Subgroups — #46845440

Order 34688.

402 conjugacy classes of subgroups.

The subgroup lattice has 186 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#31773975	C₂	1	2	yes	maximal, complemented
2	#31773969	C₂	1	4	yes	maximal, complemented
3	D₄×C₂₇₁⋊C₈	C₂	1	2	yes	maximal, complemented
4	#31773954	C₂	1	1	yes	maximal, characteristic, complemented
5	#31773950	C₂	1	1	yes	maximal, characteristic, complemented
6	#31773864	C₂	1	4	yes	maximal, complemented
7	#532×C₂₇₁	C₂	1	1	yes	maximal, characteristic
8	C₂×#16864638	C₂²	1	4	yes	—
9	C₂²×C₂₇₁⋊C₈	C₂²	1	4	yes	—
10	C₄⋊(C₂₇₁⋊C₈)	C₂²	1	2	yes	—
11	C₄⋊(C₂₇₁⋊C₈)	C₂²	1	2	yes	—
12	C₂²⋊(C₂₇₁⋊C₈)	C₂²	1	2	yes	—
13	C₂²⋊(C₂₇₁⋊C₈)	C₂²	1	4	yes	—
14	C₂²⋊(C₂₇₁⋊C₈)	C₂²	1	4	yes	—
15	C₂²⋊(C₂₇₁⋊C₈)	C₂²	1	2	yes	—
16	#22576071	C₂²	1	2	yes	—
17	C₄×C₂₇₁⋊C₈	C₂²	1	1	yes	characteristic
18	C₄×C₂₇₁⋊C₈	C₂²	1	1	yes	characteristic
19	C₂×#41×C₂₇₁	C₂²	1	2	yes	—
20	C₄⋊Q₈×C₂₇₁	C₄	1	1	yes	characteristic
21	#127×C₂₇₁	C₄	1	1	yes	characteristic
22	#124×C₂₇₁	C₄	1	2	yes	—
23	#121×C₂₇₁	C₄	1	2	yes	—
24	#121×C₂₇₁	C₄	1	2	yes	—
25	C₄×D₄×C₂₇₁	C₂²	1	2	yes	—
26	C₄×D₄×C₂₇₁	C₂²	1	2	yes	—
27	C₂×C₄²×C₂₇₁	C₂²	1	1	yes	characteristic
28	#16864638	—	4	4	no	—
29	C₂×C₂₇₁⋊C₈	—	4	4	no	—
30	C₂×C₂₇₁⋊C₈	C₂³	1	2	yes	—
31	C₂×C₂₇₁⋊C₈	C₂³	1	4	yes	—
32	C₂×C₂₇₁⋊C₈	C₂³	1	2	yes	—
33	#41×C₂₇₁	—	4	4	no	—
34	C₂×Q₈×C₂₇₁	C₂×C₄	1	2	yes	—
35	C₂×D₄×C₂₇₁	C₂×C₄	1	2	yes	—
36	C₂×D₄×C₂₇₁	C₂×C₄	1	2	yes	—
37	C₂×D₄×C₂₇₁	C₂×C₄	1	4	yes	—
38	C₂×D₄×C₂₇₁	C₂×C₄	1	2	yes	—
39	C₂²×C₄×C₂₇₁	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₄×C₂₇₁	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₄×C₂₇₁	C₂³	1	4	yes	—
42	C₂²×C₄×C₂₇₁	C₂³	1	1	yes	characteristic
43	C₄⋊C₄×C₂₇₁	C₂×C₄	1	2	yes	—
44	C₄⋊C₄×C₂₇₁	C₂×C₄	1	2	yes	—
45	C₂²⋊C₄×C₂₇₁	C₂×C₄	1	4	yes	—
46	C₂²⋊C₄×C₂₇₁	C₂×C₄	1	4	yes	—
47	C₄²×C₂₇₁	C₂×C₄	1	1	yes	characteristic
48	C₄²×C₂₇₁	C₂×C₄	1	1	yes	characteristic
49	C₄²×C₂₇₁	C₂³	1	1	yes	characteristic
50	C₄²×C₂₇₁	C₂³	1	1	yes	characteristic
51	C₂₇₁⋊C₈	—	2	2	no	—
52	C₂₇₁⋊C₈	—	2	4	no	—
53	C₂₇₁⋊C₈	—	2	2	no	—
54	C₂³×C₂₇₁	C₂²×C₄	1	4	yes	—
55	C₂³×C₂₇₁	C₂²×C₄	1	1	yes	characteristic
56	Q₈×C₂₇₁	—	4	2	no	—
57	D₄×C₂₇₁	—	2	4	no	—
58	D₄×C₂₇₁	—	4	2	no	—
59	D₄×C₂₇₁	—	4	2	no	—
