Finite Group Databasebeta

Subgroups — #46826107

Order 34432.

178 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 178 conjugacy classes of subgroups — too many to draw. Only its 78 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	#31771624	C₂	1	1	yes	maximal, characteristic
2	#31771623	C₂	1	1	yes	maximal, characteristic
3	#14524137⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	#14524137⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	#31771446	C₂	1	1	yes	maximal, characteristic, complemented
6	C₈×#14524137	C₂	1	1	yes	maximal, characteristic, complemented
7	#31770657	C₂	1	1	yes	maximal, characteristic
8	#22575520	C₄	1	2	yes	—
9	C₄⋊#14524137	C₄	1	2	yes	—
10	#22575510	C₄	1	2	yes	—
11	#22575510	C₄	1	2	yes	—
12	C₄×#14524137	C₂²	1	1	yes	characteristic
13	C₄×#14524137	C₂²	1	1	yes	characteristic
14	C₄×#14524137	C₂²	1	1	yes	characteristic
15	C₄×#14524137	C₂²	1	1	yes	characteristic
16	C₄²×D₂₆₉	C₂²	1	1	yes	characteristic
17	C₄⋊#14524130	C₄	1	1	yes	characteristic
18	#22575365	C₄	1	1	yes	characteristic
19	#22575365	C₄	1	1	yes	characteristic
20	C₈×Dic₂₆₉	C₂²	1	1	yes	characteristic
21	C₈×Dic₂₆₉	C₂²	1	1	yes	characteristic
22	C₄⋊C₈×C₂₆₉	C₄	1	1	yes	characteristic
23	C₂×#14524137	C₂×C₄	1	2	yes	—
24	C₂×#14524137	C₂×C₄	1	2	yes	—
25	C₂×#14524137	C₂×C₄	1	2	yes	—
26	C₂×#14524137	C₂×C₄	1	2	yes	—
27	C₂×C₄×D₂₆₉	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄×D₂₆₉	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄×D₂₆₉	C₂×C₄	1	1	yes	characteristic
30	C₂×#14524130	C₂×C₄	1	1	yes	characteristic
31	C₂×#14524130	C₂×C₄	1	1	yes	characteristic
32	C₄×Dic₂₆₉	C₂×C₄	1	1	yes	characteristic
33	C₄×Dic₂₆₉	C₂×C₄	1	1	yes	characteristic
34	C₄×Dic₂₆₉	C₂³	1	1	yes	characteristic
35	C₂×C₈×C₂₆₉	C₂×C₄	1	1	yes	characteristic
36	C₂×C₈×C₂₆₉	C₂×C₄	1	1	yes	characteristic
37	C₄²×C₂₆₉	C₂×C₄	1	1	yes	characteristic
38	#14524137	—	2	2	no	—
39	#14524137	—	2	2	no	—
40	#14524137	—	2	2	no	—
41	#14524137	—	2	2	no	—
42	C₂²×D₂₆₉	C₄²	1	1	yes	characteristic
43	C₂×Dic₂₆₉	C₄²	1	1	yes	characteristic
44	C₂×Dic₂₆₉	C₂²×C₄	1	1	yes	characteristic
45	C₂×Dic₂₆₉	C₂²×C₄	1	1	yes	characteristic
46	C₄×D₂₆₉	—	2	1	no	—
47	C₄×D₂₆₉	—	2	1	no	—
48	C₄×D₂₆₉	—	2	2	no	—
49	C₄×D₂₆₉	—	2	2	no	—
50	#14524130	—	2	1	no	—
51	#14524130	—	2	1	no	—
52	C₂×C₄×C₂₆₉	C₄²	1	1	yes	characteristic
53	C₂×C₄×C₂₆₉	C₄²	1	1	yes	characteristic
54	C₂×C₄×C₂₆₉	C₂²×C₄	1	1	yes	characteristic
55	C₈×C₂₆₉	—	2	1	no	—
56	C₈×C₂₆₉	—	2	1	no	—
57	C₂²×C₂₆₉	C₂×C₄²	1	1	yes	characteristic
58	C₂×D₂₆₉	—	2	1	no	—
59	C₂×D₂₆₉	—	2	1	no	—
60	C₂×D₂₆₉	—	2	1	no	—
61	C₄×C₂₆₉	—	2	1	no	—
62	C₄×C₂₆₉	—	2	1	no	—
63	C₄×C₂₆₉	#131	1	1	yes	characteristic
64	C₄×C₂₆₉	#131	1	1	yes	characteristic
65	Dic₂₆₉	—	2	1	no	—
66	Dic₂₆₉	#131	1	2	yes	—
67	Dic₂₆₉	#131	1	2	yes	—
