no	—
191	C₂³×C₆₇	—	2	1	no	—
192	C₂³×C₆₇	—	2	1	no	—
193	C₂³×C₆₇	—	2	1	no	—
194	C₂³×C₆₇	—	2	1	no	—
195	C₂³×C₆₇	—	2	1	no	—
196	C₂³×C₆₇	—	2	2	no	—
197	C₂³×C₆₇	—	4	1	no	—
198	C₂³×C₆₇	—	4	1	no	—
199	C₂³×C₆₇	—	4	1	no	—
200	C₂³×C₆₇	—	4	1	no	—
201	C₂³×C₆₇	—	4	1	no	—
202	C₂³×C₆₇	—	4	1	no	—
203	C₂³×C₆₇	—	4	2	no	—
204	C₂³×C₆₇	—	4	2	no	—
205	C₂³×C₆₇	—	4	2	no	—
206	C₂³×C₆₇	—	4	2	no	—
207	C₂³×C₆₇	—	4	2	no	—
208	C₂³×C₆₇	—	4	2	no	—
209	C₂³×C₆₇	—	4	2	no	—
210	C₂³×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
211	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
212	D₄×C₆₇	—	2	2	no	—
213	D₄×C₆₇	—	2	2	no	—
214	D₄×C₆₇	—	4	8	no	—
215	D₄×C₆₇	—	4	2	no	—
216	D₄×C₆₇	—	4	1	no	—
217	D₄×C₆₇	—	4	2	no	—
218	D₄×C₆₇	—	4	2	no	—
219	D₄×C₆₇	—	4	1	no	—
220	D₄×C₆₇	—	4	2	no	—
221	D₄×C₆₇	—	8	4	no	—
222	D₄×C₆₇	—	8	8	no	—
223	D₄×C₆₇	—	8	4	no	—
224	C₂×C₄×C₆₇	—	2	1	no	—
225	C₂×C₄×C₆₇	—	2	1	no	—
226	C₂×C₄×C₆₇	—	2	1	no	—
227	C₂×C₄×C₆₇	—	2	1	no	—
228	C₂×C₄×C₆₇	—	2	1	no	—
229	C₂×C₄×C₆₇	—	2	1	no	—
230	C₂×C₄×C₆₇	—	2	2	no	—
231	C₂×C₄×C₆₇	—	2	1	no	—
232	C₂×C₄×C₆₇	—	2	1	no	—
233	C₂×C₄×C₆₇	—	2	1	no	—
234	C₂×C₄×C₆₇	—	2	1	no	—
235	C₂×C₄×C₆₇	—	2	1	no	—
236	C₂×C₄×C₆₇	—	4	4	no	—
237	C₂×C₄×C₆₇	—	4	1	no	—
238	C₂×C₄×C₆₇	—	4	1	no	—
239	C₂×C₄×C₆₇	—	4	1	no	—
240	C₂×C₄×C₆₇	—	4	1	no	—
241	C₂×C₄×C₆₇	—	4	1	no	—
242	C₂×C₄×C₆₇	—	4	1	no	—
243	C₂×C₄×C₆₇	—	4	4	no	—
244	C₂×C₄×C₆₇	—	4	1	no	—
245	C₂×C₄×C₆₇	—	4	2	no	—
246	C₂×C₄×C₆₇	—	4	1	no	—
247	C₂×C₄×C₆₇	—	4	1	no	—
248	C₂×C₄×C₆₇	—	4	1	no	—
249	C₂×C₄×C₆₇	—	4	2	no	—
250	C₂×C₄×C₆₇	—	4	2	no	—
251	C₂×C₄×C₆₇	—	4	1	no	—
252	C₂×C₄×C₆₇	—	4	1	no	—
253	C₂×C₄×C₆₇	—	8	4	no	—
254	C₂×C₄×C₆₇	#121	1	2	yes	—
255	C₂×C₄×C₆₇	#121	1	2	yes	—
256	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
257	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
258	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
259	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
260	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
261	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
262	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
263	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
264	C₈×C₆₇	—	2	4	no	—
265	C₈×C₆₇	—	4	2	no	—
266	C₂²×C₆₇	—	2	2	no	—
267	C₂²×C₆₇	—	2	2	no	—
268	C₂²×C₆₇	—	2	2	no	—
269	C₂²×C₆₇	—	4	1	no	—
270	C₂²×C₆₇	—	4	1	no	—
271	C₂²×C₆₇	—	4	1	no	—
272	C₂²×C₆₇	—	4	1	no	—
