Subgroups — C₆₇×#25902 #30705153

Order 17152.

1306 conjugacy classes of subgroups.

The subgroup lattice has 396 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
2	C₂×#487×C₆₇	C₂	1	2	yes	maximal, complemented
3	C₆₇×#2857	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#2024	C₂	1	4	yes	maximal, complemented
5	C₆₇×#1990	C₂	1	2	yes	maximal, complemented
6	C₆₇×#1872	C₂	1	4	yes	maximal
7	C₄×C₂²⋊C₈×C₆₇	C₂	1	1	yes	maximal, characteristic
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	1	yes	characteristic
13	C₂×C₄⋊Q₈×C₆₇	C₂²	1	2	yes	—
14	C₂×C₄⋊Q₈×C₆₇	C₂²	1	2	yes	—
15	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	2	yes	—
16	C₂²×C₄⋊C₄×C₆₇	—	2	1	no	—
17	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
18	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
19	#487×C₆₇	—	2	8	no	—
20	C₂×#103×C₆₇	C₂²	1	8	yes	—
21	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
22	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
23	#391×C₆₇	C₂²	1	2	yes	—
24	#389×C₆₇	—	2	1	no	—
25	#389×C₆₇	—	2	1	no	—
26	#384×C₆₇	C₂²	1	1	yes	characteristic
27	#380×C₆₇	—	2	1	no	—
28	#380×C₆₇	—	2	1	no	—
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×Q₈×C₆₇	C₂³	1	2	yes	—
37	C₂³×C₄×C₆₇	—	2	2	no	—
38	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
39	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
40	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
41	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
42	C₄⋊Q₈×C₆₇	—	2	8	no	—
43	C₄⋊Q₈×C₆₇	—	2	8	no	—
44	C₄⋊Q₈×C₆₇	—	2	8	no	—
45	C₄⋊Q₈×C₆₇	—	2	4	no	—
46	C₄⋊Q₈×C₆₇	—	2	4	no	—
47	C₄⋊Q₈×C₆₇	D₄	1	8	yes	—
48	C₂²⋊Q₈×C₆₇	—	2	8	no	—
49	C₂×C₄⋊C₄×C₆₇	—	2	8	no	—
50	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
51	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
52	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
53	C₂×C₄⋊C₄×C₆₇	—	2	8	no	—
54	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
55	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
56	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
57	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
58	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
59	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
60	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
61	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
62	C₂×C₄²×C₆₇	—	2	4	no	—
63	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
64	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
65	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
66	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
67	#103×C₆₇	—	2	16	no	—
68	C₂²⋊C₈×C₆₇	—	2	4	no	—
