Subgroups — C₆₇×#17901 #30697152

Order 17152.

686 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#3103	C₂	1	1	yes	maximal, characteristic
2	C₂×C₈×Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#2931	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#1583	C₂	1	4	yes	maximal, complemented
5	C₈×SD₁₆×C₆₇	C₂	1	4	yes	maximal, complemented
6	C₆₇×#1570	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#1487	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₇×#1480	C₂	1	1	yes	maximal, characteristic
9	C₆₇×#1450	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#131×C₆₇	—	2	1	no	—
11	#532×C₆₇	C₂²	1	1	yes	characteristic
12	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
13	#481×C₆₇	C₄	1	2	yes	—
14	#475×C₆₇	C₄	1	2	yes	—
15	#460×C₆₇	C₄	1	2	yes	—
16	#448×C₆₇	C₄	1	2	yes	—
17	C₈×Q₈×C₆₇	—	2	1	no	—
18	C₈×Q₈×C₆₇	—	2	1	no	—
19	C₈×Q₈×C₆₇	C₂²	1	2	yes	—
20	C₈×Q₈×C₆₇	C₂²	1	2	yes	—
21	C₄×SD₁₆×C₆₇	C₂²	1	4	yes	—
22	#436×C₆₇	—	2	1	no	—
23	#436×C₆₇	C₂²	1	2	yes	—
24	C₈×D₄×C₆₇	—	2	1	no	—
25	C₈×D₄×C₆₇	C₂²	1	2	yes	—
26	#431×C₆₇	C₂²	1	1	yes	characteristic
27	#431×C₆₇	C₂²	1	1	yes	characteristic
28	#423×C₆₇	C₂²	1	1	yes	characteristic
29	C₂×C₄⋊C₈×C₆₇	—	2	1	no	—
30	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
31	#408×C₆₇	—	2	1	no	—
32	C₂×C₄×C₈×C₆₇	—	2	1	no	—
33	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
34	#334×C₆₇	C₂²	1	4	yes	—
35	Q₈⋊C₈×C₆₇	C₂²	1	4	yes	—
36	D₄⋊C₈×C₆₇	C₂²	1	4	yes	—
37	#322×C₆₇	C₂²	1	2	yes	—
38	C₈²×C₆₇	C₂²	1	2	yes	—
39	C₂×#41×C₆₇	—	2	1	no	—
40	C₂²×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
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	2	no	—
45	C₂²×C₈×C₆₇	—	2	1	no	—
46	C₂²×C₈×C₆₇	—	2	2	no	—
47	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
48	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
49	C₄⋊Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
50	#127×C₆₇	C₂×C₄	1	1	yes	characteristic
51	#124×C₆₇	—	2	1	no	—
52	#121×C₆₇	—	2	1	no	—
53	#121×C₆₇	C₂×C₄	1	2	yes	—
54	C₄×Q₈×C₆₇	—	2	1	no	—
55	C₄×Q₈×C₆₇	—	2	1	no	—
56	C₄×Q₈×C₆₇	C₂×C₄	1	2	yes	—
57	C₄×Q₈×C₆₇	C₂³	1	2	yes	—
58	C₄×D₄×C₆₇	—	2	1	no	—
59	C₄×D₄×C₆₇	C₂³	1	2	yes	—
60	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
61	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
62	C₂×C₄²×C₆₇	—	2	1	no	—
63	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
64	#106×C₆₇	C₂×C₄	1	4	yes	—
65	C₄⋊C₈×C₆₇	—	2	2	no	—
66	C₄⋊C₈×C₆₇	—	2	2	no	—
67	C₄⋊C₈×C₆₇	—	2	1	no	—
68	C₄⋊C₈×C₆₇	—	2	1	no	—
69	C₄⋊C₈×C₆₇	—	2	1	no	—
70	C₄⋊C₈×C₆₇	C₂×C₄	1	2	yes	—
71	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
72	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
