Subgroups — #31908752

Order 18176.

310 conjugacy classes of subgroups.

The subgroup lattice has 232 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
2	C₂×#17985752	C₂	1	1	yes	maximal, characteristic
3	#22586398	C₂	1	2	yes	maximal, complemented
4	#22586395	C₂	1	2	yes	maximal, complemented
5	C₇₁×#3160	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556068	—	2	1	no	—
7	C₂×C₄⋊(C₇₁⋊C₈)	—	2	1	no	—
8	C₄⋊(C₇₁⋊C₁₆)	—	2	1	no	—
9	C₄⋊(C₇₁⋊C₁₆)	C₂²	1	2	yes	—
10	#17985752	—	2	1	no	—
11	#17985752	C₂²	1	2	yes	—
12	#533×C₇₁	—	2	1	no	—
13	#500×C₇₁	C₂²	1	1	yes	characteristic
14	#500×C₇₁	C₂²	1	1	yes	characteristic
15	#499×C₇₁	—	2	1	no	—
16	#499×C₇₁	C₄	1	2	yes	—
17	#498×C₇₁	—	2	1	no	—
18	#427×C₇₁	—	2	1	no	—
19	#427×C₇₁	C₄	1	2	yes	—
20	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
21	C₂²×C₇₁⋊C₈	—	4	1	no	—
22	#14556068	—	4	2	no	—
23	C₂×C₇₁⋊C₁₆	—	2	2	no	—
24	C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
25	C₄⋊(C₇₁⋊C₈)	—	4	1	no	—
26	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
28	C₄⋊Q₈×C₇₁	—	2	1	no	—
29	C₄⋊Q₈×C₇₁	—	2	1	no	—
30	#125×C₇₁	—	2	2	no	—
31	C₂²⋊Q₈×C₇₁	—	4	1	no	—
32	C₂²⋊Q₈×C₇₁	—	4	1	no	—
33	C₄×Q₈×C₇₁	—	4	1	no	—
34	C₄×Q₈×C₇₁	—	4	1	no	—
35	#117×C₇₁	—	2	2	no	—
36	C₂×C₄²×C₇₁	D₄	1	1	yes	characteristic
37	#107×C₇₁	—	2	1	no	—
38	#107×C₇₁	—	2	1	no	—
39	#107×C₇₁	C₂×C₄	1	2	yes	—
40	#107×C₇₁	C₂×C₄	1	2	yes	—
41	#106×C₇₁	—	2	2	no	—
42	#106×C₇₁	—	2	2	no	—
43	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
44	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
45	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
46	C₄×C₈×C₇₁	C₂³	1	1	yes	characteristic
47	C₂×C₇₁⋊C₈	—	4	2	no	—
48	C₂×C₇₁⋊C₈	—	8	2	no	—
49	C₇₁⋊C₁₆	—	4	2	no	—
50	C₂×Q₈×C₇₁	—	4	1	no	—
51	C₂×Q₈×C₇₁	—	4	1	no	—
52	C₂²×C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
53	C₂²×C₄×C₇₁	D₈	1	1	yes	characteristic
54	C₂²×C₄×C₇₁	SD₁₆	1	1	yes	characteristic
55	C₂×C₈×C₇₁	—	2	2	no	—
56	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
57	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
58	C₂×C₈×C₇₁	SD₁₆	1	2	yes	—
59	C₂×C₈×C₇₁	Q₁₆	1	2	yes	—
60	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
62	C₄⋊C₄×C₇₁	—	2	2	no	—
63	C₄⋊C₄×C₇₁	—	2	2	no	—
64	C₄⋊C₄×C₇₁	—	4	2	no	—
65	C₄⋊C₄×C₇₁	—	4	2	no	—
66	C₄⋊C₄×C₇₁	—	4	1	no	—
67	C₄⋊C₄×C₇₁	—	4	1	no	—
68	C₂²⋊C₄×C₇₁	—	4	2	no	—
69	C₄²×C₇₁	—	4	2	no	—
70	C₄²×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
71	C₄²×C₇₁	D₈	1	1	yes	characteristic
72	C₄²×C₇₁	SD₁₆	1	1	yes	characteristic
73	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
74	C₂³×C₇₁	#102	1	1	yes	characteristic
75	Q₈×C₇₁	—	8	2	no	—
76	Q₈×C₇₁	—	8	1	no	—
77	Q₈×C₇₁	—	8	1	no	—
78	C₂×C₄×C₇₁	—	2	1	no	—
79	C₂×C₄×C₇₁	—	2	1	no	—