69	C₄×C₈×C₆₇	—	2	4	no	—
70	#95×C₆₇	—	2	2	no	—
71	#95×C₆₇	—	2	2	no	—
72	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
73	C₂×Q₈×C₆₇	—	2	8	no	—
74	C₂×Q₈×C₆₇	—	2	8	no	—
75	C₂×Q₈×C₆₇	—	2	4	no	—
76	C₂×Q₈×C₆₇	—	2	4	no	—
77	C₂×Q₈×C₆₇	—	2	4	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	C₂²×C₄×C₆₇	—	2	2	no	—
84	C₂²×C₄×C₆₇	—	2	2	no	—
85	C₂²×C₄×C₆₇	—	2	2	no	—
86	C₂²×C₄×C₆₇	—	2	2	no	—
87	C₂²×C₄×C₆₇	—	2	2	no	—
88	C₂²×C₄×C₆₇	—	2	2	no	—
89	C₂²×C₄×C₆₇	—	2	8	no	—
90	C₂²×C₄×C₆₇	—	2	4	no	—
91	C₂²×C₄×C₆₇	—	2	2	no	—
92	C₂²×C₄×C₆₇	—	2	2	no	—
93	C₂²×C₄×C₆₇	—	4	4	no	—
94	C₂²×C₄×C₆₇	—	4	4	no	—
95	C₂²×C₄×C₆₇	SD₁₆	1	4	yes	—
96	C₂²×C₄×C₆₇	Q₁₆	1	4	yes	—
97	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
98	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
99	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
100	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
101	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
102	C₂²×C₄×C₆₇	#41	1	4	yes	—
103	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
104	C₂×C₈×C₆₇	—	2	4	no	—
105	C₂×C₈×C₆₇	—	2	4	no	—
106	C₂×C₈×C₆₇	—	2	4	no	—
107	C₄⋊C₄×C₆₇	—	2	8	no	—
108	C₄⋊C₄×C₆₇	—	2	8	no	—
109	C₄⋊C₄×C₆₇	—	2	8	no	—
110	C₄⋊C₄×C₆₇	—	2	4	no	—
111	C₄⋊C₄×C₆₇	—	2	4	no	—
112	C₄⋊C₄×C₆₇	—	4	2	no	—
113	C₄⋊C₄×C₆₇	—	4	2	no	—
114	C₄⋊C₄×C₆₇	—	4	4	no	—
115	C₄⋊C₄×C₆₇	—	4	2	no	—
116	C₄⋊C₄×C₆₇	—	4	2	no	—
117	C₄⋊C₄×C₆₇	C₂×D₄	1	8	yes	—
118	C₂²⋊C₄×C₆₇	—	4	4	no	—
119	C₄²×C₆₇	—	2	2	no	—
120	C₄²×C₆₇	—	2	2	no	—
121	C₄²×C₆₇	—	2	2	no	—
122	C₄²×C₆₇	C₂×D₄	1	4	yes	—
123	C₂³×C₆₇	—	2	1	no	—
124	C₂³×C₆₇	—	2	1	no	—
125	C₂³×C₆₇	—	2	1	no	—
126	C₂³×C₆₇	—	2	1	no	—
127	C₂³×C₆₇	#124	1	2	yes	—
128	C₂³×C₆₇	C₂×SD₁₆	1	2	yes	—
129	C₂³×C₆₇	C₂×Q₁₆	1	2	yes	—
130	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
131	Q₈×C₆₇	—	4	16	no	—
132	Q₈×C₆₇	—	4	16	no	—
133	Q₈×C₆₇	—	4	8	no	—
134	C₂×C₄×C₆₇	—	2	1	no	—
135	C₂×C₄×C₆₇	—	2	1	no	—
136	C₂×C₄×C₆₇	—	2	1	no	—
137	C₂×C₄×C₆₇	—	2	1	no	—
138	C₂×C₄×C₆₇	—	2	1	no	—
139	C₂×C₄×C₆₇	—	2	1	no	—
140	C₂×C₄×C₆₇	—	2	8	no	—
141	C₂×C₄×C₆₇	—	2	8	no	—
142	C₂×C₄×C₆₇	—	2	8	no	—
143	C₂×C₄×C₆₇	—	2	8	no	—
144	C₂×C₄×C₆₇	—	4	16	no	—
145	C₂×C₄×C₆₇	—	4	4	no	—
146	C₂×C₄×C₆₇	—	4	4	no	—
147	C₂×C₄×C₆₇	—	4	2	no	—
148	C₂×C₄×C₆₇	—	4	2	no	—
149	C₂×C₄×C₆₇	—	4	2	no	—
150	C₂×C₄×C₆₇	—	4	2	no	—
151	C₂×C₄×C₆₇	—	4	2	no	—
152	C₂×C₄×C₆₇	—	4	2	no	—
153	C₂×C₄×C₆₇	#120	1	2	yes	—
154	C₂×C₄×C₆₇	#120	1	2	yes	—
155	C₂×C₄×C₆₇	#124	1	2	yes	—
156	C₂×C₄×C₆₇	C₂×SD₁₆	1	2	yes	—
157	C₂×C₄×C₆₇	C₂×SD₁₆	1	2	yes	—
158	C₂×C₄×C₆₇	C₂×Q₁₆	1	2	yes	—
159	C₂×C₄×C₆₇	C₂×Q₁₆	1	2	yes	—
160	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
161	C₂×C₄×C₆₇	#137	1	2	yes	—