73	#103×C₆₇	C₂×C₄	1	4	yes	—
74	#102×C₆₇	C₂×C₄	1	4	yes	—
75	C₂²⋊C₈×C₆₇	—	2	1	no	—
76	C₂²⋊C₈×C₆₇	—	2	2	no	—
77	C₂²⋊C₈×C₆₇	—	2	2	no	—
78	C₂²⋊C₈×C₆₇	C₂×C₄	1	2	yes	—
79	C₄×C₈×C₆₇	—	2	2	no	—
80	C₄×C₈×C₆₇	—	2	2	no	—
81	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
82	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
83	C₄×C₈×C₆₇	C₂³	1	2	yes	—
84	C₄×C₈×C₆₇	C₂³	1	2	yes	—
85	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
86	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
87	#41×C₆₇	—	4	2	no	—
88	#41×C₆₇	—	4	2	no	—
89	C₂×Q₈×C₆₇	—	2	4	no	—
90	C₂×Q₈×C₆₇	—	2	1	no	—
91	C₂×Q₈×C₆₇	—	2	1	no	—
92	C₂×Q₈×C₆₇	C₂²×C₄	1	2	yes	—
93	C₂×D₄×C₆₇	—	2	2	no	—
94	C₂×D₄×C₆₇	—	2	1	no	—
95	C₂×D₄×C₆₇	—	2	1	no	—
96	C₂×D₄×C₆₇	C₂²×C₄	1	2	yes	—
97	C₂²×C₄×C₆₇	—	2	1	no	—
98	C₂²×C₄×C₆₇	—	2	1	no	—
99	C₂²×C₄×C₆₇	—	2	2	no	—
100	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
101	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
102	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
103	SD₁₆×C₆₇	—	2	8	no	—
104	M₁₆×C₆₇	—	4	2	no	—
105	M₁₆×C₆₇	—	4	4	no	—
106	C₂×C₈×C₆₇	—	2	2	no	—
107	C₂×C₈×C₆₇	—	2	2	no	—
108	C₂×C₈×C₆₇	—	2	2	no	—
109	C₂×C₈×C₆₇	—	2	2	no	—
110	C₂×C₈×C₆₇	—	4	2	no	—
111	C₂×C₈×C₆₇	—	4	4	no	—
112	C₂×C₈×C₆₇	SD₁₆	1	4	yes	complemented
113	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
114	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
115	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
116	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
117	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
118	C₄⋊C₄×C₆₇	—	2	2	no	—
119	C₄⋊C₄×C₆₇	—	2	1	no	—
120	C₄⋊C₄×C₆₇	—	2	1	no	—
121	C₄⋊C₄×C₆₇	—	2	2	no	—
122	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
123	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
124	C₂²⋊C₄×C₆₇	—	2	2	no	—
125	C₂²⋊C₄×C₆₇	—	2	2	no	—
126	C₄²×C₆₇	—	2	2	no	—
127	C₄²×C₆₇	—	2	2	no	—
128	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
129	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
130	C₄²×C₆₇	#41	1	1	yes	characteristic
131	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
132	C₂³×C₆₇	—	2	2	no	—
133	C₂³×C₆₇	C₄×D₄	1	1	yes	characteristic
134	Q₈×C₆₇	—	2	4	no	—
135	Q₈×C₆₇	—	2	2	no	—
136	Q₈×C₆₇	—	4	2	no	—
137	Q₈×C₆₇	#131	1	4	yes	complemented
138	D₄×C₆₇	—	2	4	no	—
139	D₄×C₆₇	—	4	2	no	—
140	D₄×C₆₇	—	4	4	no	—
141	D₄×C₆₇	—	4	2	no	—
142	C₂×C₄×C₆₇	—	2	4	no	—
143	C₂×C₄×C₆₇	—	2	2	no	—
144	C₂×C₄×C₆₇	—	2	2	no	—
145	C₂×C₄×C₆₇	—	2	2	no	—
146	C₂×C₄×C₆₇	—	2	2	no	—
147	C₂×C₄×C₆₇	—	4	1	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	C₄×D₄	1	1	yes	characteristic
152	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
