Subgroups — C₆₇×#32936 #30712187

Order 17152.

1658 conjugacy classes of subgroups.

The subgroup lattice has 444 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×M₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#469×C₆₇	C₂	1	2	yes	maximal, complemented
3	C₂×#466×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#2847	C₂	1	2	yes	maximal, complemented
5	C₆₇×#1895	C₂	1	4	yes	maximal, complemented
6	C₆₇×#1893	C₂	1	2	yes	maximal, complemented
7	C₆₇×#1809	C₂	1	2	yes	maximal, complemented
8	C₆₇×#1807	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂⁴×C₄×C₆₇	C₂²	1	1	yes	characteristic
10	C₂²×D₈×C₆₇	C₂²	1	1	yes	characteristic, complemented
11	C₂²×M₁₆×C₆₇	—	2	8	no	—
12	C₂²×M₁₆×C₆₇	—	2	4	no	—
13	C₂²×M₁₆×C₆₇	C₂²	1	2	yes	—
14	C₂²×M₁₆×C₆₇	C₂²	1	2	yes	—
15	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
16	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
17	C₂×#121×C₆₇	C₂²	1	2	yes	—
18	C₂×#121×C₆₇	C₂²	1	4	yes	—
19	C₂×#120×C₆₇	—	2	2	no	—
20	#469×C₆₇	—	2	8	no	—
21	#466×C₆₇	—	2	4	no	—
22	C₂×#107×C₆₇	C₂²	1	1	yes	characteristic, complemented
23	C₂×#106×C₆₇	C₂²	1	2	yes	complemented
24	C₂×#102×C₆₇	C₂²	1	2	yes	complemented
25	C₂×#102×C₆₇	C₂²	1	4	yes	complemented
26	#386×C₆₇	—	2	4	no	—
27	#386×C₆₇	—	2	2	no	—
28	#380×C₆₇	C₂²	1	2	yes	—
29	#380×C₆₇	C₂²	1	4	yes	—
30	#340×C₆₇	C₂²	1	2	yes	complemented
31	#340×C₆₇	C₂²	1	4	yes	complemented
32	C₂⁵×C₆₇	D₄	1	1	yes	characteristic
33	C₂²×D₄×C₆₇	—	2	4	no	—
34	C₂²×D₄×C₆₇	—	2	2	no	—
35	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
36	C₂³×C₄×C₆₇	—	2	4	no	—
37	C₂³×C₄×C₆₇	—	2	2	no	—
38	C₂³×C₄×C₆₇	—	2	4	no	—
39	C₂³×C₄×C₆₇	—	2	2	no	—
40	C₂³×C₄×C₆₇	D₄	1	2	yes	—
41	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
42	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
43	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
44	C₂×D₈×C₆₇	—	2	2	no	—
45	C₂×D₈×C₆₇	—	2	4	no	—
46	C₂×M₁₆×C₆₇	—	2	32	no	—
47	C₂×M₁₆×C₆₇	—	2	4	no	—
48	C₂×M₁₆×C₆₇	—	2	4	no	—
49	C₂×M₁₆×C₆₇	—	2	4	no	—
50	C₂×M₁₆×C₆₇	—	2	4	no	—
51	C₂×M₁₆×C₆₇	—	2	4	no	—
52	C₂×M₁₆×C₆₇	D₄	1	8	yes	complemented
53	C₂²×C₈×C₆₇	—	2	2	no	—
54	C₂²×C₈×C₆₇	—	2	4	no	—
55	C₂²×C₈×C₆₇	—	2	4	no	—
56	C₂²×C₈×C₆₇	D₄	1	4	yes	complemented
57	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
58	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
59	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
60	#121×C₆₇	—	2	4	no	—
61	#121×C₆₇	—	2	8	no	—
62	#121×C₆₇	—	2	4	no	—
63	#121×C₆₇	—	2	8	no	—
64	#120×C₆₇	—	4	8	no	—
65	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
