Subgroups — #31915973

Order 18176.

386 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	C₂×C₈×Dic₁₄₂	C₂	1	1	yes	maximal, characteristic
2	C₂×Dic₇₁⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	C₂×#17985774	C₂	1	1	yes	maximal, characteristic
4	#22586418	C₂	1	1	yes	maximal, characteristic
5	#22586417	C₂	1	1	yes	maximal, characteristic
6	#22586417	C₂	1	1	yes	maximal, characteristic
7	C₇₁×#1374	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄×Dic₁₄₂	C₄	1	1	yes	characteristic
9	C₂²×C₇₁⋊C₁₆	C₂²	1	1	yes	characteristic
10	C₂²×C₇₁⋊C₁₆	C₂²	1	1	yes	characteristic
11	C₈×Dic₁₄₂	—	2	2	no	—
12	C₈×Dic₁₄₂	—	2	2	no	—
13	C₂×C₄⋊(C₇₁⋊C₈)	C₄	1	1	yes	characteristic
14	C₂×Dic₇₁⋊C₈	C₄	1	1	yes	characteristic
15	C₂×Dic₇₁⋊C₈	C₄	1	1	yes	characteristic
16	C₂×C₈×Dic₇₁	C₂²	1	1	yes	characteristic
17	C₂×C₈×Dic₇₁	C₂²	1	1	yes	characteristic
18	Dic₇₁⋊C₁₆	—	2	2	no	—
19	#17985774	—	2	2	no	—
20	C₂²×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
21	C₂²×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
23	C₂²×Dic₁₄₂	C₈	1	1	yes	characteristic
24	C₂²×C₇₁⋊C₈	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₇₁⋊C₈	C₂×C₄	1	1	yes	characteristic
26	C₄×Dic₁₄₂	—	2	2	no	—
27	C₄×Dic₁₄₂	—	2	2	no	—
28	C₂×C₄⋊Dic₇₁	C₈	1	1	yes	characteristic
29	C₂×Dic₇₁⋊C₄	C₈	1	1	yes	characteristic
30	C₂×Dic₇₁⋊C₄	C₈	1	1	yes	characteristic
31	C₂×C₄×Dic₇₁	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄×Dic₇₁	C₂×C₄	1	1	yes	characteristic
33	C₂×C₇₁⋊C₁₆	—	2	1	no	—
34	C₂×C₇₁⋊C₁₆	—	2	1	no	—
35	C₂×C₇₁⋊C₁₆	—	2	1	no	—
36	C₂×C₇₁⋊C₁₆	—	2	1	no	—
37	C₂×C₇₁⋊C₁₆	—	2	1	no	—
38	C₂×C₇₁⋊C₁₆	Q₈	1	2	yes	complemented
39	C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
40	Dic₇₁⋊C₈	—	2	2	no	—
41	Dic₇₁⋊C₈	—	2	2	no	—
42	C₈×Dic₇₁	—	2	1	no	—
43	C₈×Dic₇₁	—	2	1	no	—
44	C₈×Dic₇₁	D₄	1	2	yes	—
45	C₈×Dic₇₁	D₄	1	2	yes	—
46	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
47	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
48	C₂×C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
49	C₂×C₁₆×C₇₁	—	2	1	no	—
50	C₂×C₁₆×C₇₁	—	2	1	no	—
51	C₂×C₁₆×C₇₁	—	2	1	no	—
