Subgroups — #31908739

Order 18176.

306 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊(C₇₁⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
2	#22587088	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	C₂²×C₇₁⋊C₁₆	—	2	1	no	—
7	#17986052	—	2	1	no	—
8	C₄⋊(C₇₁⋊C₁₆)	—	2	1	no	—
9	C₄⋊(C₇₁⋊C₁₆)	C₂²	1	2	yes	—
10	#17985752	C₂²	1	2	yes	—
11	#17985750	—	2	1	no	—
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	C₂×C₇₁⋊C₁₆	—	2	2	no	—
22	C₂×C₇₁⋊C₁₆	—	4	2	no	—
23	C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
24	C₂²⋊(C₇₁⋊C₈)	—	4	1	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	C₇₁⋊C₁₆	—	4	2	no	—
49	C₂×Q₈×C₇₁	—	4	1	no	—
50	C₂×Q₈×C₇₁	—	4	1	no	—
51	C₂²×C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
52	C₂²×C₄×C₇₁	D₈	1	1	yes	characteristic
53	C₂²×C₄×C₇₁	SD₁₆	1	1	yes	characteristic
54	C₂×C₈×C₇₁	—	2	2	no	—
55	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
56	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
57	C₂×C₈×C₇₁	SD₁₆	1	2	yes	—
58	C₂×C₈×C₇₁	Q₁₆	1	2	yes	—
59	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
61	C₄⋊C₄×C₇₁	—	2	2	no	—
62	C₄⋊C₄×C₇₁	—	2	2	no	—
63	C₄⋊C₄×C₇₁	—	4	2	no	—
64	C₄⋊C₄×C₇₁	—	4	2	no	—
65	C₄⋊C₄×C₇₁	—	4	1	no	—
66	C₄⋊C₄×C₇₁	—	4	1	no	—
67	C₂²⋊C₄×C₇₁	—	4	2	no	—
68	C₄²×C₇₁	—	4	2	no	—
69	C₄²×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
70	C₄²×C₇₁	D₈	1	1	yes	characteristic
71	C₄²×C₇₁	SD₁₆	1	1	yes	characteristic
72	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
73	C₂³×C₇₁	#102	1	1	yes	characteristic
74	Q₈×C₇₁	—	8	2	no	—
75	Q₈×C₇₁	—	8	2	no	—
76	C₂×C₄×C₇₁	—	2	2	no	—
77	C₂×C₄×C₇₁	—	4	1	no	—
78	C₂×C₄×C₇₁	—	4	2	no	—
79	C₂×C₄×C₇₁	—	4	2	no	—
80	C₂×C₄×C₇₁	#101	1	2	yes	—
81	C₂×C₄×C₇₁	#102	1	1	yes	characteristic
82	C₂×C₄×C₇₁	#102	1	1	yes	characteristic
83	C₂×C₄×C₇₁	#102	1	1	yes	characteristic
84	C₂×C₄×C₇₁	#103	1	2	yes	—
85	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
86	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
87	C₂×C₄×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
88	C₈×C₇₁	—	2	1	no	—
89	C₈×C₇₁	—	2	1	no	—
90	C₈×C₇₁	#103	1	2	yes	—
91	C₈×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
92	C₈×C₇₁	C₂×Q₁₆	1	1	yes	characteristic
93	C₇₁⋊C₈	—	8	2	no	—
94	C₂²×C₇₁	—	2	1	no	—
95	C₂²×C₇₁	—	2	1	no	—
96	C₂²×C₇₁	#332	1	2	yes	—
97	C₂²×C₇₁	C₂×#102	1	1	yes	characteristic
98	C₄×C₇₁	—	2	1	no	—
99	C₄×C₇₁	—	2	1	no	—
100	C₄×C₇₁	—	2	1	no	—
101	C₄×C₇₁	—	8	2	no	—
102	C₄×C₇₁	—	8	2	no	—
103	C₄×C₇₁	#332	1	2	yes	—
104	C₄×C₇₁	C₂×#101	1	1	yes	characteristic
105	C₄×C₇₁	C₂×#103	1	1	yes	characteristic
106	C₄×C₇₁	#508	1	2	yes	—
107	#11939	—	71	1	no	maximal, Sylow
108	C₂×C₇₁	—	2	2	no	—
109	C₂×C₇₁	C₂×#332	1	1	yes	characteristic
110	C₂×C₇₁	#2138	1	1	yes	characteristic
111	C₂×C₇₁	#2148	1	1	yes	characteristic
112	#3160	D₇₁	1	1	yes	characteristic
113	C₂×C₄⋊C₁₆	—	71	1	no	—
114	#1564	—	71	1	no	—
115	#1365	—	71	2	no	—
116	#1364	—	71	2	no	—
117	C₇₁	#11939	1	1	yes	characteristic, complemented, Sylow
118	#533	—	2	1	no	—
119	C₂²×C₁₆	—	142	1	no	—
120	#500	C₂×D₇₁	1	1	yes	characteristic
121	#500	C₂×D₇₁	1	1	yes	characteristic
122	#499	—	2	1	no	—
123	#499	Dic₇₁	1	2	yes	—
124	#498	—	2	1	no	—
125	#431	—	142	1	no	—
126	#427	—	2	1	no	—
127	#427	Dic₇₁	1	2	yes	—
128	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
