Subgroups — #2932×C₂₇₁ #46833518

Order 34688.

484 conjugacy classes of subgroups.

The subgroup lattice has 160 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#131×C₂₇₁	C₂	1	2	yes	maximal, complemented
2	#532×C₂₇₁	C₂	1	1	yes	maximal, characteristic
3	#435×C₂₇₁	C₂	1	8	yes	maximal, complemented
4	#423×C₂₇₁	C₂	1	1	yes	maximal, characteristic, complemented
5	#408×C₂₇₁	C₂	1	2	yes	maximal, complemented
6	C₂×#97×C₂₇₁	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#41×C₂₇₁	C₂²	1	2	yes	—
8	#131×C₂₇₁	—	2	8	no	—
9	C₂×M₁₆×C₂₇₁	C₂²	1	2	yes	—
10	C₂×M₁₆×C₂₇₁	C₂²	1	4	yes	—
11	C₂²×C₈×C₂₇₁	C₂²	1	4	yes	—
12	C₂²×C₈×C₂₇₁	C₂²	1	1	yes	characteristic
13	C₂²×C₈×C₂₇₁	C₂²	1	1	yes	characteristic
14	C₄⋊Q₈×C₂₇₁	C₄	1	1	yes	characteristic
15	#127×C₂₇₁	C₄	1	1	yes	characteristic
16	#124×C₂₇₁	C₄	1	2	yes	—
17	#121×C₂₇₁	C₄	1	4	yes	—
18	C₄×D₄×C₂₇₁	C₂²	1	4	yes	—
19	C₂×C₄²×C₂₇₁	C₂²	1	1	yes	characteristic
20	C₄⋊C₈×C₂₇₁	C₂²	1	4	yes	—
21	C₂²⋊C₈×C₂₇₁	C₂²	1	8	yes	—
22	#97×C₂₇₁	C₂²	1	4	yes	—
23	#41×C₂₇₁	—	2	4	no	—
24	#41×C₂₇₁	—	2	4	no	—
25	C₂×Q₈×C₂₇₁	C₂×C₄	1	2	yes	—
26	C₂×D₄×C₂₇₁	C₂×C₄	1	4	yes	—
27	C₂×D₄×C₂₇₁	C₂×C₄	1	2	yes	—
28	C₂×D₄×C₂₇₁	C₂×C₄	1	4	yes	—
29	C₂²×C₄×C₂₇₁	C₂×C₄	1	2	yes	—
30	C₂²×C₄×C₂₇₁	C₂³	1	4	yes	—
31	C₂²×C₄×C₂₇₁	C₂³	1	1	yes	characteristic
32	M₁₆×C₂₇₁	—	2	4	no	—
33	M₁₆×C₂₇₁	—	2	8	no	—
34	C₂×C₈×C₂₇₁	—	2	2	no	—
35	C₂×C₈×C₂₇₁	—	2	8	no	—
36	C₂×C₈×C₂₇₁	D₄	1	4	yes	complemented
37	C₂×C₈×C₂₇₁	C₂³	1	2	yes	—
38	C₂×C₈×C₂₇₁	C₂³	1	4	yes	—
39	C₂×C₈×C₂₇₁	C₂³	1	2	yes	—
40	C₄⋊C₄×C₂₇₁	C₂×C₄	1	4	yes	—
41	C₂²⋊C₄×C₂₇₁	C₂×C₄	1	8	yes	—
42	C₄²×C₂₇₁	C₂×C₄	1	1	yes	characteristic
43	C₄²×C₂₇₁	C₂×C₄	1	1	yes	characteristic
44	C₄²×C₂₇₁	C₂³	1	2	yes	—
45	C₂³×C₂₇₁	C₂²×C₄	1	4	yes	—
46	C₂³×C₂₇₁	C₂²×C₄	1	1	yes	characteristic
47	Q₈×C₂₇₁	—	2	4	no	—
48	D₄×C₂₇₁	—	2	8	no	—
49	D₄×C₂₇₁	—	2	4	no	—
50	D₄×C₂₇₁	—	4	4	no	—
51	C₂×C₄×C₂₇₁	—	2	4	no	—
52	C₂×C₄×C₂₇₁	—	2	4	no	—
53	C₂×C₄×C₂₇₁	—	4	2	no	—
54	C₂×C₄×C₂₇₁	C₂²×C₄	1	2	yes	—
55	C₂×C₄×C₂₇₁	C₂²×C₄	1	2	yes	—