52	C₂×C₁₆×C₇₁	—	2	1	no	—
53	C₂×C₁₆×C₇₁	—	2	1	no	—
54	C₂×C₁₆×C₇₁	Q₈	1	2	yes	complemented
55	C₄×C₈×C₇₁	—	2	1	no	—
56	C₄×C₈×C₇₁	—	2	1	no	—
57	C₂×Dic₁₄₂	—	2	2	no	—
58	C₂×Dic₁₄₂	—	2	2	no	—
59	C₂×Dic₁₄₂	—	2	2	no	—
60	C₂²×Dic₇₁	C₂×C₈	1	1	yes	characteristic
61	C₂²×Dic₇₁	C₂×C₈	1	1	yes	characteristic
62	C₂²×Dic₇₁	C₂×C₈	1	1	yes	characteristic
63	C₂²×Dic₇₁	C₂×C₈	1	1	yes	characteristic
64	C₂×C₇₁⋊C₈	—	2	1	no	—
65	C₂×C₇₁⋊C₈	—	2	1	no	—
66	C₂×C₇₁⋊C₈	—	2	1	no	—
67	C₂×C₇₁⋊C₈	—	2	1	no	—
68	C₂×C₇₁⋊C₈	—	2	1	no	—
69	C₂×C₇₁⋊C₈	C₂²⋊C₄	1	2	yes	—
70	C₄⋊Dic₇₁	—	2	2	no	—
71	Dic₇₁⋊C₄	—	2	2	no	—
72	Dic₇₁⋊C₄	—	2	2	no	—
73	C₄×Dic₇₁	—	2	1	no	—
74	C₄×Dic₇₁	—	2	1	no	—
75	C₄×Dic₇₁	C₂²⋊C₄	1	2	yes	—
76	C₄×Dic₇₁	M₁₆	1	2	yes	—
77	C₇₁⋊C₁₆	—	2	2	no	—
78	C₇₁⋊C₁₆	—	4	1	no	—
79	C₂²×C₄×C₇₁	C₂×C₈	1	1	yes	characteristic
80	C₂²×C₄×C₇₁	C₂×C₈	1	1	yes	characteristic
81	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₈×C₇₁	—	2	1	no	—
83	C₂×C₈×C₇₁	—	2	1	no	—
84	C₂×C₈×C₇₁	—	2	1	no	—
85	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
86	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
87	C₂×C₈×C₇₁	C₂×Q₈	1	1	yes	characteristic
88	C₂×C₈×C₇₁	C₂×Q₈	1	1	yes	characteristic
89	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
90	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
91	C₄²×C₇₁	—	2	1	no	—
92	C₄²×C₇₁	—	2	1	no	—
93	C₁₆×C₇₁	—	2	2	no	—
94	C₁₆×C₇₁	—	4	1	no	—
95	C₂³×C₇₁	C₂²×C₈	1	1	yes	characteristic
96	C₂×C₄×C₇₁	—	2	1	no	—
97	C₂×C₄×C₇₁	—	2	1	no	—
98	C₂×C₄×C₇₁	—	2	1	no	—
99	C₂×C₄×C₇₁	—	2	1	no	—
100	C₂×C₄×C₇₁	—	2	1	no	—
101	C₂×C₄×C₇₁	M₃₂	1	2	yes	—
102	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
103	C₂×C₄×C₇₁	C₄×Q₈	1	1	yes	characteristic
104	C₂×C₄×C₇₁	C₄×Q₈	1	1	yes	characteristic
105	C₂×C₄×C₇₁	C₂×M₁₆	1	1	yes	characteristic
106	C₂×C₄×C₇₁	#131	1	1	yes	characteristic
107	C₂×C₄×C₇₁	#131	1	1	yes	characteristic
108	C₂×Dic₇₁	—	2	1	no	—
109	C₂×Dic₇₁	—	2	1	no	—
110	C₂×Dic₇₁	—	2	1	no	—
111	C₂×Dic₇₁	—	2	1	no	—
112	C₂×Dic₇₁	—	2	1	no	—
113	C₂×Dic₇₁	—	2	1	no	—
114	C₂×Dic₇₁	—	2	1	no	—
115	C₂×Dic₇₁	—	2	1	no	—
116	C₂×Dic₇₁	—	2	1	no	—
117	C₂×Dic₇₁	C₂²⋊C₈	1	2	yes	—
