Subgroups — C₆₇×#17960 #30697211

Order 17152.

666 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#3167	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#2960	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#2931	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#1592	C₂	1	4	yes	maximal, complemented
5	C₆₇×SD₁₆⋊C₈	C₂	1	4	yes	maximal, complemented
6	C₆₇×#1567	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#1493	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₇×#1492	C₂	1	1	yes	maximal, characteristic
9	C₂×#322×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#131×C₆₇	—	2	1	no	—
11	#533×C₆₇	C₂²	1	1	yes	characteristic
12	#532×C₆₇	C₂²	1	1	yes	characteristic
13	#498×C₆₇	C₄	1	1	yes	characteristic
14	#492×C₆₇	C₄	1	1	yes	characteristic
15	#487×C₆₇	C₄	1	1	yes	characteristic
16	#486×C₆₇	C₄	1	1	yes	characteristic
17	#465×C₆₇	C₄	1	4	yes	—
18	#446×C₆₇	—	2	1	no	—
19	#446×C₆₇	C₂²	1	2	yes	—
20	C₈×Q₈×C₆₇	—	2	1	no	—
21	C₈×Q₈×C₆₇	C₂²	1	2	yes	—
22	C₄×SD₁₆×C₆₇	C₂²	1	4	yes	—
23	#436×C₆₇	—	2	1	no	—
24	#436×C₆₇	C₂²	1	2	yes	—
25	C₈×D₄×C₆₇	—	2	1	no	—
26	C₈×D₄×C₆₇	C₂²	1	2	yes	—
27	#424×C₆₇	—	2	1	no	—
28	#423×C₆₇	C₂²	1	1	yes	characteristic
29	#423×C₆₇	C₂²	1	1	yes	characteristic
30	#408×C₆₇	—	2	1	no	—
31	#405×C₆₇	—	2	1	no	—
32	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
33	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
34	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
35	#335×C₆₇	C₂²	1	4	yes	—
36	Q₈⋊C₈×C₆₇	C₂²	1	4	yes	—
37	D₄⋊C₈×C₆₇	C₂²	1	4	yes	—
38	#322×C₆₇	C₂²	1	2	yes	—
39	#322×C₆₇	C₂²	1	2	yes	—
40	C₂×#41×C₆₇	—	2	1	no	—
41	C₂×SD₁₆×C₆₇	C₂×C₄	1	4	yes	—
42	#131×C₆₇	—	4	4	no	—
43	C₂×M₁₆×C₆₇	—	2	1	no	—
44	C₂×M₁₆×C₆₇	—	2	1	no	—
45	C₂×M₁₆×C₆₇	—	2	2	no	—
46	C₂²×C₈×C₆₇	—	2	1	no	—
47	C₂²×C₈×C₆₇	—	2	2	no	—
48	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
49	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
50	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
51	C₄⋊Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
52	C₄⋊Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
53	C₄⋊Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
54	#127×C₆₇	C₂×C₄	1	1	yes	characteristic
55	#125×C₆₇	—	2	1	no	—
56	#124×C₆₇	—	2	1	no	—
57	C₂²⋊Q₈×C₆₇	—	2	1	no	—
58	C₂²⋊Q₈×C₆₇	C₂×C₄	1	2	yes	—
59	#121×C₆₇	—	2	1	no	—
60	#121×C₆₇	C₂×C₄	1	2	yes	—
61	C₄×Q₈×C₆₇	—	2	1	no	—
62	C₄×Q₈×C₆₇	C₂³	1	2	yes	—
63	C₄×D₄×C₆₇	—	2	1	no	—
64	C₄×D₄×C₆₇	C₂³	1	2	yes	—
65	#117×C₆₇	—	2	1	no	—
66	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
67	#106×C₆₇	C₂×C₄	1	4	yes	—
68	C₄⋊C₈×C₆₇	—	2	2	no	—
69	C₄⋊C₈×C₆₇	—	2	2	no	—
70	C₄⋊C₈×C₆₇	—	2	1	no	—
71	C₄⋊C₈×C₆₇	—	2	1	no	—
72	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
73	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
