Subgroups — C₆₇×#25903 #30705154

Order 17152.

1446 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	C₂²×C₄⋊Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#486×C₆₇	C₂	1	2	yes	maximal, complemented
3	C₆₇×#2856	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#2023	C₂	1	4	yes	maximal, complemented
5	C₆₇×#1990	C₂	1	2	yes	maximal, complemented
6	C₆₇×#1874	C₂	1	4	yes	maximal, complemented
7	C₄×C₂²⋊C₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂³×Q₈×C₆₇	—	2	1	no	—
9	C₂×C₄⋊Q₈×C₆₇	—	2	8	no	—
10	C₂×C₄⋊Q₈×C₆₇	—	2	1	no	—
11	C₂×C₄⋊Q₈×C₆₇	—	2	1	no	—
12	C₂×C₄⋊Q₈×C₆₇	C₂²	1	2	yes	—
13	C₂×C₄⋊Q₈×C₆₇	C₂²	1	2	yes	—
14	C₂×#127×C₆₇	C₂²	1	1	yes	characteristic
15	C₂×#121×C₆₇	—	2	1	no	—
16	C₂×#121×C₆₇	—	2	1	no	—
17	C₂×#121×C₆₇	C₂²	1	2	yes	—
18	C₂²×C₄⋊C₄×C₆₇	—	2	1	no	—
19	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
20	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
21	#486×C₆₇	—	2	8	no	—
22	C₂×#102×C₆₇	C₂²	1	8	yes	—
23	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
24	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
25	#390×C₆₇	—	2	1	no	—
26	#390×C₆₇	—	2	1	no	—
27	#390×C₆₇	C₂²	1	2	yes	—
28	#384×C₆₇	C₂²	1	1	yes	characteristic
29	#340×C₆₇	C₂²	1	8	yes	—
30	#336×C₆₇	C₂²	1	2	yes	—
31	C₂²×Q₈×C₆₇	—	2	8	no	—
32	C₂²×Q₈×C₆₇	—	2	2	no	—
33	C₂²×Q₈×C₆₇	—	2	2	no	—
34	C₂²×Q₈×C₆₇	—	4	4	no	—
35	C₂²×Q₈×C₆₇	—	4	4	no	—
36	C₂²×D₄×C₆₇	—	2	2	no	—
37	C₂²×D₄×C₆₇	—	2	1	no	—
38	C₂²×D₄×C₆₇	—	2	1	no	—
39	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
40	C₂³×C₄×C₆₇	—	2	2	no	—
41	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
42	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
43	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
44	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
45	C₄⋊Q₈×C₆₇	—	2	8	no	—
46	C₄⋊Q₈×C₆₇	—	2	8	no	—
47	C₄⋊Q₈×C₆₇	—	2	8	no	—
48	C₄⋊Q₈×C₆₇	—	2	4	no	—
49	C₄⋊Q₈×C₆₇	D₄	1	8	yes	complemented
50	#127×C₆₇	—	2	4	no	—
51	#121×C₆₇	—	2	8	no	—
52	#121×C₆₇	—	4	4	no	—
53	#121×C₆₇	—	4	4	no	—
54	C₂×C₄⋊C₄×C₆₇	—	2	8	no	—
55	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
56	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
57	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
58	C₂×C₄⋊C₄×C₆₇	—	2	8	no	—
59	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
60	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
61	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
62	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