68	C₂×C₂₆₉	#405	1	1	yes	characteristic
69	C₂×C₂₆₉	#405	1	1	yes	characteristic
70	C₂×C₂₆₉	#405	1	1	yes	characteristic
71	D₂₆₉	—	4	1	no	—
72	C₂₆₉	#1454	1	1	yes	characteristic, complemented, Sylow
73	#1454	—	269	1	no	maximal, Sylow
74	#431	—	269	1	no	—
75	#431	—	269	1	no	—
76	#431	—	269	1	no	—
77	#322	—	269	1	no	—
78	#322	—	269	1	no	—
79	#322	—	269	1	no	—
80	C₈²	—	269	1	no	—
81	C₂×C₄²	—	269	1	no	—
82	C₄⋊C₈	—	269	2	no	—
83	C₄⋊C₈	—	269	2	no	—
84	C₄⋊C₈	—	269	1	no	—
85	C₄⋊C₈	#11761640	1	1	yes	characteristic
86	C₂²⋊C₈	—	269	2	no	—
87	C₂²⋊C₈	—	269	2	no	—
88	C₂²⋊C₈	—	269	1	no	—
89	C₂²⋊C₈	—	269	1	no	—
90	C₄×C₈	—	269	1	no	—
91	C₄×C₈	—	269	1	no	—
92	C₄×C₈	—	269	1	no	—
93	C₄×C₈	—	269	1	no	—
94	C₄×C₈	—	269	1	no	—
95	C₄×C₈	—	269	1	no	—
96	C₂²×C₄	—	269	1	no	—
97	C₂²×C₄	—	269	1	no	—
98	C₂²×C₄	—	269	1	no	—
99	C₂×C₈	—	269	2	no	—
100	C₂×C₈	—	269	2	no	—
101	C₂×C₈	—	269	2	no	—
102	C₂×C₈	—	269	2	no	—
103	C₂×C₈	—	269	1	no	—
104	C₂×C₈	—	269	1	no	—
105	C₂×C₈	C₂×#11761640	1	1	yes	characteristic
106	C₂×C₈	C₂×#11761640	1	1	yes	characteristic
107	C₄²	—	269	1	no	—
108	C₄²	—	269	1	no	—
109	C₄²	—	269	1	no	—
110	C₄²	C₂×#11761640	1	1	yes	characteristic
111	C₂³	—	269	1	no	—
112	C₂×C₄	—	269	1	no	—
113	C₂×C₄	—	269	1	no	—
114	C₂×C₄	—	269	1	no	—
115	C₂×C₄	—	538	2	no	—
116	C₂×C₄	—	538	2	no	—
117	C₂×C₄	—	538	1	no	—
118	C₂×C₄	—	538	1	no	—
119	C₂×C₄	C₄×#11761640	1	1	yes	characteristic
120	C₂×C₄	C₄×#11761640	1	1	yes	characteristic
121	C₂×C₄	C₂²×#11761640	1	1	yes	characteristic
122	C₈	—	2	1	no	—
123	C₈	—	2	1	no	—
124	C₈	—	538	2	no	—
125	C₈	—	538	2	no	—
126	C₈	—	538	2	no	—
127	C₈	—	538	2	no	—
128	C₈	—	538	1	no	—
129	C₈	—	538	1	no	—
130	C₂²	—	538	1	no	—
131	C₂²	—	538	1	no	—
132	C₂²	—	538	1	no	—
133	C₂²	C₂×C₄×#11761640	1	1	yes	characteristic
134	C₄	—	2	1	no	—
135	C₄	—	2	1	no	—
136	C₄	—	269	2	no	—
137	C₄	—	269	2	no	—
138	C₄	—	538	1	no	—
139	C₄	#22575540	1	1	yes	characteristic
140	C₄	#22575541	1	1	yes	characteristic
141	C₂	—	1076	1	no	—
142	C₂	#31771568	1	1	yes	characteristic
143	C₂	#31771570	1	1	yes	characteristic
144	C₂	#31771571	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×#11761640
commutator, socle	C₂²×C₂₆₉	C₂×C₄²
Fitting	C₄⋊C₈×C₂₆₉	C₄
prime radical	C₂²×D₂₆₉	C₄²
cosocle	C₄×Dic₂₆₉	C₂³
hypercenter	C₄⋊C₈	#11761640
hypocenter	C₂₆₉	#1454
LCS	G ▷ C₂²×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄⋊C₈

Complements

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

N	H ≅ G/N	type
3. #14524137⋊C₈	C₂	semidirect
4. #14524137⋊C₈	C₂	semidirect
5. #31771446	C₂	semidirect
6. C₈×#14524137	C₂	semidirect
72. C₂₆₉	#1454	semidirect

Maximal subgroups

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

subgroup	index
#31771624	2
#31771623	2
#14524137⋊C₈	2
#31771446	2
C₈×#14524137	2
#31770657	2
#1454	269