Subgroups — #31009358

Order 17152.

558 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22562118	C₂	1	2	yes	maximal, complemented
2	#22562041	C₂	1	2	yes	maximal, complemented
3	#22562035	C₂	1	4	yes	maximal, complemented
4	#22562008	C₂	1	2	yes	maximal, complemented
5	#22562008	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×#14523983	C₂	1	2	yes	maximal, complemented
7	C₂×C₄×C₆₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
8	C₈×#41×C₆₇	C₂	1	1	yes	maximal, characteristic
9	C₂×#14523983	C₂²	1	4	yes	—
10	C₂²×C₆₇⋊C₁₆	C₂²	1	2	yes	—
11	C₄⋊(C₆₇⋊C₁₆)	C₂²	1	4	yes	—
12	C₄⋊(C₆₇⋊C₁₆)	C₂²	1	2	yes	—
13	C₂²⋊(C₆₇⋊C₁₆)	C₂²	1	2	yes	—
14	C₂²⋊(C₆₇⋊C₁₆)	C₂²	1	4	yes	—
15	#16863172	C₂²	1	4	yes	—
16	C₄×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
17	C₄×C₆₇⋊C₁₆	C₂²	1	2	yes	—
18	C₄×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
19	C₄×C₆₇⋊C₁₆	C₂²	1	2	yes	—
20	C₄×#41×C₆₇	C₄	1	1	yes	characteristic
21	C₈×Q₈×C₆₇	C₄	1	1	yes	characteristic
22	C₈×Q₈×C₆₇	C₂²	1	1	yes	characteristic
23	C₈×D₄×C₆₇	C₄	1	2	yes	—
24	C₈×D₄×C₆₇	C₄	1	1	yes	characteristic
25	C₈×D₄×C₆₇	C₂²	1	2	yes	—
26	C₈×D₄×C₆₇	C₂²	1	1	yes	characteristic
27	#431×C₆₇	C₄	1	2	yes	—
28	#431×C₆₇	C₄	1	1	yes	characteristic
29	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
30	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
31	#14523983	—	2	8	no	—
32	C₂×C₆₇⋊C₁₆	—	2	4	no	—
33	C₂×C₆₇⋊C₁₆	C₂³	1	2	yes	—
34	C₂×C₆₇⋊C₁₆	C₂³	1	4	yes	—
35	C₂×C₆₇⋊C₁₆	C₂³	1	2	yes	—
36	C₂×#41×C₆₇	C₈	1	1	yes	characteristic
37	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
39	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
40	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
41	C₄×Q₈×C₆₇	C₈	1	1	yes	characteristic
42	C₄×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
43	C₄×D₄×C₆₇	C₈	1	2	yes	—
44	C₄×D₄×C₆₇	C₈	1	1	yes	characteristic
45	C₄×D₄×C₆₇	C₂×C₄	1	2	yes	—
46	C₄×D₄×C₆₇	C₂×C₄	1	1	yes	characteristic
47	#117×C₆₇	C₈	1	2	yes	—
48	#117×C₆₇	C₈	1	1	yes	characteristic
49	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
50	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
51	C₄⋊C₈×C₆₇	C₂×C₄	1	2	yes	—
52	C₄⋊C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
53	C₄⋊C₈×C₆₇	C₂×C₄	1	2	yes	—
54	C₄⋊C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