153	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
154	C₂×C₄×C₆₇	C₂×SD₁₆	1	2	yes	—
155	C₂×C₄×C₆₇	#135	1	2	yes	—
156	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
157	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
158	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
159	C₈×C₆₇	—	2	4	no	—
160	C₈×C₆₇	—	2	1	no	—
161	C₈×C₆₇	—	2	1	no	—
162	C₈×C₆₇	—	4	2	no	—
163	C₈×C₆₇	—	4	2	no	—
164	C₈×C₆₇	—	4	2	no	—
165	C₈×C₆₇	C₂×SD₁₆	1	2	yes	—
166	C₈×C₆₇	C₂×SD₁₆	1	2	yes	—
167	C₂²×C₆₇	—	2	1	no	—
168	C₂²×C₆₇	—	2	1	no	—
169	C₂²×C₆₇	—	4	2	no	—
170	C₂²×C₆₇	—	4	2	no	—
171	C₂²×C₆₇	—	4	2	no	—
172	C₂²×C₆₇	C₄×SD₁₆	1	2	yes	—
173	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
174	C₄×C₆₇	—	2	4	no	—
175	C₄×C₆₇	—	2	1	no	—
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	C₄×SD₁₆	1	2	yes	—
181	C₄×C₆₇	C₂×#131	1	2	yes	—
182	C₄×C₆₇	C₂²×SD₁₆	1	1	yes	characteristic
183	C₄×C₆₇	C₂×#135	1	1	yes	characteristic
184	#17901	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
185	C₂×C₆₇	—	2	2	no	—
186	C₂×C₆₇	—	8	2	no	—
187	C₂×C₆₇	#2929	1	1	yes	characteristic
188	C₂×C₆₇	C₂×C₄×SD₁₆	1	1	yes	characteristic
189	C₂×C₆₇	#2955	1	1	yes	characteristic
190	#3103	C₂×C₆₇	1	1	yes	characteristic
191	C₂×C₈×Q₈	C₂×C₆₇	1	1	yes	characteristic, complemented
192	#2931	C₂×C₆₇	1	1	yes	characteristic
193	#1583	C₂×C₆₇	1	4	yes	complemented
194	C₈×SD₁₆	C₂×C₆₇	1	4	yes	complemented
195	#1570	C₂×C₆₇	1	1	yes	characteristic, complemented
196	#1487	C₂×C₆₇	1	1	yes	characteristic, complemented
197	#1480	C₂×C₆₇	1	1	yes	characteristic
198	#1450	C₂×C₆₇	1	1	yes	characteristic, complemented
199	C₆₇	#17901	1	1	yes	characteristic, direct product factor, Sylow
200	C₂×#131	—	2	1	no	—
201	#532	C₂²×C₆₇	1	1	yes	characteristic
202	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
203	#481	C₄×C₆₇	1	2	yes	—
204	#475	C₄×C₆₇	1	2	yes	—
205	#460	C₄×C₆₇	1	2	yes	—
206	#448	C₄×C₆₇	1	2	yes	—
207	C₈×Q₈	—	2	1	no	—
208	C₈×Q₈	—	2	1	no	—
209	C₈×Q₈	C₂²×C₆₇	1	2	yes	—
210	C₈×Q₈	C₂²×C₆₇	1	2	yes	—
211	C₄×SD₁₆	C₂²×C₆₇	1	4	yes	—
212	#436	—	2	1	no	—
213	#436	C₂²×C₆₇	1	2	yes	—
214	C₈×D₄	—	2	1	no	—
215	C₈×D₄	C₂²×C₆₇	1	2	yes	—
216	#431	C₂²×C₆₇	1	1	yes	characteristic
217	#431	C₂²×C₆₇	1	1	yes	characteristic
218	#423	C₂²×C₆₇	1	1	yes	characteristic
219	C₂×C₄⋊C₈	—	2	1	no	—
220	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
221	#408	—	2	1	no	—
222	C₂×C₄×C₈	—	2	1	no	—
223	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
224	#334	C₂²×C₆₇	1	4	yes	—
225	Q₈⋊C₈	C₂²×C₆₇	1	4	yes	—
226	D₄⋊C₈	C₂²×C₆₇	1	4	yes	—
227	#322	C₂²×C₆₇	1	2	yes	—
228	C₈²	C₂²×C₆₇	1	2	yes	—
229	C₂×#41	—	2	1	no	—
230	C₂²×Q₈	C₂×C₄×C₆₇	1	1	yes	characteristic
231	C₂×SD₁₆	C₂×C₄×C₆₇	1	4	yes	—