273	C₂²×C₆₇	—	4	2	no	—
274	C₂²×C₆₇	—	4	2	no	—
275	C₂²×C₆₇	—	4	2	no	—
276	C₂²×C₆₇	—	8	1	no	—
277	C₂²×C₆₇	—	8	1	no	—
278	C₂²×C₆₇	—	8	1	no	—
279	C₂²×C₆₇	—	8	1	no	—
280	C₂²×C₆₇	—	8	2	no	—
281	C₂²×C₆₇	—	8	2	no	—
282	C₂²×C₆₇	—	8	2	no	—
283	C₂²×C₆₇	—	8	2	no	—
284	C₂²×C₆₇	—	8	2	no	—
285	C₂²×C₆₇	—	8	2	no	—
286	C₂²×C₆₇	—	8	2	no	—
287	C₂²×C₆₇	C₂×#121	1	1	yes	characteristic
288	C₂²×C₆₇	#525	1	1	yes	characteristic
289	C₂²×C₆₇	#535	1	1	yes	characteristic
290	C₂²×C₆₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
291	C₂²×C₆₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
292	C₂²×C₆₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
293	C₂²×C₆₇	#575	1	1	yes	characteristic
294	C₄×C₆₇	—	2	2	no	—
295	C₄×C₆₇	—	4	2	no	—
296	C₄×C₆₇	—	4	1	no	—
297	C₄×C₆₇	—	4	1	no	—
298	C₄×C₆₇	—	4	1	no	—
299	C₄×C₆₇	—	8	1	no	—
300	C₄×C₆₇	—	8	2	no	—
301	C₄×C₆₇	—	8	1	no	—
302	C₄×C₆₇	#469	1	4	yes	—
303	C₄×C₆₇	#546	1	1	yes	characteristic
304	C₄×C₆₇	#550	1	1	yes	characteristic
305	C₄×C₆₇	#553	1	1	yes	characteristic
306	C₄×C₆₇	#555	1	1	yes	characteristic
307	#24658	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
308	C₂×C₆₇	—	4	2	no	—
309	C₂×C₆₇	—	8	2	no	—
310	C₂×C₆₇	—	16	2	no	—
311	C₂×C₆₇	#2414	1	1	yes	characteristic
312	C₂×C₆₇	C₂×#469	1	1	yes	characteristic
313	C₂×C₆₇	#3055	1	1	yes	characteristic
314	C₂×C₆₇	#3109	1	1	yes	characteristic
315	C₂×C₆₇	#3115	1	1	yes	characteristic
316	C₂×C₆₇	#3214	1	1	yes	characteristic
317	C₂×C₆₇	#3217	1	1	yes	characteristic
318	C₂×#459	C₂×C₆₇	1	1	yes	characteristic, complemented
319	C₂×#432	C₂×C₆₇	1	1	yes	characteristic, complemented
320	#2856	C₂×C₆₇	1	1	yes	characteristic, complemented
321	C₄×#121	C₂×C₆₇	1	1	yes	characteristic, complemented
322	#2034	C₂×C₆₇	1	2	yes	complemented
323	#1999	C₂×C₆₇	1	2	yes	complemented
324	#1982	C₂×C₆₇	1	1	yes	characteristic, complemented
325	#1972	C₂×C₆₇	1	2	yes	complemented
326	#1948	C₂×C₆₇	1	1	yes	characteristic, complemented
327	#1936	C₂×C₆₇	1	2	yes	complemented
328	#1802	C₂×C₆₇	1	1	yes	characteristic, complemented
329	C₆₇	#24658	1	1	yes	characteristic, direct product factor, Sylow
330	C₂²×D₈	C₂²×C₆₇	1	2	yes	—
331	C₂×#127	C₂²×C₆₇	1	1	yes	characteristic
332	C₂×#121	—	2	1	no	—
333	C₂×#121	—	2	1	no	—
334	C₂×#121	C₂²×C₆₇	1	1	yes	characteristic
335	C₂×#121	C₂²×C₆₇	1	2	yes	—
336	C₂×C₄×D₄	—	2	1	no	—
