Finite Group Databasebeta

Subgroups — #46835652

Order 34688.

262 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 45 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#31774639	C₂	1	1	yes	maximal, characteristic, complemented
2	#31774284	C₂	1	2	yes	maximal, complemented
3	#31773637	C₂	1	1	yes	maximal, characteristic, complemented
4	#31773636	C₂	1	2	yes	maximal, complemented
5	#496×C₂₇₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#16864642	—	2	2	no	—
7	C₂×#16864639	—	2	2	no	—
8	#22576149	—	2	1	no	—
9	#22576148	C₂²	1	2	yes	—
10	#22576146	—	2	2	no	—
11	#22576130	—	2	2	no	—
12	#22576128	—	2	1	no	—
13	#22576071	C₂²	1	2	yes	complemented
14	C₂×SD₁₆×C₂₇₁	—	2	1	no	—
15	C₂×D₈×C₂₇₁	—	2	1	no	—
16	#127×C₂₇₁	C₂²	1	1	yes	characteristic, complemented
17	#124×C₂₇₁	C₂²	1	1	yes	characteristic
18	#97×C₂₇₁	C₂²	1	1	yes	characteristic, complemented
19	C₂³×D₂₇₁	—	2	1	no	—
20	C₂×Dic₅₄₂	D₄	1	2	yes	—
21	C₂×D₁₀₈₄	—	2	1	no	—
22	C₂×D₁₀₈₄	D₄	1	2	yes	—
23	C₂×C₂²⋊D₂₇₁	—	2	2	no	—
24	#16864642	—	4	4	no	—
25	#16864639	—	4	4	no	—
26	C₂×C₂₇₁⋊C₈	—	2	2	no	—
27	C₂²⋊Dic₂₇₁	—	2	1	no	—
28	(C₂×D₂₇₁)⋊C₄	—	2	2	no	—
29	(C₂×D₂₇₁)⋊C₄	—	2	2	no	—
30	(C₂×D₂₇₁)⋊C₄	—	2	2	no	—
31	C₄×Dic₂₇₁	—	2	1	no	—
32	C₂×Q₈×C₂₇₁	D₄	1	1	yes	characteristic
33	C₂×D₄×C₂₇₁	—	2	2	no	—
34	C₂×D₄×C₂₇₁	—	2	1	no	—
35	C₂×D₄×C₂₇₁	D₄	1	1	yes	characteristic
36	SD₁₆×C₂₇₁	—	4	2	no	—
37	D₈×C₂₇₁	—	4	2	no	—
38	C₂×C₈×C₂₇₁	—	2	1	no	—
39	C₂²⋊C₄×C₂₇₁	—	2	2	no	—
40	C₄²×C₂₇₁	C₂³	1	1	yes	characteristic
41	C₂²×D₂₇₁	—	2	2	no	—
42	C₂²×D₂₇₁	—	8	2	no	—
43	C₂²×D₂₇₁	—	8	1	no	—
44	Dic₅₄₂	—	4	2	no	—
45	D₁₀₈₄	—	4	2	no	—
46	D₁₀₈₄	—	8	1	no	—
47	C₂²⋊D₂₇₁	—	8	2	no	—
48	C₂×Dic₂₇₁	—	2	2	no	—
49	C₂₇₁⋊C₈	—	2	4	no	—
50	C₂³×C₂₇₁	—	2	1	no	—
51	C₂³×C₂₇₁	—	4	1	no	—
52	Q₈×C₂₇₁	—	4	1	no	—
53	D₄×C₂₇₁	—	4	1	no	—
54	D₄×C₂₇₁	—	4	1	no	—
55	D₄×C₂₇₁	—	4	1	no	—
56	D₄×C₂₇₁	—	4	2	no	—
57	D₄×C₂₇₁	—	4	2	no	—
58	C₂×C₄×C₂₇₁	—	2	1	no	—
59	C₂×C₄×C₂₇₁	C₂×D₄	1	2	yes	—
60	C₂×C₄×C₂₇₁	C₂×D₄	1	1	yes	characteristic
61	C₈×C₂₇₁	—	2	2	no	—
62	C₂²×C₂₇₁	—	4	1	no	—
63	C₂²×C₂₇₁	—	4	2	no	—
64	C₂²×C₂₇₁	—	8	2	no	—
65	C₂²×C₂₇₁	—	8	1	no	—
66	C₂²×C₂₇₁	#120	1	1	yes	characteristic
67	C₂×D₂₇₁	—	4	2	no	—
68	C₂×D₂₇₁	—	4	2	no	—
69	C₂×D₂₇₁	—	4	2	no	—
70	C₂×D₂₇₁	—	8	4	no	—
71	C₄×C₂₇₁	—	2	1	no	—
72	C₄×C₂₇₁	—	2	2	no	—
73	C₄×C₂₇₁	—	4	1	no	—
74	Dic₂₇₁	—	4	2	no	—
75	C₂×C₂₇₁	—	8	1	no	—
76	C₂×C₂₇₁	—	16	1	no	—
77	C₂×C₂₇₁	#453	1	2	yes	—
78	C₂×C₂₇₁	#453	1	1	yes	characteristic
79	D₂₇₁	—	8	2	no	—
80	C₂₇₁	#1655	1	1	yes	characteristic, complemented, Sylow
81	#1655	—	271	1	no	maximal, Sylow
82	H(3,𝔽₄)	—	271	1	no	—
83	#496	—	271	2	no	—
84	#496	D₂₇₁	1	1	yes	characteristic, complemented
85	#329	—	271	1	no	—
86	#329	—	271	2	no	—
