Subgroups — C₆₇×#33031 #30712282

Order 17152.

978 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#477×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#460×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#2858	C₂	1	1	yes	maximal, characteristic
4	C₄²×Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#1987	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₇×#1982	C₂	1	1	yes	maximal, characteristic
7	C₆₇×#1925	C₂	1	4	yes	maximal, complemented
8	C₆₇×#1880	C₂	1	4	yes	maximal, complemented
9	C₆₇×#1831	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×SD₁₆×C₆₇	C₂²	1	2	yes	—
11	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
12	C₂×#127×C₆₇	C₂²	1	1	yes	characteristic
13	C₂×#124×C₆₇	—	2	2	no	—
14	C₂×C₄×Q₈×C₆₇	C₂²	1	2	yes	—
15	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
16	#477×C₆₇	—	2	4	no	—
17	#460×C₆₇	—	2	4	no	—
18	C₂×#107×C₆₇	C₂²	1	2	yes	—
19	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
21	C₂×#103×C₆₇	C₂²	1	2	yes	—
22	C₂×#102×C₆₇	C₂²	1	2	yes	—
23	C₂×#102×C₆₇	C₂²	1	4	yes	—
24	#390×C₆₇	—	2	1	no	—
25	#390×C₆₇	—	2	1	no	—
26	#390×C₆₇	C₂²	1	4	yes	—
27	C₄×C₄⋊C₄×C₆₇	—	2	2	no	—
28	C₄×C₄⋊C₄×C₆₇	—	2	1	no	—
29	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
30	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
31	C₄³×C₆₇	—	2	1	no	—
32	C₄³×C₆₇	C₂²	1	1	yes	characteristic
33	#340×C₆₇	C₂²	1	4	yes	—
34	#336×C₆₇	C₂²	1	4	yes	—
35	C₂²×Q₈×C₆₇	—	2	1	no	—
36	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
37	C₂²×D₄×C₆₇	—	2	1	no	—
38	C₂²×D₄×C₆₇	—	2	1	no	—
39	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
40	C₂×SD₁₆×C₆₇	—	2	4	no	—
41	C₂×SD₁₆×C₆₇	—	2	8	no	—
42	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
43	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
44	C₄⋊Q₈×C₆₇	—	2	2	no	—
45	C₄⋊Q₈×C₆₇	—	2	2	no	—
46	#127×C₆₇	—	2	2	no	—
47	#127×C₆₇	—	2	2	no	—
48	#124×C₆₇	—	4	8	no	—
49	C₄×Q₈×C₆₇	—	2	4	no	—
50	C₄×Q₈×C₆₇	—	2	4	no	—
51	C₄×Q₈×C₆₇	—	2	2	no	—
52	C₄×Q₈×C₆₇	D₄	1	4	yes	—
53	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
54	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
55	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
56	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
57	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
58	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
59	C₂×C₂²⋊C₄×C₆₇	—	2	4	no	—
60	C₂×C₂²⋊C₄×C₆₇	—	2	4	no	—
61	C₂×C₄²×C₆₇	—	2	2	no	—
62	C₂×C₄²×C₆₇	—	2	1	no	—
63	C₂×C₄²×C₆₇	—	2	1	no	—
64	C₂×C₄²×C₆₇	—	2	2	no	—
65	C₂×C₄²×C₆₇	D₄	1	2	yes	—
66	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
67	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