55	C₂²⋊C₈×C₆₇	C₂×C₄	1	2	yes	—
56	C₂²⋊C₈×C₆₇	C₂×C₄	1	2	yes	—
57	C₂²⋊C₈×C₆₇	C₂×C₄	1	2	yes	—
58	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
59	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
60	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
61	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
62	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
63	C₄×C₈×C₆₇	C₂³	1	2	yes	—
64	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
65	C₆₇⋊C₁₆	—	2	2	no	—
66	C₆₇⋊C₁₆	—	2	4	no	—
67	C₆₇⋊C₁₆	#41	1	4	yes	complemented
68	#41×C₆₇	—	2	4	no	—
69	C₂×Q₈×C₆₇	C₂×C₈	1	1	yes	characteristic
70	C₂×D₄×C₆₇	C₂×C₈	1	2	yes	—
71	C₂×D₄×C₆₇	C₂×C₈	1	1	yes	characteristic
72	C₂²×C₄×C₆₇	C₂×C₈	1	2	yes	—
73	C₂²×C₄×C₆₇	C₂×C₈	1	2	yes	—
74	C₂²×C₄×C₆₇	C₂×C₈	1	1	yes	characteristic
75	C₂²×C₄×C₆₇	C₂×C₈	1	1	yes	characteristic
76	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
77	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₈×C₆₇	—	2	2	no	—
79	C₂×C₈×C₆₇	—	2	4	no	—
80	C₂×C₈×C₆₇	—	2	2	no	—
81	C₂×C₈×C₆₇	—	2	4	no	—
82	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
84	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
86	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
88	C₄⋊C₄×C₆₇	C₂×C₈	1	2	yes	—
89	C₄⋊C₄×C₆₇	C₂×C₈	1	1	yes	characteristic
90	C₄⋊C₄×C₆₇	C₂×C₈	1	2	yes	—
91	C₄⋊C₄×C₆₇	C₂×C₈	1	1	yes	characteristic
92	C₂²⋊C₄×C₆₇	C₂×C₈	1	2	yes	—
93	C₂²⋊C₄×C₆₇	C₂×C₈	1	2	yes	—
94	C₂²⋊C₄×C₆₇	C₂×C₈	1	2	yes	—
95	C₄²×C₆₇	C₂×C₈	1	2	yes	—
96	C₄²×C₆₇	C₂×C₈	1	1	yes	characteristic
97	C₄²×C₆₇	C₂×C₈	1	2	yes	—
98	C₄²×C₆₇	C₂×C₈	1	1	yes	characteristic
99	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
100	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
101	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
102	C₂³×C₆₇	C₂²×C₈	1	2	yes	—
103	C₂³×C₆₇	C₂²×C₈	1	1	yes	characteristic
104	Q₈×C₆₇	—	2	2	no	—
105	D₄×C₆₇	—	2	4	no	—
106	D₄×C₆₇	—	2	2	no	—
107	C₂×C₄×C₆₇	—	2	1	no	—
108	C₂×C₄×C₆₇	—	2	2	no	—
109	C₂×C₄×C₆₇	—	2	1	no	—
110	C₂×C₄×C₆₇	—	2	2	no	—
111	C₂×C₄×C₆₇	—	2	4	no	—
112	C₂×C₄×C₆₇	—	2	2	no	—
113	C₂×C₄×C₆₇	—	2	4	no	—
114	C₂×C₄×C₆₇	M₃₂	1	4	yes	complemented
115	C₂×C₄×C₆₇	C₂²×C₈	1	2	yes	—
116	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
117	C₂×C₄×C₆₇	C₂²×C₈	1	2	yes	—
118	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