80	C₂×C₄×C₇₁	—	4	1	no	—
81	C₂×C₄×C₇₁	—	4	2	no	—
82	C₂×C₄×C₇₁	—	4	2	no	—
83	C₂×C₄×C₇₁	#101	1	2	yes	—
84	C₂×C₄×C₇₁	#102	1	1	yes	characteristic
85	C₂×C₄×C₇₁	#102	1	1	yes	characteristic
86	C₂×C₄×C₇₁	#102	1	1	yes	characteristic
87	C₂×C₄×C₇₁	#103	1	2	yes	—
88	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
89	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
90	C₂×C₄×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
91	C₈×C₇₁	—	2	1	no	—
92	C₈×C₇₁	—	2	1	no	—
93	C₈×C₇₁	#103	1	2	yes	—
94	C₈×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
95	C₈×C₇₁	C₂×Q₁₆	1	1	yes	characteristic
96	C₇₁⋊C₈	—	8	2	no	—
97	C₂²×C₇₁	—	2	1	no	—
98	C₂²×C₇₁	—	2	1	no	—
99	C₂²×C₇₁	#332	1	2	yes	—
100	C₂²×C₇₁	C₂×#102	1	1	yes	characteristic
101	C₄×C₇₁	—	2	1	no	—
102	C₄×C₇₁	—	2	1	no	—
103	C₄×C₇₁	—	2	1	no	—
104	C₄×C₇₁	—	8	2	no	—
105	C₄×C₇₁	—	8	2	no	—
106	C₄×C₇₁	#332	1	2	yes	—
107	C₄×C₇₁	C₂×#101	1	1	yes	characteristic
108	C₄×C₇₁	C₂×#103	1	1	yes	characteristic
109	C₄×C₇₁	#509	1	1	yes	characteristic
110	C₄×C₇₁	#510	1	1	yes	characteristic
111	#11942	—	71	1	no	maximal, Sylow
112	C₂×C₇₁	—	2	2	no	—
113	C₂×C₇₁	C₂×#332	1	1	yes	characteristic
114	C₂×C₇₁	#2139	1	1	yes	characteristic
115	C₂×C₇₁	#2148	1	1	yes	characteristic
116	#3160	D₇₁	1	1	yes	characteristic
117	#2150	—	71	1	no	—
118	C₂×#335	—	71	1	no	—
119	#1365	—	71	2	no	—
120	#1364	—	71	2	no	—
121	C₇₁	#11942	1	1	yes	characteristic, complemented, Sylow
122	#533	—	2	1	no	—
123	C₂×M₃₂	—	142	1	no	—
124	#500	C₂×D₇₁	1	1	yes	characteristic
125	#500	C₂×D₇₁	1	1	yes	characteristic
126	#499	—	2	1	no	—
127	#499	Dic₇₁	1	2	yes	—
128	#498	—	2	1	no	—
129	#427	—	2	1	no	—
130	#427	Dic₇₁	1	2	yes	—
131	C₂×C₄⋊C₈	—	142	1	no	—
132	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
133	C₄⋊C₁₆	—	71	2	no	—
134	C₄⋊C₁₆	—	142	1	no	—
135	#335	—	71	2	no	—
136	#335	—	142	1	no	—
137	C₂²×C₈	—	284	1	no	—
138	C₂²×C₈	C₂×Dic₇₁	1	1	yes	characteristic
139	C₂²×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
140	C₄⋊Q₈	—	2	1	no	—
141	C₄⋊Q₈	—	2	1	no	—
142	#125	—	2	2	no	—
143	C₂²⋊Q₈	—	4	1	no	—
144	C₂²⋊Q₈	—	4	1	no	—
145	C₄×Q₈	—	4	1	no	—
146	C₄×Q₈	—	4	1	no	—
147	#117	—	2	2	no	—
148	C₂×C₄²	C₂²⋊D₇₁	1	1	yes	characteristic
149	M₃₂	—	284	2	no	—
150	C₂×C₁₆	—	142	2	no	—
151	#107	—	2	1	no	—
152	#107	—	2	1	no	—
153	#107	C₂×Dic₇₁	1	2	yes	—
154	#107	C₂×Dic₇₁	1	2	yes	—
155	#106	—	2	2	no	—
156	#106	—	2	2	no	—
157	C₄⋊C₈	—	284	1	no	—
158	C₄⋊C₈	—	142	2	no	—
159	C₄×C₈	C₂×Dic₇₁	1	1	yes	characteristic
160	C₄×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
161	C₄×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
162	C₄×C₈	C₂²×D₇₁	1	1	yes	characteristic
163	C₂×Q₈	—	4	1	no	—
164	C₂×Q₈	—	4	1	no	—
165	C₂²×C₄	C₂²⋊Dic₇₁	1	1	yes	characteristic