162	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
163	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
164	C₂×C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
165	C₈×C₆₇	—	4	4	no	—
166	C₂²×C₆₇	—	2	2	no	—
167	C₂²×C₆₇	—	2	2	no	—
168	C₂²×C₆₇	—	2	2	no	—
169	C₂²×C₆₇	—	2	2	no	—
170	C₂²×C₆₇	—	2	2	no	—
171	C₂²×C₆₇	—	2	2	no	—
172	C₂²×C₆₇	#487	1	4	yes	complemented
173	C₂²×C₆₇	C₂×#120	1	1	yes	characteristic
174	C₂²×C₆₇	C₂×#124	1	1	yes	characteristic
175	C₂²×C₆₇	#536	1	1	yes	characteristic
176	C₂²×C₆₇	C₂²×SD₁₆	1	1	yes	characteristic
177	C₂²×C₆₇	C₂²×Q₁₆	1	1	yes	characteristic
178	C₂²×C₆₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
179	C₂²×C₆₇	C₂×#137	1	1	yes	characteristic
180	C₄×C₆₇	—	2	2	no	—
181	C₄×C₆₇	—	2	2	no	—
182	C₄×C₆₇	—	2	2	no	—
183	C₄×C₆₇	—	4	8	no	—
184	C₄×C₆₇	—	8	2	no	—
185	C₄×C₆₇	—	8	2	no	—
186	C₄×C₆₇	#450	1	4	yes	—
187	C₄×C₆₇	#451	1	4	yes	—
188	C₄×C₆₇	#548	1	4	yes	—
189	#25902	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
190	C₂×C₆₇	—	2	4	no	—
191	C₂×C₆₇	#2436	1	1	yes	characteristic
192	C₂×C₆₇	C₂×#450	1	1	yes	characteristic
193	C₂×C₆₇	C₂×#451	1	1	yes	characteristic
194	C₂×C₆₇	C₂×#487	1	1	yes	characteristic
195	C₂×C₆₇	#3141	1	1	yes	characteristic
196	C₂×C₆₇	#3229	1	1	yes	characteristic
197	C₂×C₆₇	#3230	1	1	yes	characteristic
198	C₂²×C₄⋊Q₈	C₂×C₆₇	1	1	yes	characteristic
199	C₂×#487	C₂×C₆₇	1	2	yes	complemented
200	#2857	C₂×C₆₇	1	1	yes	characteristic
201	#2024	C₂×C₆₇	1	4	yes	complemented
202	#1990	C₂×C₆₇	1	2	yes	complemented
203	#1872	C₂×C₆₇	1	4	yes	—
204	C₄×C₂²⋊C₈	C₂×C₆₇	1	1	yes	characteristic
205	C₆₇	#25902	1	1	yes	characteristic, direct product factor, Sylow
206	C₂³×Q₈	—	2	1	no	—
207	C₂×C₄⋊Q₈	—	2	8	no	—
208	C₂×C₄⋊Q₈	—	2	1	no	—
209	C₂×C₄⋊Q₈	—	2	1	no	—
210	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
211	C₂×C₄⋊Q₈	C₂²×C₆₇	1	2	yes	—
212	C₂×C₄⋊Q₈	C₂²×C₆₇	1	2	yes	—
213	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	2	yes	—
214	C₂²×C₄⋊C₄	—	2	1	no	—
215	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
216	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
217	#487	—	2	8	no	—
218	C₂×#103	C₂²×C₆₇	1	8	yes	—
219	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
220	C₂×C₄×C₈	C₂²×C₆₇	1	2	yes	—
221	#391	C₂²×C₆₇	1	2	yes	—
222	#389	—	2	1	no	—
223	#389	—	2	1	no	—
224	#384	C₂²×C₆₇	1	1	yes	characteristic
225	#380	—	2	1	no	—
226	#380	—	2	1	no	—
227	#340	C₂²×C₆₇	1	8	yes	—
228	#336	C₂²×C₆₇	1	2	yes	—
229	C₂²×Q₈	—	2	8	no	—
230	C₂²×Q₈	—	2	2	no	—
231	C₂²×Q₈	—	2	2	no	—
232	C₂²×Q₈	—	4	4	no	—
233	C₂²×Q₈	—	4	4	no	—
234	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
235	C₂³×C₄	—	2	2	no	—