119	C₂×C₄×C₆₇	C₂²×C₈	1	2	yes	—
120	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
121	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
122	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
123	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
124	C₈×C₆₇	—	2	1	no	—
125	C₈×C₆₇	—	2	2	no	—
126	C₈×C₆₇	—	2	1	no	—
127	C₈×C₆₇	—	2	2	no	—
128	C₈×C₆₇	#131	1	2	yes	—
129	C₈×C₆₇	C₂×#41	1	2	yes	—
130	C₂²×C₆₇	—	2	1	no	—
131	C₂²×C₆₇	—	2	1	no	—
132	C₂²×C₆₇	—	2	2	no	—
133	C₂²×C₆₇	—	2	2	no	—
134	C₂²×C₆₇	—	2	2	no	—
135	C₂²×C₆₇	C₂×M₃₂	1	2	yes	—
136	C₂²×C₆₇	C₂³×C₈	1	1	yes	characteristic
137	C₄×C₆₇	—	2	2	no	—
138	C₄×C₆₇	—	2	1	no	—
139	C₄×C₆₇	—	2	2	no	—
140	C₄×C₆₇	—	2	1	no	—
141	C₄×C₆₇	—	2	2	no	—
142	C₄×C₆₇	C₂×M₃₂	1	2	yes	—
143	C₄×C₆₇	C₂×M₃₂	1	2	yes	—
144	C₄×C₆₇	#504	1	2	yes	—
145	C₄×C₆₇	C₄×#41	1	1	yes	characteristic
146	C₄×C₆₇	C₂×#131	1	1	yes	characteristic
147	#33479	—	67	1	no	maximal, Sylow
148	C₂×C₆₇	—	2	2	no	—
149	C₂×C₆₇	—	4	2	no	—
150	C₂×C₆₇	C₈×#41	1	1	yes	characteristic
151	C₂×C₆₇	C₂²×M₃₂	1	1	yes	characteristic
152	C₂×C₆₇	C₂×#504	1	1	yes	characteristic
153	C₈×#41	D₆₇	1	1	yes	characteristic
154	#2183	—	67	2	no	—
155	#2169	—	67	2	no	—
156	#2168	—	67	4	no	—
157	#2162	—	67	2	no	—
158	#2162	—	67	1	no	—
159	C₄×M₃₂	—	67	2	no	—
160	C₂×C₄×C₁₆	—	67	1	no	—
161	C₆₇	#33479	1	1	yes	characteristic, complemented, Sylow
162	C₄×#41	Dic₆₇	1	1	yes	characteristic
163	C₂×M₃₂	—	67	4	no	—
164	C₂²×C₁₆	—	67	2	no	—
165	C₈×Q₈	Dic₆₇	1	1	yes	characteristic
166	C₈×Q₈	C₂×D₆₇	1	1	yes	characteristic
167	C₈×D₄	Dic₆₇	1	1	yes	characteristic
168	C₈×D₄	Dic₆₇	1	2	yes	—
169	C₈×D₄	C₂×D₆₇	1	2	yes	—
170	C₈×D₄	C₂×D₆₇	1	1	yes	characteristic
171	#431	Dic₆₇	1	2	yes	—
172	#431	Dic₆₇	1	1	yes	characteristic
173	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
174	C₂×C₄×C₈	C₂×D₆₇	1	2	yes	—
175	C₄⋊C₁₆	—	67	4	no	—
176	C₄⋊C₁₆	—	67	2	no	—
177	C₂²⋊C₁₆	—	67	2	no	—
178	C₂²⋊C₁₆	—	67	4	no	—
179	#346	—	67	4	no	—
180	C₄×C₁₆	—	67	2	no	—
181	C₄×C₁₆	—	67	1	no	—
182	C₄×C₁₆	—	67	1	no	—
183	C₄×C₁₆	—	67	2	no	—
184	C₂×#41	C₆₇⋊C₈	1	1	yes	characteristic
185	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
186	C₂²×C₈	C₂×Dic₆₇	1	2	yes	—
187	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
188	C₂²×C₈	C₂²×D₆₇	1	2	yes	—
189	C₄×Q₈	C₆₇⋊C₈	1	1	yes	characteristic
