Subgroups — #31908719

Order 18176.

306 conjugacy classes of subgroups.

The subgroup lattice has 222 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22598854	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊(C₇₁⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
3	#22586393	C₂	1	4	yes	maximal, complemented
4	C₇₁×#1564	C₂	1	1	yes	maximal, characteristic
5	#17986556	—	2	1	no	—
6	C₂²×C₇₁⋊C₁₆	—	2	1	no	—
7	#17986404	C₄	1	1	yes	characteristic
8	#17986404	C₄	1	1	yes	characteristic
9	#17986401	—	2	1	no	—
10	#17986401	C₂²	1	2	yes	—
11	#17986397	—	2	1	no	—
12	#17986037	—	2	1	no	—
13	#17986037	C₄	1	2	yes	—
14	C₄⋊(C₇₁⋊C₁₆)	—	2	1	no	—
15	C₄⋊(C₇₁⋊C₁₆)	C₂²	1	2	yes	—
16	#431×C₇₁	—	2	1	no	—
17	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
18	#335×C₇₁	—	2	1	no	—
19	#334×C₇₁	C₂²	1	2	yes	—
20	#14556128	—	2	1	no	—
21	#14556128	—	2	1	no	—
22	#14556120	—	2	2	no	—
23	#14556107	—	4	1	no	—
24	#14556107	—	4	1	no	—
25	C₄×Dic₁₄₂	—	4	1	no	—
26	C₄×Dic₁₄₂	—	4	1	no	—
27	#14556087	—	2	2	no	—
28	C₂×C₇₁⋊C₁₆	—	2	2	no	—
29	C₂×C₇₁⋊C₁₆	—	4	2	no	—
30	#14556062	—	2	1	no	—
31	#14556062	—	2	1	no	—
32	#14556062	C₂×C₄	1	2	yes	—
33	#14556062	C₂×C₄	1	2	yes	—
34	#14556060	—	2	2	no	—
35	#14556060	—	2	2	no	—
36	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
38	C₂×C₄²×C₇₁	D₄	1	1	yes	characteristic
39	C₄⋊C₈×C₇₁	—	2	2	no	—
40	C₂²⋊C₈×C₇₁	—	4	1	no	—
41	C₄×C₈×C₇₁	—	4	1	no	—
42	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
43	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
44	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
45	C₄×C₈×C₇₁	C₂³	1	1	yes	characteristic
46	C₂×Dic₁₄₂	—	4	1	no	—
47	C₂×Dic₁₄₂	—	4	1	no	—
48	C₄⋊Dic₇₁	—	2	2	no	—
49	C₄⋊Dic₇₁	—	2	2	no	—
50	C₄⋊Dic₇₁	—	4	1	no	—
51	C₄⋊Dic₇₁	—	4	1	no	—
52	Dic₇₁⋊C₄	—	4	2	no	—
53	Dic₇₁⋊C₄	—	4	2	no	—
54	C₂²⋊Dic₇₁	—	4	2	no	—
55	C₄×Dic₇₁	—	4	2	no	—
56	C₇₁⋊C₁₆	—	4	2	no	—
57	C₂²×C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
58	C₂²×C₄×C₇₁	D₈	1	1	yes	characteristic
59	C₂²×C₄×C₇₁	SD₁₆	1	1	yes	characteristic
60	C₂×C₈×C₇₁	—	2	2	no	—
61	C₂×C₈×C₇₁	—	4	2	no	—
62	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
63	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
64	C₂×C₈×C₇₁	SD₁₆	1	2	yes	—
65	C₂×C₈×C₇₁	Q₁₆	1	2	yes	—
66	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
67	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
68	C₄²×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
69	C₄²×C₇₁	D₈	1	1	yes	characteristic
70	C₄²×C₇₁	SD₁₆	1	1	yes	characteristic
71	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
72	C₂³×C₇₁	#102	1	1	yes	characteristic