236	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
237	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
238	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
239	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
240	C₄⋊Q₈	—	2	8	no	—
241	C₄⋊Q₈	—	2	8	no	—
242	C₄⋊Q₈	—	2	8	no	—
243	C₄⋊Q₈	—	2	4	no	—
244	C₄⋊Q₈	—	2	4	no	—
245	C₄⋊Q₈	D₄×C₆₇	1	8	yes	—
246	C₂²⋊Q₈	—	2	8	no	—
247	C₂×C₄⋊C₄	—	2	8	no	—
248	C₂×C₄⋊C₄	—	2	2	no	—
249	C₂×C₄⋊C₄	—	2	2	no	—
250	C₂×C₄⋊C₄	—	2	2	no	—
251	C₂×C₄⋊C₄	—	2	8	no	—
252	C₂×C₄⋊C₄	—	2	2	no	—
253	C₂×C₄⋊C₄	—	2	1	no	—
254	C₂×C₄⋊C₄	—	2	1	no	—
255	C₂×C₄⋊C₄	—	2	1	no	—
256	C₂×C₄⋊C₄	—	2	1	no	—
257	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
258	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
259	C₂×C₂²⋊C₄	—	2	2	no	—
260	C₂×C₄²	—	2	4	no	—
261	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
262	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
263	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
264	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
265	#103	—	2	16	no	—
266	C₂²⋊C₈	—	2	4	no	—
267	C₄×C₈	—	2	4	no	—
268	#95	—	2	2	no	—
269	#95	—	2	2	no	—
270	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
271	C₂×Q₈	—	2	8	no	—
272	C₂×Q₈	—	2	8	no	—
273	C₂×Q₈	—	2	4	no	—
274	C₂×Q₈	—	2	4	no	—
275	C₂×Q₈	—	2	4	no	—
276	C₂×Q₈	—	4	4	no	—
277	C₂×Q₈	—	4	32	no	—
278	C₂×Q₈	—	4	4	no	—
279	C₂×Q₈	—	4	4	no	—
280	C₂×Q₈	—	4	4	no	—
281	C₂²×C₄	—	2	2	no	—
282	C₂²×C₄	—	2	2	no	—
283	C₂²×C₄	—	2	2	no	—
284	C₂²×C₄	—	2	2	no	—
285	C₂²×C₄	—	2	2	no	—
286	C₂²×C₄	—	2	2	no	—
287	C₂²×C₄	—	2	8	no	—
288	C₂²×C₄	—	2	4	no	—
289	C₂²×C₄	—	2	2	no	—
290	C₂²×C₄	—	2	2	no	—
291	C₂²×C₄	—	4	4	no	—
292	C₂²×C₄	—	4	4	no	—
293	C₂²×C₄	SD₁₆×C₆₇	1	4	yes	—
294	C₂²×C₄	Q₁₆×C₆₇	1	4	yes	—
295	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
296	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
297	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
298	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
299	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
300	C₂²×C₄	#41×C₆₇	1	4	yes	—
301	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
302	C₂×C₈	—	2	4	no	—
303	C₂×C₈	—	2	4	no	—
304	C₂×C₈	—	2	4	no	—
305	C₄⋊C₄	—	2	8	no	—
306	C₄⋊C₄	—	2	8	no	—
307	C₄⋊C₄	—	2	8	no	—
308	C₄⋊C₄	—	2	4	no	—
309	C₄⋊C₄	—	2	4	no	—
310	C₄⋊C₄	—	4	2	no	—
311	C₄⋊C₄	—	4	2	no	—
312	C₄⋊C₄	—	4	4	no	—
313	C₄⋊C₄	—	4	2	no	—
314	C₄⋊C₄	—	4	2	no	—
315	C₄⋊C₄	C₂×D₄×C₆₇	1	8	yes	—
316	C₂²⋊C₄	—	4	4	no	—
317	C₄²	—	2	2	no	—
318	C₄²	—	2	2	no	—
319	C₄²	—	2	2	no	—
320	C₄²	C₂×D₄×C₆₇	1	4	yes	—
321	C₂³	—	2	1	no	—
322	C₂³	—	2	1	no	—
323	C₂³	—	2	1	no	—
324	C₂³	—	2	1	no	—
325	C₂³	#124×C₆₇	1	2	yes	—