63	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
64	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
65	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
66	C₂×C₄²×C₆₇	—	2	4	no	—
67	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
68	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
69	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
70	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
71	#102×C₆₇	—	2	16	no	—
72	C₂²⋊C₈×C₆₇	—	2	4	no	—
73	C₄×C₈×C₆₇	—	2	4	no	—
74	C₂⁴×C₆₇	—	2	2	no	—
75	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
76	C₂×Q₈×C₆₇	—	2	8	no	—
77	C₂×Q₈×C₆₇	—	2	8	no	—
78	C₂×Q₈×C₆₇	—	4	4	no	—
79	C₂×Q₈×C₆₇	—	4	32	no	—
80	C₂×Q₈×C₆₇	—	4	4	no	—
81	C₂×Q₈×C₆₇	—	4	4	no	—
82	C₂×Q₈×C₆₇	—	4	4	no	—
83	C₂×D₄×C₆₇	—	2	4	no	—
84	C₂×D₄×C₆₇	—	2	4	no	—
85	C₂×D₄×C₆₇	—	2	4	no	—
86	C₂×D₄×C₆₇	—	4	4	no	—
87	C₂×D₄×C₆₇	—	4	4	no	—
88	C₂×D₄×C₆₇	—	4	4	no	—
89	C₂×D₄×C₆₇	—	4	2	no	—
90	C₂×D₄×C₆₇	—	4	2	no	—
91	C₂×D₄×C₆₇	—	4	2	no	—
92	C₂×D₄×C₆₇	—	4	2	no	—
93	C₂×D₄×C₆₇	—	4	2	no	—
94	C₂×D₄×C₆₇	—	4	2	no	—
95	C₂²×C₄×C₆₇	—	2	2	no	—
96	C₂²×C₄×C₆₇	—	2	2	no	—
97	C₂²×C₄×C₆₇	—	2	2	no	—
98	C₂²×C₄×C₆₇	—	2	2	no	—
99	C₂²×C₄×C₆₇	—	2	2	no	—
100	C₂²×C₄×C₆₇	—	2	2	no	—
101	C₂²×C₄×C₆₇	—	2	8	no	—
102	C₂²×C₄×C₆₇	—	2	4	no	—
103	C₂²×C₄×C₆₇	—	2	2	no	—
104	C₂²×C₄×C₆₇	—	4	4	no	—
105	C₂²×C₄×C₆₇	—	4	4	no	—
106	C₂²×C₄×C₆₇	D₈	1	4	yes	—
107	C₂²×C₄×C₆₇	SD₁₆	1	4	yes	—
108	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
109	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
110	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
111	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
112	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
113	C₂²×C₄×C₆₇	#41	1	4	yes	—
114	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
115	C₂×C₈×C₆₇	—	2	4	no	—
116	C₂×C₈×C₆₇	—	2	4	no	—
117	C₂×C₈×C₆₇	—	2	4	no	—
118	C₄⋊C₄×C₆₇	—	2	8	no	—
119	C₄⋊C₄×C₆₇	—	2	8	no	—
120	C₄⋊C₄×C₆₇	—	2	8	no	—
121	C₄⋊C₄×C₆₇	—	2	4	no	—
122	C₄⋊C₄×C₆₇	—	2	4	no	—
123	C₄⋊C₄×C₆₇	—	4	2	no	—
124	C₄⋊C₄×C₆₇	—	4	2	no	—
125	C₄⋊C₄×C₆₇	C₂×D₄	1	8	yes	—
126	C₂²⋊C₄×C₆₇	—	4	4	no	—
127	C₂²⋊C₄×C₆₇	—	4	4	no	—
128	C₂²⋊C₄×C₆₇	—	4	4	no	—
129	C₄²×C₆₇	—	2	2	no	—
130	C₄²×C₆₇	—	2	2	no	—
131	C₄²×C₆₇	—	2	2	no	—
132	C₄²×C₆₇	C₂×D₄	1	4	yes	—
133	C₂³×C₆₇	—	2	1	no	—
134	C₂³×C₆₇	—	2	1	no	—
135	C₂³×C₆₇	—	2	1	no	—
136	C₂³×C₆₇	—	2	1	no	—
137	C₂³×C₆₇	—	4	2	no	—
138	C₂³×C₆₇	—	4	2	no	—
139	C₂³×C₆₇	—	4	2	no	—
140	C₂³×C₆₇	—	4	2	no	—
141	C₂³×C₆₇	—	4	2	no	—
142	C₂³×C₆₇	—	4	2	no	—
