Subgroups — #31908706

Order 18176.

502 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22590096	C₂	1	1	yes	maximal, characteristic, complemented
2	#22587088	C₂	1	1	yes	maximal, characteristic, complemented
3	#22586386	C₂	1	4	yes	maximal, complemented
4	C₂×#493×C₇₁	C₂	1	1	yes	maximal, characteristic
5	C₂×#14556068	—	2	1	no	—
6	#17986052	—	2	1	no	—
7	C₄⋊(C₇₁⋊C₁₆)	—	2	1	no	—
8	C₄⋊(C₇₁⋊C₁₆)	C₂²	1	2	yes	complemented
9	#17985752	C₂²	1	2	yes	complemented
10	#17985750	—	2	1	no	—
11	C₂²×D₈×C₇₁	—	2	1	no	—
12	C₂²×D₈×C₇₁	C₄	1	2	yes	—
13	C₂×#127×C₇₁	—	2	1	no	—
14	#493×C₇₁	—	2	1	no	—
15	#493×C₇₁	—	2	1	no	—
16	#493×C₇₁	C₄	1	2	yes	—
17	#493×C₇₁	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
19	#14556068	—	4	2	no	—
20	C₂×C₇₁⋊C₁₆	—	2	2	no	—
21	C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
22	C₂²⋊(C₇₁⋊C₈)	—	4	1	no	—
23	C₄×C₇₁⋊C₈	—	4	1	no	—
24	C₂²×D₄×C₇₁	—	2	2	no	—
25	C₂²×D₄×C₇₁	—	4	1	no	—
26	C₂²×D₄×C₇₁	—	4	1	no	—
27	C₂×D₈×C₇₁	—	2	2	no	—
28	C₂×D₈×C₇₁	—	2	2	no	—
29	C₂×D₈×C₇₁	—	2	2	no	—
30	C₂×D₈×C₇₁	—	4	4	no	—
31	C₂×D₈×C₇₁	—	4	2	no	—
32	C₂×D₈×C₇₁	—	4	2	no	—
33	C₂×D₈×C₇₁	C₂×C₄	1	4	yes	—
34	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
36	#127×C₇₁	—	2	2	no	—
37	#127×C₇₁	—	2	2	no	—
38	#127×C₇₁	—	4	1	no	—
39	#127×C₇₁	—	4	1	no	—
40	C₂×C₄²×C₇₁	D₄	1	1	yes	characteristic
41	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
42	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
43	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
44	C₄×C₈×C₇₁	C₂³	1	1	yes	characteristic
45	C₂×C₇₁⋊C₈	—	4	2	no	—
46	C₇₁⋊C₁₆	—	4	2	no	—
47	C₂⁴×C₇₁	—	4	2	no	—
48	C₂×D₄×C₇₁	—	2	4	no	—
49	C₂×D₄×C₇₁	—	4	4	no	—
50	C₂×D₄×C₇₁	—	4	4	no	—
51	C₂×D₄×C₇₁	—	4	2	no	—
52	C₂×D₄×C₇₁	—	4	2	no	—
53	C₂×D₄×C₇₁	—	4	2	no	—
54	C₂×D₄×C₇₁	—	4	2	no	—
55	C₂×D₄×C₇₁	—	4	2	no	—
56	C₂×D₄×C₇₁	—	8	4	no	—
57	C₂×D₄×C₇₁	—	8	4	no	—
58	C₂²×C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
59	C₂²×C₄×C₇₁	D₈	1	1	yes	characteristic
60	C₂²×C₄×C₇₁	SD₁₆	1	1	yes	characteristic
61	D₈×C₇₁	—	4	4	no	—
62	D₈×C₇₁	—	4	4	no	—
63	D₈×C₇₁	—	4	4	no	—
64	D₈×C₇₁	—	4	2	no	—
65	D₈×C₇₁	—	4	2	no	—
66	C₂×C₈×C₇₁	—	2	2	no	—
67	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
68	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
69	C₂×C₈×C₇₁	D₈	1	2	yes	—
70	C₂×C₈×C₇₁	SD₁₆	1	2	yes	—
71	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
73	C₄²×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
74	C₄²×C₇₁	D₈	1	1	yes	characteristic
75	C₄²×C₇₁	SD₁₆	1	1	yes	characteristic
76	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
77	C₂³×C₇₁	—	4	4	no	—
78	C₂³×C₇₁	—	8	4	no	—
79	C₂³×C₇₁	—	8	4	no	—
80	C₂³×C₇₁	—	8	4	no	—
81	C₂³×C₇₁	#102	1	1	yes	characteristic
82	D₄×C₇₁	—	4	4	no	—
83	D₄×C₇₁	—	4	4	no	—
84	D₄×C₇₁	—	8	4	no	—
85	D₄×C₇₁	—	8	4	no	—
86	D₄×C₇₁	—	8	4	no	—
87	D₄×C₇₁	—	8	4	no	—