337	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
338	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
339	#459	—	2	4	no	—
340	#432	—	2	4	no	—
341	C₂×#106	C₂²×C₆₇	1	2	yes	—
342	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
343	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
344	C₂×#102	C₂²×C₆₇	1	2	yes	—
345	C₂×#102	C₂²×C₆₇	1	2	yes	—
346	C₂×#102	C₂²×C₆₇	1	2	yes	—
347	C₂×#102	C₂²×C₆₇	1	2	yes	—
348	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
349	C₂×#97	C₂²×C₆₇	1	2	yes	—
350	#390	—	2	1	no	—
351	#390	—	2	1	no	—
352	#390	C₂²×C₆₇	1	2	yes	—
353	#390	C₂²×C₆₇	1	1	yes	characteristic
354	#389	C₂²×C₆₇	1	1	yes	characteristic
355	#388	—	2	1	no	—
356	#386	—	2	1	no	—
357	#384	C₂²×C₆₇	1	1	yes	characteristic
358	#380	C₂²×C₆₇	1	1	yes	characteristic
359	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
360	C₄×C₂²⋊C₄	—	2	1	no	—
361	#340	C₂²×C₆₇	1	2	yes	—
362	#340	C₂²×C₆₇	1	2	yes	—
363	C₂²×D₄	—	2	1	no	—
364	C₂²×D₄	—	2	2	no	—
365	C₂²×D₄	—	2	1	no	—
366	C₂²×D₄	—	2	1	no	—
367	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
368	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
369	C₂³×C₄	—	2	1	no	—
370	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
371	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
372	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
373	C₂×D₈	—	2	4	no	—
374	C₂×D₈	—	2	4	no	—
375	C₂×D₈	—	4	2	no	—
376	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
377	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
378	#127	—	2	2	no	—
379	#127	—	2	2	no	—
380	#121	—	2	2	no	—
381	#121	—	2	2	no	—
382	#121	—	4	2	no	—
383	#121	—	4	2	no	—
384	#121	—	4	4	no	—
385	#121	—	4	4	no	—
386	C₄×D₄	—	2	2	no	—
387	C₄×D₄	—	2	2	no	—
388	C₄×D₄	—	4	4	no	—
389	C₂×C₄⋊C₄	—	2	1	no	—
390	C₂×C₄⋊C₄	—	2	1	no	—
391	C₂×C₄⋊C₄	—	2	1	no	—
392	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
393	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
394	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
395	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
396	C₂×C₂²⋊C₄	—	2	1	no	—
397	C₂×C₂²⋊C₄	—	2	1	no	—
398	C₂×C₂²⋊C₄	—	2	1	no	—
399	C₂×C₂²⋊C₄	—	2	1	no	—
400	C₂×C₂²⋊C₄	—	2	2	no	—
401	C₂×C₂²⋊C₄	—	2	2	no	—
402	C₂×C₂²⋊C₄	—	2	2	no	—
403	C₂×C₂²⋊C₄	—	2	1	no	—
404	C₂×C₄²	—	2	2	no	—
405	C₂×C₄²	—	2	2	no	—
406	C₂×C₄²	—	2	1	no	—