143	C₂³×C₆₇	—	4	2	no	—
144	C₂³×C₆₇	#124	1	2	yes	—
145	C₂³×C₆₇	C₂×D₈	1	2	yes	—
146	C₂³×C₆₇	C₂×SD₁₆	1	2	yes	—
147	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
148	Q₈×C₆₇	—	4	16	no	—
149	Q₈×C₆₇	—	4	16	no	—
150	D₄×C₆₇	—	4	8	no	—
151	D₄×C₆₇	—	8	8	no	—
152	D₄×C₆₇	—	8	4	no	—
153	D₄×C₆₇	—	8	4	no	—
154	C₂×C₄×C₆₇	—	2	1	no	—
155	C₂×C₄×C₆₇	—	2	1	no	—
156	C₂×C₄×C₆₇	—	2	1	no	—
157	C₂×C₄×C₆₇	—	2	1	no	—
158	C₂×C₄×C₆₇	—	2	1	no	—
159	C₂×C₄×C₆₇	—	2	1	no	—
160	C₂×C₄×C₆₇	—	2	8	no	—
161	C₂×C₄×C₆₇	—	2	8	no	—
162	C₂×C₄×C₆₇	—	2	8	no	—
163	C₂×C₄×C₆₇	—	2	8	no	—
164	C₂×C₄×C₆₇	—	4	16	no	—
165	C₂×C₄×C₆₇	—	4	4	no	—
166	C₂×C₄×C₆₇	—	4	4	no	—
167	C₂×C₄×C₆₇	—	4	2	no	—
168	C₂×C₄×C₆₇	—	4	2	no	—
169	C₂×C₄×C₆₇	—	4	2	no	—
170	C₂×C₄×C₆₇	#120	1	2	yes	—
171	C₂×C₄×C₆₇	#120	1	2	yes	—
172	C₂×C₄×C₆₇	#124	1	2	yes	—
173	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
174	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
175	C₂×C₄×C₆₇	C₂×SD₁₆	1	2	yes	—
176	C₂×C₄×C₆₇	C₂×SD₁₆	1	2	yes	—
177	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
178	C₂×C₄×C₆₇	#137	1	2	yes	—
179	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
180	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
181	C₂×C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
182	C₈×C₆₇	—	4	4	no	—
183	C₂²×C₆₇	—	2	2	no	—
184	C₂²×C₆₇	—	2	2	no	—
185	C₂²×C₆₇	—	2	2	no	—
186	C₂²×C₆₇	—	2	2	no	—
187	C₂²×C₆₇	—	2	2	no	—
188	C₂²×C₆₇	—	2	2	no	—
189	C₂²×C₆₇	—	8	2	no	—
190	C₂²×C₆₇	—	8	2	no	—
191	C₂²×C₆₇	—	8	2	no	—
192	C₂²×C₆₇	—	8	2	no	—
193	C₂²×C₆₇	—	8	2	no	—
194	C₂²×C₆₇	—	8	2	no	—
195	C₂²×C₆₇	—	8	2	no	—
196	C₂²×C₆₇	#486	1	4	yes	complemented
197	C₂²×C₆₇	C₂×#120	1	1	yes	characteristic
198	C₂²×C₆₇	C₂×#124	1	1	yes	characteristic
199	C₂²×C₆₇	#536	1	1	yes	characteristic
200	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
201	C₂²×C₆₇	C₂²×SD₁₆	1	1	yes	characteristic
202	C₂²×C₆₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
203	C₂²×C₆₇	C₂×#137	1	1	yes	characteristic
204	C₄×C₆₇	—	2	2	no	—
205	C₄×C₆₇	—	2	2	no	—
206	C₄×C₆₇	—	2	2	no	—
207	C₄×C₆₇	—	4	8	no	—
208	C₄×C₆₇	—	8	2	no	—
209	C₄×C₆₇	#447	1	4	yes	—
210	C₄×C₆₇	#448	1	4	yes	—
211	C₄×C₆₇	#548	1	4	yes	—
212	#25903	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
213	C₂×C₆₇	—	2	4	no	—
214	C₂×C₆₇	—	16	2	no	—
215	C₂×C₆₇	#2436	1	1	yes	characteristic
216	C₂×C₆₇	C₂×#447	1	1	yes	characteristic
217	C₂×C₆₇	C₂×#448	1	1	yes	characteristic
218	C₂×C₆₇	C₂×#486	1	1	yes	characteristic
219	C₂×C₆₇	#3141	1	1	yes	characteristic