60	D₄×C₂₇₁	—	4	4	no	—
61	C₂×C₄×C₂₇₁	—	2	2	no	—
62	C₂×C₄×C₂₇₁	—	2	2	no	—
63	C₂×C₄×C₂₇₁	—	2	4	no	—
64	C₂×C₄×C₂₇₁	—	2	4	no	—
65	C₂×C₄×C₂₇₁	—	4	1	no	—
66	C₂×C₄×C₂₇₁	C₂²×C₄	1	1	yes	characteristic
67	C₂×C₄×C₂₇₁	C₂²×C₄	1	1	yes	characteristic
68	C₂×C₄×C₂₇₁	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₄×C₂₇₁	C₂²×C₄	1	1	yes	characteristic
70	C₂×C₄×C₂₇₁	C₂²×C₄	1	4	yes	—
71	C₂×C₄×C₂₇₁	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₄×C₂₇₁	C₂⁴	1	1	yes	characteristic
73	C₂²×C₂₇₁	—	2	1	no	—
74	C₂²×C₂₇₁	—	2	4	no	—
75	C₂²×C₂₇₁	—	2	1	no	—
76	C₂²×C₂₇₁	—	2	4	no	—
77	C₂²×C₂₇₁	—	2	1	no	—
78	C₂²×C₂₇₁	—	2	4	no	—
79	C₂²×C₂₇₁	C₂³×C₄	1	1	yes	characteristic
80	C₄×C₂₇₁	—	2	1	no	—
81	C₄×C₂₇₁	—	2	4	no	—
82	C₄×C₂₇₁	—	2	1	no	—
83	C₄×C₂₇₁	—	2	1	no	—
84	C₄×C₂₇₁	—	2	1	no	—
85	C₄×C₂₇₁	#131	1	2	yes	—
86	C₄×C₂₇₁	H(5,𝔽₂)	1	2	yes	—
87	C₂×C₂₇₁	—	4	1	no	—
88	C₂×C₂₇₁	—	4	4	no	—
89	C₂×C₂₇₁	#518	1	1	yes	characteristic
90	C₂×C₂₇₁	C₂×#131	1	1	yes	characteristic
91	C₂×C₂₇₁	#568	1	1	yes	characteristic
92	C₂₇₁	#2981	1	1	yes	characteristic, complemented, Sylow
93	#2981	—	271	1	no	maximal, Sylow
94	#532	D₂₇₁	1	1	yes	characteristic
95	#436	—	271	2	no	—
96	#435	—	271	4	no	—
97	C₈×D₄	—	271	2	no	—
98	#432	—	271	1	no	—
99	#431	—	271	1	no	—
100	#408	—	271	4	no	—
101	C₂×#41	C₂×D₂₇₁	1	2	yes	—
102	C₂×M₁₆	—	271	4	no	—
103	C₂²×C₈	—	271	4	no	—
104	C₄⋊Q₈	Dic₂₇₁	1	1	yes	characteristic
105	#127	Dic₂₇₁	1	1	yes	characteristic
106	#124	Dic₂₇₁	1	2	yes	—
107	#121	Dic₂₇₁	1	2	yes	—
108	#121	Dic₂₇₁	1	2	yes	—
109	C₄×D₄	C₂×D₂₇₁	1	2	yes	—
110	C₄×D₄	C₂×D₂₇₁	1	2	yes	—
111	C₂×C₄²	C₂×D₂₇₁	1	1	yes	characteristic
112	C₄⋊C₈	—	271	2	no	—
113	C₄⋊C₈	—	271	2	no	—
114	C₂²⋊C₈	—	271	2	no	—
115	C₂²⋊C₈	—	271	2	no	—
116	C₂²⋊C₈	—	271	4	no	—
117	C₂²⋊C₈	—	271	4	no	—
118	#97	—	271	2	no	—
119	C₄×C₈	—	271	1	no	—
120	C₄×C₈	—	271	1	no	—
121	#41	—	4	4	no	—
122	C₂×Q₈	C₂×Dic₂₇₁	1	2	yes	—
123	C₂×D₄	C₂×Dic₂₇₁	1	2	yes	—
124	C₂×D₄	C₂×Dic₂₇₁	1	2	yes	—
125	C₂×D₄	C₂×Dic₂₇₁	1	4	yes	—
126	C₂×D₄	C₂×Dic₂₇₁	1	2	yes	—
127	C₂²×C₄	C₂×Dic₂₇₁	1	1	yes	characteristic
128	C₂²×C₄	C₂×Dic₂₇₁	1	1	yes	characteristic
129	C₂²×C₄	C₂²×D₂₇₁	1	1	yes	characteristic
130	C₂²×C₄	C₂²×D₂₇₁	1	4	yes	—
131	M₁₆	—	1084	4	no	—
132	C₂×C₈	—	271	2	no	—
