Subgroups — #31915867

Order 18176.

410 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22590124	C₂	1	1	yes	maximal, characteristic, complemented
2	#22590076	C₂	1	2	yes	maximal, complemented
3	#22589936	C₂	1	1	yes	maximal, characteristic, complemented
4	#22589239	C₂	1	2	yes	maximal, complemented
5	C₂×#508×C₇₁	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556068	C₂²	1	1	yes	characteristic
7	#17986037	C₂²	1	2	yes	—
8	#17986016	—	2	2	no	—
9	#17985973	—	2	2	no	—
10	#17985954	C₂²	1	1	yes	characteristic
11	#17985827	—	2	1	no	—
12	#17985825	—	2	1	no	—
13	#17985809	—	2	4	no	—
14	#17985774	—	2	1	no	—
15	C₁₆×Dic₇₁	—	2	1	no	—
16	C₂×#135×C₇₁	C₂²	1	2	yes	—
17	#508×C₇₁	—	2	1	no	—
18	#508×C₇₁	—	2	1	no	—
19	#508×C₇₁	—	2	1	no	—
20	#508×C₇₁	—	2	1	no	—
21	C₂×Q₃₂×C₇₁	C₄	1	1	yes	characteristic, complemented
22	C₂×SD₃₂×C₇₁	C₄	1	2	yes	complemented
23	C₂×D₁₆×C₇₁	C₄	1	1	yes	characteristic, complemented
24	C₂²×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
25	C₂×#11765849	D₄	1	1	yes	characteristic
26	#14556087	—	2	2	no	—
27	#14556087	D₄	1	1	yes	characteristic
28	#14556068	—	2	1	no	—
29	#14556068	D₄	1	2	yes	—
30	C₂×C₇₁⋊C₁₆	—	2	1	no	—
31	#14556062	—	2	2	no	—
32	#14556060	—	2	2	no	—
33	#14556055	—	4	4	no	—
34	#14556043	—	4	4	no	—
35	#14556038	—	2	1	no	—
36	C₈×Dic₇₁	—	2	1	no	—
37	C₂×#41×C₇₁	—	2	2	no	—
38	#135×C₇₁	—	2	2	no	—
39	#135×C₇₁	—	2	2	no	—
40	#135×C₇₁	—	2	2	no	—
41	#135×C₇₁	—	2	2	no	—
42	C₂×Q₁₆×C₇₁	C₂×C₄	1	2	yes	—
43	C₂×SD₁₆×C₇₁	—	2	2	no	—
44	C₂×D₈×C₇₁	C₂×C₄	1	2	yes	—
45	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
46	Q₃₂×C₇₁	—	2	1	no	—
47	Q₃₂×C₇₁	—	2	1	no	—
48	SD₃₂×C₇₁	—	2	2	no	—
49	SD₃₂×C₇₁	—	2	2	no	—
50	D₁₆×C₇₁	—	2	1	no	—
51	D₁₆×C₇₁	—	2	1	no	—
52	C₂×C₁₆×C₇₁	C₂×C₄	1	1	yes	characteristic
53	C₂×C₁₆×C₇₁	C₂×C₄	1	1	yes	characteristic
54	C₂×C₁₆×C₇₁	D₄	1	1	yes	characteristic
55	C₂×C₁₆×C₇₁	D₄	1	1	yes	characteristic
56	C₂×C₁₆×C₇₁	D₄	1	1	yes	characteristic
57	C₂×C₁₆×C₇₁	D₄	1	1	yes	characteristic
58	#11765849	—	2	1	no	—
59	#11765849	D₈	1	2	yes	—
60	C₂×C₇₁⋊C₈	—	2	1	no	—
61	C₄⋊Dic₇₁	—	2	1	no	—
62	C₄⋊Dic₇₁	—	4	2	no	—
63	C₂²⋊Dic₇₁	—	4	2	no	—
64	C₂²⋊Dic₇₁	D₈	1	2	yes	—
65	C₄×Dic₇₁	—	2	1	no	—
66	C₄×Dic₇₁	—	4	2	no	—
67	C₇₁⋊C₁₆	—	2	2	no	—
68	#41×C₇₁	—	4	2	no	—
69	#41×C₇₁	—	4	2	no	—
70	#41×C₇₁	—	4	2	no	—
71	#41×C₇₁	—	4	2	no	—
72	C₂×Q₈×C₇₁	—	2	2	no	—
73	C₂×D₄×C₇₁	—	2	2	no	—