220	C₂×C₆₇	#3226	1	1	yes	characteristic
221	C₂×C₆₇	#3227	1	1	yes	characteristic
222	C₂²×C₄⋊Q₈	C₂×C₆₇	1	1	yes	characteristic, complemented
223	C₂×#486	C₂×C₆₇	1	2	yes	complemented
224	#2856	C₂×C₆₇	1	1	yes	characteristic
225	#2023	C₂×C₆₇	1	4	yes	complemented
226	#1990	C₂×C₆₇	1	2	yes	complemented
227	#1874	C₂×C₆₇	1	4	yes	complemented
228	C₄×C₂²⋊C₈	C₂×C₆₇	1	1	yes	characteristic, complemented
229	C₆₇	#25903	1	1	yes	characteristic, direct product factor, Sylow
230	C₂³×Q₈	—	2	1	no	—
231	C₂×C₄⋊Q₈	—	2	8	no	—
232	C₂×C₄⋊Q₈	—	2	1	no	—
233	C₂×C₄⋊Q₈	—	2	1	no	—
234	C₂×C₄⋊Q₈	C₂²×C₆₇	1	2	yes	—
235	C₂×C₄⋊Q₈	C₂²×C₆₇	1	2	yes	—
236	C₂×#127	C₂²×C₆₇	1	1	yes	characteristic
237	C₂×#121	—	2	1	no	—
238	C₂×#121	—	2	1	no	—
239	C₂×#121	C₂²×C₆₇	1	2	yes	—
240	C₂²×C₄⋊C₄	—	2	1	no	—
241	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
242	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
243	#486	—	2	8	no	—
244	C₂×#102	C₂²×C₆₇	1	8	yes	—
245	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
246	C₂×C₄×C₈	C₂²×C₆₇	1	2	yes	—
247	#390	—	2	1	no	—
248	#390	—	2	1	no	—
249	#390	C₂²×C₆₇	1	2	yes	—
250	#384	C₂²×C₆₇	1	1	yes	characteristic
251	#340	C₂²×C₆₇	1	8	yes	—
252	#336	C₂²×C₆₇	1	2	yes	—
253	C₂²×Q₈	—	2	8	no	—
254	C₂²×Q₈	—	2	2	no	—
255	C₂²×Q₈	—	2	2	no	—
256	C₂²×Q₈	—	4	4	no	—
257	C₂²×Q₈	—	4	4	no	—
258	C₂²×D₄	—	2	2	no	—
259	C₂²×D₄	—	2	1	no	—
260	C₂²×D₄	—	2	1	no	—
261	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
262	C₂³×C₄	—	2	2	no	—
263	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
264	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
265	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
266	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
267	C₄⋊Q₈	—	2	8	no	—
268	C₄⋊Q₈	—	2	8	no	—
269	C₄⋊Q₈	—	2	8	no	—
270	C₄⋊Q₈	—	2	4	no	—
271	C₄⋊Q₈	D₄×C₆₇	1	8	yes	complemented
272	#127	—	2	4	no	—
273	#121	—	2	8	no	—
274	#121	—	4	4	no	—
275	#121	—	4	4	no	—
276	C₂×C₄⋊C₄	—	2	8	no	—
277	C₂×C₄⋊C₄	—	2	2	no	—
278	C₂×C₄⋊C₄	—	2	2	no	—
279	C₂×C₄⋊C₄	—	2	2	no	—
280	C₂×C₄⋊C₄	—	2	8	no	—
281	C₂×C₄⋊C₄	—	2	1	no	—
282	C₂×C₄⋊C₄	—	2	1	no	—
283	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
284	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
285	C₂×C₂²⋊C₄	—	2	2	no	—
286	C₂×C₂²⋊C₄	—	2	2	no	—
287	C₂×C₂²⋊C₄	—	2	2	no	—
288	C₂×C₄²	—	2	4	no	—
289	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
290	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
291	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