56	C₂×C₄×C₂₇₁	C₂²×C₄	1	4	yes	—
57	C₂×C₄×C₂₇₁	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₄×C₂₇₁	C₂×D₄	1	2	yes	—
59	C₂×C₄×C₂₇₁	#41	1	2	yes	—
60	C₂×C₄×C₂₇₁	C₂⁴	1	1	yes	characteristic
61	C₈×C₂₇₁	—	2	4	no	—
62	C₈×C₂₇₁	—	2	2	no	—
63	C₈×C₂₇₁	C₂×D₄	1	4	yes	—
64	C₂²×C₂₇₁	—	2	2	no	—
65	C₂²×C₂₇₁	—	2	4	no	—
66	C₂²×C₂₇₁	—	2	8	no	—
67	C₂²×C₂₇₁	C₄×D₄	1	2	yes	—
68	C₂²×C₂₇₁	C₂³×C₄	1	1	yes	characteristic
69	C₄×C₂₇₁	—	2	4	no	—
70	C₄×C₂₇₁	—	2	2	no	—
71	C₄×C₂₇₁	—	2	2	no	—
72	C₄×C₂₇₁	C₄×D₄	1	2	yes	—
73	C₄×C₂₇₁	C₂²×D₄	1	1	yes	characteristic
74	C₄×C₂₇₁	C₂×#41	1	1	yes	characteristic
75	C₂×C₂₇₁	—	2	2	no	—
76	C₂×C₂₇₁	—	4	4	no	—
77	C₂×C₂₇₁	C₂×C₄×D₄	1	1	yes	characteristic
78	C₂×C₂₇₁	#568	1	2	yes	—
79	C₂₇₁	#2932	1	1	yes	characteristic, direct product factor, Sylow
80	#2932	C₂₇₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
81	C₂×#131	C₂×C₂₇₁	1	2	yes	complemented
82	#532	C₂×C₂₇₁	1	1	yes	characteristic
83	#435	C₂×C₂₇₁	1	8	yes	complemented
84	#423	C₂×C₂₇₁	1	1	yes	characteristic, complemented
85	#408	C₂×C₂₇₁	1	2	yes	complemented
86	C₂×#97	C₂×C₂₇₁	1	1	yes	characteristic, complemented
87	C₂×#41	C₂²×C₂₇₁	1	2	yes	—
88	#131	—	2	8	no	—
89	C₂×M₁₆	C₂²×C₂₇₁	1	2	yes	—
90	C₂×M₁₆	C₂²×C₂₇₁	1	4	yes	—
91	C₂²×C₈	C₂²×C₂₇₁	1	4	yes	—
92	C₂²×C₈	C₂²×C₂₇₁	1	1	yes	characteristic
93	C₂²×C₈	C₂²×C₂₇₁	1	1	yes	characteristic
94	C₄⋊Q₈	C₄×C₂₇₁	1	1	yes	characteristic
95	#127	C₄×C₂₇₁	1	1	yes	characteristic
96	#124	C₄×C₂₇₁	1	2	yes	—
97	#121	C₄×C₂₇₁	1	4	yes	—
98	C₄×D₄	C₂²×C₂₇₁	1	4	yes	—
99	C₂×C₄²	C₂²×C₂₇₁	1	1	yes	characteristic
100	C₄⋊C₈	C₂²×C₂₇₁	1	4	yes	—
101	C₂²⋊C₈	C₂²×C₂₇₁	1	8	yes	—
102	#97	C₂²×C₂₇₁	1	4	yes	—
103	#41	—	2	4	no	—
104	#41	—	2	4	no	—
105	C₂×Q₈	C₂×C₄×C₂₇₁	1	2	yes	—
106	C₂×D₄	C₂×C₄×C₂₇₁	1	4	yes	—
107	C₂×D₄	C₂×C₄×C₂₇₁	1	2	yes	—
108	C₂×D₄	C₂×C₄×C₂₇₁	1	4	yes	—
109	C₂²×C₄	C₂×C₄×C₂₇₁	1	2	yes	—
110	C₂²×C₄	C₂³×C₂₇₁	1	4	yes	—
111	C₂²×C₄	C₂³×C₂₇₁	1	1	yes	characteristic
112	M₁₆	—	2	4	no	—
113	M₁₆	—	2	8	no	—
114	C₂×C₈	—	2	2	no	—
