Finite Group Databasebeta

Subgroups — #31782689

Order 17536.

154 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 154 conjugacy classes of subgroups — too many to draw. Only its 53 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	#22577841	C₂	1	1	yes	maximal, characteristic
2	#22577526	C₂	1	1	yes	maximal, characteristic, complemented
3	#22577516	C₂	1	1	yes	maximal, characteristic, complemented
4	#22577226	C₂	1	1	yes	maximal, characteristic, complemented
5	#22577217	C₂	1	1	yes	maximal, characteristic, complemented
6	#22577214	C₂	1	1	yes	maximal, characteristic, complemented
7	#347×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
8	#17984110	C₂²	1	1	yes	characteristic
9	C₂×D₁₀₉₆	C₂²	1	1	yes	characteristic
10	#17984094	—	2	1	no	—
11	C₄×D₅₄₈	—	2	1	no	—
12	#17984033	C₄	1	2	yes	complemented
13	D₂₁₉₂	C₄	1	2	yes	complemented
14	#17984024	C₂²	1	1	yes	characteristic
15	#17984020	C₂²	1	1	yes	characteristic
16	#17984016	—	2	1	no	—
17	#17984010	—	2	1	no	—
18	M₃₂×C₁₃₇	C₂²	1	1	yes	characteristic
19	M₃₂×C₁₃₇	C₂²	1	1	yes	characteristic
20	#97×C₁₃₇	C₂²	1	1	yes	characteristic
21	#14555510	—	2	1	no	—
22	C₂×D₅₄₈	—	2	1	no	—
23	C₂×C₄×D₁₃₇	—	4	1	no	—
24	Dic₅₄₈	C₂×C₄	1	1	yes	characteristic, complemented
25	#14555499	—	2	1	no	—
26	D₁₀₉₆	—	2	1	no	—
27	D₁₀₉₆	C₂×C₄	1	2	yes	—
28	D₁₀₉₆	C₂×C₄	1	1	yes	characteristic, complemented
29	#14555494	—	2	1	no	—
30	#14555493	—	4	1	no	—
31	C₈×D₁₃₇	—	4	1	no	—
32	C₄⋊Dic₁₃₇	—	2	1	no	—
33	#14555487	—	4	1	no	—
34	C₂×C₈×C₁₃₇	D₄	1	1	yes	characteristic
35	C₂×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
36	C₄²×C₁₃₇	D₄	1	1	yes	characteristic
37	C₁₆×C₁₃₇	—	2	1	no	—
38	C₁₆×C₁₃₇	C₂×C₄	1	2	yes	complemented
39	C₂²×D₁₃₇	—	4	1	no	—
40	Dic₂₇₄	—	2	1	no	—
41	D₅₄₈	—	2	2	no	—
42	D₅₄₈	—	2	1	no	—
43	D₅₄₈	—	4	1	no	—
44	C₂²⋊D₁₃₇	—	4	1	no	—
45	C₂×Dic₁₃₇	—	4	1	no	—
46	C₄×D₁₃₇	—	4	1	no	—
47	C₄×D₁₃₇	—	8	2	no	—
48	#11764040	—	4	1	no	—
49	C₂×C₄×C₁₃₇	—	2	1	no	—
50	C₂×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
51	C₈×C₁₃₇	—	2	1	no	—
52	C₈×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
53	C₈×C₁₃₇	#41	1	1	yes	characteristic
54	C₂²×C₁₃₇	AGL(1,ℤ/8)	1	1	yes	characteristic
55	C₂×D₁₃₇	—	4	1	no	—
56	C₂×D₁₃₇	—	4	2	no	—
57	C₂×D₁₃₇	—	8	1	no	—
58	C₂×D₁₃₇	—	8	1	no	—
59	C₄×C₁₃₇	—	4	1	no	—
60	C₄×C₁₃₇	C₄×D₄	1	1	yes	characteristic
61	C₄×C₁₃₇	AGL(1,ℤ/8)	1	1	yes	characteristic
62	Dic₁₃₇	—	4	1	no	—
63	Dic₁₃₇	—	8	1	no	—
64	C₂×C₁₃₇	—	2	1	no	—
65	C₂×C₁₃₇	#442	1	1	yes	characteristic
66	D₁₃₇	—	8	1	no	—
67	D₁₃₇	—	8	2	no	—
68	C₁₃₇	AGL(1,ℤ/16)	1	1	yes	characteristic, complemented, Sylow
69	AGL(1,ℤ/16)	—	137	1	no	maximal, Sylow
70	#509	—	137	1	no	—