74	C₂×D₄×C₇₁	—	4	2	no	—
75	C₂²×C₄×C₇₁	—	4	2	no	—
76	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
77	Q₁₆×C₇₁	—	2	2	no	—
78	Q₁₆×C₇₁	—	2	2	no	—
79	SD₁₆×C₇₁	—	4	2	no	—
80	SD₁₆×C₇₁	—	4	2	no	—
81	D₈×C₇₁	—	2	2	no	—
82	D₈×C₇₁	—	2	2	no	—
83	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
84	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
85	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
86	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
87	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
88	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
89	C₁₆×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
90	C₁₆×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
91	C₁₆×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
92	C₁₆×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
93	C₂³×C₇₁	—	4	2	no	—
94	C₂³×C₇₁	C₂×D₈	1	1	yes	characteristic
95	Q₈×C₇₁	—	4	2	no	—
96	Q₈×C₇₁	—	4	2	no	—
97	D₄×C₇₁	—	4	2	no	—
98	D₄×C₇₁	—	4	2	no	—
99	D₄×C₇₁	—	8	2	no	—
100	D₄×C₇₁	—	8	2	no	—
101	C₂×C₄×C₇₁	—	4	2	no	—
102	C₂×C₄×C₇₁	—	8	2	no	—
103	C₂×C₄×C₇₁	—	8	2	no	—
104	C₂×C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
105	C₂×C₄×C₇₁	#121	1	1	yes	characteristic
106	C₂×C₄×C₇₁	#127	1	1	yes	characteristic
107	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
108	C₂×C₄×C₇₁	#135	1	1	yes	characteristic
109	C₂×C₄×C₇₁	#135	1	1	yes	characteristic
110	C₂×Dic₇₁	—	2	2	no	—
111	C₂×Dic₇₁	—	8	2	no	—
112	C₈×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
113	C₈×C₇₁	C₄×D₄	1	1	yes	characteristic
114	C₈×C₇₁	#121	1	1	yes	characteristic
115	C₈×C₇₁	#124	1	1	yes	characteristic
116	C₇₁⋊C₈	—	2	2	no	—
117	C₂²×C₇₁	—	4	1	no	—
118	C₂²×C₇₁	—	8	2	no	—
119	C₂²×C₇₁	—	16	2	no	—
120	C₂²×C₇₁	C₄×D₈	1	1	yes	characteristic
121	C₂²×C₇₁	#466	1	1	yes	characteristic
122	C₂²×C₇₁	#493	1	1	yes	characteristic
123	C₄×C₇₁	—	8	2	no	—
124	C₄×C₇₁	#390	1	1	yes	characteristic
125	C₄×C₇₁	C₄×D₈	1	1	yes	characteristic
126	C₄×C₇₁	#466	1	1	yes	characteristic
127	C₄×C₇₁	#495	1	1	yes	characteristic
128	Dic₇₁	—	4	2	no	—
129	Dic₇₁	—	16	2	no	—
130	#13243	—	71	1	no	maximal, Sylow
131	C₂×C₇₁	—	2	1	no	—
132	C₂×C₇₁	—	2	1	no	—
133	C₂×C₇₁	—	2	1	no	—
134	C₂×C₇₁	—	16	2	no	—
135	C₂×C₇₁	#1970	1	1	yes	characteristic
136	C₂×#508	D₇₁	1	1	yes	characteristic
137	#2158	—	71	1	no	—
138	#2145	—	71	2	no	—