66	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
67	C₂×C₂²⋊C₄×C₆₇	—	2	4	no	—
68	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
69	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
70	C₂×C₂²⋊C₄×C₆₇	—	2	4	no	—
71	#107×C₆₇	—	2	2	no	—
72	#106×C₆₇	—	2	4	no	—
73	#102×C₆₇	—	2	4	no	—
74	#102×C₆₇	—	2	8	no	—
75	#95×C₆₇	—	2	2	no	—
76	#95×C₆₇	—	2	4	no	—
77	C₂⁴×C₆₇	—	2	4	no	—
78	C₂⁴×C₆₇	—	2	2	no	—
79	C₂⁴×C₆₇	—	2	4	no	—
80	C₂⁴×C₆₇	—	2	2	no	—
81	C₂⁴×C₆₇	—	4	2	no	—
82	C₂⁴×C₆₇	C₂×D₄	1	2	yes	—
83	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
84	C₂×D₄×C₆₇	—	2	2	no	—
85	C₂×D₄×C₆₇	—	2	4	no	—
86	C₂×D₄×C₆₇	—	2	2	no	—
87	C₂×D₄×C₆₇	—	2	4	no	—
88	C₂×D₄×C₆₇	—	4	8	no	—
89	C₂×D₄×C₆₇	—	4	8	no	—
90	C₂×D₄×C₆₇	—	4	8	no	—
91	C₂×D₄×C₆₇	—	4	4	no	—
92	C₂×D₄×C₆₇	—	4	4	no	—
93	C₂×D₄×C₆₇	—	4	4	no	—
94	C₂²×C₄×C₆₇	—	2	8	no	—
95	C₂²×C₄×C₆₇	—	2	8	no	—
96	C₂²×C₄×C₆₇	—	2	16	no	—
97	C₂²×C₄×C₆₇	—	2	4	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	4	no	—
102	C₂²×C₄×C₆₇	—	2	2	no	—
103	C₂²×C₄×C₆₇	—	2	2	no	—
104	C₂²×C₄×C₆₇	—	2	2	no	—
105	C₂²×C₄×C₆₇	—	2	2	no	—
106	C₂²×C₄×C₆₇	—	2	2	no	—
107	C₂²×C₄×C₆₇	—	2	2	no	—
108	C₂²×C₄×C₆₇	—	2	2	no	—
109	C₂²×C₄×C₆₇	—	2	4	no	—
110	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
111	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
112	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
113	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
114	C₂²×C₄×C₆₇	#41	1	4	yes	—
115	C₂²×C₄×C₆₇	#41	1	2	yes	—
116	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
117	D₈×C₆₇	—	4	4	no	—
118	M₁₆×C₆₇	—	2	16	no	—
119	M₁₆×C₆₇	—	2	16	no	—
120	C₂×C₈×C₆₇	—	2	8	no	—
121	C₂×C₈×C₆₇	—	2	2	no	—
122	C₂×C₈×C₆₇	—	2	2	no	—
123	C₂×C₈×C₆₇	—	2	2	no	—
124	C₂×C₈×C₆₇	—	2	8	no	—
125	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
126	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
127	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
128	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
129	C₄⋊C₄×C₆₇	—	2	4	no	—
130	C₄⋊C₄×C₆₇	—	2	2	no	—
131	C₄⋊C₄×C₆₇	—	2	4	no	—
132	C₄⋊C₄×C₆₇	—	2	2	no	—
133	C₂²⋊C₄×C₆₇	—	4	8	no	—
134	C₂²⋊C₄×C₆₇	—	4	4	no	—
135	C₂²⋊C₄×C₆₇	—	4	4	no	—
136	C₂²⋊C₄×C₆₇	—	4	8	no	—
137	C₂³×C₆₇	—	2	2	no	—
138	C₂³×C₆₇	—	2	2	no	—
139	C₂³×C₆₇	—	2	8	no	—
140	C₂³×C₆₇	—	2	2	no	—
141	C₂³×C₆₇	—	2	2	no	—
142	C₂³×C₆₇	—	2	4	no	—
143	C₂³×C₆₇	—	2	4	no	—
144	C₂³×C₆₇	—	4	16	no	—
145	C₂³×C₆₇	—	4	4	no	—
146	C₂³×C₆₇	—	4	4	no	—
147	C₂³×C₆₇	—	4	2	no	—
148	C₂³×C₆₇	—	4	4	no	—
149	C₂³×C₆₇	—	4	2	no	—