407	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
408	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
409	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
410	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
411	#106	—	4	2	no	—
412	C₄⋊C₈	—	2	2	no	—
413	C₄⋊C₈	D₄×C₆₇	1	4	yes	complemented
414	#102	—	2	4	no	—
415	#102	—	2	4	no	—
416	#102	—	4	2	no	—
417	#102	—	4	2	no	—
418	C₂²⋊C₈	—	2	4	no	—
419	#97	—	2	4	no	—
420	#95	—	2	1	no	—
421	#95	—	2	1	no	—
422	#95	—	2	1	no	—
423	C₂⁴	—	2	1	no	—
424	C₂⁴	—	2	2	no	—
425	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
426	C₂×D₄	—	2	4	no	—
427	C₂×D₄	—	2	1	no	—
428	C₂×D₄	—	2	2	no	—
429	C₂×D₄	—	2	1	no	—
430	C₂×D₄	—	2	1	no	—
431	C₂×D₄	—	2	2	no	—
432	C₂×D₄	—	2	2	no	—
433	C₂×D₄	—	2	2	no	—
434	C₂×D₄	—	2	1	no	—
435	C₂×D₄	—	2	1	no	—
436	C₂×D₄	—	2	2	no	—
437	C₂×D₄	—	2	2	no	—
438	C₂×D₄	—	4	2	no	—
439	C₂×D₄	—	4	2	no	—
440	C₂×D₄	—	4	2	no	—
441	C₂×D₄	—	4	2	no	—
442	C₂×D₄	—	4	4	no	—
443	C₂×D₄	—	4	4	no	—
444	C₂×D₄	—	4	4	no	—
445	C₂×D₄	—	4	2	no	—
446	C₂×D₄	—	4	2	no	—
447	C₂×D₄	—	4	2	no	—
448	C₂×D₄	—	4	2	no	—
449	C₂×D₄	#41×C₆₇	1	2	yes	—
450	C₂²×C₄	—	2	2	no	—
451	C₂²×C₄	—	2	1	no	—
452	C₂²×C₄	—	2	2	no	—
453	C₂²×C₄	—	2	2	no	—
454	C₂²×C₄	—	2	2	no	—
455	C₂²×C₄	—	2	2	no	—
456	C₂²×C₄	—	2	1	no	—
457	C₂²×C₄	—	2	1	no	—
458	C₂²×C₄	—	2	1	no	—
459	C₂²×C₄	—	2	1	no	—
460	C₂²×C₄	—	2	1	no	—
461	C₂²×C₄	—	2	1	no	—
462	C₂²×C₄	—	2	1	no	—
463	C₂²×C₄	—	2	1	no	—
464	C₂²×C₄	—	2	2	no	—
465	C₂²×C₄	—	4	1	no	—
466	C₂²×C₄	—	4	2	no	—
467	C₂²×C₄	—	4	2	no	—
468	C₂²×C₄	—	4	2	no	—
469	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
470	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
471	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
472	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
473	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
474	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
475	D₈	—	4	8	no	—
476	C₂×C₈	—	2	2	no	—
477	C₂×C₈	—	2	2	no	—
478	C₂×C₈	—	2	2	no	—
479	C₂×C₈	—	2	2	no	—
480	C₂×C₈	—	2	2	no	—
481	C₂×C₈	C₂×D₄×C₆₇	1	4	yes	—
482	C₄⋊C₄	—	2	1	no	—
483	C₄⋊C₄	—	2	1	no	—
484	C₄⋊C₄	—	2	1	no	—
485	C₄⋊C₄	—	2	1	no	—
486	C₄⋊C₄	—	2	2	no	—
487	C₄⋊C₄	—	2	1	no	—
488	C₄⋊C₄	—	2	2	no	—
489	C₄⋊C₄	—	4	2	no	—
490	C₄⋊C₄	—	4	2	no	—
491	C₄⋊C₄	—	4	2	no	—
492	C₄⋊C₄	#41×C₆₇	1	2	yes	—
493	C₂²⋊C₄	—	2	4	no	—
494	C₂²⋊C₄	—	4	2	no	—
495	C₂²⋊C₄	—	4	4	no	—
