Subgroups — #32150942

Order 18176.

506 conjugacy classes of subgroups.

The subgroup lattice has 194 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22597274	C₂	1	1	yes	maximal, characteristic
2	#22590245	C₂	1	4	yes	maximal, complemented
3	#22590242	C₂	1	4	yes	maximal, complemented
4	#22590100	C₂	1	2	yes	maximal
5	#22589939	C₂	1	2	yes	maximal
6	C₂×C₄×C₇₁⋊C₁₆	C₂	1	1	yes	maximal, characteristic
7	C₇₁×#2150	C₂	1	1	yes	maximal, characteristic
8	#17986556	C₄	1	1	yes	characteristic
9	C₂²×C₇₁⋊C₁₆	C₂²	1	1	yes	characteristic
10	C₂²×C₇₁⋊C₁₆	C₂²	1	1	yes	characteristic
11	#17986129	C₄	1	4	yes	—
12	C₈×Dic₁₄₂	C₂²	1	4	yes	—
13	#17986025	C₄	1	2	yes	—
14	#17986023	C₄	1	1	yes	characteristic
15	#17985954	C₂²	1	2	yes	—
16	Dic₇₁⋊C₁₆	C₂²	1	8	yes	—
17	#17985774	C₂²	1	4	yes	—
18	C₄×C₇₁⋊C₁₆	C₂²	1	2	yes	—
19	C₄×C₇₁⋊C₁₆	C₂²	1	2	yes	—
20	C₁₆×Dic₇₁	C₂²	1	4	yes	—
21	C₂×M₃₂×C₇₁	C₂²	1	2	yes	—
22	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
23	C₄⋊C₁₆×C₇₁	C₂²	1	4	yes	—
24	C₂×#11765849	C₂×C₄	1	2	yes	—
25	#14556128	C₈	1	2	yes	—
26	#14556120	C₈	1	2	yes	—
27	#14556107	C₈	1	4	yes	—
28	C₄×Dic₁₄₂	C₂×C₄	1	4	yes	—
29	#14556087	C₂×C₄	1	2	yes	—
30	C₂×C₇₁⋊C₁₆	—	2	2	no	—
31	C₂×C₇₁⋊C₁₆	Q₈	1	4	yes	complemented
32	C₂×C₇₁⋊C₁₆	C₂³	1	2	yes	—
33	C₂×C₇₁⋊C₁₆	C₂³	1	1	yes	characteristic
34	C₂×C₇₁⋊C₁₆	C₂³	1	1	yes	characteristic
35	C₄⋊(C₇₁⋊C₈)	C₂×C₄	1	4	yes	—
36	Dic₇₁⋊C₈	C₂×C₄	1	8	yes	—
37	#14556038	C₂×C₄	1	4	yes	—
38	C₈×Dic₇₁	C₂³	1	4	yes	—
39	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
41	C₂×C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
42	M₃₂×C₇₁	—	2	4	no	—
43	C₂×C₁₆×C₇₁	C₂³	1	4	yes	—
44	C₄×C₈×C₇₁	C₂×C₄	1	2	yes	—
45	C₄×C₈×C₇₁	C₂³	1	2	yes	—
46	C₂×Dic₁₄₂	C₂×C₈	1	4	yes	—
47	#11765849	—	2	4	no	—
48	C₂×C₇₁⋊C₈	C₂²×C₄	1	4	yes	—
49	C₄⋊Dic₇₁	C₂×C₈	1	4	yes	—
50	C₄⋊Dic₇₁	C₂×C₈	1	4	yes	—
51	Dic₇₁⋊C₄	C₂×C₈	1	8	yes	—
52	C₂²⋊Dic₇₁	C₂×C₈	1	4	yes	—
53	C₄×Dic₇₁	C₂²×C₄	1	4	yes	—
54	C₇₁⋊C₁₆	—	2	2	no	—
55	C₇₁⋊C₁₆	C₂×Q₈	1	2	yes	—
56	C₇₁⋊C₁₆	C₂×Q₈	1	2	yes	—
57	C₂²×C₄×C₇₁	C₂×C₈	1	2	yes	—
58	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₈×C₇₁	—	2	2	no	—
60	C₂×C₈×C₇₁	C₂²×C₄	1	2	yes	—
61	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₈×C₇₁	C₂×Q₈	1	2	yes	—
63	C₂×C₈×C₇₁	#41	1	2	yes	—