88	D₄×C₇₁	—	8	4	no	—
89	C₂×C₄×C₇₁	—	2	2	no	—
90	C₂×C₄×C₇₁	—	2	2	no	—
91	C₂×C₄×C₇₁	#100	1	2	yes	—
92	C₂×C₄×C₇₁	#102	1	1	yes	characteristic
93	C₂×C₄×C₇₁	#102	1	1	yes	characteristic
94	C₂×C₄×C₇₁	#102	1	2	yes	—
95	C₂×C₄×C₇₁	#102	1	1	yes	characteristic
96	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
97	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
98	C₂×C₄×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
99	C₈×C₇₁	—	2	1	no	—
100	C₈×C₇₁	—	2	1	no	—
101	C₈×C₇₁	#102	1	2	yes	—
102	C₈×C₇₁	C₂×D₈	1	1	yes	characteristic
103	C₈×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
104	C₇₁⋊C₈	—	8	2	no	—
105	C₂²×C₇₁	—	2	1	no	—
106	C₂²×C₇₁	—	2	1	no	—
107	C₂²×C₇₁	—	8	4	no	—
108	C₂²×C₇₁	—	8	4	no	—
109	C₂²×C₇₁	—	8	4	no	—
110	C₂²×C₇₁	—	16	8	no	—
111	C₂²×C₇₁	#331	1	2	yes	—
112	C₂²×C₇₁	C₂×#102	1	1	yes	characteristic
113	C₄×C₇₁	—	2	1	no	—
114	C₄×C₇₁	—	2	1	no	—
115	C₄×C₇₁	#331	1	2	yes	—
116	C₄×C₇₁	#361	1	2	yes	—
117	C₄×C₇₁	C₂×#100	1	1	yes	characteristic
118	C₄×C₇₁	C₂×#102	1	1	yes	characteristic
119	C₄×C₇₁	#509	1	2	yes	—
120	#11931	—	71	1	no	maximal, Sylow
121	C₂×C₇₁	—	2	2	no	—
122	C₂×C₇₁	—	16	4	no	—
123	C₂×C₇₁	C₂×#331	1	1	yes	characteristic
124	C₂×C₇₁	#2140	1	1	yes	characteristic
125	C₂×C₇₁	C₂×#361	1	1	yes	characteristic
126	C₂×#493	D₇₁	1	1	yes	characteristic
127	#2150	—	71	1	no	—
128	#1564	—	71	1	no	—
129	#1361	—	71	4	no	—
130	C₇₁	#11931	1	1	yes	characteristic, complemented, Sylow
131	C₂²×D₈	—	2	1	no	—
132	C₂²×D₈	Dic₇₁	1	2	yes	—
133	C₂×#127	—	2	1	no	—
134	C₂×M₃₂	—	142	1	no	—
135	#493	—	2	1	no	—
136	#493	—	2	1	no	—
137	#493	Dic₇₁	1	2	yes	—
138	#493	C₂×D₇₁	1	2	yes	—
139	#431	—	142	1	no	—
140	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
141	C₄⋊C₁₆	—	71	2	no	—
142	C₄⋊C₁₆	—	142	1	no	—
143	#335	—	71	2	no	—
144	#334	—	142	1	no	—
145	C₂²×D₄	—	2	2	no	—
146	C₂²×D₄	—	4	1	no	—
147	C₂²×D₄	—	4	1	no	—
148	C₂×D₈	—	2	2	no	—
149	C₂×D₈	—	2	2	no	—
150	C₂×D₈	—	2	2	no	—
151	C₂×D₈	—	4	4	no	—
152	C₂×D₈	—	4	2	no	—
153	C₂×D₈	—	4	2	no	—
154	C₂×D₈	C₂×Dic₇₁	1	4	yes	—
155	C₂²×C₈	C₂×Dic₇₁	1	1	yes	characteristic
156	C₂²×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
157	#127	—	2	2	no	—
158	#127	—	2	2	no	—
159	#127	—	4	1	no	—
160	#127	—	4	1	no	—
161	C₂×C₄²	C₂²⋊D₇₁	1	1	yes	characteristic
162	M₃₂	—	284	2	no	—
163	C₂×C₁₆	—	142	2	no	—
164	C₄⋊C₈	—	142	2	no	—
165	C₂²⋊C₈	—	284	1	no	—
166	C₄×C₈	—	284	1	no	—
167	C₄×C₈	C₂×Dic₇₁	1	1	yes	characteristic
168	C₄×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
169	C₄×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
170	C₄×C₈	C₂²×D₇₁	1	1	yes	characteristic
171	C₂⁴	—	4	2	no	—
172	C₂×D₄	—	2	4	no	—
173	C₂×D₄	—	4	4	no	—
174	C₂×D₄	—	4	4	no	—
175	C₂×D₄	—	4	2	no	—
176	C₂×D₄	—	4	2	no	—
177	C₂×D₄	—	4	2	no	—
178	C₂×D₄	—	4	2	no	—
179	C₂×D₄	—	4	2	no	—
180	C₂×D₄	—	8	4	no	—
181	C₂×D₄	—	8	4	no	—