232	#131	—	4	4	no	—
233	C₂×M₁₆	—	2	1	no	—
234	C₂×M₁₆	—	2	2	no	—
235	C₂²×C₈	—	2	1	no	—
236	C₂²×C₈	—	2	2	no	—
237	C₂²×C₈	D₄×C₆₇	1	1	yes	characteristic
238	C₂²×C₈	D₄×C₆₇	1	1	yes	characteristic
239	C₄⋊Q₈	C₂×C₄×C₆₇	1	1	yes	characteristic
240	#127	C₂×C₄×C₆₇	1	1	yes	characteristic
241	#124	—	2	1	no	—
242	#121	—	2	1	no	—
243	#121	C₂×C₄×C₆₇	1	2	yes	—
244	C₄×Q₈	—	2	1	no	—
245	C₄×Q₈	—	2	1	no	—
246	C₄×Q₈	C₂×C₄×C₆₇	1	2	yes	—
247	C₄×Q₈	C₂³×C₆₇	1	2	yes	—
248	C₄×D₄	—	2	1	no	—
249	C₄×D₄	C₂³×C₆₇	1	2	yes	—
250	C₂×C₄⋊C₄	—	2	1	no	—
251	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
252	C₂×C₄²	—	2	1	no	—
253	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
254	#106	C₂×C₄×C₆₇	1	4	yes	—
255	C₄⋊C₈	—	2	2	no	—
256	C₄⋊C₈	—	2	2	no	—
257	C₄⋊C₈	—	2	1	no	—
258	C₄⋊C₈	—	2	1	no	—
259	C₄⋊C₈	—	2	1	no	—
260	C₄⋊C₈	C₂×C₄×C₆₇	1	2	yes	—
261	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
262	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
263	#103	C₂×C₄×C₆₇	1	4	yes	—
264	#102	C₂×C₄×C₆₇	1	4	yes	—
265	C₂²⋊C₈	—	2	1	no	—
266	C₂²⋊C₈	—	2	2	no	—
267	C₂²⋊C₈	—	2	2	no	—
268	C₂²⋊C₈	C₂×C₄×C₆₇	1	2	yes	—
269	C₄×C₈	—	2	2	no	—
270	C₄×C₈	—	2	2	no	—
271	C₄×C₈	D₄×C₆₇	1	1	yes	characteristic
272	C₄×C₈	D₄×C₆₇	1	1	yes	characteristic
273	C₄×C₈	C₂³×C₆₇	1	2	yes	—
274	C₄×C₈	C₂³×C₆₇	1	2	yes	—
275	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
276	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
277	#41	—	4	2	no	—
278	#41	—	4	2	no	—
279	C₂×Q₈	—	2	4	no	—
280	C₂×Q₈	—	2	1	no	—
281	C₂×Q₈	—	2	1	no	—
282	C₂×Q₈	C₂²×C₄×C₆₇	1	2	yes	—
283	C₂×D₄	—	2	2	no	—
284	C₂×D₄	—	2	1	no	—
285	C₂×D₄	—	2	1	no	—
286	C₂×D₄	C₂²×C₄×C₆₇	1	2	yes	—
287	C₂²×C₄	—	2	1	no	—
288	C₂²×C₄	—	2	1	no	—
289	C₂²×C₄	—	2	2	no	—
290	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
291	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
292	C₂²×C₄	#41×C₆₇	1	1	yes	characteristic
293	SD₁₆	—	2	8	no	—
294	M₁₆	—	4	2	no	—
295	M₁₆	—	4	4	no	—
296	C₂×C₈	—	2	2	no	—
297	C₂×C₈	—	2	2	no	—
298	C₂×C₈	—	2	2	no	—
299	C₂×C₈	—	2	2	no	—
300	C₂×C₈	—	4	2	no	—
301	C₂×C₈	—	4	4	no	—
302	C₂×C₈	SD₁₆×C₆₇	1	4	yes	complemented
303	C₂×C₈	C₂²×C₄×C₆₇	1	4	yes	—
304	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
305	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
306	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
307	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
308	C₄⋊C₄	—	2	2	no	—
309	C₄⋊C₄	—	2	1	no	—
310	C₄⋊C₄	—	2	1	no	—
311	C₄⋊C₄	—	2	2	no	—
312	C₄⋊C₄	C₂²×C₄×C₆₇	1	2	yes	—
313	C₄⋊C₄	C₂²×C₄×C₆₇	1	2	yes	—
314	C₂²⋊C₄	—	2	2	no	—
315	C₂²⋊C₄	—	2	2	no	—
316	C₄²	—	2	2	no	—
317	C₄²	—	2	2	no	—
318	C₄²	C₂²×C₄×C₆₇	1	1	yes	characteristic