64	C₂×C₈×C₇₁	C₂⁴	1	1	yes	characteristic
65	C₄²×C₇₁	C₂×C₈	1	1	yes	characteristic
66	C₄²×C₇₁	C₂×C₈	1	1	yes	characteristic
67	C₄²×C₇₁	C₂²×C₄	1	2	yes	—
68	C₁₆×C₇₁	—	2	4	no	—
69	C₂³×C₇₁	C₂²×C₈	1	1	yes	characteristic
70	C₂×C₄×C₇₁	—	2	4	no	—
71	C₂×C₄×C₇₁	C₄×Q₈	1	2	yes	—
72	C₂×C₄×C₇₁	C₂²×C₈	1	2	yes	—
73	C₂×C₄×C₇₁	C₂²×C₈	1	1	yes	characteristic
74	C₂×C₄×C₇₁	C₂²×C₈	1	2	yes	—
75	C₂×C₄×C₇₁	#131	1	2	yes	—
76	C₂×C₄×C₇₁	C₂³×C₄	1	1	yes	characteristic
77	C₂×Dic₇₁	C₂²×C₈	1	8	yes	—
78	Dic₁₄₂	—	2	8	no	—
79	C₈×C₇₁	—	2	2	no	—
80	C₈×C₇₁	C₄×Q₈	1	2	yes	—
81	C₈×C₇₁	C₂²×Q₈	1	1	yes	characteristic
82	C₈×C₇₁	C₂×#41	1	1	yes	characteristic
83	C₇₁⋊C₈	—	2	4	no	—
84	C₂²×C₇₁	—	2	2	no	—
85	C₂²×C₇₁	C₈×Q₈	1	2	yes	—
86	C₂²×C₇₁	C₂³×C₈	1	1	yes	characteristic
87	C₄×C₇₁	—	2	2	no	—
88	C₄×C₇₁	C₈×Q₈	1	2	yes	—
89	C₄×C₇₁	#504	1	4	yes	—
90	C₄×C₇₁	C₂×C₄×Q₈	1	1	yes	characteristic
91	C₄×C₇₁	C₂×#131	1	1	yes	characteristic
92	Dic₇₁	—	2	8	no	—
93	#33474	—	71	1	no	maximal, Sylow
94	C₂×C₇₁	—	2	2	no	—
95	C₂×C₇₁	C₂×C₈×Q₈	1	1	yes	characteristic
96	C₂×C₇₁	C₂×#504	1	2	yes	—
97	#2960	—	71	1	no	—
98	#2183	—	71	4	no	—
99	Q₈×C₁₆	—	71	4	no	—
100	#2151	—	71	2	no	—
101	#2150	D₇₁	1	1	yes	characteristic
102	#2108	—	71	2	no	—
103	C₂×C₄×C₁₆	—	71	1	no	—
104	C₇₁	#33474	1	1	yes	characteristic, complemented, Sylow
105	#533	—	71	1	no	—
106	C₂×M₃₂	C₂×D₇₁	1	2	yes	—
107	C₂²×C₁₆	—	71	1	no	—
108	C₂²×C₁₆	—	71	1	no	—
109	#446	—	71	4	no	—
110	C₈×Q₈	—	71	4	no	—
111	#424	—	71	2	no	—
112	#423	—	71	1	no	—
113	#405	—	71	2	no	—
114	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
115	C₄⋊C₁₆	—	71	8	no	—
116	C₄⋊C₁₆	C₂×D₇₁	1	4	yes	—
117	#346	—	71	4	no	—
118	C₄×C₁₆	—	71	4	no	—
119	C₄×C₁₆	—	71	2	no	—
120	C₄×C₁₆	—	71	2	no	—
121	C₂×M₁₆	—	71	2	no	—
122	C₂²×C₈	C₄×D₇₁	1	1	yes	characteristic
123	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
124	C₄⋊Q₈	—	71	2	no	—
125	#125	—	71	2	no	—
126	C₂²⋊Q₈	—	71	4	no	—
127	C₄×Q₈	—	71	4	no	—
128	#117	—	71	2	no	—
129	C₂×C₄²	C₄×D₇₁	1	1	yes	characteristic
130	M₃₂	—	2	4	no	—
131	C₂×C₁₆	—	71	4	no	—
132	C₂×C₁₆	—	71	1	no	—
133	C₂×C₁₆	—	71	1	no	—
134	C₂×C₁₆	—	71	2	no	—
135	C₂×C₁₆	—	142	2	no	—
136	C₂×C₁₆	C₂²×D₇₁	1	4	yes	—
137	C₄⋊C₈	—	71	8	no	—
138	C₄⋊C₈	—	71	4	no	—
139	#97	—	71	4	no	—