118	C₂×Dic₇₁	C₂²⋊C₈	1	2	yes	—
119	C₂×Dic₇₁	M₃₂	1	2	yes	—
120	Dic₁₄₂	—	2	8	no	—
121	C₈×C₇₁	—	4	1	no	—
122	C₈×C₇₁	C₂²⋊Q₈	1	1	yes	characteristic
123	C₈×C₇₁	C₂²⋊Q₈	1	1	yes	characteristic
124	C₈×C₇₁	#124	1	1	yes	characteristic
125	C₈×C₇₁	C₄⋊Q₈	1	1	yes	characteristic
126	C₇₁⋊C₈	—	2	2	no	—
127	C₇₁⋊C₈	—	4	1	no	—
128	C₂²×C₇₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
129	C₂²×C₇₁	C₈×Q₈	1	1	yes	characteristic
130	C₂²×C₇₁	C₈×Q₈	1	1	yes	characteristic
131	C₂²×C₇₁	C₂×M₃₂	1	1	yes	characteristic
132	C₂²×C₇₁	C₂×M₃₂	1	1	yes	characteristic
133	C₂²×C₇₁	#504	1	1	yes	characteristic
134	C₂²×C₇₁	#504	1	1	yes	characteristic
135	C₄×C₇₁	—	2	2	no	—
136	C₄×C₇₁	—	4	1	no	—
137	C₄×C₇₁	#391	1	1	yes	characteristic
138	C₄×C₇₁	#408	1	1	yes	characteristic
139	C₄×C₇₁	#446	1	1	yes	characteristic
140	C₄×C₇₁	#446	1	1	yes	characteristic
141	Dic₇₁	—	2	2	no	—
142	Dic₇₁	—	2	2	no	—
143	Dic₇₁	—	2	2	no	—
144	Dic₇₁	—	4	1	no	—
145	#13264	—	71	1	no	maximal, Sylow
146	C₂×C₇₁	#1984	1	1	yes	characteristic
147	C₂×C₇₁	#2112	1	1	yes	characteristic
148	C₂×C₇₁	#2113	1	1	yes	characteristic
149	C₂×C₇₁	#2183	1	1	yes	characteristic
150	C₂×C₇₁	#2183	1	1	yes	characteristic
151	C₂×C₇₁	#2183	1	1	yes	characteristic
152	C₂×C₇₁	#2183	1	1	yes	characteristic
153	C₂×C₈×Q₈	—	71	1	no	—
154	C₂×C₄⋊C₁₆	—	71	1	no	—
155	C₂×#346	—	71	1	no	—
156	#1374	—	71	1	no	—
157	#1374	—	71	1	no	—
158	#1374	—	71	1	no	—
159	#1374	D₇₁	1	1	yes	characteristic
160	C₇₁	#13264	1	1	yes	characteristic, complemented, Sylow
161	C₂×C₄×Q₈	—	71	1	no	—
162	C₂²×C₁₆	—	71	1	no	—
163	C₂²×C₁₆	—	71	1	no	—
164	C₂²×C₁₆	C₂×D₇₁	1	1	yes	characteristic
165	C₂²×C₁₆	C₂×D₇₁	1	1	yes	characteristic
166	C₈×Q₈	—	142	2	no	—
167	C₈×Q₈	—	142	2	no	—
168	C₂×C₄⋊C₈	—	71	1	no	—
169	C₂×C₄⋊C₈	—	71	1	no	—
170	C₂×C₄⋊C₈	—	71	1	no	—
171	C₂×C₄×C₈	—	71	1	no	—
172	C₂×C₄×C₈	—	71	1	no	—
173	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
174	C₄⋊C₁₆	—	142	2	no	—
175	#346	—	142	2	no	—
176	C₂²×Q₈	—	71	1	no	—
177	C₂²×C₈	—	71	1	no	—
178	C₂²×C₈	—	71	1	no	—
179	C₂²×C₈	C₄×D₇₁	1	1	yes	characteristic
180	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