319	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
320	C₄²	#41×C₆₇	1	1	yes	characteristic
321	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
322	C₂³	—	2	2	no	—
323	C₂³	C₄×D₄×C₆₇	1	1	yes	characteristic
324	Q₈	—	2	4	no	—
325	Q₈	—	2	2	no	—
326	Q₈	—	4	2	no	—
327	Q₈	#131×C₆₇	1	4	yes	complemented
328	D₄	—	2	4	no	—
329	D₄	—	4	2	no	—
330	D₄	—	4	4	no	—
331	D₄	—	4	2	no	—
332	C₂×C₄	—	2	4	no	—
333	C₂×C₄	—	2	2	no	—
334	C₂×C₄	—	2	2	no	—
335	C₂×C₄	—	2	2	no	—
336	C₂×C₄	—	2	2	no	—
337	C₂×C₄	—	4	1	no	—
338	C₂×C₄	—	4	2	no	—
339	C₂×C₄	—	4	2	no	—
340	C₂×C₄	—	4	2	no	—
341	C₂×C₄	C₄×D₄×C₆₇	1	1	yes	characteristic
342	C₂×C₄	C₄×D₄×C₆₇	1	1	yes	characteristic
343	C₂×C₄	C₄×D₄×C₆₇	1	1	yes	characteristic
344	C₂×C₄	C₂×SD₁₆×C₆₇	1	2	yes	—
345	C₂×C₄	#135×C₆₇	1	2	yes	—
346	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
347	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
348	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
349	C₈	—	2	4	no	—
350	C₈	—	2	1	no	—
351	C₈	—	2	1	no	—
352	C₈	—	4	2	no	—
353	C₈	—	4	2	no	—
354	C₈	—	4	2	no	—
355	C₈	C₂×SD₁₆×C₆₇	1	2	yes	—
356	C₈	C₂×SD₁₆×C₆₇	1	2	yes	—
357	C₂²	—	2	1	no	—
358	C₂²	—	2	1	no	—
359	C₂²	—	4	2	no	—
360	C₂²	—	4	2	no	—
361	C₂²	—	4	2	no	—
362	C₂²	C₄×SD₁₆×C₆₇	1	2	yes	—
363	C₂²	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
364	C₄	—	2	4	no	—
365	C₄	—	2	1	no	—
366	C₄	—	2	1	no	—
367	C₄	—	2	1	no	—
368	C₄	—	4	2	no	—
369	C₄	—	4	2	no	—
370	C₄	C₄×SD₁₆×C₆₇	1	2	yes	—
371	C₄	C₂×#131×C₆₇	1	2	yes	—
372	C₄	C₂²×SD₁₆×C₆₇	1	1	yes	characteristic
373	C₄	C₂×#135×C₆₇	1	1	yes	characteristic
374	C₂	—	2	2	no	—
375	C₂	—	8	2	no	—
376	C₂	C₆₇×#2929	1	1	yes	characteristic
377	C₂	C₂×C₄×SD₁₆×C₆₇	1	1	yes	characteristic
378	C₂	C₆₇×#2955	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈×C₆₇	C₂×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×C₈×C₆₇ ◁ C₂×C₄×C₈×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₈×Q₈×C₆₇	C₂	semidirect
4. C₆₇×#1583	C₂	semidirect
5. C₈×SD₁₆×C₆₇	C₂	semidirect
6. C₆₇×#1570	C₂	semidirect
7. C₆₇×#1487	C₂	semidirect
9. C₆₇×#1450	C₂	semidirect
112. C₂×C₈×C₆₇	SD₁₆	semidirect
137. Q₈×C₆₇	#131	semidirect
184. #17901	C₆₇	direct (N × H)
191. C₂×C₈×Q₈	C₂×C₆₇	semidirect
193. #1583	C₂×C₆₇	semidirect
194. C₈×SD₁₆	C₂×C₆₇	semidirect
195. #1570	C₂×C₆₇	semidirect
196. #1487	C₂×C₆₇	semidirect
198. #1450	C₂×C₆₇	semidirect
199. C₆₇	#17901	direct (N × H)
302. C₂×C₈	SD₁₆×C₆₇	semidirect
327. Q₈	#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×#3103	2
C₂×C₈×Q₈×C₆₇	2
C₆₇×#2931	2
C₆₇×#1583	2
C₈×SD₁₆×C₆₇	2
C₆₇×#1570	2
C₆₇×#1487	2
C₆₇×#1480	2
C₆₇×#1450	2
#17901	67

Subgroups — C67×#17901 #30697152

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#17901 #30697152