190	C₄×Q₈	C₂×Dic₆₇	1	1	yes	characteristic
191	C₄×D₄	C₆₇⋊C₈	1	2	yes	—
192	C₄×D₄	C₆₇⋊C₈	1	1	yes	characteristic
193	C₄×D₄	C₂×Dic₆₇	1	2	yes	—
194	C₄×D₄	C₂×Dic₆₇	1	1	yes	characteristic
195	#117	C₆₇⋊C₈	1	2	yes	—
196	#117	C₆₇⋊C₈	1	1	yes	characteristic
197	C₂×C₄²	C₂×Dic₆₇	1	2	yes	—
198	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
199	M₃₂	—	134	8	no	—
200	C₂×C₁₆	—	67	2	no	—
201	C₂×C₁₆	—	67	2	no	—
202	C₂×C₁₆	—	67	4	no	—
203	C₂×C₁₆	—	134	4	no	—
204	C₄⋊C₈	C₂×Dic₆₇	1	2	yes	—
205	C₄⋊C₈	C₂×Dic₆₇	1	1	yes	characteristic
206	C₄⋊C₈	C₂×Dic₆₇	1	2	yes	—
207	C₄⋊C₈	C₂×Dic₆₇	1	1	yes	characteristic
208	C₂²⋊C₈	C₂×Dic₆₇	1	2	yes	—
209	C₂²⋊C₈	C₂×Dic₆₇	1	2	yes	—
210	C₂²⋊C₈	C₂×Dic₆₇	1	2	yes	—
211	C₄×C₈	C₂×Dic₆₇	1	1	yes	characteristic
212	C₄×C₈	C₂×Dic₆₇	1	2	yes	—
213	C₄×C₈	C₂×Dic₆₇	1	1	yes	characteristic
214	C₄×C₈	C₂×Dic₆₇	1	2	yes	—
215	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
216	C₄×C₈	C₂²×D₆₇	1	2	yes	—
217	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
218	#41	—	2	4	no	—
219	C₂×Q₈	C₂×C₆₇⋊C₈	1	1	yes	characteristic
220	C₂×D₄	C₂×C₆₇⋊C₈	1	2	yes	—
221	C₂×D₄	C₂×C₆₇⋊C₈	1	1	yes	characteristic
222	C₂²×C₄	C₂×C₆₇⋊C₈	1	2	yes	—
223	C₂²×C₄	C₂×C₆₇⋊C₈	1	2	yes	—
224	C₂²×C₄	C₂×C₆₇⋊C₈	1	1	yes	characteristic
225	C₂²×C₄	C₂×C₆₇⋊C₈	1	1	yes	characteristic
226	C₂²×C₄	C₂²×Dic₆₇	1	2	yes	—
227	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
228	C₂×C₈	—	2	2	no	—
229	C₂×C₈	—	2	4	no	—
230	C₂×C₈	—	2	2	no	—
231	C₂×C₈	—	2	4	no	—
232	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
233	C₂×C₈	C₂²×Dic₆₇	1	2	yes	—
234	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
235	C₂×C₈	C₂²×Dic₆₇	1	2	yes	—
236	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
237	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
238	C₄⋊C₄	C₂×C₆₇⋊C₈	1	2	yes	—
239	C₄⋊C₄	C₂×C₆₇⋊C₈	1	1	yes	characteristic
240	C₄⋊C₄	C₂×C₆₇⋊C₈	1	2	yes	—
241	C₄⋊C₄	C₂×C₆₇⋊C₈	1	1	yes	characteristic
242	C₂²⋊C₄	C₂×C₆₇⋊C₈	1	2	yes	—
243	C₂²⋊C₄	C₂×C₆₇⋊C₈	1	2	yes	—
244	C₂²⋊C₄	C₂×C₆₇⋊C₈	1	2	yes	—
245	C₄²	C₂×C₆₇⋊C₈	1	2	yes	—
246	C₄²	C₂×C₆₇⋊C₈	1	1	yes	characteristic
247	C₄²	C₂×C₆₇⋊C₈	1	2	yes	—
248	C₄²	C₂×C₆₇⋊C₈	1	1	yes	characteristic
249	C₄²	C₂²×Dic₆₇	1	1	yes	characteristic
250	C₄²	C₂²×Dic₆₇	1	2	yes	—
251	C₄²	C₂²×Dic₆₇	1	1	yes	characteristic
252	C₁₆	—	67	4	no	—
253	C₁₆	—	134	2	no	—