115	C₂×C₈	—	2	8	no	—
116	C₂×C₈	D₄×C₂₇₁	1	4	yes	complemented
117	C₂×C₈	C₂³×C₂₇₁	1	2	yes	—
118	C₂×C₈	C₂³×C₂₇₁	1	4	yes	—
119	C₂×C₈	C₂³×C₂₇₁	1	2	yes	—
120	C₄⋊C₄	C₂×C₄×C₂₇₁	1	4	yes	—
121	C₂²⋊C₄	C₂×C₄×C₂₇₁	1	8	yes	—
122	C₄²	C₂×C₄×C₂₇₁	1	1	yes	characteristic
123	C₄²	C₂×C₄×C₂₇₁	1	1	yes	characteristic
124	C₄²	C₂³×C₂₇₁	1	2	yes	—
125	C₂³	C₂²×C₄×C₂₇₁	1	4	yes	—
126	C₂³	C₂²×C₄×C₂₇₁	1	1	yes	characteristic
127	Q₈	—	2	4	no	—
128	D₄	—	2	8	no	—
129	D₄	—	2	4	no	—
130	D₄	—	4	4	no	—
131	C₂×C₄	—	2	4	no	—
132	C₂×C₄	—	2	4	no	—
133	C₂×C₄	—	4	2	no	—
134	C₂×C₄	C₂²×C₄×C₂₇₁	1	2	yes	—
135	C₂×C₄	C₂²×C₄×C₂₇₁	1	2	yes	—
136	C₂×C₄	C₂²×C₄×C₂₇₁	1	4	yes	—
137	C₂×C₄	C₂²×C₄×C₂₇₁	1	1	yes	characteristic
138	C₂×C₄	C₂×D₄×C₂₇₁	1	2	yes	—
139	C₂×C₄	#41×C₂₇₁	1	2	yes	—
140	C₂×C₄	C₂⁴×C₂₇₁	1	1	yes	characteristic
141	C₈	—	2	4	no	—
142	C₈	—	2	2	no	—
143	C₈	C₂×D₄×C₂₇₁	1	4	yes	—
144	C₂²	—	2	2	no	—
145	C₂²	—	2	4	no	—
146	C₂²	—	2	8	no	—
147	C₂²	C₄×D₄×C₂₇₁	1	2	yes	—
148	C₂²	C₂³×C₄×C₂₇₁	1	1	yes	characteristic
149	C₄	—	2	4	no	—
150	C₄	—	2	2	no	—
151	C₄	—	2	2	no	—
152	C₄	C₄×D₄×C₂₇₁	1	2	yes	—
153	C₄	C₂²×D₄×C₂₇₁	1	1	yes	characteristic
154	C₄	C₂×#41×C₂₇₁	1	1	yes	characteristic
155	C₂	—	2	2	no	—
156	C₂	—	4	4	no	—
157	C₂	C₂×C₄×D₄×C₂₇₁	1	1	yes	characteristic
158	C₂	#568×C₂₇₁	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
1. C₂×#131×C₂₇₁	C₂	semidirect
3. #435×C₂₇₁	C₂	semidirect
4. #423×C₂₇₁	C₂	semidirect
5. #408×C₂₇₁	C₂	semidirect
6. C₂×#97×C₂₇₁	C₂	semidirect
36. C₂×C₈×C₂₇₁	D₄	semidirect
79. C₂₇₁	#2932	direct (N × H)
80. #2932	C₂₇₁	direct (N × H)
81. C₂×#131	C₂×C₂₇₁	semidirect
83. #435	C₂×C₂₇₁	semidirect
84. #423	C₂×C₂₇₁	semidirect
85. #408	C₂×C₂₇₁	semidirect
86. C₂×#97	C₂×C₂₇₁	semidirect
116. C₂×C₈	D₄×C₂₇₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×#131×C₂₇₁	2
#532×C₂₇₁	2
#435×C₂₇₁	2
#423×C₂₇₁	2
#408×C₂₇₁	2
C₂×#97×C₂₇₁	2
#2932	271

Subgroups — #2932×C271 #46833518

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #2932×C₂₇₁ #46833518