87	C₂×SD₁₆	—	2	1	no	—
88	C₂×SD₁₆	—	542	2	no	—
89	C₂×D₈	—	2	1	no	—
90	C₂×D₈	—	542	2	no	—
91	#127	C₂×D₂₇₁	1	1	yes	characteristic, complemented
92	#124	—	271	2	no	—
93	#124	—	542	2	no	—
94	#124	—	542	1	no	—
95	#124	C₂×D₂₇₁	1	1	yes	characteristic
96	#120	—	542	2	no	—
97	#120	—	542	1	no	—
98	#97	—	271	2	no	—
99	#97	C₂×D₂₇₁	1	1	yes	characteristic, complemented
100	C₂⁴	—	542	1	no	—
101	C₂×Q₈	—	271	2	no	—
102	C₂×Q₈	C₂²⋊D₂₇₁	1	1	yes	characteristic
103	C₂×D₄	—	2	2	no	—
104	C₂×D₄	—	2	1	no	—
105	C₂×D₄	—	271	2	no	—
106	C₂×D₄	—	542	1	no	—
107	C₂×D₄	—	542	2	no	—
108	C₂×D₄	C₂²⋊D₂₇₁	1	1	yes	characteristic
109	SD₁₆	—	4	2	no	—
110	SD₁₆	—	1084	4	no	—
111	D₈	—	4	2	no	—
112	D₈	—	1084	4	no	—
113	C₂×C₈	—	2	1	no	—
114	C₂×C₈	—	542	2	no	—
115	C₂²⋊C₄	—	2	2	no	—
116	C₂²⋊C₄	—	542	2	no	—
117	C₂²⋊C₄	—	542	1	no	—
118	C₂²⋊C₄	—	542	2	no	—
119	C₂²⋊C₄	—	542	2	no	—
120	C₄²	—	542	1	no	—
121	C₄²	C₂²×D₂₇₁	1	1	yes	characteristic
122	C₂³	—	2	1	no	—
123	C₂³	—	4	1	no	—
124	C₂³	—	542	2	no	—
125	C₂³	—	2168	2	no	—
126	C₂³	—	2168	1	no	—
127	Q₈	—	4	1	no	—
128	Q₈	—	1084	2	no	—
129	D₄	—	4	1	no	—
130	D₄	—	4	1	no	—
131	D₄	—	4	2	no	—
132	D₄	—	4	1	no	—
133	D₄	—	4	2	no	—
134	D₄	—	1084	2	no	—
135	D₄	—	2168	1	no	—
136	D₄	—	2168	2	no	—
137	C₂×C₄	—	2	1	no	—
138	C₂×C₄	—	542	2	no	—
139	C₂×C₄	D₄×D₂₇₁	1	2	yes	—
140	C₂×C₄	C₂×C₂²⋊D₂₇₁	1	1	yes	characteristic
141	C₈	—	2	2	no	—
142	C₈	—	542	4	no	—
143	C₂²	—	4	2	no	—
144	C₂²	—	4	1	no	—
145	C₂²	—	8	2	no	—
146	C₂²	—	8	1	no	—
147	C₂²	—	1084	2	no	—
148	C₂²	—	1084	2	no	—
149	C₂²	—	1084	2	no	—
150	C₂²	—	2168	4	no	—
151	C₂²	#22576128	1	1	yes	characteristic
152	C₄	—	2	2	no	—
153	C₄	—	2	1	no	—
154	C₄	—	4	1	no	—
155	C₄	—	1084	2	no	—
156	C₂	—	8	1	no	—
157	C₂	—	16	1	no	—
158	C₂	—	2168	2	no	—
159	C₂	#31774063	1	2	yes	—
160	C₂	#31774064	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#22576128
commutator, cosocle	C₄²×C₂₇₁	C₂³
Frattini	C₄²	C₂²×D₂₇₁
Fitting	#496×C₂₇₁	C₂
socle	C₂²×C₂₇₁	#120
hypercenter	#496	D₂₇₁
hypocenter	C₂₇₁	#1655
LCS	G ▷ C₄²×C₂₇₁ ▷ C₂²×C₂₇₁ ▷ C₂₇₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #496

Complements

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

N	H ≅ G/N	type
1. #31774639	C₂	semidirect
2. #31774284	C₂	semidirect
3. #31773637	C₂	semidirect
4. #31773636	C₂	semidirect
5. #496×C₂₇₁	C₂	semidirect
13. #22576071	C₂²	semidirect
16. #127×C₂₇₁	C₂²	semidirect
18. #97×C₂₇₁	C₂²	semidirect
80. C₂₇₁	#1655	semidirect
84. #496	D₂₇₁	semidirect
91. #127	C₂×D₂₇₁	semidirect
99. #97	C₂×D₂₇₁	semidirect

Maximal subgroups

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

subgroup	index
#31774639	2
#31774284	2
#31773637	2
#31773636	2
#496×C₂₇₁	2
#1655	271