Finite Group Databasebeta

Subgroups — #46813089

Order 34432.

140 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 140 conjugacy classes of subgroups — too many to draw. Only its 47 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	#31770847	C₂	1	1	yes	maximal, characteristic
2	#107×D₂₆₉	C₂	1	1	yes	maximal, characteristic
3	#31770430	C₂	1	1	yes	maximal, characteristic, complemented
4	#31770410	C₂	1	1	yes	maximal, characteristic, complemented
5	#31770408	C₂	1	1	yes	maximal, characteristic, complemented
6	#31770384	C₂	1	1	yes	maximal, characteristic
7	#366×C₂₆₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×Dic₁₀₇₆	C₂²	1	1	yes	characteristic
9	C₂×C₈×D₂₆₉	C₂²	1	1	yes	characteristic
10	#22575433	—	2	1	no	—
11	C₄⋊C₄×D₂₆₉	—	2	1	no	—
12	C₂×#16864377	C₂²	1	1	yes	characteristic
13	#22575387	C₂²	1	1	yes	characteristic
14	#22575386	—	2	1	no	—
15	#22575376	—	2	1	no	—
16	C₂×C₁₆×C₂₆₉	C₂²	1	1	yes	characteristic
17	#107×C₂₆₉	C₂²	1	1	yes	characteristic
18	#107×C₂₆₉	C₂²	1	1	yes	characteristic
19	C₂×Dic₅₃₈	—	2	1	no	—
20	C₂×C₄×D₂₆₉	—	4	1	no	—
21	C₂×C₄×D₂₆₉	D₄	1	1	yes	characteristic
22	Dic₁₀₇₆	—	2	1	no	—
23	Dic₁₀₇₆	—	2	1	no	—
24	C₂×#14524130	D₄	1	1	yes	characteristic
25	C₈×D₂₆₉	—	2	1	no	—
26	C₈×D₂₆₉	—	2	1	no	—
27	C₄⋊Dic₂₆₉	—	2	1	no	—
28	C₄⋊Dic₂₆₉	—	4	1	no	—
29	#16864381	—	4	1	no	—
30	#16864379	—	4	1	no	—
31	#16864377	—	2	1	no	—
32	C₂×C₈×C₂₆₉	C₂³	1	1	yes	characteristic
33	C₄⋊C₄×C₂₆₉	—	2	1	no	—
34	C₄⋊C₄×C₂₆₉	—	2	1	no	—
35	C₁₆×C₂₆₉	—	2	1	no	—
36	C₂²×D₂₆₉	D₈	1	1	yes	characteristic
37	Dic₅₃₈	—	4	1	no	—
38	Dic₅₃₈	—	4	1	no	—
39	C₂×Dic₂₆₉	—	4	1	no	—
40	C₂×Dic₂₆₉	—	4	1	no	—
41	C₂×Dic₂₆₉	D₈	1	1	yes	characteristic
42	C₄×D₂₆₉	—	2	1	no	—
43	C₄×D₂₆₉	—	2	1	no	—
44	C₄×D₂₆₉	—	8	2	no	—
45	#14524130	—	2	1	no	—
46	C₂×C₄×C₂₆₉	—	4	1	no	—
47	C₂×C₄×C₂₆₉	—	4	1	no	—
48	C₂×C₄×C₂₆₉	C₂×D₄	1	1	yes	characteristic
49	C₈×C₂₆₉	#41	1	1	yes	characteristic
50	C₈×C₂₆₉	#41	1	1	yes	characteristic
51	C₂²×C₂₆₉	C₂×D₈	1	1	yes	characteristic
52	C₂×D₂₆₉	—	2	1	no	—
53	C₂×D₂₆₉	—	2	1	no	—
54	C₂×D₂₆₉	Q₃₂	1	2	yes	—
55	C₄×C₂₆₉	—	8	1	no	—
56	C₄×C₂₆₉	—	8	1	no	—
57	C₄×C₂₆₉	#123	1	1	yes	characteristic
58	C₄×C₂₆₉	#137	1	1	yes	characteristic
59	Dic₂₆₉	—	2	1	no	—
60	Dic₂₆₉	—	8	1	no	—
61	Dic₂₆₉	—	8	1	no	—
62	C₂×C₂₆₉	#480	1	1	yes	characteristic
63	C₂×C₂₆₉	C₂×Q₃₂	1	1	yes	characteristic
64	C₂×C₂₆₉	#509	1	1	yes	characteristic
65	D₂₆₉	—	2	2	no	—
66	C₂₆₉	#2236	1	1	yes	characteristic, complemented, Sylow
67	#2236	—	269	1	no	maximal, Sylow
