Finite Group Databasebeta

Subgroups — #31781578

Order 17536.

184 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 79 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	#22577452	C₂	1	1	yes	maximal, characteristic
2	#22577451	C₂	1	1	yes	maximal, characteristic
3	C₂×#17984023	C₂	1	1	yes	maximal, characteristic
4	C₂×#17984023	C₂	1	1	yes	maximal, characteristic
5	#22577359	C₂	1	1	yes	maximal, characteristic
6	#22577173	C₂	1	1	yes	maximal, characteristic
7	#344×C₁₃₇	C₂	1	1	yes	maximal, characteristic
8	C₂×#14555494	C₂²	1	1	yes	characteristic
9	C₂×#14555494	C₂²	1	1	yes	characteristic
10	C₂×#14555494	C₂²	1	1	yes	characteristic
11	C₂²×#11764040	C₂²	1	1	yes	characteristic
12	#17984024	—	2	2	no	—
13	#17984023	—	2	2	no	—
14	#17984023	—	2	1	no	—
15	#17984023	—	2	1	no	—
16	#17984023	C₄	1	4	yes	complemented
17	#17983999	—	2	1	no	—
18	C₄×#11764040	—	2	1	no	—
19	C₂×M₁₆×C₁₃₇	C₂²	1	1	yes	characteristic
20	C₂×M₁₆×C₁₃₇	C₂²	1	1	yes	characteristic
21	#117×C₁₃₇	C₂²	1	1	yes	characteristic
22	#14555494	—	2	1	no	—
23	#14555494	—	2	1	no	—
24	#14555494	—	2	1	no	—
25	#14555494	—	2	1	no	—
26	#14555494	C₂×C₄	1	2	yes	—
27	#14555494	D₄	1	2	yes	—
28	C₂×#11764040	—	2	1	no	—
29	C₂×#11764040	—	2	1	no	—
30	C₂×#11764040	—	2	1	no	—
31	C₂×#11764040	C₂×C₄	1	2	yes	—
32	C₂×#11764040	D₄	1	1	yes	characteristic
33	C₂×#11764040	D₄	1	1	yes	characteristic
34	C₂×#11764040	Q₈	1	2	yes	—
35	C₂²×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
36	M₁₆×C₁₃₇	—	2	1	no	—
37	M₁₆×C₁₃₇	—	2	1	no	—
38	M₁₆×C₁₃₇	—	2	1	no	—
39	M₁₆×C₁₃₇	C₂×C₄	1	2	yes	—
40	C₂×C₈×C₁₃₇	—	2	1	no	—
41	C₂×C₈×C₁₃₇	—	2	1	no	—
42	C₄⋊C₄×C₁₃₇	—	2	1	no	—
43	C₂²⋊C₄×C₁₃₇	Q₈	1	2	yes	—
44	C₄²×C₁₃₇	—	2	1	no	—
45	#11764040	—	2	2	no	—
46	#11764040	—	2	2	no	—
47	#11764040	—	4	1	no	—
48	#11764040	C₄⋊C₄	1	2	yes	—
49	#11764040	C₄⋊C₄	1	2	yes	—
50	C₂³×C₁₃₇	C₂×Q₈	1	1	yes	characteristic
51	C₂×C₄×C₁₃₇	—	2	2	no	—
52	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
55	C₂×C₄×C₁₃₇	C₂×Q₈	1	1	yes	characteristic
56	C₂×C₄×C₁₃₇	#41	1	1	yes	characteristic
57	C₂×C₄×C₁₃₇	#41	1	1	yes	characteristic
58	C₈×C₁₃₇	—	2	2	no	—
59	C₈×C₁₃₇	—	4	1	no	—
60	C₂²×C₁₃₇	—	4	1	no	—
61	C₂²×C₁₃₇	C₄×Q₈	1	1	yes	characteristic
62	C₂²×C₁₃₇	C₂²⋊Q₈	1	1	yes	characteristic
63	C₂²×C₁₃₇	C₄⋊Q₈	1	1	yes	characteristic
64	C₄×C₁₃₇	—	4	2	no	—
65	C₄×C₁₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
66	C₄×C₁₃₇	C₄×D₄	1	1	yes	characteristic
67	C₄×C₁₃₇	#121	1	1	yes	characteristic
68	C₄×C₁₃₇	#125	1	1	yes	characteristic
69	C₂×C₁₃₇	—	2	1	no	—
70	C₂×C₁₃₇	—	2	1	no	—
71	C₂×C₁₃₇	—	2	1	no	—
72	C₂×C₁₃₇	#389	1	1	yes	characteristic
73	C₁₃₇	#1953	1	1	yes	characteristic, complemented, Sylow