71	#444	—	137	1	no	—
72	#442	—	137	1	no	—
73	#365	—	137	1	no	—
74	#361	—	137	1	no	—
75	#360	—	137	1	no	—
76	#347	D₁₃₇	1	1	yes	characteristic, complemented
77	#135	—	137	1	no	—
78	C₂×D₈	—	137	1	no	—
79	#131	—	274	1	no	—
80	C₄×D₄	—	274	1	no	—
81	SD₃₂	—	137	2	no	—
82	D₁₆	—	137	2	no	—
83	M₃₂	C₂×D₁₃₇	1	1	yes	characteristic
84	M₃₂	C₂×D₁₃₇	1	1	yes	characteristic
85	#108	—	137	1	no	—
86	#106	—	137	1	no	—
87	#104	—	274	1	no	—
88	#102	—	274	1	no	—
89	#97	C₂×D₁₃₇	1	1	yes	characteristic
90	#41	—	274	1	no	—
91	C₂×D₄	—	274	1	no	—
92	C₂²×C₄	—	548	1	no	—
93	Q₁₆	—	137	1	no	—
94	SD₁₆	—	274	1	no	—
95	D₈	—	274	1	no	—
96	D₈	—	137	2	no	—
97	D₈	—	137	1	no	—
98	M₁₆	—	274	1	no	—
99	M₁₆	—	548	1	no	—
100	C₂×C₈	—	548	1	no	—
101	C₂×C₈	D₅₄₈	1	1	yes	characteristic
102	C₂×C₈	C₂²×D₁₃₇	1	1	yes	characteristic
103	C₄⋊C₄	—	274	1	no	—
104	C₂²⋊C₄	—	548	1	no	—
105	C₄²	D₅₄₈	1	1	yes	characteristic
106	C₁₆	—	2	1	no	—
107	C₁₆	C₄×D₁₃₇	1	2	yes	complemented
108	C₂³	—	548	1	no	—
109	Q₈	—	274	1	no	—
110	D₄	—	274	2	no	—
111	D₄	—	274	1	no	—
112	D₄	—	548	1	no	—
113	D₄	—	548	1	no	—
114	C₂×C₄	—	2	1	no	—
115	C₂×C₄	—	548	1	no	—
116	C₂×C₄	—	548	1	no	—
117	C₂×C₄	—	1096	2	no	—
118	C₂×C₄	C₂×D₅₄₈	1	1	yes	characteristic
119	C₈	—	2	1	no	—
120	C₈	—	548	1	no	—
121	C₈	C₂×C₄×D₁₃₇	1	1	yes	characteristic
122	C₈	#14555510	1	1	yes	characteristic
123	C₂²	—	548	1	no	—
124	C₂²	—	548	2	no	—
125	C₂²	—	1096	1	no	—
126	C₂²	—	1096	1	no	—
127	C₂²	#17984116	1	1	yes	characteristic
128	C₄	—	4	1	no	—
129	C₄	—	548	1	no	—
130	C₄	—	1096	1	no	—
131	C₄	C₄×D₅₄₈	1	1	yes	characteristic
132	C₄	#17984116	1	1	yes	characteristic
133	C₂	—	2	1	no	—
134	C₂	—	1096	2	no	—
135	C₂	—	1096	1	no	—
136	C₂	#22577516	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#22577516
commutator	C₈×C₁₃₇	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₁₃₇
Fitting	#347×C₁₃₇	C₂
socle	C₂×C₁₃₇	#442
prime radical	#22577841	C₂
cosocle	C₂×C₈×C₁₃₇	C₂³
hypercenter	#347	D₁₃₇
hypocenter	C₁₃₇	AGL(1,ℤ/16)
LCS	G ▷ C₈×C₁₃₇ ▷ C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #97 ◁ #347

Complements

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

N	H ≅ G/N	type
2. #22577526	C₂	semidirect
3. #22577516	C₂	semidirect
4. #22577226	C₂	semidirect
5. #22577217	C₂	semidirect
6. #22577214	C₂	semidirect
7. #347×C₁₃₇	C₂	semidirect
12. #17984033	C₄	semidirect
13. D₂₁₉₂	C₄	semidirect
24. Dic₅₄₈	C₂×C₄	semidirect
28. D₁₀₉₆	C₂×C₄	semidirect
38. C₁₆×C₁₃₇	C₂×C₄	semidirect
68. C₁₃₇	AGL(1,ℤ/16)	semidirect
76. #347	D₁₃₇	semidirect
107. 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
#22577841	2
#22577526	2
#22577516	2
#22577226	2
#22577217	2
#22577214	2
#347×C₁₃₇	2
AGL(1,ℤ/16)	137