133	C₂×C₈	—	271	2	no	—
134	C₂×C₈	—	271	4	no	—
135	C₂×C₈	—	1084	4	no	—
136	C₄⋊C₄	C₂×Dic₂₇₁	1	2	yes	—
137	C₄⋊C₄	C₂×Dic₂₇₁	1	2	yes	—
138	C₂²⋊C₄	C₂×Dic₂₇₁	1	4	yes	—
139	C₂²⋊C₄	C₂×Dic₂₇₁	1	4	yes	—
140	C₄²	C₂×Dic₂₇₁	1	1	yes	characteristic
141	C₄²	C₂×Dic₂₇₁	1	1	yes	characteristic
142	C₄²	C₂²×D₂₇₁	1	1	yes	characteristic
143	C₄²	C₂²×D₂₇₁	1	1	yes	characteristic
144	C₂³	C₂²×Dic₂₇₁	1	4	yes	—
145	C₂³	C₂²×Dic₂₇₁	1	1	yes	characteristic
146	Q₈	—	4	2	no	—
147	D₄	—	2	4	no	—
148	D₄	—	4	2	no	—
149	D₄	—	4	4	no	—
150	D₄	—	4	2	no	—
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	2	4	no	—
153	C₂×C₄	—	2	4	no	—
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	—	4	1	no	—
156	C₂×C₄	C₂²×Dic₂₇₁	1	1	yes	characteristic
157	C₂×C₄	C₂²×Dic₂₇₁	1	1	yes	characteristic
158	C₂×C₄	C₂²×Dic₂₇₁	1	1	yes	characteristic
159	C₂×C₄	C₂²×Dic₂₇₁	1	1	yes	characteristic
160	C₂×C₄	C₂²×Dic₂₇₁	1	4	yes	—
161	C₂×C₄	C₂²×Dic₂₇₁	1	1	yes	characteristic
162	C₂×C₄	C₂³×D₂₇₁	1	1	yes	characteristic
163	C₈	—	542	2	no	—
164	C₈	—	542	4	no	—
165	C₈	—	542	2	no	—
166	C₂²	—	2	1	no	—
167	C₂²	—	2	4	no	—
168	C₂²	—	2	1	no	—
169	C₂²	—	2	4	no	—
170	C₂²	—	2	1	no	—
171	C₂²	—	2	4	no	—
172	C₂²	C₂³×Dic₂₇₁	1	1	yes	characteristic
173	C₄	—	2	1	no	—
174	C₄	—	2	4	no	—
175	C₄	—	2	1	no	—
176	C₄	—	2	1	no	—
177	C₄	—	2	1	no	—
178	C₄	#22576167	1	2	yes	—
179	C₄	#22576205	1	2	yes	—
180	C₂	—	4	1	no	—
181	C₂	—	4	4	no	—
182	C₂	#31774372	1	1	yes	characteristic
183	C₂	C₂×#22576167	1	1	yes	characteristic
184	C₂	#31774664	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₂₇₁
commutator, socle	C₂²×C₂₇₁	C₂³×C₄
Fitting, prime radical	#532×C₂₇₁	C₂
cosocle	C₂×C₄×C₂₇₁	C₂⁴
hypercenter	#532	D₂₇₁
hypocenter	C₂₇₁	#2981
LCS	G ▷ C₂²×C₂₇₁ ▷ C₂₇₁
UCS	C₁ ◁ C₂×C₄ ◁ #532

Complements

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

N	H ≅ G/N	type
1. #31773975	C₂	semidirect
2. #31773969	C₂	semidirect
3. D₄×C₂₇₁⋊C₈	C₂	semidirect
4. #31773954	C₂	semidirect
5. #31773950	C₂	semidirect
6. #31773864	C₂	semidirect
92. C₂₇₁	#2981	semidirect

Maximal subgroups

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

subgroup	index
#31773975	2
#31773969	2
D₄×C₂₇₁⋊C₈	2
#31773954	2
#31773950	2
#31773864	2
#532×C₂₇₁	2
#2981	271