73	C₂×C₄×C₇₁	—	2	2	no	—
74	C₂×C₄×C₇₁	—	4	1	no	—
75	C₂×C₄×C₇₁	#101	1	2	yes	—
76	C₂×C₄×C₇₁	#102	1	1	yes	characteristic
77	C₂×C₄×C₇₁	#102	1	1	yes	characteristic
78	C₂×C₄×C₇₁	#102	1	1	yes	characteristic
79	C₂×C₄×C₇₁	#103	1	2	yes	—
80	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
81	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
82	C₂×C₄×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
83	C₂×Dic₇₁	—	4	2	no	—
84	C₂×Dic₇₁	—	4	2	no	—
85	Dic₁₄₂	—	8	2	no	—
86	Dic₁₄₂	—	8	2	no	—
87	C₈×C₇₁	—	2	1	no	—
88	C₈×C₇₁	—	2	1	no	—
89	C₈×C₇₁	—	8	2	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	C₂²×C₇₁	—	2	1	no	—
94	C₂²×C₇₁	—	2	1	no	—
95	C₂²×C₇₁	#332	1	2	yes	—
96	C₂²×C₇₁	C₂×#102	1	1	yes	characteristic
97	C₄×C₇₁	—	2	1	no	—
98	C₄×C₇₁	—	2	1	no	—
99	C₄×C₇₁	—	2	1	no	—
100	C₄×C₇₁	#332	1	2	yes	—
101	C₄×C₇₁	#359	1	2	yes	—
102	C₄×C₇₁	C₂×#101	1	1	yes	characteristic
103	C₄×C₇₁	C₂×#103	1	1	yes	characteristic
104	Dic₇₁	—	8	2	no	—
105	Dic₇₁	—	8	2	no	—
106	#11934	—	71	1	no	maximal, Sylow
107	C₂×C₇₁	—	2	2	no	—
108	C₂×C₇₁	C₂×#332	1	1	yes	characteristic
109	C₂×C₇₁	C₂×#359	1	1	yes	characteristic
110	C₂×C₇₁	#2145	1	1	yes	characteristic
111	#3160	—	71	1	no	—
112	C₂×C₄⋊C₁₆	—	71	1	no	—
113	#1564	D₇₁	1	1	yes	characteristic
114	#1363	—	71	4	no	—
115	C₇₁	#11934	1	1	yes	characteristic, complemented, Sylow
116	#533	—	142	1	no	—
117	C₂²×C₁₆	—	142	1	no	—
118	#500	—	71	1	no	—
119	#500	—	71	1	no	—
120	#499	—	71	2	no	—
121	#499	—	142	1	no	—
122	#498	—	142	1	no	—
123	#431	—	2	1	no	—
124	#427	—	71	2	no	—
125	#427	—	142	1	no	—
126	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
127	C₄⋊C₁₆	—	71	2	no	—
128	C₄⋊C₁₆	—	142	1	no	—
129	#335	—	2	1	no	—
130	#334	C₂×D₇₁	1	2	yes	—
131	C₂²×C₈	C₄×D₇₁	1	1	yes	characteristic
132	C₂²×C₈	D₂₈₄	1	1	yes	characteristic
133	C₄⋊Q₈	—	142	1	no	—
134	C₄⋊Q₈	—	142	1	no	—
135	#125	—	142	2	no	—
136	C₂²⋊Q₈	—	284	1	no	—
137	C₂²⋊Q₈	—	284	1	no	—
138	C₄×Q₈	—	284	1	no	—
139	C₄×Q₈	—	284	1	no	—
140	#117	—	142	2	no	—
141	C₂×C₄²	C₂²⋊D₇₁	1	1	yes	characteristic
142	C₂×C₁₆	—	284	2	no	—
143	C₂×C₁₆	—	142	2	no	—
144	#107	—	71	2	no	—
145	#107	—	71	2	no	—
146	#107	—	142	1	no	—
147	#107	—	142	1	no	—
148	#106	—	142	2	no	—
149	#106	—	142	2	no	—
150	C₄⋊C₈	—	2	2	no	—
151	C₂²⋊C₈	—	4	1	no	—
152	C₄×C₈	—	4	1	no	—
153	C₄×C₈	C₄×D₇₁	1	1	yes	characteristic
154	C₄×C₈	D₂₈₄	1	1	yes	characteristic
155	C₄×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
156	C₄×C₈	C₂²×D₇₁	1	1	yes	characteristic
157	C₂×Q₈	—	284	1	no	—
158	C₂×Q₈	—	284	1	no	—
159	C₂²×C₄	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
160	C₂²×C₄	#11765850	1	1	yes	characteristic