496	C₂²⋊C₄	—	4	4	no	—
497	C₂²⋊C₄	—	4	4	no	—
498	C₂²⋊C₄	—	4	2	no	—
499	C₂²⋊C₄	—	4	2	no	—
500	C₂²⋊C₄	—	4	2	no	—
501	C₄²	—	2	2	no	—
502	C₄²	—	2	2	no	—
503	C₄²	—	2	1	no	—
504	C₄²	—	2	1	no	—
505	C₄²	—	4	2	no	—
506	C₄²	C₂×D₄×C₆₇	1	2	yes	—
507	C₄²	#41×C₆₇	1	2	yes	—
508	C₂³	—	2	1	no	—
509	C₂³	—	2	1	no	—
510	C₂³	—	2	1	no	—
511	C₂³	—	2	1	no	—
512	C₂³	—	2	1	no	—
513	C₂³	—	2	1	no	—
514	C₂³	—	2	2	no	—
515	C₂³	—	4	1	no	—
516	C₂³	—	4	1	no	—
517	C₂³	—	4	1	no	—
518	C₂³	—	4	1	no	—
519	C₂³	—	4	1	no	—
520	C₂³	—	4	1	no	—
521	C₂³	—	4	2	no	—
522	C₂³	—	4	2	no	—
523	C₂³	—	4	2	no	—
524	C₂³	—	4	2	no	—
525	C₂³	—	4	2	no	—
526	C₂³	—	4	2	no	—
527	C₂³	—	4	2	no	—
528	C₂³	AGL(1,ℤ/8)×C₆₇	1	2	yes	—
529	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
530	D₄	—	2	2	no	—
531	D₄	—	2	2	no	—
532	D₄	—	4	8	no	—
533	D₄	—	4	2	no	—
534	D₄	—	4	1	no	—
535	D₄	—	4	2	no	—
536	D₄	—	4	2	no	—
537	D₄	—	4	1	no	—
538	D₄	—	4	2	no	—
539	D₄	—	8	4	no	—
540	D₄	—	8	8	no	—
541	D₄	—	8	4	no	—
542	C₂×C₄	—	2	1	no	—
543	C₂×C₄	—	2	1	no	—
544	C₂×C₄	—	2	1	no	—
545	C₂×C₄	—	2	1	no	—
546	C₂×C₄	—	2	1	no	—
547	C₂×C₄	—	2	1	no	—
548	C₂×C₄	—	2	2	no	—
549	C₂×C₄	—	2	1	no	—
550	C₂×C₄	—	2	1	no	—
551	C₂×C₄	—	2	1	no	—
552	C₂×C₄	—	2	1	no	—
553	C₂×C₄	—	2	1	no	—
554	C₂×C₄	—	4	4	no	—
555	C₂×C₄	—	4	1	no	—
556	C₂×C₄	—	4	1	no	—
557	C₂×C₄	—	4	4	no	—
558	C₂×C₄	—	4	1	no	—
559	C₂×C₄	—	4	1	no	—
560	C₂×C₄	—	4	1	no	—
561	C₂×C₄	—	4	1	no	—
562	C₂×C₄	—	4	1	no	—
563	C₂×C₄	—	4	2	no	—
564	C₂×C₄	—	4	1	no	—
565	C₂×C₄	—	4	1	no	—
566	C₂×C₄	—	4	1	no	—
567	C₂×C₄	—	4	2	no	—
568	C₂×C₄	—	4	2	no	—
569	C₂×C₄	—	4	1	no	—
570	C₂×C₄	—	4	1	no	—
571	C₂×C₄	—	8	4	no	—
572	C₂×C₄	#121×C₆₇	1	2	yes	—
573	C₂×C₄	#121×C₆₇	1	2	yes	—
574	C₂×C₄	AGL(1,ℤ/8)×C₆₇	1	2	yes	—
575	C₂×C₄	AGL(1,ℤ/8)×C₆₇	1	2	yes	—
576	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
577	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
578	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
579	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
580	C₂×C₄	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
581	C₂×C₄	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
582	C₈	—	2	4	no	—
583	C₈	—	4	2	no	—
584	C₂²	—	2	2	no	—
585	C₂²	—	2	2	no	—
586	C₂²	—	2	2	no	—
587	C₂²	—	4	1	no	—
588	C₂²	—	4	1	no	—
589	C₂²	—	4	1	no	—
590	C₂²	—	4	1	no	—
591	C₂²	—	4	2	no	—
592	C₂²	—	4	2	no	—
593	C₂²	—	4	2	no	—