129	C₄⋊C₁₆	—	71	2	no	—
130	C₄⋊C₁₆	—	142	1	no	—
131	#335	—	71	2	no	—
132	#334	—	142	1	no	—
133	C₂²×C₈	C₂×Dic₇₁	1	1	yes	characteristic
134	C₂²×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
135	C₄⋊Q₈	—	2	1	no	—
136	C₄⋊Q₈	—	2	1	no	—
137	#125	—	2	2	no	—
138	C₂²⋊Q₈	—	4	1	no	—
139	C₂²⋊Q₈	—	4	1	no	—
140	C₄×Q₈	—	4	1	no	—
141	C₄×Q₈	—	4	1	no	—
142	#117	—	2	2	no	—
143	C₂×C₄²	C₂²⋊D₇₁	1	1	yes	characteristic
144	C₂×C₁₆	—	284	2	no	—
145	C₂×C₁₆	—	142	2	no	—
146	#107	—	2	1	no	—
147	#107	—	2	1	no	—
148	#107	C₂×Dic₇₁	1	2	yes	—
149	#107	C₂×Dic₇₁	1	2	yes	—
150	#106	—	2	2	no	—
151	#106	—	2	2	no	—
152	C₄⋊C₈	—	142	2	no	—
153	C₂²⋊C₈	—	284	1	no	—
154	C₄×C₈	—	284	1	no	—
155	C₄×C₈	C₂×Dic₇₁	1	1	yes	characteristic
156	C₄×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
157	C₄×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
158	C₄×C₈	C₂²×D₇₁	1	1	yes	characteristic
159	C₂×Q₈	—	4	1	no	—
160	C₂×Q₈	—	4	1	no	—
161	C₂²×C₄	C₂²⋊Dic₇₁	1	1	yes	characteristic
162	C₂²×C₄	#11765850	1	1	yes	characteristic
163	C₂²×C₄	#11765853	1	1	yes	characteristic
164	C₂×C₈	—	2	2	no	—
165	C₂×C₈	—	284	2	no	—
166	C₂×C₈	C₂²⋊Dic₇₁	1	1	yes	characteristic
167	C₂×C₈	C₂²⋊Dic₇₁	1	1	yes	characteristic
168	C₂×C₈	#11765852	1	2	yes	—
169	C₂×C₈	#11765855	1	2	yes	—
170	C₂×C₈	C₂²×Dic₇₁	1	1	yes	characteristic
171	C₂×C₈	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
172	C₄⋊C₄	—	2	2	no	—
173	C₄⋊C₄	—	2	2	no	—
174	C₄⋊C₄	—	4	2	no	—
175	C₄⋊C₄	—	4	2	no	—
176	C₄⋊C₄	—	4	1	no	—
177	C₄⋊C₄	—	4	1	no	—
178	C₂²⋊C₄	—	4	2	no	—
179	C₄²	—	4	2	no	—
180	C₄²	C₂²⋊Dic₇₁	1	1	yes	characteristic
181	C₄²	#11765850	1	1	yes	characteristic
182	C₄²	#11765853	1	1	yes	characteristic
183	C₄²	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
184	C₁₆	—	284	2	no	—
185	C₂³	#14556049	1	1	yes	characteristic
186	Q₈	—	8	2	no	—
187	Q₈	—	8	2	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	4	1	no	—
190	C₂×C₄	—	4	2	no	—
191	C₂×C₄	—	4	2	no	—
192	C₂×C₄	#14556047	1	2	yes	—
193	C₂×C₄	#14556049	1	1	yes	characteristic
194	C₂×C₄	#14556049	1	1	yes	characteristic
195	C₂×C₄	#14556049	1	1	yes	characteristic
196	C₂×C₄	#14556052	1	2	yes	—
197	C₂×C₄	C₂×C₂²⋊Dic₇₁	1	1	yes	characteristic
198	C₂×C₄	C₂×#11765850	1	1	yes	characteristic
199	C₂×C₄	C₂×#11765853	1	1	yes	characteristic
200	C₈	—	2	1	no	—
201	C₈	—	2	1	no	—
202	C₈	—	568	2	no	—
203	C₈	#14556052	1	2	yes	—
204	C₈	C₂×#11765852	1	1	yes	characteristic
205	C₈	C₂×#11765855	1	1	yes	characteristic
206	C₂²	—	2	1	no	—
207	C₂²	—	2	1	no	—
208	C₂²	#17985745	1	2	yes	—
209	C₂²	C₂×#14556049	1	1	yes	characteristic
210	C₄	—	2	1	no	—
211	C₄	—	2	1	no	—
212	C₄	—	2	1	no	—
213	C₄	—	8	2	no	—
214	C₄	—	8	2	no	—
215	C₄	#17985745	1	2	yes	—
216	C₄	C₂×#14556047	1	1	yes	characteristic
217	C₄	C₂×#14556052	1	1	yes	characteristic
218	C₄	#17986429	1	2	yes	—
219	C₂	—	2	2	no	—
220	C₂	C₂×#17985745	1	1	yes	characteristic
221	C₂	#22590047	1	1	yes	characteristic
222	C₂	#22590089	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#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	#11939
LCS	G ▷ C₂×C₈×C₇₁ ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₄ ◁ 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
117. C₇₁	#11939	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊(C₇₁⋊C₁₆)	2
#22587088	2
#22586398	2
#22586395	2
C₇₁×#3160	2
#11939	71