161	C₂²×C₄	#11765852	1	1	yes	characteristic
162	C₂×C₈	—	2	2	no	—
163	C₂×C₈	—	4	2	no	—
164	C₂×C₈	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
165	C₂×C₈	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
166	C₂×C₈	#11765854	1	2	yes	—
167	C₂×C₈	Dic₂₈₄	1	2	yes	—
168	C₂×C₈	C₂×C₄×D₇₁	1	1	yes	characteristic
169	C₂×C₈	C₂×D₂₈₄	1	1	yes	characteristic
170	C₄⋊C₄	—	284	1	no	—
171	C₄⋊C₄	—	284	1	no	—
172	C₄⋊C₄	—	284	2	no	—
173	C₄⋊C₄	—	284	2	no	—
174	C₄⋊C₄	—	142	2	no	—
175	C₄⋊C₄	—	142	2	no	—
176	C₂²⋊C₄	—	284	2	no	—
177	C₄²	—	284	2	no	—
178	C₄²	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
179	C₄²	#11765850	1	1	yes	characteristic
180	C₄²	#11765852	1	1	yes	characteristic
181	C₄²	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
182	C₁₆	—	284	2	no	—
183	C₂³	#14556048	1	1	yes	characteristic
184	Q₈	—	568	2	no	—
185	Q₈	—	568	2	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	4	1	no	—
188	C₂×C₄	—	284	2	no	—
189	C₂×C₄	—	284	2	no	—
190	C₂×C₄	#14556046	1	2	yes	—
191	C₂×C₄	#14556048	1	1	yes	characteristic
192	C₂×C₄	#14556048	1	1	yes	characteristic
193	C₂×C₄	#14556048	1	1	yes	characteristic
194	C₂×C₄	#14556053	1	2	yes	—
195	C₂×C₄	C₂×(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
196	C₂×C₄	C₂×#11765850	1	1	yes	characteristic
197	C₂×C₄	C₂×#11765852	1	1	yes	characteristic
198	C₈	—	2	1	no	—
199	C₈	—	2	1	no	—
200	C₈	—	8	2	no	—
201	C₈	#14556053	1	2	yes	—
202	C₈	C₂×#11765854	1	1	yes	characteristic
203	C₈	C₂×Dic₂₈₄	1	1	yes	characteristic
204	C₂²	—	2	1	no	—
205	C₂²	—	2	1	no	—
206	C₂²	#17985746	1	2	yes	—
207	C₂²	C₂×#14556048	1	1	yes	characteristic
208	C₄	—	2	1	no	—
209	C₄	—	2	1	no	—
210	C₄	—	2	1	no	—
211	C₄	—	568	2	no	—
212	C₄	—	568	2	no	—
213	C₄	#17985746	1	2	yes	—
214	C₄	#17985805	1	2	yes	—
215	C₄	C₂×#14556046	1	1	yes	characteristic
216	C₄	C₂×#14556053	1	1	yes	characteristic
217	C₂	—	2	2	no	—
218	C₂	C₂×#17985746	1	1	yes	characteristic
219	C₂	C₂×#17985805	1	1	yes	characteristic
220	C₂	#22590075	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#14556048
commutator	C₂×C₈×C₇₁	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₇₁
Fitting	C₇₁×#1564	C₂
socle	C₂²×C₇₁	C₂×#102
prime radical	C₂³×C₇₁	#102
cosocle	C₄×C₈×C₇₁	C₂³
hypercenter	#1564	D₇₁
hypocenter	C₇₁	#11934
LCS	G ▷ C₂×C₈×C₇₁ ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1564

Complements

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

N	H ≅ G/N	type
3. #22586393	C₂	semidirect
115. C₇₁	#11934	semidirect

Maximal subgroups

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

subgroup	index
#22598854	2
C₂×C₄⋊(C₇₁⋊C₁₆)	2
#22586393	2
C₇₁×#1564	2
#11934	71