139	#2108	—	71	1	no	—
140	#1976	—	71	2	no	—
141	C₇₁	#13243	1	1	yes	characteristic, complemented, Sylow
142	C₂×#135	C₂×D₇₁	1	2	yes	—
143	#508	—	2	1	no	—
144	#508	—	2	1	no	—
145	#508	—	2	1	no	—
146	#508	—	2	1	no	—
147	C₂×Q₃₂	Dic₇₁	1	1	yes	characteristic, complemented
148	C₂×SD₃₂	Dic₇₁	1	2	yes	complemented
149	C₂×D₁₆	Dic₇₁	1	1	yes	characteristic, complemented
150	C₂×M₃₂	—	71	1	no	—
151	C₂²×C₁₆	C₂×D₇₁	1	1	yes	characteristic
152	#427	—	71	2	no	—
153	#421	—	142	2	no	—
154	#410	—	142	2	no	—
155	#405	—	71	1	no	—
156	#367	—	142	1	no	—
157	#366	—	142	1	no	—
158	#360	—	142	4	no	—
159	#346	—	142	1	no	—
160	C₄×C₁₆	—	142	1	no	—
161	C₂×#41	—	2	2	no	—
162	#135	—	2	2	no	—
163	#135	—	2	2	no	—
164	#135	—	2	2	no	—
165	#135	—	2	2	no	—
166	C₂×Q₁₆	C₂×Dic₇₁	1	2	yes	—
167	C₂×SD₁₆	—	2	2	no	—
168	C₂×D₈	C₂×Dic₇₁	1	2	yes	—
169	C₂×M₁₆	—	71	1	no	—
170	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
171	#117	—	71	1	no	—
172	#117	—	142	2	no	—
173	Q₃₂	—	2	1	no	—
174	Q₃₂	—	2	1	no	—
175	SD₃₂	—	2	2	no	—
176	SD₃₂	—	2	2	no	—
177	D₁₆	—	2	1	no	—
178	D₁₆	—	2	1	no	—
179	M₃₂	—	71	2	no	—
180	M₃₂	—	142	1	no	—
181	C₂×C₁₆	—	142	1	no	—
182	C₂×C₁₆	C₂×Dic₇₁	1	1	yes	characteristic
183	C₂×C₁₆	C₂×Dic₇₁	1	1	yes	characteristic
184	C₂×C₁₆	C₂²⋊D₇₁	1	1	yes	characteristic
185	C₂×C₁₆	C₂²⋊D₇₁	1	1	yes	characteristic
186	C₂×C₁₆	C₂²⋊D₇₁	1	1	yes	characteristic
187	C₂×C₁₆	C₂²⋊D₇₁	1	1	yes	characteristic
188	#107	—	142	2	no	—
189	#106	—	142	2	no	—
190	#104	—	284	4	no	—
191	#99	—	284	4	no	—
192	#97	—	142	1	no	—
193	C₄×C₈	—	142	1	no	—
194	#41	—	4	2	no	—
195	#41	—	4	2	no	—
196	#41	—	4	2	no	—
197	#41	—	4	2	no	—
198	C₂×Q₈	—	2	2	no	—
199	C₂×D₄	—	2	2	no	—
200	C₂×D₄	—	4	2	no	—
201	C₂²×C₄	—	4	2	no	—
202	C₂²×C₄	D₄×D₇₁	1	1	yes	characteristic
203	Q₁₆	—	2	2	no	—
204	Q₁₆	—	2	2	no	—
205	SD₁₆	—	4	2	no	—
206	SD₁₆	—	4	2	no	—
207	D₈	—	2	2	no	—
208	D₈	—	2	2	no	—
209	M₁₆	—	71	2	no	—
210	M₁₆	—	142	1	no	—
211	C₂×C₈	—	142	1	no	—
212	C₂×C₈	C₂²×Dic₇₁	1	1	yes	characteristic
213	C₂×C₈	D₄×D₇₁	1	1	yes	characteristic
214	C₂×C₈	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
215	C₂×C₈	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
216	C₂×C₈	#11765864	1	1	yes	characteristic
217	C₂×C₈	#11765864	1	1	yes	characteristic
218	C₄⋊C₄	—	284	2	no	—
219	C₄⋊C₄	—	142	1	no	—
220	C₂²⋊C₄	—	284	2	no	—
221	C₂²⋊C₄	—	71	2	no	—
222	C₄²	—	284	2	no	—