166	C₂²×C₄	#11765850	1	1	yes	characteristic
167	C₂²×C₄	#11765853	1	1	yes	characteristic
168	C₂×C₈	—	2	2	no	—
169	C₂×C₈	—	284	2	no	—
170	C₂×C₈	—	568	2	no	—
171	C₂×C₈	C₂²⋊Dic₇₁	1	1	yes	characteristic
172	C₂×C₈	C₂²⋊Dic₇₁	1	1	yes	characteristic
173	C₂×C₈	#11765852	1	2	yes	—
174	C₂×C₈	#11765855	1	2	yes	—
175	C₂×C₈	C₂²×Dic₇₁	1	1	yes	characteristic
176	C₂×C₈	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
177	C₄⋊C₄	—	2	2	no	—
178	C₄⋊C₄	—	2	2	no	—
179	C₄⋊C₄	—	4	2	no	—
180	C₄⋊C₄	—	4	2	no	—
181	C₄⋊C₄	—	4	1	no	—
182	C₄⋊C₄	—	4	1	no	—
183	C₂²⋊C₄	—	4	2	no	—
184	C₄²	—	4	2	no	—
185	C₄²	C₂²⋊Dic₇₁	1	1	yes	characteristic
186	C₄²	#11765850	1	1	yes	characteristic
187	C₄²	#11765853	1	1	yes	characteristic
188	C₄²	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
189	C₁₆	—	284	2	no	—
190	C₂³	#14556049	1	1	yes	characteristic
191	Q₈	—	8	2	no	—
192	Q₈	—	8	1	no	—
193	Q₈	—	8	1	no	—
194	C₂×C₄	—	2	1	no	—
195	C₂×C₄	—	2	1	no	—
196	C₂×C₄	—	4	1	no	—
197	C₂×C₄	—	4	2	no	—
198	C₂×C₄	—	4	2	no	—
199	C₂×C₄	#14556047	1	2	yes	—
200	C₂×C₄	#14556049	1	1	yes	characteristic
201	C₂×C₄	#14556049	1	1	yes	characteristic
202	C₂×C₄	#14556049	1	1	yes	characteristic
203	C₂×C₄	#14556052	1	2	yes	—
204	C₂×C₄	C₂×C₂²⋊Dic₇₁	1	1	yes	characteristic
205	C₂×C₄	C₂×#11765850	1	1	yes	characteristic
206	C₂×C₄	C₂×#11765853	1	1	yes	characteristic
207	C₈	—	2	1	no	—
208	C₈	—	2	1	no	—
209	C₈	—	568	2	no	—
210	C₈	#14556052	1	2	yes	—
211	C₈	C₂×#11765852	1	1	yes	characteristic
212	C₈	C₂×#11765855	1	1	yes	characteristic
213	C₂²	—	2	1	no	—
214	C₂²	—	2	1	no	—
215	C₂²	#17985745	1	2	yes	—
216	C₂²	C₂×#14556049	1	1	yes	characteristic
217	C₄	—	2	1	no	—
218	C₄	—	2	1	no	—
219	C₄	—	2	1	no	—
220	C₄	—	8	2	no	—
221	C₄	—	8	2	no	—
222	C₄	#17985745	1	2	yes	—
223	C₄	C₂×#14556047	1	1	yes	characteristic
224	C₄	C₂×#14556052	1	1	yes	characteristic
225	C₄	#17986434	1	1	yes	characteristic
226	C₄	#17986439	1	1	yes	characteristic
227	C₂	—	2	2	no	—
228	C₂	C₂×#17985745	1	1	yes	characteristic
229	C₂	#22590052	1	1	yes	characteristic
230	C₂	#22590089	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#14556049
commutator	C₂×C₈×C₇₁	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₇₁
Fitting	C₇₁×#3160	C₂
socle	C₂²×C₇₁	C₂×#102
prime radical	C₂³×C₇₁	#102
cosocle	C₄×C₈×C₇₁	C₂³
hypercenter	#3160	D₇₁
hypocenter	C₇₁	#11942
LCS	G ▷ C₂×C₈×C₇₁ ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #3160

Complements

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

N	H ≅ G/N	type
3. #22586398	C₂	semidirect
4. #22586395	C₂	semidirect
121. C₇₁	#11942	semidirect

Maximal subgroups

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

subgroup	index
#22590096	2
C₂×#17985752	2
#22586398	2
#22586395	2
C₇₁×#3160	2
#11942	71