182	C₂²×C₄	C₂²⋊Dic₇₁	1	1	yes	characteristic
183	C₂²×C₄	#11765850	1	1	yes	characteristic
184	C₂²×C₄	#11765853	1	1	yes	characteristic
185	D₈	—	4	4	no	—
186	D₈	—	4	4	no	—
187	D₈	—	4	4	no	—
188	D₈	—	4	2	no	—
189	D₈	—	4	2	no	—
190	C₂×C₈	—	2	2	no	—
191	C₂×C₈	—	284	2	no	—
192	C₂×C₈	C₂²⋊Dic₇₁	1	1	yes	characteristic
193	C₂×C₈	C₂²⋊Dic₇₁	1	1	yes	characteristic
194	C₂×C₈	#11765850	1	2	yes	—
195	C₂×C₈	#11765853	1	2	yes	—
196	C₂×C₈	C₂²×Dic₇₁	1	1	yes	characteristic
197	C₂×C₈	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
198	C₄²	C₂²⋊Dic₇₁	1	1	yes	characteristic
199	C₄²	#11765850	1	1	yes	characteristic
200	C₄²	#11765853	1	1	yes	characteristic
201	C₄²	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
202	C₁₆	—	284	2	no	—
203	C₂³	—	4	4	no	—
204	C₂³	—	8	4	no	—
205	C₂³	—	8	4	no	—
206	C₂³	—	8	4	no	—
207	C₂³	#14556049	1	1	yes	characteristic
208	D₄	—	4	4	no	—
209	D₄	—	4	4	no	—
210	D₄	—	8	4	no	—
211	D₄	—	8	4	no	—
212	D₄	—	8	4	no	—
213	D₄	—	8	4	no	—
214	D₄	—	8	4	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	#14556045	1	2	yes	—
218	C₂×C₄	#14556049	1	1	yes	characteristic
219	C₂×C₄	#14556049	1	1	yes	characteristic
220	C₂×C₄	#14556049	1	2	yes	—
221	C₂×C₄	#14556049	1	1	yes	characteristic
222	C₂×C₄	C₂×C₂²⋊Dic₇₁	1	1	yes	characteristic
223	C₂×C₄	C₂×#11765850	1	1	yes	characteristic
224	C₂×C₄	C₂×#11765853	1	1	yes	characteristic
225	C₈	—	2	1	no	—
226	C₈	—	2	1	no	—
227	C₈	—	568	2	no	—
228	C₈	#14556049	1	2	yes	—
229	C₈	C₂×#11765850	1	1	yes	characteristic
230	C₈	C₂×#11765853	1	1	yes	characteristic
231	C₂²	—	2	1	no	—
232	C₂²	—	2	1	no	—
233	C₂²	—	8	4	no	—
234	C₂²	—	8	4	no	—
235	C₂²	—	8	4	no	—
236	C₂²	—	16	8	no	—
237	C₂²	#17985743	1	2	yes	—
238	C₂²	C₂×#14556049	1	1	yes	characteristic
239	C₄	—	2	1	no	—
240	C₄	—	2	1	no	—
241	C₄	#17985743	1	2	yes	—
242	C₄	#17985812	1	2	yes	—
243	C₄	C₂×#14556045	1	1	yes	characteristic
244	C₄	C₂×#14556049	1	1	yes	characteristic
245	C₄	#17986435	1	2	yes	—
246	C₂	—	2	2	no	—
247	C₂	—	16	4	no	—
248	C₂	C₂×#17985743	1	1	yes	characteristic
249	C₂	#22590056	1	1	yes	characteristic
250	C₂	C₂×#17985812	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#14556049
commutator	C₂×C₈×C₇₁	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₇₁
Fitting, prime radical	C₂×#493×C₇₁	C₂
socle	C₂²×C₇₁	C₂×#102
cosocle	C₄×C₈×C₇₁	C₂³
hypercenter	C₂×#493	D₇₁
hypocenter	C₇₁	#11931
LCS	G ▷ C₂×C₈×C₇₁ ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#493

Complements

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

N	H ≅ G/N	type
1. #22590096	C₂	semidirect
2. #22587088	C₂	semidirect
3. #22586386	C₂	semidirect
8. C₄⋊(C₇₁⋊C₁₆)	C₂²	semidirect
9. #17985752	C₂²	semidirect
130. C₇₁	#11931	semidirect

Maximal subgroups

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

subgroup	index
#22590096	2
#22587088	2
#22586386	2
C₂×#493×C₇₁	2
#11931	71