150	C₂³×C₆₇	—	4	2	no	—
151	C₂³×C₆₇	—	4	4	no	—
152	C₂³×C₆₇	#121	1	4	yes	—
153	C₂³×C₆₇	#121	1	2	yes	—
154	C₂³×C₆₇	AGL(1,ℤ/8)	1	4	yes	—
155	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
156	D₄×C₆₇	—	4	2	no	—
157	D₄×C₆₇	—	4	2	no	—
158	D₄×C₆₇	—	4	4	no	—
159	D₄×C₆₇	—	8	16	no	—
160	D₄×C₆₇	—	8	8	no	—
161	C₂×C₄×C₆₇	—	2	8	no	—
162	C₂×C₄×C₆₇	—	2	16	no	—
163	C₂×C₄×C₆₇	—	2	2	no	—
164	C₂×C₄×C₆₇	—	2	2	no	—
165	C₂×C₄×C₆₇	—	2	2	no	—
166	C₂×C₄×C₆₇	—	2	2	no	—
167	C₂×C₄×C₆₇	—	2	4	no	—
168	C₂×C₄×C₆₇	—	2	2	no	—
169	C₂×C₄×C₆₇	—	2	4	no	—
170	C₂×C₄×C₆₇	—	2	4	no	—
171	C₂×C₄×C₆₇	—	2	4	no	—
172	C₂×C₄×C₆₇	—	2	4	no	—
173	C₂×C₄×C₆₇	—	4	2	no	—
174	C₂×C₄×C₆₇	—	4	4	no	—
175	C₂×C₄×C₆₇	—	4	2	no	—
176	C₂×C₄×C₆₇	—	4	4	no	—
177	C₂×C₄×C₆₇	—	4	2	no	—
178	C₂×C₄×C₆₇	—	4	4	no	—
179	C₂×C₄×C₆₇	#121	1	2	yes	—
180	C₂×C₄×C₆₇	#121	1	4	yes	—
181	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
182	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
183	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
184	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
185	C₈×C₆₇	—	2	4	no	—
186	C₈×C₆₇	—	2	2	no	—
187	C₈×C₆₇	#120	1	4	yes	complemented
188	C₂²×C₆₇	—	2	2	no	—
189	C₂²×C₆₇	—	2	2	no	—
190	C₂²×C₆₇	—	2	8	no	—
191	C₂²×C₆₇	—	2	4	no	—
192	C₂²×C₆₇	—	2	8	no	—
193	C₂²×C₆₇	—	4	16	no	—
194	C₂²×C₆₇	—	4	2	no	—
195	C₂²×C₆₇	—	4	2	no	—
196	C₂²×C₆₇	—	4	4	no	—
197	C₂²×C₆₇	—	8	2	no	—
198	C₂²×C₆₇	—	8	2	no	—
199	C₂²×C₆₇	—	8	2	no	—
200	C₂²×C₆₇	—	8	4	no	—
201	C₂²×C₆₇	—	8	4	no	—
202	C₂²×C₆₇	#469	1	4	yes	complemented
203	C₂²×C₆₇	C₂×#121	1	1	yes	characteristic
204	C₂²×C₆₇	C₂×#121	1	2	yes	—
205	C₂²×C₆₇	C₂×AGL(1,ℤ/8)	1	2	yes	—
206	C₂²×C₆₇	#575	1	2	yes	—
207	C₄×C₆₇	—	2	2	no	—
208	C₄×C₆₇	—	2	4	no	—
209	C₄×C₆₇	—	8	2	no	—
210	C₄×C₆₇	—	8	4	no	—
211	C₄×C₆₇	C₂×#120	1	1	yes	characteristic
212	C₄×C₆₇	#525	1	1	yes	characteristic
213	C₄×C₆₇	#525	1	2	yes	—
214	#32936	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
215	C₂×C₆₇	—	2	4	no	—
216	C₂×C₆₇	—	4	4	no	—
217	C₂×C₆₇	—	16	2	no	—
218	C₂×C₆₇	#2847	1	1	yes	characteristic
219	C₂×C₆₇	C₂×#469	1	1	yes	characteristic
220	C₂×C₆₇	#3054	1	1	yes	characteristic
221	C₂×C₆₇	#3055	1	4	yes	—
222	C₂³×M₁₆	C₂×C₆₇	1	1	yes	characteristic, complemented
223	C₂×#469	C₂×C₆₇	1	2	yes	complemented
224	C₂×#466	C₂×C₆₇	1	1	yes	characteristic, complemented
225	#2847	C₂×C₆₇	1	2	yes	complemented
226	#1895	C₂×C₆₇	1	4	yes	complemented
227	#1893	C₂×C₆₇	1	2	yes	complemented
228	#1809	C₂×C₆₇	1	2	yes	complemented
229	#1807	C₂×C₆₇	1	1	yes	characteristic, complemented