74	#1953	—	137	1	no	maximal, Sylow
75	#430	—	137	1	no	—
76	#430	—	137	1	no	—
77	C₂×#108	—	137	1	no	—
78	C₂×#108	—	137	1	no	—
79	#405	—	137	1	no	—
80	#344	—	137	1	no	—
81	#344	D₁₃₇	1	1	yes	characteristic
82	C₂×M₁₆	—	137	1	no	—
83	C₂×M₁₆	—	137	1	no	—
84	C₂×M₁₆	—	137	1	no	—
85	C₂×M₁₆	C₂×D₁₃₇	1	1	yes	characteristic
86	C₂×M₁₆	C₂×D₁₃₇	1	1	yes	characteristic
87	C₂²×C₈	—	137	1	no	—
88	#117	C₂×D₁₃₇	1	1	yes	characteristic
89	#108	—	274	2	no	—
90	#108	—	274	2	no	—
91	#108	—	274	1	no	—
92	#108	—	274	1	no	—
93	#108	—	137	4	no	—
94	#97	—	274	1	no	—
95	C₄×C₈	—	274	1	no	—
96	C₂²×C₄	C₂²×D₁₃₇	1	1	yes	characteristic
97	M₁₆	—	2	1	no	—
98	M₁₆	—	2	1	no	—
99	M₁₆	—	2	1	no	—
100	M₁₆	—	274	1	no	—
101	M₁₆	—	274	1	no	—
102	M₁₆	—	274	1	no	—
103	M₁₆	—	274	1	no	—
104	M₁₆	—	137	2	no	—
105	M₁₆	—	137	2	no	—
106	M₁₆	C₄×D₁₃₇	1	2	yes	—
107	C₂×C₈	—	2	1	no	—
108	C₂×C₈	—	2	1	no	—
109	C₂×C₈	—	274	1	no	—
110	C₂×C₈	—	274	1	no	—
111	C₂×C₈	—	274	1	no	—
112	C₂×C₈	—	137	2	no	—
113	C₂×C₈	—	137	1	no	—
114	C₂×C₈	—	137	1	no	—
115	C₂×C₈	—	137	2	no	—
116	C₄⋊C₄	—	2	1	no	—
117	C₂²⋊C₄	Dic₂₇₄	1	2	yes	—
118	C₄²	—	2	1	no	—
119	C₂³	C₂×Dic₂₇₄	1	1	yes	characteristic
120	C₂×C₄	—	2	2	no	—
121	C₂×C₄	C₂×C₄×D₁₃₇	1	1	yes	characteristic
122	C₂×C₄	D₄×D₁₃₇	1	1	yes	characteristic
123	C₂×C₄	D₄×D₁₃₇	1	1	yes	characteristic
124	C₂×C₄	Q₈×D₁₃₇	1	1	yes	characteristic
125	C₂×C₄	#14555509	1	1	yes	characteristic
126	C₂×C₄	#14555510	1	1	yes	characteristic
127	C₈	—	2	2	no	—
128	C₈	—	4	1	no	—
129	C₈	—	274	2	no	—
130	C₈	—	274	2	no	—
131	C₈	—	548	1	no	—
132	C₈	—	137	2	no	—
133	C₈	—	137	2	no	—
134	C₂²	—	4	1	no	—
135	C₂²	C₄×Dic₂₇₄	1	1	yes	characteristic
136	C₂²	#17984065	1	1	yes	characteristic
137	C₂²	#17984087	1	1	yes	characteristic
138	C₄	—	4	2	no	—
139	C₄	C₄⋊C₄×D₁₃₇	1	1	yes	characteristic
140	C₄	#17984048	1	1	yes	characteristic
141	C₄	#17984061	1	1	yes	characteristic
142	C₄	#17984078	1	1	yes	characteristic
143	C₂	—	2	1	no	—
144	C₂	—	2	1	no	—
145	C₂	—	2	1	no	—
146	C₂	#22577300	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#17984061
commutator	C₂×C₄×C₁₃₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₃₇
Fitting	#344×C₁₃₇	C₂
socle	C₂×C₁₃₇	#389
prime radical	C₂³×C₁₃₇	C₂×Q₈
cosocle	C₂²×C₄×C₁₃₇	C₂³
hypercenter	#344	D₁₃₇
hypocenter	C₁₃₇	#1953
LCS	G ▷ C₂×C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #344

Complements

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

N	H ≅ G/N	type
16. #17984023	C₄	semidirect
73. C₁₃₇	#1953	semidirect

Maximal subgroups

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

subgroup	index
#22577452	2
#22577451	2
C₂×#17984023	2
#22577359	2
#22577173	2
#344×C₁₃₇	2
#1953	137