74	#103×C₆₇	C₂×C₄	1	4	yes	—
75	#102×C₆₇	C₂×C₄	1	4	yes	—
76	C₂²⋊C₈×C₆₇	—	2	2	no	—
77	C₂²⋊C₈×C₆₇	—	2	2	no	—
78	#97×C₆₇	—	2	2	no	—
79	C₄×C₈×C₆₇	—	2	2	no	—
80	C₄×C₈×C₆₇	—	2	1	no	—
81	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
82	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
83	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
84	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
85	C₄×C₈×C₆₇	C₂³	1	2	yes	—
86	C₄×C₈×C₆₇	C₂³	1	2	yes	—
87	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
88	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
89	#41×C₆₇	—	4	2	no	—
90	#41×C₆₇	—	4	2	no	—
91	C₂×Q₈×C₆₇	—	2	1	no	—
92	C₂×Q₈×C₆₇	—	2	1	no	—
93	C₂×Q₈×C₆₇	C₂²×C₄	1	2	yes	—
94	C₂×D₄×C₆₇	—	2	2	no	—
95	C₂×D₄×C₆₇	—	2	1	no	—
96	C₂×D₄×C₆₇	—	2	1	no	—
97	C₂×D₄×C₆₇	C₂²×C₄	1	2	yes	—
98	C₂²×C₄×C₆₇	—	2	2	no	—
99	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
100	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
101	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
102	SD₁₆×C₆₇	—	2	8	no	—
103	M₁₆×C₆₇	—	4	4	no	—
104	M₁₆×C₆₇	—	4	2	no	—
105	M₁₆×C₆₇	—	4	2	no	—
106	C₂×C₈×C₆₇	—	2	2	no	—
107	C₂×C₈×C₆₇	—	2	2	no	—
108	C₂×C₈×C₆₇	—	2	4	no	—
109	C₂×C₈×C₆₇	—	2	2	no	—
110	C₂×C₈×C₆₇	—	2	2	no	—
111	C₂×C₈×C₆₇	—	2	2	no	—
112	C₂×C₈×C₆₇	—	4	4	no	—
113	C₂×C₈×C₆₇	—	4	2	no	—
114	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
115	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
116	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
117	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
118	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
119	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
120	C₄⋊C₄×C₆₇	—	2	1	no	—
121	C₄⋊C₄×C₆₇	—	2	1	no	—
122	C₄⋊C₄×C₆₇	—	2	1	no	—
123	C₄⋊C₄×C₆₇	—	2	1	no	—
124	C₄⋊C₄×C₆₇	—	2	2	no	—
125	C₄⋊C₄×C₆₇	—	2	2	no	—
126	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
127	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
128	C₂²⋊C₄×C₆₇	—	2	2	no	—
129	C₂²⋊C₄×C₆₇	—	2	2	no	—
130	C₂²⋊C₄×C₆₇	—	2	2	no	—
131	C₄²×C₆₇	—	2	2	no	—
132	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
133	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
134	C₄²×C₆₇	#41	1	1	yes	characteristic
135	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
136	C₂³×C₆₇	—	2	2	no	—
137	C₂³×C₆₇	C₄×D₄	1	1	yes	characteristic
138	Q₈×C₆₇	—	2	4	no	—
139	Q₈×C₆₇	—	4	2	no	—
140	Q₈×C₆₇	—	4	2	no	—
141	D₄×C₆₇	—	2	4	no	—
142	D₄×C₆₇	—	4	4	no	—
143	D₄×C₆₇	—	4	2	no	—
144	D₄×C₆₇	—	4	2	no	—
145	C₂×C₄×C₆₇	—	2	2	no	—
146	C₂×C₄×C₆₇	—	2	2	no	—
147	C₂×C₄×C₆₇	—	2	2	no	—
148	C₂×C₄×C₆₇	—	2	2	no	—
149	C₂×C₄×C₆₇	—	2	2	no	—
150	C₂×C₄×C₆₇	—	4	2	no	—
151	C₂×C₄×C₆₇	—	4	2	no	—
152	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
153	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
