Finite Group Databasebeta

Subgroups — #31782655

Order 17536.

160 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 160 conjugacy classes of subgroups — too many to draw. Only its 61 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	C₂×#17984033	C₂	1	1	yes	maximal, characteristic
2	C₄×Dic₅₄₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×D₁₀₉₆	C₂	1	1	yes	maximal, characteristic
4	#22577229	C₂	1	1	yes	maximal, characteristic, complemented
5	#22577208	C₂	1	1	yes	maximal, characteristic, complemented
6	#22577205	C₂	1	1	yes	maximal, characteristic
7	#346×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×Dic₅₄₈	C₂²	1	1	yes	characteristic
9	C₂×D₁₀₉₆	C₂²	1	1	yes	characteristic
10	C₄×Dic₂₇₄	—	2	1	no	—
11	C₄×D₅₄₈	—	2	1	no	—
12	#17984033	C₄	1	4	yes	complemented
13	#17984022	C₂²	1	1	yes	characteristic
14	#17984022	C₂²	1	1	yes	characteristic
15	#17984013	—	2	1	no	—
16	#17984010	—	2	1	no	—
17	C₂×C₁₆×C₁₃₇	C₂²	1	1	yes	characteristic
18	C₂×C₁₆×C₁₃₇	C₂²	1	1	yes	characteristic
19	C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
20	C₂×Dic₂₇₄	—	2	1	no	—
21	C₂×D₅₄₈	—	2	1	no	—
22	C₂×C₄×D₁₃₇	—	4	1	no	—
23	Dic₅₄₈	—	2	1	no	—
24	Dic₅₄₈	C₂×C₄	1	2	yes	complemented
25	D₁₀₉₆	—	2	1	no	—
26	D₁₀₉₆	C₂×C₄	1	2	yes	—
27	C₄⋊Dic₁₃₇	—	2	1	no	—
28	C₄⋊Dic₁₃₇	—	2	1	no	—
29	#14555489	—	4	1	no	—
30	#14555487	—	4	1	no	—
31	C₄×Dic₁₃₇	—	4	1	no	—
32	C₂×C₈×C₁₃₇	D₄	1	1	yes	characteristic
33	C₂×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
34	C₄²×C₁₃₇	D₄	1	1	yes	characteristic
35	C₁₆×C₁₃₇	—	2	1	no	—
36	C₁₆×C₁₃₇	C₂×C₄	1	2	yes	complemented
37	C₂²×D₁₃₇	—	4	1	no	—
38	Dic₂₇₄	—	2	2	no	—
39	Dic₂₇₄	—	4	1	no	—
40	D₅₄₈	—	2	2	no	—
41	D₅₄₈	—	4	1	no	—
42	C₂×Dic₁₃₇	—	4	1	no	—
43	C₂×Dic₁₃₇	—	4	1	no	—
44	C₂×Dic₁₃₇	—	4	1	no	—
45	C₄×D₁₃₇	—	8	2	no	—
46	C₂×C₄×C₁₃₇	D₈	1	1	yes	characteristic
47	C₂×C₄×C₁₃₇	D₈	1	1	yes	characteristic
48	C₂×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
49	C₈×C₁₃₇	—	2	1	no	—
50	C₈×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
51	C₈×C₁₃₇	#41	1	1	yes	characteristic
52	C₂²×C₁₃₇	C₂×D₈	1	1	yes	characteristic
53	C₂×D₁₃₇	—	4	2	no	—
54	C₂×D₁₃₇	—	8	1	no	—
55	C₂×D₁₃₇	—	8	1	no	—
56	C₄×C₁₃₇	—	2	1	no	—
57	C₄×C₁₃₇	—	2	1	no	—
58	C₄×C₁₃₇	C₄×D₄	1	1	yes	characteristic
59	C₄×C₁₃₇	#135	1	1	yes	characteristic
60	Dic₁₃₇	—	4	2	no	—
61	Dic₁₃₇	—	8	1	no	—
62	Dic₁₃₇	—	8	1	no	—
63	C₂×C₁₃₇	C₄×D₈	1	1	yes	characteristic
64	C₂×C₁₃₇	#509	1	1	yes	characteristic
65	C₂×C₁₃₇	#510	1	1	yes	characteristic
66	D₁₃₇	—	8	2	no	—
67	C₁₃₇	#2175	1	1	yes	characteristic, complemented, Sylow
68	#2175	—	137	1	no	maximal, Sylow
69	C₂×SD₃₂	—	137	1	no	—