230	C₆₇	#32936	1	1	yes	characteristic, direct product factor, Sylow
231	C₂⁴×C₄	C₂²×C₆₇	1	1	yes	characteristic
232	C₂²×D₈	C₂²×C₆₇	1	1	yes	characteristic, complemented
233	C₂²×M₁₆	—	2	8	no	—
234	C₂²×M₁₆	—	2	4	no	—
235	C₂²×M₁₆	C₂²×C₆₇	1	2	yes	—
236	C₂²×M₁₆	C₂²×C₆₇	1	2	yes	—
237	C₂³×C₈	C₂²×C₆₇	1	1	yes	characteristic
238	C₂³×C₈	C₂²×C₆₇	1	1	yes	characteristic
239	C₂×#121	C₂²×C₆₇	1	2	yes	—
240	C₂×#121	C₂²×C₆₇	1	4	yes	—
241	C₂×#120	—	2	2	no	—
242	#469	—	2	8	no	—
243	#466	—	2	4	no	—
244	C₂×#107	C₂²×C₆₇	1	1	yes	characteristic, complemented
245	C₂×#106	C₂²×C₆₇	1	2	yes	complemented
246	C₂×#102	C₂²×C₆₇	1	4	yes	complemented
247	C₂×#102	C₂²×C₆₇	1	2	yes	complemented
248	#386	—	2	4	no	—
249	#386	—	2	2	no	—
250	#380	C₂²×C₆₇	1	2	yes	—
251	#380	C₂²×C₆₇	1	4	yes	—
252	#340	C₂²×C₆₇	1	2	yes	complemented
253	#340	C₂²×C₆₇	1	4	yes	complemented
254	C₂⁵	D₄×C₆₇	1	1	yes	characteristic
255	C₂²×D₄	—	2	4	no	—
256	C₂²×D₄	—	2	2	no	—
257	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
258	C₂³×C₄	—	2	4	no	—
259	C₂³×C₄	—	2	2	no	—
260	C₂³×C₄	—	2	4	no	—
261	C₂³×C₄	—	2	2	no	—
262	C₂³×C₄	D₄×C₆₇	1	2	yes	—
263	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
264	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
265	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
266	C₂×D₈	—	2	2	no	—
267	C₂×D₈	—	2	4	no	—
268	C₂×M₁₆	—	2	32	no	—
269	C₂×M₁₆	—	2	4	no	—
270	C₂×M₁₆	—	2	4	no	—
271	C₂×M₁₆	—	2	4	no	—
272	C₂×M₁₆	—	2	4	no	—
273	C₂×M₁₆	—	2	4	no	—
274	C₂×M₁₆	D₄×C₆₇	1	8	yes	complemented
275	C₂²×C₈	—	2	2	no	—
276	C₂²×C₈	—	2	4	no	—
277	C₂²×C₈	—	2	4	no	—
278	C₂²×C₈	D₄×C₆₇	1	4	yes	complemented
279	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
280	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
281	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
282	#121	—	2	4	no	—
283	#121	—	2	8	no	—
284	#121	—	2	4	no	—
285	#121	—	2	8	no	—
286	#120	—	4	8	no	—
287	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
288	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
289	C₂×C₂²⋊C₄	—	2	4	no	—
290	C₂×C₂²⋊C₄	—	2	2	no	—
291	C₂×C₂²⋊C₄	—	2	2	no	—
292	C₂×C₂²⋊C₄	—	2	4	no	—
293	#107	—	2	2	no	—
294	#106	—	2	4	no	—
295	#102	—	2	4	no	—
296	#102	—	2	8	no	—
297	#95	—	2	2	no	—
298	#95	—	2	4	no	—
299	C₂⁴	—	2	4	no	—
300	C₂⁴	—	2	2	no	—
301	C₂⁴	—	2	4	no	—
302	C₂⁴	—	2	2	no	—
303	C₂⁴	—	4	2	no	—
304	C₂⁴	C₂×D₄×C₆₇	1	2	yes	—
305	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
306	C₂×D₄	—	2	2	no	—
307	C₂×D₄	—	2	4	no	—
308	C₂×D₄	—	2	2	no	—
309	C₂×D₄	—	2	4	no	—
310	C₂×D₄	—	4	8	no	—
311	C₂×D₄	—	4	8	no	—