292	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
293	#102	—	2	16	no	—
294	C₂²⋊C₈	—	2	4	no	—
295	C₄×C₈	—	2	4	no	—
296	C₂⁴	—	2	2	no	—
297	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
298	C₂×Q₈	—	2	8	no	—
299	C₂×Q₈	—	2	8	no	—
300	C₂×Q₈	—	4	4	no	—
301	C₂×Q₈	—	4	32	no	—
302	C₂×Q₈	—	4	4	no	—
303	C₂×Q₈	—	4	4	no	—
304	C₂×Q₈	—	4	4	no	—
305	C₂×D₄	—	2	4	no	—
306	C₂×D₄	—	2	4	no	—
307	C₂×D₄	—	2	4	no	—
308	C₂×D₄	—	4	4	no	—
309	C₂×D₄	—	4	4	no	—
310	C₂×D₄	—	4	4	no	—
311	C₂×D₄	—	4	2	no	—
312	C₂×D₄	—	4	2	no	—
313	C₂×D₄	—	4	2	no	—
314	C₂×D₄	—	4	2	no	—
315	C₂×D₄	—	4	2	no	—
316	C₂×D₄	—	4	2	no	—
317	C₂²×C₄	—	2	2	no	—
318	C₂²×C₄	—	2	2	no	—
319	C₂²×C₄	—	2	2	no	—
320	C₂²×C₄	—	2	2	no	—
321	C₂²×C₄	—	2	2	no	—
322	C₂²×C₄	—	2	2	no	—
323	C₂²×C₄	—	2	8	no	—
324	C₂²×C₄	—	2	4	no	—
325	C₂²×C₄	—	2	2	no	—
326	C₂²×C₄	—	4	4	no	—
327	C₂²×C₄	—	4	4	no	—
328	C₂²×C₄	D₈×C₆₇	1	4	yes	—
329	C₂²×C₄	SD₁₆×C₆₇	1	4	yes	—
330	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
331	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
332	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
333	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
334	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
335	C₂²×C₄	#41×C₆₇	1	4	yes	—
336	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
337	C₂×C₈	—	2	4	no	—
338	C₂×C₈	—	2	4	no	—
339	C₂×C₈	—	2	4	no	—
340	C₄⋊C₄	—	2	8	no	—
341	C₄⋊C₄	—	2	8	no	—
342	C₄⋊C₄	—	2	8	no	—
343	C₄⋊C₄	—	2	4	no	—
344	C₄⋊C₄	—	2	4	no	—
345	C₄⋊C₄	—	4	2	no	—
346	C₄⋊C₄	—	4	2	no	—
347	C₄⋊C₄	C₂×D₄×C₆₇	1	8	yes	—
348	C₂²⋊C₄	—	4	4	no	—
349	C₂²⋊C₄	—	4	4	no	—
350	C₂²⋊C₄	—	4	4	no	—
351	C₄²	—	2	2	no	—
352	C₄²	—	2	2	no	—
353	C₄²	—	2	2	no	—
354	C₄²	C₂×D₄×C₆₇	1	4	yes	—
355	C₂³	—	2	1	no	—
356	C₂³	—	2	1	no	—
357	C₂³	—	2	1	no	—
358	C₂³	—	2	1	no	—
359	C₂³	—	4	2	no	—
360	C₂³	—	4	2	no	—
361	C₂³	—	4	2	no	—
362	C₂³	—	4	2	no	—
363	C₂³	—	4	2	no	—
364	C₂³	—	4	2	no	—
365	C₂³	—	4	2	no	—
366	C₂³	#124×C₆₇	1	2	yes	—
367	C₂³	C₂×D₈×C₆₇	1	2	yes	—
368	C₂³	C₂×SD₁₆×C₆₇	1	2	yes	—
369	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
370	Q₈	—	4	16	no	—
371	Q₈	—	4	16	no	—
372	D₄	—	4	8	no	—
373	D₄	—	8	8	no	—
374	D₄	—	8	4	no	—
375	D₄	—	8	4	no	—
376	C₂×C₄	—	2	1	no	—
377	C₂×C₄	—	2	1	no	—
378	C₂×C₄	—	2	1	no	—
379	C₂×C₄	—	2	1	no	—
380	C₂×C₄	—	2	1	no	—
381	C₂×C₄	—	2	1	no	—
382	C₂×C₄	—	2	8	no	—
383	C₂×C₄	—	2	8	no	—
384	C₂×C₄	—	2	8	no	—
385	C₂×C₄	—	2	8	no	—
386	C₂×C₄	—	4	16	no	—
387	C₂×C₄	—	4	4	no	—
388	C₂×C₄	—	4	4	no	—
389	C₂×C₄	—	4	2	no	—