223	C₄²	—	142	1	no	—
224	C₁₆	—	142	2	no	—
225	C₁₆	C₂²⋊Dic₇₁	1	1	yes	characteristic
226	C₁₆	C₂²⋊Dic₇₁	1	1	yes	characteristic
227	C₁₆	C₂²⋊Dic₇₁	1	1	yes	characteristic
228	C₁₆	C₂²⋊Dic₇₁	1	1	yes	characteristic
229	C₂³	—	4	2	no	—
230	C₂³	D₈×D₇₁	1	1	yes	characteristic
231	Q₈	—	4	2	no	—
232	Q₈	—	4	2	no	—
233	D₄	—	4	2	no	—
234	D₄	—	4	2	no	—
235	D₄	—	8	2	no	—
236	D₄	—	8	2	no	—
237	C₂×C₄	—	4	2	no	—
238	C₂×C₄	—	8	2	no	—
239	C₂×C₄	—	8	2	no	—
240	C₂×C₄	—	568	2	no	—
241	C₂×C₄	—	142	2	no	—
242	C₂×C₄	D₄×Dic₇₁	1	1	yes	characteristic
243	C₂×C₄	#14556099	1	1	yes	characteristic
244	C₂×C₄	#14556124	1	1	yes	characteristic
245	C₂×C₄	D₈×D₇₁	1	1	yes	characteristic
246	C₂×C₄	#14556147	1	1	yes	characteristic
247	C₂×C₄	#14556147	1	1	yes	characteristic
248	C₈	—	142	2	no	—
249	C₈	C₂×C₂²⋊Dic₇₁	1	1	yes	characteristic
250	C₈	D₄×Dic₇₁	1	1	yes	characteristic
251	C₈	#14556099	1	1	yes	characteristic
252	C₈	#14556115	1	1	yes	characteristic
253	C₂²	—	4	1	no	—
254	C₂²	—	8	2	no	—
255	C₂²	—	16	2	no	—
256	C₂²	D₈×Dic₇₁	1	1	yes	characteristic
257	C₂²	#17986232	1	1	yes	characteristic
258	C₂²	#17986374	1	1	yes	characteristic
259	C₄	—	8	2	no	—
260	C₄	—	284	2	no	—
261	C₄	—	1136	2	no	—
262	C₄	#17985902	1	1	yes	characteristic
263	C₄	D₈×Dic₇₁	1	1	yes	characteristic
264	C₄	#17986232	1	1	yes	characteristic
265	C₄	#17986382	1	1	yes	characteristic
266	C₂	—	2	1	no	—
267	C₂	—	2	1	no	—
268	C₂	—	2	1	no	—
269	C₂	—	16	2	no	—
270	C₂	#22589207	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#17986232
commutator	C₂×C₈×C₇₁	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇₁
Fitting, prime radical	C₂×#508×C₇₁	C₂
socle	C₂×C₇₁	#1970
cosocle	C₂²×C₈×C₇₁	C₂³
hypercenter	C₂×#508	D₇₁
hypocenter	C₇₁	#13243
LCS	G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#508

Complements

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

N	H ≅ G/N	type
1. #22590124	C₂	semidirect
2. #22590076	C₂	semidirect
3. #22589936	C₂	semidirect
4. #22589239	C₂	semidirect
21. C₂×Q₃₂×C₇₁	C₄	semidirect
22. C₂×SD₃₂×C₇₁	C₄	semidirect
23. C₂×D₁₆×C₇₁	C₄	semidirect
141. C₇₁	#13243	semidirect
147. C₂×Q₃₂	Dic₇₁	semidirect
148. C₂×SD₃₂	Dic₇₁	semidirect
149. C₂×D₁₆	Dic₇₁	semidirect

Maximal subgroups

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

subgroup	index
#22590124	2
#22590076	2
#22589936	2
#22589239	2
C₂×#508×C₇₁	2
#13243	71