181	C₄×Q₈	—	142	2	no	—
182	C₄×Q₈	—	142	2	no	—
183	C₂×C₄⋊C₄	—	71	1	no	—
184	C₂×C₄⋊C₄	—	71	1	no	—
185	C₂×C₄⋊C₄	—	71	1	no	—
186	C₂×C₄²	—	71	1	no	—
187	C₂×C₄²	—	71	1	no	—
188	C₂×C₄²	C₄×D₇₁	1	1	yes	characteristic
189	C₂×C₁₆	—	2	1	no	—
190	C₂×C₁₆	—	2	1	no	—
191	C₂×C₁₆	—	2	1	no	—
192	C₂×C₁₆	—	2	1	no	—
193	C₂×C₁₆	—	2	1	no	—
194	C₂×C₁₆	—	71	2	no	—
195	C₂×C₁₆	—	142	1	no	—
196	C₂×C₁₆	—	142	1	no	—
197	C₂×C₁₆	—	142	1	no	—
198	C₂×C₁₆	—	142	1	no	—
199	C₂×C₁₆	—	142	1	no	—
200	C₂×C₁₆	Dic₁₄₂	1	2	yes	complemented
201	C₄⋊C₈	—	142	2	no	—
202	C₄⋊C₈	—	142	2	no	—
203	C₄⋊C₈	—	142	2	no	—
204	C₄×C₈	—	2	1	no	—
205	C₄×C₈	—	2	1	no	—
206	C₄×C₈	—	71	2	no	—
207	C₄×C₈	—	71	2	no	—
208	C₄×C₈	—	142	1	no	—
209	C₄×C₈	—	142	1	no	—
210	C₂×Q₈	—	142	2	no	—
211	C₂×Q₈	—	142	2	no	—
212	C₂×Q₈	—	142	2	no	—
213	C₂²×C₄	—	71	1	no	—
214	C₂²×C₄	—	71	1	no	—
215	C₂²×C₄	—	71	1	no	—
216	C₂²×C₄	—	71	1	no	—
217	C₂²×C₄	C₈×D₇₁	1	1	yes	characteristic
218	C₂²×C₄	C₈×D₇₁	1	1	yes	characteristic
219	C₂²×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
220	C₂×C₈	—	2	1	no	—
221	C₂×C₈	—	2	1	no	—
222	C₂×C₈	—	2	1	no	—
223	C₂×C₈	—	71	2	no	—
224	C₂×C₈	—	142	1	no	—
225	C₂×C₈	—	142	1	no	—
226	C₂×C₈	—	142	1	no	—
227	C₂×C₈	—	142	1	no	—
228	C₂×C₈	—	142	1	no	—
229	C₂×C₈	D₄×D₇₁	1	1	yes	characteristic
230	C₂×C₈	D₄×D₇₁	1	1	yes	characteristic
231	C₂×C₈	Q₈×D₇₁	1	1	yes	characteristic
232	C₂×C₈	C₂×Dic₁₄₂	1	1	yes	characteristic
233	C₂×C₈	#11765864	1	1	yes	characteristic
234	C₂×C₈	#11765865	1	1	yes	characteristic
235	C₄⋊C₄	—	142	2	no	—
236	C₄⋊C₄	—	142	2	no	—
237	C₄⋊C₄	—	142	2	no	—
238	C₄²	—	2	1	no	—
239	C₄²	—	2	1	no	—
240	C₄²	—	71	2	no	—
241	C₄²	—	71	2	no	—
242	C₄²	—	142	1	no	—
243	C₄²	—	142	1	no	—
244	C₁₆	—	2	2	no	—
245	C₁₆	—	4	1	no	—
246	C₁₆	—	284	1	no	—
247	C₁₆	—	142	2	no	—
248	C₂³	C₂×C₈×D₇₁	1	1	yes	characteristic
249	Q₈	—	142	8	no	—
250	C₂×C₄	—	2	1	no	—
251	C₂×C₄	—	2	1	no	—
252	C₂×C₄	—	2	1	no	—
253	C₂×C₄	—	2	1	no	—
254	C₂×C₄	—	2	1	no	—
255	C₂×C₄	—	71	2	no	—
256	C₂×C₄	—	71	2	no	—
257	C₂×C₄	—	71	2	no	—
258	C₂×C₄	—	142	1	no	—
259	C₂×C₄	—	142	1	no	—
260	C₂×C₄	—	142	1	no	—
261	C₂×C₄	—	142	1	no	—