68	#467	—	269	1	no	—
69	C₂×#107	—	269	1	no	—
70	#366	—	269	1	no	—
71	#366	D₂₆₉	1	1	yes	characteristic, complemented
72	#358	—	269	1	no	—
73	#358	—	269	1	no	—
74	C₂²⋊C₁₆	—	269	1	no	—
75	C₂×Q₁₆	—	269	1	no	—
76	C₂²×C₈	—	269	1	no	—
77	C₂²⋊Q₈	—	538	1	no	—
78	C₂×C₄⋊C₄	—	538	1	no	—
79	C₂×C₁₆	—	269	1	no	—
80	C₂×C₁₆	C₂×D₂₆₉	1	1	yes	characteristic
81	#107	—	269	1	no	—
82	#107	—	538	1	no	—
83	#107	C₂×D₂₆₉	1	1	yes	characteristic
84	#107	C₂×D₂₆₉	1	1	yes	characteristic
85	#103	—	538	1	no	—
86	C₂×Q₈	—	538	1	no	—
87	C₂²×C₄	—	269	1	no	—
88	C₂²×C₄	—	1076	1	no	—
89	Q₁₆	—	538	1	no	—
90	Q₁₆	—	538	1	no	—
91	C₂×C₈	—	269	1	no	—
92	C₂×C₈	—	538	1	no	—
93	C₂×C₈	—	538	1	no	—
94	C₂×C₈	C₂²×D₂₆₉	1	1	yes	characteristic
95	C₄⋊C₄	—	2	1	no	—
96	C₄⋊C₄	—	2	1	no	—
97	C₄⋊C₄	—	538	1	no	—
98	C₄⋊C₄	—	1076	1	no	—
99	C₄⋊C₄	—	1076	1	no	—
100	C₂²⋊C₄	—	1076	1	no	—
101	C₁₆	—	2	1	no	—
102	C₁₆	—	538	1	no	—
103	C₂³	—	269	1	no	—
104	Q₈	—	1076	1	no	—
105	Q₈	—	1076	1	no	—
106	C₂×C₄	—	4	1	no	—
107	C₂×C₄	—	4	1	no	—
108	C₂×C₄	—	269	1	no	—
109	C₂×C₄	—	538	1	no	—
110	C₂×C₄	—	538	1	no	—
111	C₂×C₄	—	1076	1	no	—
112	C₂×C₄	—	1076	1	no	—
113	C₂×C₄	—	2152	2	no	—
114	C₂×C₄	D₄×D₂₆₉	1	1	yes	characteristic
115	C₈	—	538	1	no	—
116	C₈	#16864402	1	1	yes	characteristic
117	C₈	#16864403	1	1	yes	characteristic
118	C₂²	—	269	2	no	—
119	C₂²	—	538	1	no	—
120	C₂²	—	538	1	no	—
121	C₂²	D₈×D₂₆₉	1	1	yes	characteristic
122	C₄	—	8	1	no	—
123	C₄	—	8	1	no	—
124	C₄	—	538	1	no	—
125	C₄	—	2152	1	no	—
126	C₄	—	2152	1	no	—
127	C₄	#22575438	1	1	yes	characteristic
128	C₄	#22575485	1	1	yes	characteristic
129	C₂	—	538	2	no	—
130	C₂	#31770920	1	1	yes	characteristic
131	C₂	Q₃₂×D₂₆₉	1	1	yes	characteristic
132	C₂	#31771039	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₈×D₂₆₉
commutator, cosocle	C₂×C₈×C₂₆₉	C₂³
Frattini	C₂×C₈	C₂²×D₂₆₉
Fitting	#366×C₂₆₉	C₂
socle	C₂²×C₂₆₉	C₂×D₈
prime radical	C₂²×D₂₆₉	D₈
hypercenter	#366	D₂₆₉
hypocenter	C₂₆₉	#2236
LCS	G ▷ C₂×C₈×C₂₆₉ ▷ C₄×C₂₆₉ ▷ C₂×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #366

Complements

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

N	H ≅ G/N	type
3. #31770430	C₂	semidirect
4. #31770410	C₂	semidirect
5. #31770408	C₂	semidirect
7. #366×C₂₆₉	C₂	semidirect
66. C₂₆₉	#2236	semidirect
71. #366	D₂₆₉	semidirect

Maximal subgroups

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

subgroup	index
#31770847	2
#107×D₂₆₉	2
#31770430	2
#31770410	2
#31770408	2
#31770384	2
#366×C₂₆₉	2
#2236	269