154	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
155	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
156	C₂×C₄×C₆₇	#137	1	2	yes	—
157	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
158	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
159	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
160	C₈×C₆₇	—	2	2	no	—
161	C₈×C₆₇	—	2	1	no	—
162	C₈×C₆₇	—	2	1	no	—
163	C₈×C₆₇	—	2	1	no	—
164	C₈×C₆₇	—	2	1	no	—
165	C₈×C₆₇	—	4	2	no	—
166	C₈×C₆₇	—	4	2	no	—
167	C₈×C₆₇	—	4	2	no	—
168	C₈×C₆₇	#131	1	4	yes	complemented
169	C₂²×C₆₇	—	2	1	no	—
170	C₂²×C₆₇	—	2	1	no	—
171	C₂²×C₆₇	—	4	2	no	—
172	C₂²×C₆₇	—	4	2	no	—
173	C₂²×C₆₇	—	4	2	no	—
174	C₂²×C₆₇	#440	1	2	yes	—
175	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
176	C₄×C₆₇	—	2	1	no	—
177	C₄×C₆₇	—	2	1	no	—
178	C₄×C₆₇	—	4	2	no	—
179	C₄×C₆₇	—	4	2	no	—
180	C₄×C₆₇	—	4	2	no	—
181	C₄×C₆₇	#440	1	2	yes	—
182	C₄×C₆₇	#444	1	2	yes	—
183	C₄×C₆₇	C₂×#131	1	2	yes	—
184	C₄×C₆₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
185	C₄×C₆₇	C₂×#137	1	1	yes	characteristic
186	#17960	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
187	C₂×C₆₇	—	2	2	no	—
188	C₂×C₆₇	—	8	2	no	—
189	C₂×C₆₇	#2931	1	1	yes	characteristic
190	C₂×C₆₇	C₂×#440	1	1	yes	characteristic
191	C₂×C₆₇	C₂×#444	1	1	yes	characteristic
192	#3167	C₂×C₆₇	1	1	yes	characteristic
193	#2960	C₂×C₆₇	1	1	yes	characteristic, complemented
194	#2931	C₂×C₆₇	1	1	yes	characteristic
195	#1592	C₂×C₆₇	1	4	yes	complemented
196	SD₁₆⋊C₈	C₂×C₆₇	1	4	yes	complemented
197	#1567	C₂×C₆₇	1	1	yes	characteristic, complemented
198	#1493	C₂×C₆₇	1	1	yes	characteristic, complemented
199	#1492	C₂×C₆₇	1	1	yes	characteristic
200	C₂×#322	C₂×C₆₇	1	1	yes	characteristic, complemented
201	C₆₇	#17960	1	1	yes	characteristic, direct product factor, Sylow
202	C₂×#131	—	2	1	no	—
203	#533	C₂²×C₆₇	1	1	yes	characteristic
204	#532	C₂²×C₆₇	1	1	yes	characteristic
205	#498	C₄×C₆₇	1	1	yes	characteristic
206	#492	C₄×C₆₇	1	1	yes	characteristic
207	#487	C₄×C₆₇	1	1	yes	characteristic
208	#486	C₄×C₆₇	1	1	yes	characteristic
209	#465	C₄×C₆₇	1	4	yes	—
210	#446	—	2	1	no	—
211	#446	C₂²×C₆₇	1	2	yes	—
212	C₈×Q₈	—	2	1	no	—
213	C₈×Q₈	C₂²×C₆₇	1	2	yes	—
214	C₄×SD₁₆	C₂²×C₆₇	1	4	yes	—
215	#436	—	2	1	no	—
216	#436	C₂²×C₆₇	1	2	yes	—
217	C₈×D₄	—	2	1	no	—
218	C₈×D₄	C₂²×C₆₇	1	2	yes	—
219	#424	—	2	1	no	—
220	#423	C₂²×C₆₇	1	1	yes	characteristic
221	#423	C₂²×C₆₇	1	1	yes	characteristic
222	#408	—	2	1	no	—
223	#405	—	2	1	no	—
224	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
225	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
226	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
227	#335	C₂²×C₆₇	1	4	yes	—
228	Q₈⋊C₈	C₂²×C₆₇	1	4	yes	—
229	D₄⋊C₈	C₂²×C₆₇	1	4	yes	—
230	#322	C₂²×C₆₇	1	2	yes	—
231	#322	C₂²×C₆₇	1	2	yes	—
232	C₂×#41	—	2	1	no	—