594	C₂²	—	8	1	no	—
595	C₂²	—	8	1	no	—
596	C₂²	—	8	1	no	—
597	C₂²	—	8	1	no	—
598	C₂²	—	8	2	no	—
599	C₂²	—	8	2	no	—
600	C₂²	—	8	2	no	—
601	C₂²	—	8	2	no	—
602	C₂²	—	8	2	no	—
603	C₂²	—	8	2	no	—
604	C₂²	—	8	2	no	—
605	C₂²	C₂×#121×C₆₇	1	1	yes	characteristic
606	C₂²	#525×C₆₇	1	1	yes	characteristic
607	C₂²	#535×C₆₇	1	1	yes	characteristic
608	C₂²	C₂×AGL(1,ℤ/8)×C₆₇	1	1	yes	characteristic
609	C₂²	C₂×AGL(1,ℤ/8)×C₆₇	1	1	yes	characteristic
610	C₂²	C₂×AGL(1,ℤ/8)×C₆₇	1	1	yes	characteristic
611	C₂²	#575×C₆₇	1	1	yes	characteristic
612	C₄	—	2	2	no	—
613	C₄	—	4	1	no	—
614	C₄	—	4	1	no	—
615	C₄	—	4	2	no	—
616	C₄	—	4	1	no	—
617	C₄	—	8	1	no	—
618	C₄	—	8	1	no	—
619	C₄	—	8	2	no	—
620	C₄	#469×C₆₇	1	4	yes	—
621	C₄	#546×C₆₇	1	1	yes	characteristic
622	C₄	#550×C₆₇	1	1	yes	characteristic
623	C₄	#553×C₆₇	1	1	yes	characteristic
624	C₄	#555×C₆₇	1	1	yes	characteristic
625	C₂	—	4	2	no	—
626	C₂	—	8	2	no	—
627	C₂	—	16	2	no	—
628	C₂	C₆₇×#2414	1	1	yes	characteristic
629	C₂	C₂×#469×C₆₇	1	1	yes	characteristic
630	C₂	C₆₇×#3055	1	1	yes	characteristic
631	C₂	C₆₇×#3109	1	1	yes	characteristic
632	C₂	C₆₇×#3115	1	1	yes	characteristic
633	C₂	C₆₇×#3214	1	1	yes	characteristic
634	C₂	C₆₇×#3217	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₂²×C₄²×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#459×C₆₇	C₂	semidirect
2. C₂×#432×C₆₇	C₂	semidirect
3. C₆₇×#2856	C₂	semidirect
4. C₄×#121×C₆₇	C₂	semidirect
5. C₆₇×#2034	C₂	semidirect
6. C₆₇×#1999	C₂	semidirect
7. C₆₇×#1982	C₂	semidirect
8. C₆₇×#1972	C₂	semidirect
9. C₆₇×#1948	C₂	semidirect
10. C₆₇×#1936	C₂	semidirect
11. C₆₇×#1802	C₂	semidirect
95. C₄⋊C₈×C₆₇	D₄	semidirect
307. #24658	C₆₇	direct (N × H)
318. C₂×#459	C₂×C₆₇	semidirect
319. C₂×#432	C₂×C₆₇	semidirect
320. #2856	C₂×C₆₇	semidirect
321. C₄×#121	C₂×C₆₇	semidirect
322. #2034	C₂×C₆₇	semidirect
323. #1999	C₂×C₆₇	semidirect
324. #1982	C₂×C₆₇	semidirect
325. #1972	C₂×C₆₇	semidirect
326. #1948	C₂×C₆₇	semidirect
327. #1936	C₂×C₆₇	semidirect
328. #1802	C₂×C₆₇	semidirect
329. C₆₇	#24658	direct (N × H)
413. C₄⋊C₈	D₄×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#459×C₆₇	2
C₂×#432×C₆₇	2
C₆₇×#2856	2
C₄×#121×C₆₇	2
C₆₇×#2034	2
C₆₇×#1999	2
C₆₇×#1982	2
C₆₇×#1972	2
C₆₇×#1948	2
C₆₇×#1936	2
C₆₇×#1802	2
#24658	67

Subgroups — C67×#24658 #30703909

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#24658 #30703909