390	C₂×C₄	—	4	2	no	—
391	C₂×C₄	—	4	2	no	—
392	C₂×C₄	#120×C₆₇	1	2	yes	—
393	C₂×C₄	#120×C₆₇	1	2	yes	—
394	C₂×C₄	#124×C₆₇	1	2	yes	—
395	C₂×C₄	C₂×D₈×C₆₇	1	2	yes	—
396	C₂×C₄	C₂×D₈×C₆₇	1	2	yes	—
397	C₂×C₄	C₂×SD₁₆×C₆₇	1	2	yes	—
398	C₂×C₄	C₂×SD₁₆×C₆₇	1	2	yes	—
399	C₂×C₄	AGL(1,ℤ/8)×C₆₇	1	2	yes	—
400	C₂×C₄	#137×C₆₇	1	2	yes	—
401	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
402	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
403	C₂×C₄	2₋¹⁺⁴×C₆₇	1	2	yes	—
404	C₈	—	4	4	no	—
405	C₂²	—	2	2	no	—
406	C₂²	—	2	2	no	—
407	C₂²	—	2	2	no	—
408	C₂²	—	2	2	no	—
409	C₂²	—	2	2	no	—
410	C₂²	—	2	2	no	—
411	C₂²	—	8	2	no	—
412	C₂²	—	8	2	no	—
413	C₂²	—	8	2	no	—
414	C₂²	—	8	2	no	—
415	C₂²	—	8	2	no	—
416	C₂²	—	8	2	no	—
417	C₂²	—	8	2	no	—
418	C₂²	#486×C₆₇	1	4	yes	complemented
419	C₂²	C₂×#120×C₆₇	1	1	yes	characteristic
420	C₂²	C₂×#124×C₆₇	1	1	yes	characteristic
421	C₂²	#536×C₆₇	1	1	yes	characteristic
422	C₂²	C₂²×D₈×C₆₇	1	1	yes	characteristic
423	C₂²	C₂²×SD₁₆×C₆₇	1	1	yes	characteristic
424	C₂²	C₂×AGL(1,ℤ/8)×C₆₇	1	1	yes	characteristic
425	C₂²	C₂×#137×C₆₇	1	1	yes	characteristic
426	C₄	—	2	2	no	—
427	C₄	—	2	2	no	—
428	C₄	—	2	2	no	—
429	C₄	—	4	8	no	—
430	C₄	—	8	2	no	—
431	C₄	#447×C₆₇	1	4	yes	—
432	C₄	#448×C₆₇	1	4	yes	—
433	C₄	#548×C₆₇	1	4	yes	—
434	C₂	—	2	4	no	—
435	C₂	—	16	2	no	—
436	C₂	C₆₇×#2436	1	1	yes	characteristic
437	C₂	C₂×#447×C₆₇	1	1	yes	characteristic
438	C₂	C₂×#448×C₆₇	1	1	yes	characteristic
439	C₂	C₂×#486×C₆₇	1	1	yes	characteristic
440	C₂	C₆₇×#3141	1	1	yes	characteristic
441	C₂	C₆₇×#3226	1	1	yes	characteristic
442	C₂	C₆₇×#3227	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₆₇×#2856	C₂
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. C₂²×C₄⋊Q₈×C₆₇	C₂	semidirect
2. C₂×#486×C₆₇	C₂	semidirect
4. C₆₇×#2023	C₂	semidirect
5. C₆₇×#1990	C₂	semidirect
6. C₆₇×#1874	C₂	semidirect
7. C₄×C₂²⋊C₈×C₆₇	C₂	semidirect
49. C₄⋊Q₈×C₆₇	D₄	semidirect
196. C₂²×C₆₇	#486	semidirect
212. #25903	C₆₇	direct (N × H)
222. C₂²×C₄⋊Q₈	C₂×C₆₇	semidirect
223. C₂×#486	C₂×C₆₇	semidirect
225. #2023	C₂×C₆₇	semidirect
226. #1990	C₂×C₆₇	semidirect
227. #1874	C₂×C₆₇	semidirect
228. C₄×C₂²⋊C₈	C₂×C₆₇	semidirect
229. C₆₇	#25903	direct (N × H)
271. C₄⋊Q₈	D₄×C₆₇	semidirect
418. C₂²	#486×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₄⋊Q₈×C₆₇	2
C₂×#486×C₆₇	2
C₆₇×#2856	2
C₆₇×#2023	2
C₆₇×#1990	2
C₆₇×#1874	2
C₄×C₂²⋊C₈×C₆₇	2
#25903	67

Subgroups — C67×#25903 #30705154

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#25903 #30705154