326	C₂³	C₂×SD₁₆×C₆₇	1	2	yes	—
327	C₂³	C₂×Q₁₆×C₆₇	1	2	yes	—
328	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
329	Q₈	—	4	16	no	—
330	Q₈	—	4	16	no	—
331	Q₈	—	4	8	no	—
332	C₂×C₄	—	2	1	no	—
333	C₂×C₄	—	2	1	no	—
334	C₂×C₄	—	2	1	no	—
335	C₂×C₄	—	2	1	no	—
336	C₂×C₄	—	2	1	no	—
337	C₂×C₄	—	2	1	no	—
338	C₂×C₄	—	2	8	no	—
339	C₂×C₄	—	2	8	no	—
340	C₂×C₄	—	2	8	no	—
341	C₂×C₄	—	2	8	no	—
342	C₂×C₄	—	4	16	no	—
343	C₂×C₄	—	4	4	no	—
344	C₂×C₄	—	4	4	no	—
345	C₂×C₄	—	4	2	no	—
346	C₂×C₄	—	4	2	no	—
347	C₂×C₄	—	4	2	no	—
348	C₂×C₄	—	4	2	no	—
349	C₂×C₄	—	4	2	no	—
350	C₂×C₄	—	4	2	no	—
351	C₂×C₄	#120×C₆₇	1	2	yes	—
352	C₂×C₄	#120×C₆₇	1	2	yes	—
353	C₂×C₄	#124×C₆₇	1	2	yes	—
354	C₂×C₄	C₂×SD₁₆×C₆₇	1	2	yes	—
355	C₂×C₄	C₂×SD₁₆×C₆₇	1	2	yes	—
356	C₂×C₄	C₂×Q₁₆×C₆₇	1	2	yes	—
357	C₂×C₄	C₂×Q₁₆×C₆₇	1	2	yes	—
358	C₂×C₄	AGL(1,ℤ/8)×C₆₇	1	2	yes	—
359	C₂×C₄	#137×C₆₇	1	2	yes	—
360	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
361	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
362	C₂×C₄	2₋¹⁺⁴×C₆₇	1	2	yes	—
363	C₈	—	4	4	no	—
364	C₂²	—	2	2	no	—
365	C₂²	—	2	2	no	—
366	C₂²	—	2	2	no	—
367	C₂²	—	2	2	no	—
368	C₂²	—	2	2	no	—
369	C₂²	—	2	2	no	—
370	C₂²	#487×C₆₇	1	4	yes	complemented
371	C₂²	C₂×#120×C₆₇	1	1	yes	characteristic
372	C₂²	C₂×#124×C₆₇	1	1	yes	characteristic
373	C₂²	#536×C₆₇	1	1	yes	characteristic
374	C₂²	C₂²×SD₁₆×C₆₇	1	1	yes	characteristic
375	C₂²	C₂²×Q₁₆×C₆₇	1	1	yes	characteristic
376	C₂²	C₂×AGL(1,ℤ/8)×C₆₇	1	1	yes	characteristic
377	C₂²	C₂×#137×C₆₇	1	1	yes	characteristic
378	C₄	—	2	2	no	—
379	C₄	—	2	2	no	—
380	C₄	—	2	2	no	—
381	C₄	—	4	8	no	—
382	C₄	—	8	2	no	—
383	C₄	—	8	2	no	—
384	C₄	#450×C₆₇	1	4	yes	—
385	C₄	#451×C₆₇	1	4	yes	—
386	C₄	#548×C₆₇	1	4	yes	—
387	C₂	—	2	4	no	—
388	C₂	C₆₇×#2436	1	1	yes	characteristic
389	C₂	C₂×#450×C₆₇	1	1	yes	characteristic
390	C₂	C₂×#451×C₆₇	1	1	yes	characteristic
391	C₂	C₂×#487×C₆₇	1	1	yes	characteristic
392	C₂	C₆₇×#3141	1	1	yes	characteristic
393	C₂	C₆₇×#3229	1	1	yes	characteristic
394	C₂	C₆₇×#3230	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₂⁴×C₆₇	C₂×D₄
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
2. C₂×#487×C₆₇	C₂	semidirect
4. C₆₇×#2024	C₂	semidirect
5. C₆₇×#1990	C₂	semidirect
172. C₂²×C₆₇	#487	semidirect
189. #25902	C₆₇	direct (N × H)
199. C₂×#487	C₂×C₆₇	semidirect
201. #2024	C₂×C₆₇	semidirect
202. #1990	C₂×C₆₇	semidirect
205. C₆₇	#25902	direct (N × H)
370. C₂²	#487×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×#487×C₆₇	2
C₆₇×#2857	2
C₆₇×#2024	2
C₆₇×#1990	2
C₆₇×#1872	2
C₄×C₂²⋊C₈×C₆₇	2
#25902	67

Subgroups — C67×#25902 #30705153

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#25902 #30705153