233	C₂×SD₁₆	C₂×C₄×C₆₇	1	4	yes	—
234	#131	—	4	4	no	—
235	C₂×M₁₆	—	2	1	no	—
236	C₂×M₁₆	—	2	1	no	—
237	C₂×M₁₆	—	2	2	no	—
238	C₂²×C₈	—	2	1	no	—
239	C₂²×C₈	—	2	2	no	—
240	C₂²×C₈	C₂×C₄×C₆₇	1	2	yes	—
241	C₂²×C₈	D₄×C₆₇	1	1	yes	characteristic
242	C₂²×C₈	D₄×C₆₇	1	1	yes	characteristic
243	C₄⋊Q₈	C₂×C₄×C₆₇	1	1	yes	characteristic
244	C₄⋊Q₈	C₂×C₄×C₆₇	1	1	yes	characteristic
245	C₄⋊Q₈	C₂×C₄×C₆₇	1	1	yes	characteristic
246	#127	C₂×C₄×C₆₇	1	1	yes	characteristic
247	#125	—	2	1	no	—
248	#124	—	2	1	no	—
249	C₂²⋊Q₈	—	2	1	no	—
250	C₂²⋊Q₈	C₂×C₄×C₆₇	1	2	yes	—
251	#121	—	2	1	no	—
252	#121	C₂×C₄×C₆₇	1	2	yes	—
253	C₄×Q₈	—	2	1	no	—
254	C₄×Q₈	C₂³×C₆₇	1	2	yes	—
255	C₄×D₄	—	2	1	no	—
256	C₄×D₄	C₂³×C₆₇	1	2	yes	—
257	#117	—	2	1	no	—
258	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
259	#106	C₂×C₄×C₆₇	1	4	yes	—
260	C₄⋊C₈	—	2	2	no	—
261	C₄⋊C₈	—	2	2	no	—
262	C₄⋊C₈	—	2	1	no	—
263	C₄⋊C₈	—	2	1	no	—
264	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
265	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
266	#103	C₂×C₄×C₆₇	1	4	yes	—
267	#102	C₂×C₄×C₆₇	1	4	yes	—
268	C₂²⋊C₈	—	2	2	no	—
269	C₂²⋊C₈	—	2	2	no	—
270	#97	—	2	2	no	—
271	C₄×C₈	—	2	2	no	—
272	C₄×C₈	—	2	1	no	—
273	C₄×C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
274	C₄×C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
275	C₄×C₈	D₄×C₆₇	1	1	yes	characteristic
276	C₄×C₈	D₄×C₆₇	1	1	yes	characteristic
277	C₄×C₈	C₂³×C₆₇	1	2	yes	—
278	C₄×C₈	C₂³×C₆₇	1	2	yes	—
279	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
280	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
281	#41	—	4	2	no	—
282	#41	—	4	2	no	—
283	C₂×Q₈	—	2	1	no	—
284	C₂×Q₈	—	2	1	no	—
285	C₂×Q₈	C₂²×C₄×C₆₇	1	2	yes	—
286	C₂×D₄	—	2	2	no	—
287	C₂×D₄	—	2	1	no	—
288	C₂×D₄	—	2	1	no	—
289	C₂×D₄	C₂²×C₄×C₆₇	1	2	yes	—
290	C₂²×C₄	—	2	2	no	—
291	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
292	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
293	C₂²×C₄	#41×C₆₇	1	1	yes	characteristic
294	SD₁₆	—	2	8	no	—
295	M₁₆	—	4	4	no	—
296	M₁₆	—	4	2	no	—
297	M₁₆	—	4	2	no	—
298	C₂×C₈	—	2	2	no	—
299	C₂×C₈	—	2	2	no	—
300	C₂×C₈	—	2	4	no	—
301	C₂×C₈	—	2	2	no	—
302	C₂×C₈	—	2	2	no	—
303	C₂×C₈	—	2	2	no	—
304	C₂×C₈	—	4	4	no	—
305	C₂×C₈	—	4	2	no	—
306	C₂×C₈	C₂²×C₄×C₆₇	1	2	yes	—
307	C₂×C₈	C₂²×C₄×C₆₇	1	2	yes	—
308	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
309	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
310	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
311	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
312	C₄⋊C₄	—	2	1	no	—
313	C₄⋊C₄	—	2	1	no	—
314	C₄⋊C₄	—	2	1	no	—
315	C₄⋊C₄	—	2	1	no	—
316	C₄⋊C₄	—	2	2	no	—
317	C₄⋊C₄	—	2	2	no	—
318	C₄⋊C₄	C₂²×C₄×C₆₇	1	2	yes	—
319	C₄⋊C₄	C₂²×C₄×C₆₇	1	2	yes	—
320	C₂²⋊C₄	—	2	2	no	—
321	C₂²⋊C₄	—	2	2	no	—
322	C₂²⋊C₄	—	2	2	no	—