70	C₄×Q₁₆	—	137	1	no	—
71	C₄×D₈	—	137	1	no	—
72	#366	—	137	1	no	—
73	#358	—	137	1	no	—
74	#357	—	137	1	no	—
75	#346	D₁₃₇	1	1	yes	characteristic, complemented
76	C₂×Q₁₆	—	137	1	no	—
77	C₂×D₈	—	137	1	no	—
78	C₄×Q₈	—	274	1	no	—
79	C₄×D₄	—	274	1	no	—
80	SD₃₂	—	137	4	no	—
81	C₂×C₁₆	C₂×D₁₃₇	1	1	yes	characteristic
82	C₂×C₁₆	C₂×D₁₃₇	1	1	yes	characteristic
83	#107	—	137	1	no	—
84	#107	—	137	1	no	—
85	#103	—	274	1	no	—
86	#102	—	274	1	no	—
87	C₄×C₈	C₂×D₁₃₇	1	1	yes	characteristic
88	C₂×Q₈	—	274	1	no	—
89	C₂×D₄	—	274	1	no	—
90	C₂²×C₄	—	548	1	no	—
91	Q₁₆	—	274	1	no	—
92	Q₁₆	—	137	2	no	—
93	D₈	—	274	1	no	—
94	D₈	—	137	2	no	—
95	C₂×C₈	D₅₄₈	1	1	yes	characteristic
96	C₂×C₈	C₂²×D₁₃₇	1	1	yes	characteristic
97	C₄⋊C₄	—	274	1	no	—
98	C₄⋊C₄	—	274	1	no	—
99	C₄⋊C₄	—	548	1	no	—
100	C₂²⋊C₄	—	548	1	no	—
101	C₄²	—	548	1	no	—
102	C₄²	D₅₄₈	1	1	yes	characteristic
103	C₁₆	—	2	1	no	—
104	C₁₆	C₄×D₁₃₇	1	2	yes	complemented
105	C₂³	—	548	1	no	—
106	Q₈	—	274	2	no	—
107	Q₈	—	548	1	no	—
108	D₄	—	274	2	no	—
109	D₄	—	548	1	no	—
110	C₂×C₄	—	548	1	no	—
111	C₂×C₄	—	548	1	no	—
112	C₂×C₄	—	548	1	no	—
113	C₂×C₄	—	1096	2	no	—
114	C₂×C₄	D₁₀₉₆	1	1	yes	characteristic
115	C₂×C₄	D₁₀₉₆	1	1	yes	characteristic
116	C₂×C₄	C₂×D₅₄₈	1	1	yes	characteristic
117	C₈	—	2	1	no	—
118	C₈	C₂×C₄×D₁₃₇	1	1	yes	characteristic
119	C₈	#14555510	1	1	yes	characteristic
120	C₂²	—	548	2	no	—
121	C₂²	—	1096	1	no	—
122	C₂²	—	1096	1	no	—
123	C₂²	C₂×D₁₀₉₆	1	1	yes	characteristic
124	C₄	—	2	1	no	—
125	C₄	—	2	1	no	—
126	C₄	—	548	2	no	—
127	C₄	—	1096	1	no	—
128	C₄	—	1096	1	no	—
129	C₄	C₄×D₅₄₈	1	1	yes	characteristic
130	C₄	#17984110	1	1	yes	characteristic
131	C₂	—	1096	2	no	—
132	C₂	C₄×D₁₀₉₆	1	1	yes	characteristic
133	C₂	#22577841	1	1	yes	characteristic
134	C₂	#22577846	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₁₀₉₆
commutator	C₈×C₁₃₇	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₁₃₇
Fitting	#346×C₁₃₇	C₂
socle	C₂²×C₁₃₇	C₂×D₈
prime radical	C₂×D₁₀₉₆	C₂²
cosocle	C₂×C₈×C₁₃₇	C₂³
hypercenter	#346	D₁₃₇
hypocenter	C₁₃₇	#2175
LCS	G ▷ C₈×C₁₃₇ ▷ C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #346

Complements

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

N	H ≅ G/N	type
2. C₄×Dic₅₄₈	C₂	semidirect
4. #22577229	C₂	semidirect
5. #22577208	C₂	semidirect
7. #346×C₁₃₇	C₂	semidirect
12. #17984033	C₄	semidirect
24. Dic₅₄₈	C₂×C₄	semidirect
36. C₁₆×C₁₃₇	C₂×C₄	semidirect
67. C₁₃₇	#2175	semidirect
75. #346	D₁₃₇	semidirect
104. C₁₆	C₄×D₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17984033	2
C₄×Dic₅₄₈	2
C₄×D₁₀₉₆	2
#22577229	2
#22577208	2
#22577205	2
#346×C₁₃₇	2
#2175	137