312	C₂×D₄	—	4	8	no	—
313	C₂×D₄	—	4	4	no	—
314	C₂×D₄	—	4	4	no	—
315	C₂×D₄	—	4	4	no	—
316	C₂²×C₄	—	2	8	no	—
317	C₂²×C₄	—	2	8	no	—
318	C₂²×C₄	—	2	16	no	—
319	C₂²×C₄	—	2	4	no	—
320	C₂²×C₄	—	2	2	no	—
321	C₂²×C₄	—	2	4	no	—
322	C₂²×C₄	—	2	4	no	—
323	C₂²×C₄	—	2	4	no	—
324	C₂²×C₄	—	2	2	no	—
325	C₂²×C₄	—	2	2	no	—
326	C₂²×C₄	—	2	2	no	—
327	C₂²×C₄	—	2	2	no	—
328	C₂²×C₄	—	2	2	no	—
329	C₂²×C₄	—	2	2	no	—
330	C₂²×C₄	—	2	2	no	—
331	C₂²×C₄	—	2	4	no	—
332	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
333	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
334	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
335	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
336	C₂²×C₄	#41×C₆₇	1	4	yes	—
337	C₂²×C₄	#41×C₆₇	1	2	yes	—
338	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
339	D₈	—	4	4	no	—
340	M₁₆	—	2	16	no	—
341	M₁₆	—	2	16	no	—
342	C₂×C₈	—	2	8	no	—
343	C₂×C₈	—	2	2	no	—
344	C₂×C₈	—	2	2	no	—
345	C₂×C₈	—	2	2	no	—
346	C₂×C₈	—	2	8	no	—
347	C₂×C₈	C₂×D₄×C₆₇	1	4	yes	—
348	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
349	C₂×C₈	C₂×D₄×C₆₇	1	4	yes	—
350	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
351	C₄⋊C₄	—	2	4	no	—
352	C₄⋊C₄	—	2	2	no	—
353	C₄⋊C₄	—	2	4	no	—
354	C₄⋊C₄	—	2	2	no	—
355	C₂²⋊C₄	—	4	8	no	—
356	C₂²⋊C₄	—	4	4	no	—
357	C₂²⋊C₄	—	4	4	no	—
358	C₂²⋊C₄	—	4	8	no	—
359	C₂³	—	2	2	no	—
360	C₂³	—	2	2	no	—
361	C₂³	—	2	8	no	—
362	C₂³	—	2	2	no	—
363	C₂³	—	2	2	no	—
364	C₂³	—	2	4	no	—
365	C₂³	—	2	4	no	—
366	C₂³	—	4	16	no	—
367	C₂³	—	4	4	no	—
368	C₂³	—	4	4	no	—
369	C₂³	—	4	2	no	—
370	C₂³	—	4	4	no	—
371	C₂³	—	4	2	no	—
372	C₂³	—	4	2	no	—
373	C₂³	—	4	4	no	—
374	C₂³	#121×C₆₇	1	4	yes	—
375	C₂³	#121×C₆₇	1	2	yes	—
376	C₂³	AGL(1,ℤ/8)×C₆₇	1	4	yes	—
377	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
378	D₄	—	4	2	no	—
379	D₄	—	4	2	no	—
380	D₄	—	4	4	no	—
381	D₄	—	8	16	no	—
382	D₄	—	8	8	no	—
383	C₂×C₄	—	2	8	no	—
384	C₂×C₄	—	2	16	no	—
385	C₂×C₄	—	2	2	no	—
386	C₂×C₄	—	2	2	no	—
387	C₂×C₄	—	2	2	no	—
388	C₂×C₄	—	2	2	no	—
389	C₂×C₄	—	2	4	no	—
390	C₂×C₄	—	2	2	no	—
391	C₂×C₄	—	2	4	no	—
392	C₂×C₄	—	2	4	no	—
393	C₂×C₄	—	2	4	no	—
394	C₂×C₄	—	2	4	no	—
395	C₂×C₄	—	4	2	no	—
396	C₂×C₄	—	4	4	no	—
397	C₂×C₄	—	4	2	no	—
398	C₂×C₄	—	4	4	no	—
399	C₂×C₄	—	4	2	no	—
400	C₂×C₄	—	4	4	no	—
401	C₂×C₄	#121×C₆₇	1	2	yes	—
402	C₂×C₄	#121×C₆₇	1	4	yes	—
403	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
404	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
405	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
406	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