68	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
69	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
70	#107×C₆₇	—	2	4	no	—
71	C₄⋊C₈×C₆₇	—	2	2	no	—
72	C₄⋊C₈×C₆₇	—	2	2	no	—
73	#103×C₆₇	—	2	4	no	—
74	#102×C₆₇	—	2	4	no	—
75	#102×C₆₇	—	2	8	no	—
76	#95×C₆₇	—	2	2	no	—
77	#95×C₆₇	—	2	1	no	—
78	C₂⁴×C₆₇	—	2	2	no	—
79	C₂×Q₈×C₆₇	—	2	1	no	—
80	C₂×Q₈×C₆₇	—	2	2	no	—
81	C₂×Q₈×C₆₇	—	4	4	no	—
82	C₂×Q₈×C₆₇	—	4	2	no	—
83	C₂×Q₈×C₆₇	C₂×D₄	1	2	yes	—
84	C₂×Q₈×C₆₇	#41	1	4	yes	—
85	C₂×D₄×C₆₇	—	2	2	no	—
86	C₂×D₄×C₆₇	—	2	4	no	—
87	C₂×D₄×C₆₇	—	2	2	no	—
88	C₂×D₄×C₆₇	—	2	4	no	—
89	C₂×D₄×C₆₇	—	4	4	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	C₂²×C₄×C₆₇	—	2	1	no	—
95	C₂²×C₄×C₆₇	—	2	1	no	—
96	C₂²×C₄×C₆₇	—	2	2	no	—
97	C₂²×C₄×C₆₇	—	2	2	no	—
98	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
99	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
100	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
101	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
102	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
103	SD₁₆×C₆₇	—	4	8	no	—
104	C₂×C₈×C₆₇	—	2	2	no	—
105	C₂×C₈×C₆₇	—	2	2	no	—
106	C₂×C₈×C₆₇	—	2	4	no	—
107	C₂×C₈×C₆₇	—	2	4	no	—
108	C₄⋊C₄×C₆₇	—	2	8	no	—
109	C₄⋊C₄×C₆₇	—	2	4	no	—
110	C₄⋊C₄×C₆₇	—	2	1	no	—
111	C₄⋊C₄×C₆₇	—	2	2	no	—
112	C₄⋊C₄×C₆₇	—	2	2	no	—
113	C₄⋊C₄×C₆₇	—	2	2	no	—
114	C₄⋊C₄×C₆₇	—	4	2	no	—
115	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
116	C₄⋊C₄×C₆₇	#41	1	4	yes	—
117	C₂²⋊C₄×C₆₇	—	4	8	no	—
118	C₂²⋊C₄×C₆₇	—	4	8	no	—
119	C₄²×C₆₇	—	2	8	no	—
120	C₄²×C₆₇	—	2	4	no	—
121	C₄²×C₆₇	—	2	2	no	—
122	C₄²×C₆₇	—	2	4	no	—
123	C₄²×C₆₇	—	2	2	no	—
124	C₄²×C₆₇	—	2	2	no	—
125	C₄²×C₆₇	—	2	2	no	—
126	C₄²×C₆₇	—	4	2	no	—
127	C₄²×C₆₇	—	4	4	no	—
128	C₄²×C₆₇	C₂×D₄	1	2	yes	—
129	C₄²×C₆₇	#41	1	2	yes	—
130	C₂³×C₆₇	—	4	2	no	—
131	C₂³×C₆₇	—	4	4	no	—
132	C₂³×C₆₇	—	4	2	no	—
133	C₂³×C₆₇	—	4	4	no	—
134	C₂³×C₆₇	—	4	2	no	—
135	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
136	Q₈×C₆₇	—	2	4	no	—
137	Q₈×C₆₇	—	2	2	no	—
138	Q₈×C₆₇	—	8	4	no	—
139	Q₈×C₆₇	#124	1	4	yes	complemented
140	D₄×C₆₇	—	4	2	no	—
141	D₄×C₆₇	—	4	4	no	—
142	D₄×C₆₇	—	4	2	no	—
143	D₄×C₆₇	—	8	4	no	—
144	D₄×C₆₇	—	8	4	no	—
145	C₂×C₄×C₆₇	—	2	2	no	—
146	C₂×C₄×C₆₇	—	2	2	no	—
147	C₂×C₄×C₆₇	—	2	4	no	—
148	C₂×C₄×C₆₇	—	2	2	no	—
149	C₂×C₄×C₆₇	—	2	2	no	—
150	C₂×C₄×C₆₇	—	2	2	no	—
151	C₂×C₄×C₆₇	—	2	2	no	—
152	C₂×C₄×C₆₇	—	2	4	no	—
153	C₂×C₄×C₆₇	—	2	2	no	—
154	C₂×C₄×C₆₇	—	2	2	no	—
155	C₂×C₄×C₆₇	—	4	2	no	—
156	C₂×C₄×C₆₇	—	4	2	no	—
157	C₂×C₄×C₆₇	—	4	2	no	—
158	C₂×C₄×C₆₇	—	4	2	no	—
159	C₂×C₄×C₆₇	—	4	4	no	—
160	C₂×C₄×C₆₇	#121	1	2	yes	—
161	C₂×C₄×C₆₇	#121	1	2	yes	—
162	C₂×C₄×C₆₇	#135	1	2	yes	—
163	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	4	yes	—