262	C₂×C₄	—	142	1	no	—
263	C₂×C₄	—	142	1	no	—
264	C₂×C₄	—	142	1	no	—
265	C₂×C₄	—	142	1	no	—
266	C₂×C₄	—	142	1	no	—
267	C₂×C₄	#14556067	1	2	yes	—
268	C₂×C₄	C₂²⋊C₄×D₇₁	1	1	yes	characteristic
269	C₂×C₄	#14556093	1	1	yes	characteristic
270	C₂×C₄	C₄×Dic₁₄₂	1	1	yes	characteristic
271	C₂×C₄	M₁₆×D₇₁	1	1	yes	characteristic
272	C₂×C₄	#14556134	1	1	yes	characteristic
273	C₂×C₄	#14556136	1	1	yes	characteristic
274	C₈	—	4	1	no	—
275	C₈	—	284	1	no	—
276	C₈	—	142	2	no	—
277	C₈	#14556105	1	1	yes	characteristic
278	C₈	#14556106	1	1	yes	characteristic
279	C₈	#14556114	1	1	yes	characteristic
280	C₈	#14556127	1	1	yes	characteristic
281	C₂²	C₂²⋊C₈×D₇₁	1	1	yes	characteristic
282	C₂²	#17986123	1	1	yes	characteristic
283	C₂²	C₈×Dic₁₄₂	1	1	yes	characteristic
284	C₂²	M₃₂×D₇₁	1	1	yes	characteristic
285	C₂²	C₂×#14556067	1	1	yes	characteristic
286	C₂²	#17986415	1	1	yes	characteristic
287	C₂²	#17986417	1	1	yes	characteristic
288	C₄	—	2	2	no	—
289	C₄	—	4	1	no	—
290	C₄	—	284	1	no	—
291	C₄	—	142	2	no	—
292	C₄	—	142	2	no	—
293	C₄	—	142	2	no	—
294	C₄	#17985903	1	1	yes	characteristic
295	C₄	#17985964	1	1	yes	characteristic
296	C₄	#17986127	1	1	yes	characteristic
297	C₄	#17986128	1	1	yes	characteristic
298	C₂	#22589274	1	1	yes	characteristic
299	C₂	#22589949	1	1	yes	characteristic
300	C₂	#22589952	1	1	yes	characteristic
301	C₂	#22590245	1	1	yes	characteristic
302	C₂	#22590246	1	1	yes	characteristic
303	C₂	#22590247	1	1	yes	characteristic
304	C₂	#22590248	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₇₁
commutator, socle, prime radical	C₂³×C₇₁	C₂²×C₈
Fitting	C₇₁×#1374	C₂
cosocle	C₂²×C₈×C₇₁	C₂³
hypercenter	#1374	D₇₁
hypocenter	C₇₁	#13264
LCS	G ▷ C₂³×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂²×C₈ ◁ #1374

Complements

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

N	H ≅ G/N	type
38. C₂×C₇₁⋊C₁₆	Q₈	semidirect
54. C₂×C₁₆×C₇₁	Q₈	semidirect
160. C₇₁	#13264	semidirect
200. C₂×C₁₆	Dic₁₄₂	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×Dic₁₄₂	2
C₂×Dic₇₁⋊C₁₆	2
C₂×#17985774	2
#22586418	2
#22586417	2
C₇₁×#1374	2
#13264	71