254	C₁₆	—	134	4	no	—
255	C₂³	C₂²×C₆₇⋊C₈	1	2	yes	—
256	C₂³	C₂²×C₆₇⋊C₈	1	1	yes	characteristic
257	Q₈	—	2	2	no	—
258	D₄	—	2	4	no	—
259	D₄	—	2	2	no	—
260	C₂×C₄	—	2	4	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	4	no	—
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	#14523983	1	4	yes	complemented
268	C₂×C₄	C₂²×C₆₇⋊C₈	1	2	yes	—
269	C₂×C₄	C₂²×C₆₇⋊C₈	1	1	yes	characteristic
270	C₂×C₄	C₂²×C₆₇⋊C₈	1	2	yes	—
271	C₂×C₄	C₂²×C₆₇⋊C₈	1	1	yes	characteristic
272	C₂×C₄	C₂²×C₆₇⋊C₈	1	2	yes	—
273	C₂×C₄	C₂²×C₆₇⋊C₈	1	1	yes	characteristic
274	C₂×C₄	C₂²×C₆₇⋊C₈	1	1	yes	characteristic
275	C₂×C₄	C₂²×C₆₇⋊C₈	1	1	yes	characteristic
276	C₂×C₄	C₂³×Dic₆₇	1	1	yes	characteristic
277	C₈	—	2	1	no	—
278	C₈	—	2	2	no	—
279	C₈	—	2	1	no	—
280	C₈	—	2	2	no	—
281	C₈	#14524050	1	2	yes	—
282	C₈	#41×D₆₇	1	2	yes	—
283	C₂²	—	2	1	no	—
284	C₂²	—	2	1	no	—
285	C₂²	—	2	2	no	—
286	C₂²	—	2	2	no	—
287	C₂²	—	2	2	no	—
288	C₂²	C₂×#14523983	1	2	yes	—
289	C₂²	C₂³×C₆₇⋊C₈	1	1	yes	characteristic
290	C₄	—	2	1	no	—
291	C₄	—	2	2	no	—
292	C₄	—	2	2	no	—
293	C₄	—	2	1	no	—
294	C₄	—	2	2	no	—
295	C₄	C₂×#14523983	1	2	yes	—
296	C₄	C₂×#14523983	1	2	yes	—
297	C₄	#16863813	1	2	yes	—
298	C₄	#41×Dic₆₇	1	1	yes	characteristic
299	C₄	C₂×#14524050	1	1	yes	characteristic
300	C₂	—	2	2	no	—
301	C₂	—	4	2	no	—
302	C₂	#41×C₆₇⋊C₈	1	1	yes	characteristic
303	C₂	C₂²×#14523983	1	1	yes	characteristic
304	C₂	C₂×#16863813	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₈	C₂²×D₆₇
commutator, socle	C₂²×C₆₇	C₂³×C₈
Frattini	C₂×C₈	C₂³×D₆₇
Fitting	C₈×#41×C₆₇	C₂
prime radical	C₂×#41×C₆₇	C₈
cosocle	C₂×C₈×C₆₇	C₂⁴
hypercenter	C₈×#41	D₆₇
hypocenter	C₆₇	#33479
LCS	G ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₄×C₈ ◁ C₈×#41

Complements

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

N	H ≅ G/N	type
1. #22562118	C₂	semidirect
2. #22562041	C₂	semidirect
3. #22562035	C₂	semidirect
4. #22562008	C₂	semidirect
5. #22562008	C₂	semidirect
6. C₄×#14523983	C₂	semidirect
7. C₂×C₄×C₆₇⋊C₁₆	C₂	semidirect
67. C₆₇⋊C₁₆	#41	semidirect
114. C₂×C₄×C₆₇	M₃₂	semidirect
161. C₆₇	#33479	semidirect
267. C₂×C₄	#14523983	semidirect

Maximal subgroups

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

subgroup	index
#22562118	2
#22562041	2
#22562035	2
#22562008	2
C₄×#14523983	2
C₂×C₄×C₆₇⋊C₁₆	2
C₈×#41×C₆₇	2
#33479	67