164	C₂×C₄×C₆₇	#137	1	2	yes	—
165	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
166	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
167	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
168	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
169	C₈×C₆₇	—	4	2	no	—
170	C₈×C₆₇	—	4	2	no	—
171	C₂²×C₆₇	—	8	2	no	—
172	C₂²×C₆₇	—	8	4	no	—
173	C₂²×C₆₇	—	8	4	no	—
174	C₂²×C₆₇	—	8	2	no	—
175	C₂²×C₆₇	—	8	2	no	—
176	C₂²×C₆₇	C₂×#121	1	1	yes	characteristic
177	C₂²×C₆₇	#525	1	2	yes	—
178	C₂²×C₆₇	C₂×#135	1	1	yes	characteristic
179	C₂²×C₆₇	C₂×AGL(1,ℤ/8)	1	2	yes	—
180	C₂²×C₆₇	C₂×#137	1	1	yes	characteristic
181	C₄×C₆₇	—	2	4	no	—
182	C₄×C₆₇	—	2	4	no	—
183	C₄×C₆₇	—	2	2	no	—
184	C₄×C₆₇	—	4	2	no	—
185	C₄×C₆₇	—	4	4	no	—
186	C₄×C₆₇	—	4	2	no	—
187	C₄×C₆₇	—	8	2	no	—
188	C₄×C₆₇	#464	1	4	yes	—
189	C₄×C₆₇	C₂×#124	1	1	yes	characteristic
190	C₄×C₆₇	#529	1	2	yes	—
191	C₄×C₆₇	#532	1	1	yes	characteristic
192	#33031	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
193	C₂×C₆₇	—	16	2	no	—
194	C₂×C₆₇	#2850	1	1	yes	characteristic
195	C₂×C₆₇	C₂×#464	1	1	yes	characteristic
196	C₂×C₆₇	#3038	1	1	yes	characteristic
197	C₂×C₆₇	#3106	1	2	yes	—
198	C₂×C₆₇	#3112	1	2	yes	—
199	C₂×#477	C₂×C₆₇	1	1	yes	characteristic, complemented
200	C₂×#460	C₂×C₆₇	1	1	yes	characteristic
201	#2858	C₂×C₆₇	1	1	yes	characteristic
202	C₄²×Q₈	C₂×C₆₇	1	1	yes	characteristic, complemented
203	#1987	C₂×C₆₇	1	1	yes	characteristic, complemented
204	#1982	C₂×C₆₇	1	1	yes	characteristic
205	#1925	C₂×C₆₇	1	4	yes	complemented
206	#1880	C₂×C₆₇	1	4	yes	complemented
207	#1831	C₂×C₆₇	1	1	yes	characteristic, complemented
208	C₆₇	#33031	1	1	yes	characteristic, direct product factor, Sylow
209	C₂²×SD₁₆	C₂²×C₆₇	1	2	yes	—
210	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
211	C₂×#127	C₂²×C₆₇	1	1	yes	characteristic
212	C₂×#124	—	2	2	no	—
213	C₂×C₄×Q₈	C₂²×C₆₇	1	2	yes	—
214	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
215	#477	—	2	4	no	—
216	#460	—	2	4	no	—
217	C₂×#107	C₂²×C₆₇	1	2	yes	—
218	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
219	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
220	C₂×#103	C₂²×C₆₇	1	2	yes	—
221	C₂×#102	C₂²×C₆₇	1	4	yes	—
222	C₂×#102	C₂²×C₆₇	1	2	yes	—
223	#390	—	2	1	no	—
224	#390	—	2	1	no	—
225	#390	C₂²×C₆₇	1	4	yes	—
226	C₄×C₄⋊C₄	—	2	2	no	—
227	C₄×C₄⋊C₄	—	2	1	no	—
228	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
229	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
230	C₄³	—	2	1	no	—
231	C₄³	C₂²×C₆₇	1	1	yes	characteristic
232	#340	C₂²×C₆₇	1	4	yes	—
233	#336	C₂²×C₆₇	1	4	yes	—
234	C₂²×Q₈	—	2	1	no	—
235	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