407	C₈	—	2	4	no	—
408	C₈	—	2	2	no	—
409	C₈	#120×C₆₇	1	4	yes	complemented
410	C₂²	—	2	2	no	—
411	C₂²	—	2	8	no	—
412	C₂²	—	2	2	no	—
413	C₂²	—	2	8	no	—
414	C₂²	—	2	4	no	—
415	C₂²	—	4	16	no	—
416	C₂²	—	4	2	no	—
417	C₂²	—	4	2	no	—
418	C₂²	—	4	4	no	—
419	C₂²	—	8	2	no	—
420	C₂²	—	8	2	no	—
421	C₂²	—	8	4	no	—
422	C₂²	—	8	4	no	—
423	C₂²	—	8	2	no	—
424	C₂²	#469×C₆₇	1	4	yes	complemented
425	C₂²	C₂×#121×C₆₇	1	1	yes	characteristic
426	C₂²	C₂×#121×C₆₇	1	2	yes	—
427	C₂²	C₂×AGL(1,ℤ/8)×C₆₇	1	2	yes	—
428	C₂²	#575×C₆₇	1	2	yes	—
429	C₄	—	2	2	no	—
430	C₄	—	2	4	no	—
431	C₄	—	8	2	no	—
432	C₄	—	8	4	no	—
433	C₄	C₂×#120×C₆₇	1	1	yes	characteristic
434	C₄	#525×C₆₇	1	1	yes	characteristic
435	C₄	#525×C₆₇	1	2	yes	—
436	C₂	—	2	4	no	—
437	C₂	—	4	4	no	—
438	C₂	—	16	2	no	—
439	C₂	C₆₇×#2847	1	1	yes	characteristic
440	C₂	C₂×#469×C₆₇	1	1	yes	characteristic
441	C₂	C₆₇×#3054	1	1	yes	characteristic
442	C₂	C₆₇×#3055	1	4	yes	—

	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 ◁ C₂⁴×C₄×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂³×M₁₆×C₆₇	C₂	semidirect
2. C₂×#469×C₆₇	C₂	semidirect
3. C₂×#466×C₆₇	C₂	semidirect
4. C₆₇×#2847	C₂	semidirect
5. C₆₇×#1895	C₂	semidirect
6. C₆₇×#1893	C₂	semidirect
7. C₆₇×#1809	C₂	semidirect
8. C₆₇×#1807	C₂	semidirect
10. C₂²×D₈×C₆₇	C₂²	semidirect
22. C₂×#107×C₆₇	C₂²	semidirect
23. C₂×#106×C₆₇	C₂²	semidirect
24. C₂×#102×C₆₇	C₂²	semidirect
25. C₂×#102×C₆₇	C₂²	semidirect
30. #340×C₆₇	C₂²	semidirect
31. #340×C₆₇	C₂²	semidirect
52. C₂×M₁₆×C₆₇	D₄	semidirect
56. C₂²×C₈×C₆₇	D₄	semidirect
187. C₈×C₆₇	#120	semidirect
202. C₂²×C₆₇	#469	semidirect
214. #32936	C₆₇	direct (N × H)
222. C₂³×M₁₆	C₂×C₆₇	semidirect
223. C₂×#469	C₂×C₆₇	semidirect
224. C₂×#466	C₂×C₆₇	semidirect
225. #2847	C₂×C₆₇	semidirect
226. #1895	C₂×C₆₇	semidirect
227. #1893	C₂×C₆₇	semidirect
228. #1809	C₂×C₆₇	semidirect
229. #1807	C₂×C₆₇	semidirect
230. C₆₇	#32936	direct (N × H)
232. C₂²×D₈	C₂²×C₆₇	semidirect
244. C₂×#107	C₂²×C₆₇	semidirect
245. C₂×#106	C₂²×C₆₇	semidirect
246. C₂×#102	C₂²×C₆₇	semidirect
247. C₂×#102	C₂²×C₆₇	semidirect
252. #340	C₂²×C₆₇	semidirect
253. #340	C₂²×C₆₇	semidirect
274. C₂×M₁₆	D₄×C₆₇	semidirect
278. C₂²×C₈	D₄×C₆₇	semidirect
409. C₈	#120×C₆₇	semidirect
424. C₂²	#469×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂³×M₁₆×C₆₇	2
C₂×#469×C₆₇	2
C₂×#466×C₆₇	2
C₆₇×#2847	2
C₆₇×#1895	2
C₆₇×#1893	2
C₆₇×#1809	2
C₆₇×#1807	2
#32936	67

Subgroups — C67×#32936 #30712187

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#32936 #30712187