323	C₄²	—	2	2	no	—
324	C₄²	C₂²×C₄×C₆₇	1	1	yes	characteristic
325	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
326	C₄²	#41×C₆₇	1	1	yes	characteristic
327	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
328	C₂³	—	2	2	no	—
329	C₂³	C₄×D₄×C₆₇	1	1	yes	characteristic
330	Q₈	—	2	4	no	—
331	Q₈	—	4	2	no	—
332	Q₈	—	4	2	no	—
333	D₄	—	2	4	no	—
334	D₄	—	4	4	no	—
335	D₄	—	4	2	no	—
336	D₄	—	4	2	no	—
337	C₂×C₄	—	2	2	no	—
338	C₂×C₄	—	2	2	no	—
339	C₂×C₄	—	2	2	no	—
340	C₂×C₄	—	2	2	no	—
341	C₂×C₄	—	2	2	no	—
342	C₂×C₄	—	4	2	no	—
343	C₂×C₄	—	4	2	no	—
344	C₂×C₄	C₄×D₄×C₆₇	1	1	yes	characteristic
345	C₂×C₄	C₄×D₄×C₆₇	1	1	yes	characteristic
346	C₂×C₄	C₄×D₄×C₆₇	1	1	yes	characteristic
347	C₂×C₄	AGL(1,ℤ/8)×C₆₇	1	2	yes	—
348	C₂×C₄	#137×C₆₇	1	2	yes	—
349	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
350	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
351	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
352	C₈	—	2	2	no	—
353	C₈	—	2	1	no	—
354	C₈	—	2	1	no	—
355	C₈	—	2	1	no	—
356	C₈	—	2	1	no	—
357	C₈	—	4	2	no	—
358	C₈	—	4	2	no	—
359	C₈	—	4	2	no	—
360	C₈	#131×C₆₇	1	4	yes	complemented
361	C₂²	—	2	1	no	—
362	C₂²	—	2	1	no	—
363	C₂²	—	4	2	no	—
364	C₂²	—	4	2	no	—
365	C₂²	—	4	2	no	—
366	C₂²	#440×C₆₇	1	2	yes	—
367	C₂²	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
368	C₄	—	2	1	no	—
369	C₄	—	2	1	no	—
370	C₄	—	4	2	no	—
371	C₄	—	4	2	no	—
372	C₄	—	4	2	no	—
373	C₄	#440×C₆₇	1	2	yes	—
374	C₄	#444×C₆₇	1	2	yes	—
375	C₄	C₂×#131×C₆₇	1	2	yes	—
376	C₄	C₂×AGL(1,ℤ/8)×C₆₇	1	1	yes	characteristic
377	C₄	C₂×#137×C₆₇	1	1	yes	characteristic
378	C₂	—	2	2	no	—
379	C₂	—	8	2	no	—
380	C₂	C₆₇×#2931	1	1	yes	characteristic
381	C₂	C₂×#440×C₆₇	1	1	yes	characteristic
382	C₂	C₂×#444×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₆₇	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₆₇
Frattini, cosocle	C₄²	C₂⁴×C₆₇
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	#532×C₆₇	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₆₇ ◁ C₂×C₄×C₈×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₆₇×#2960	C₂	semidirect
4. C₆₇×#1592	C₂	semidirect
5. C₆₇×SD₁₆⋊C₈	C₂	semidirect
6. C₆₇×#1567	C₂	semidirect
7. C₆₇×#1493	C₂	semidirect
9. C₂×#322×C₆₇	C₂	semidirect
168. C₈×C₆₇	#131	semidirect
186. #17960	C₆₇	direct (N × H)
193. #2960	C₂×C₆₇	semidirect
195. #1592	C₂×C₆₇	semidirect
196. SD₁₆⋊C₈	C₂×C₆₇	semidirect
197. #1567	C₂×C₆₇	semidirect
198. #1493	C₂×C₆₇	semidirect
200. C₂×#322	C₂×C₆₇	semidirect
201. C₆₇	#17960	direct (N × H)
360. C₈	#131×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₆₇×#3167	2
C₆₇×#2960	2
C₆₇×#2931	2
C₆₇×#1592	2
C₆₇×SD₁₆⋊C₈	2
C₆₇×#1567	2
C₆₇×#1493	2
C₆₇×#1492	2
C₂×#322×C₆₇	2
#17960	67

Subgroups — C67×#17960 #30697211

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#17960 #30697211