236	C₂²×D₄	—	2	1	no	—
237	C₂²×D₄	—	2	1	no	—
238	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
239	C₂×SD₁₆	—	2	4	no	—
240	C₂×SD₁₆	—	2	8	no	—
241	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
242	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
243	C₄⋊Q₈	—	2	2	no	—
244	C₄⋊Q₈	—	2	2	no	—
245	#127	—	2	2	no	—
246	#127	—	2	2	no	—
247	#124	—	4	8	no	—
248	C₄×Q₈	—	2	4	no	—
249	C₄×Q₈	—	2	4	no	—
250	C₄×Q₈	—	2	2	no	—
251	C₄×Q₈	D₄×C₆₇	1	4	yes	—
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	C₂³×C₆₇	1	2	yes	—
256	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
257	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
258	C₂×C₂²⋊C₄	—	2	4	no	—
259	C₂×C₂²⋊C₄	—	2	4	no	—
260	C₂×C₄²	—	2	2	no	—
261	C₂×C₄²	—	2	1	no	—
262	C₂×C₄²	—	2	1	no	—
263	C₂×C₄²	—	2	2	no	—
264	C₂×C₄²	D₄×C₆₇	1	2	yes	—
265	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
266	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
267	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
268	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
269	#107	—	2	4	no	—
270	C₄⋊C₈	—	2	2	no	—
271	C₄⋊C₈	—	2	2	no	—
272	#103	—	2	4	no	—
273	#102	—	2	4	no	—
274	#102	—	2	8	no	—
275	#95	—	2	2	no	—
276	#95	—	2	1	no	—
277	C₂⁴	—	2	2	no	—
278	C₂×Q₈	—	2	1	no	—
279	C₂×Q₈	—	2	2	no	—
280	C₂×Q₈	—	4	4	no	—
281	C₂×Q₈	—	4	2	no	—
282	C₂×Q₈	C₂×D₄×C₆₇	1	2	yes	—
283	C₂×Q₈	#41×C₆₇	1	4	yes	—
284	C₂×D₄	—	2	2	no	—
285	C₂×D₄	—	2	4	no	—
286	C₂×D₄	—	2	2	no	—
287	C₂×D₄	—	2	4	no	—
288	C₂×D₄	—	4	4	no	—
289	C₂×D₄	—	4	2	no	—
290	C₂×D₄	—	4	2	no	—
291	C₂×D₄	—	4	2	no	—
292	C₂×D₄	—	4	2	no	—
293	C₂²×C₄	—	2	1	no	—
294	C₂²×C₄	—	2	1	no	—
295	C₂²×C₄	—	2	2	no	—
296	C₂²×C₄	—	2	2	no	—
297	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
298	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
299	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
300	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
301	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
302	SD₁₆	—	4	8	no	—
303	C₂×C₈	—	2	2	no	—
304	C₂×C₈	—	2	2	no	—
305	C₂×C₈	—	2	4	no	—
306	C₂×C₈	—	2	4	no	—
307	C₄⋊C₄	—	2	8	no	—
308	C₄⋊C₄	—	2	4	no	—
309	C₄⋊C₄	—	2	1	no	—
310	C₄⋊C₄	—	2	2	no	—
311	C₄⋊C₄	—	2	2	no	—
312	C₄⋊C₄	—	2	2	no	—
313	C₄⋊C₄	—	4	2	no	—
314	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
315	C₄⋊C₄	#41×C₆₇	1	4	yes	—
316	C₂²⋊C₄	—	4	8	no	—
317	C₂²⋊C₄	—	4	8	no	—
318	C₄²	—	2	8	no	—
319	C₄²	—	2	4	no	—
320	C₄²	—	2	2	no	—
321	C₄²	—	2	4	no	—
322	C₄²	—	2	2	no	—
323	C₄²	—	2	2	no	—
324	C₄²	—	2	2	no	—
325	C₄²	—	4	2	no	—
326	C₄²	—	4	4	no	—
327	C₄²	C₂×D₄×C₆₇	1	2	yes	—
328	C₄²	#41×C₆₇	1	2	yes	—
329	C₂³	—	4	2	no	—
330	C₂³	—	4	4	no	—
331	C₂³	—	4	2	no	—
332	C₂³	—	4	4	no	—