140	C₄×C₈	—	71	4	no	—
141	C₄×C₈	C₄×D₇₁	1	2	yes	—
142	C₄×C₈	C₂²×D₇₁	1	2	yes	—
143	C₂×Q₈	—	71	4	no	—
144	C₂²×C₄	C₈×D₇₁	1	2	yes	—
145	C₂²×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
146	M₁₆	—	142	4	no	—
147	C₂×C₈	—	2	2	no	—
148	C₂×C₈	—	71	4	no	—
149	C₂×C₈	C₂×C₄×D₇₁	1	2	yes	—
150	C₂×C₈	C₂×C₄×D₇₁	1	1	yes	characteristic
151	C₂×C₈	Q₈×D₇₁	1	2	yes	—
152	C₂×C₈	#11765864	1	2	yes	—
153	C₂×C₈	C₂³×D₇₁	1	1	yes	characteristic
154	C₄⋊C₄	—	71	8	no	—
155	C₄⋊C₄	—	71	4	no	—
156	C₄⋊C₄	—	71	4	no	—
157	C₂²⋊C₄	—	71	4	no	—
158	C₄²	—	71	4	no	—
159	C₄²	C₈×D₇₁	1	1	yes	characteristic
160	C₄²	C₈×D₇₁	1	1	yes	characteristic
161	C₄²	C₂×C₄×D₇₁	1	2	yes	—
162	C₁₆	—	2	4	no	—
163	C₁₆	—	71	2	no	—
164	C₁₆	—	71	2	no	—
165	C₁₆	—	142	2	no	—
166	C₂³	C₂×C₈×D₇₁	1	1	yes	characteristic
167	Q₈	—	142	8	no	—
168	C₂×C₄	—	2	4	no	—
169	C₂×C₄	—	71	8	no	—
170	C₂×C₄	#14556093	1	2	yes	—
171	C₂×C₄	C₂×C₈×D₇₁	1	2	yes	—
172	C₂×C₄	C₂×C₈×D₇₁	1	1	yes	characteristic
173	C₂×C₄	C₂×C₈×D₇₁	1	2	yes	—
174	C₂×C₄	#14556136	1	2	yes	—
175	C₂×C₄	C₂²×C₄×D₇₁	1	1	yes	characteristic
176	C₈	—	2	2	no	—
177	C₈	—	142	4	no	—
178	C₈	#14556093	1	2	yes	—
179	C₈	C₂×Q₈×D₇₁	1	1	yes	characteristic
180	C₈	C₂×#11765864	1	1	yes	characteristic
181	C₂²	—	2	2	no	—
182	C₂²	#17986123	1	2	yes	—
183	C₂²	C₂²×C₈×D₇₁	1	1	yes	characteristic
184	C₄	—	2	2	no	—
185	C₄	—	142	8	no	—
186	C₄	#17986123	1	2	yes	—
187	C₄	#17986417	1	4	yes	—
188	C₄	C₂×#14556093	1	1	yes	characteristic
189	C₄	C₂×#14556136	1	1	yes	characteristic
190	C₂	—	2	2	no	—
191	C₂	C₂×#17986123	1	1	yes	characteristic
192	C₂	C₂×#17986417	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₇₁
commutator, socle	C₂²×C₇₁	C₂³×C₈
Fitting	C₇₁×#2150	C₂
prime radical	C₂³×C₇₁	C₂²×C₈
cosocle	C₂×C₈×C₇₁	C₂⁴
hypercenter	#2150	D₇₁
hypocenter	C₇₁	#33474
LCS	G ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₈ ◁ #2150

Complements

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

N	H ≅ G/N	type
2. #22590245	C₂	semidirect
3. #22590242	C₂	semidirect
31. C₂×C₇₁⋊C₁₆	Q₈	semidirect
104. C₇₁	#33474	semidirect

Maximal subgroups

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

subgroup	index
#22597274	2
#22590245	2
#22590242	2
#22590100	2
#22589939	2
C₂×C₄×C₇₁⋊C₁₆	2
C₇₁×#2150	2
#33474	71