333	C₂³	—	4	2	no	—
334	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
335	Q₈	—	2	4	no	—
336	Q₈	—	2	2	no	—
337	Q₈	—	8	4	no	—
338	Q₈	#124×C₆₇	1	4	yes	complemented
339	D₄	—	4	2	no	—
340	D₄	—	4	4	no	—
341	D₄	—	4	2	no	—
342	D₄	—	8	4	no	—
343	D₄	—	8	4	no	—
344	C₂×C₄	—	2	2	no	—
345	C₂×C₄	—	2	2	no	—
346	C₂×C₄	—	2	4	no	—
347	C₂×C₄	—	2	2	no	—
348	C₂×C₄	—	2	2	no	—
349	C₂×C₄	—	2	2	no	—
350	C₂×C₄	—	2	2	no	—
351	C₂×C₄	—	2	4	no	—
352	C₂×C₄	—	2	2	no	—
353	C₂×C₄	—	2	2	no	—
354	C₂×C₄	—	4	2	no	—
355	C₂×C₄	—	4	2	no	—
356	C₂×C₄	—	4	2	no	—
357	C₂×C₄	—	4	2	no	—
358	C₂×C₄	—	4	4	no	—
359	C₂×C₄	#121×C₆₇	1	2	yes	—
360	C₂×C₄	#121×C₆₇	1	2	yes	—
361	C₂×C₄	#135×C₆₇	1	2	yes	—
362	C₂×C₄	AGL(1,ℤ/8)×C₆₇	1	4	yes	—
363	C₂×C₄	#137×C₆₇	1	2	yes	—
364	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
365	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
366	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
367	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
368	C₈	—	4	2	no	—
369	C₈	—	4	2	no	—
370	C₂²	—	8	2	no	—
371	C₂²	—	8	2	no	—
372	C₂²	—	8	2	no	—
373	C₂²	—	8	4	no	—
374	C₂²	—	8	4	no	—
375	C₂²	C₂×#121×C₆₇	1	1	yes	characteristic
376	C₂²	#525×C₆₇	1	2	yes	—
377	C₂²	C₂×#135×C₆₇	1	1	yes	characteristic
378	C₂²	C₂×AGL(1,ℤ/8)×C₆₇	1	2	yes	—
379	C₂²	C₂×#137×C₆₇	1	1	yes	characteristic
380	C₄	—	2	2	no	—
381	C₄	—	2	4	no	—
382	C₄	—	2	4	no	—
383	C₄	—	4	2	no	—
384	C₄	—	4	2	no	—
385	C₄	—	4	4	no	—
386	C₄	—	8	2	no	—
387	C₄	#464×C₆₇	1	4	yes	—
388	C₄	C₂×#124×C₆₇	1	1	yes	characteristic
389	C₄	#529×C₆₇	1	2	yes	—
390	C₄	#532×C₆₇	1	1	yes	characteristic
391	C₂	—	16	2	no	—
392	C₂	C₆₇×#2850	1	1	yes	characteristic
393	C₂	C₂×#464×C₆₇	1	1	yes	characteristic
394	C₂	C₆₇×#3038	1	1	yes	characteristic
395	C₂	C₆₇×#3106	1	2	yes	—
396	C₂	C₆₇×#3112	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₂×#127×C₆₇	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₄³×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#477×C₆₇	C₂	semidirect
4. C₄²×Q₈×C₆₇	C₂	semidirect
5. C₆₇×#1987	C₂	semidirect
7. C₆₇×#1925	C₂	semidirect
8. C₆₇×#1880	C₂	semidirect
9. C₆₇×#1831	C₂	semidirect
139. Q₈×C₆₇	#124	semidirect
192. #33031	C₆₇	direct (N × H)
199. C₂×#477	C₂×C₆₇	semidirect
202. C₄²×Q₈	C₂×C₆₇	semidirect
203. #1987	C₂×C₆₇	semidirect
205. #1925	C₂×C₆₇	semidirect
206. #1880	C₂×C₆₇	semidirect
207. #1831	C₂×C₆₇	semidirect
208. C₆₇	#33031	direct (N × H)
338. Q₈	#124×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#477×C₆₇	2
C₂×#460×C₆₇	2
C₆₇×#2858	2
C₄²×Q₈×C₆₇	2
C₆₇×#1987	2
C₆₇×#1982	2
C₆₇×#1925	2
C₆₇×#1880	2
C₆₇×#1831	2
#33031	67

Subgroups — C67×#33031 #30712282

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33031 #30712282