Finite Group Databasebeta

Subgroups — #31797180

Order 17536.

144 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 144 conjugacy classes of subgroups — too many to draw. Only its 58 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	#22578462	C₂	1	1	yes	maximal, characteristic, complemented
2	#22578461	C₂	1	1	yes	maximal, characteristic, complemented
3	#22578326	C₂	1	1	yes	maximal, characteristic, complemented
4	#22578302	C₂	1	1	yes	maximal, characteristic, complemented
5	#22578301	C₂	1	1	yes	maximal, characteristic, complemented
6	#22578291	C₂	1	1	yes	maximal, characteristic, complemented
7	#22577517	C₂	1	1	yes	maximal, characteristic
8	#17984163	—	2	1	no	—
9	#17984162	—	2	1	no	—
10	#17984162	C₂²	1	1	yes	characteristic
11	#17984162	C₂²	1	1	yes	characteristic
12	#17984161	—	2	1	no	—
13	C₂×#14555513	—	2	1	no	—
14	C₂×#14555513	C₂²	1	1	yes	characteristic
15	C₂×#14555513	C₂²	1	1	yes	characteristic
16	#17984108	C₄	1	1	yes	characteristic
17	#17984096	C₂²	1	1	yes	characteristic
18	#17984095	C₂²	1	1	yes	characteristic
19	#17984023	C₄	1	1	yes	characteristic
20	#17984016	C₄	1	1	yes	characteristic
21	C₄×#11764040	C₂²	1	1	yes	characteristic
22	#104×C₁₃₇	C₄	1	1	yes	characteristic
23	#14555513	—	2	1	no	—
24	#14555513	—	2	1	no	—
25	#14555513	—	2	1	no	—
26	#14555513	D₄	1	1	yes	characteristic
27	#14555513	D₄	1	1	yes	characteristic
28	#14555510	C₂×C₄	1	1	yes	characteristic
29	#14555500	C₈	1	1	yes	characteristic
30	#14555498	C₈	1	1	yes	characteristic
31	#14555497	C₈	1	1	yes	characteristic
32	#14555495	C₈	1	1	yes	characteristic
33	#14555494	—	2	1	no	—
34	#14555494	C₂×C₄	1	1	yes	characteristic
35	#14555493	—	2	1	no	—
36	C₂×#11764040	—	2	1	no	—
37	C₂×#11764040	C₂×C₄	1	1	yes	characteristic
38	C₂×#11764040	C₂³	1	1	yes	characteristic
39	C₈×D₁₃₇	—	2	1	no	—
40	#41×C₁₃₇	C₂×C₄	1	1	yes	characteristic
41	M₁₆×C₁₃₇	C₂×C₄	1	1	yes	characteristic
42	C₄²×C₁₃₇	C₂×C₄	1	1	yes	characteristic
43	Dic₂₇₄	C₂×C₈	1	1	yes	characteristic
44	D₅₄₈	C₂×C₈	1	1	yes	characteristic
45	C₂²⋊D₁₃₇	—	2	1	no	—
46	C₄×D₁₃₇	—	2	1	no	—
47	#11764040	—	2	1	no	—
48	#11764040	C₂×C₈	1	1	yes	characteristic
49	#11764040	C₂×C₈	1	1	yes	characteristic
50	#11764040	C₂×D₄	1	1	yes	characteristic
51	#11764040	#41	1	1	yes	characteristic
52	Q₈×C₁₃₇	C₂×C₈	1	1	yes	characteristic
53	D₄×C₁₃₇	—	2	1	no	—
54	D₄×C₁₃₇	C₂×C₈	1	1	yes	characteristic
55	C₂×C₄×C₁₃₇	—	2	1	no	—
56	C₂×C₄×C₁₃₇	—	2	1	no	—
57	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
58	C₈×C₁₃₇	—	2	1	no	—
59	C₂²×C₁₃₇	—	2	1	no	—
60	C₂²×C₁₃₇	#131	1	1	yes	characteristic
61	C₂×D₁₃₇	—	2	1	no	—
62	C₄×C₁₃₇	—	2	1	no	—
63	C₄×C₁₃₇	—	4	1	no	—
64	C₄×C₁₃₇	C₄×D₄	1	1	yes	characteristic
65	C₄×C₁₃₇	C₂²×C₈	1	1	yes	characteristic
66	Dic₁₃₇	—	2	1	no	—
67	C₂×C₁₃₇	—	2	1	no	—
68	C₂×C₁₃₇	—	4	1	no	—
69	C₂×C₁₃₇	C₈×D₄	1	1	yes	characteristic
70	D₁₃₇	—	4	1	no	—
71	C₁₃₇	#2171	1	1	yes	characteristic, complemented, Sylow
72	#2171	—	137	1	no	maximal, Sylow
73	#504	—	137	1	no	—
74	#504	—	137	1	no	—
75	#443	—	137	1	no	—
76	#364	—	137	1	no	—
77	#350	—	137	1	no	—
78	#350	—	137	1	no	—
79	#346	—	137	1	no	—
80	#135	—	137	1	no	—
81	#131	—	137	1	no	—
82	#131	—	137	1	no	—
83	M₃₂	—	274	1	no	—
84	M₃₂	—	274	1	no	—
85	M₃₂	—	137	1	no	—
86	M₃₂	—	137	1	no	—
87	C₂×C₁₆	—	274	1	no	—
88	C₂×C₁₆	—	274	1	no	—
89	C₂×C₁₆	—	137	1	no	—
90	C₂×C₁₆	—	137	1	no	—
91	#108	—	137	1	no	—
92	#104	—	137	1	no	—
93	#104	#10587739	1	1	yes	characteristic
94	C₄×C₈	—	137	1	no	—
95	#41	—	137	1	no	—
96	#41	C₂×#10587739	1	1	yes	characteristic
97	Q₁₆	—	137	1	no	—
98	SD₁₆	—	137	1	no	—
99	SD₁₆	—	137	1	no	—
100	D₈	—	137	1	no	—
101	M₁₆	—	274	1	no	—
102	M₁₆	—	274	1	no	—
103	M₁₆	—	137	1	no	—
104	M₁₆	C₂×#10587739	1	1	yes	characteristic
105	C₂×C₈	—	274	1	no	—
106	C₂×C₈	—	274	1	no	—
107	C₂×C₈	—	137	1	no	—
108	C₂×C₈	—	137	1	no	—
109	C₄²	C₂×#10587739	1	1	yes	characteristic
110	C₁₆	—	274	1	no	—
111	C₁₆	—	274	1	no	—
112	C₁₆	—	274	1	no	—
113	C₁₆	—	137	1	no	—
114	C₁₆	—	137	1	no	—
115	Q₈	—	137	1	no	—
116	Q₈	#14555518	1	1	yes	characteristic
117	D₄	—	2	1	no	—
118	D₄	—	274	1	no	—
119	D₄	—	137	1	no	—
120	D₄	#14555518	1	1	yes	characteristic
121	C₂×C₄	—	2	1	no	—
122	C₂×C₄	—	2	1	no	—
123	C₂×C₄	—	274	1	no	—
124	C₂×C₄	C₂²×#10587739	1	1	yes	characteristic
125	C₈	—	2	1	no	—
126	C₈	—	274	1	no	—
127	C₈	—	137	1	no	—
128	C₈	—	137	1	no	—
129	C₈	—	137	1	no	—
130	C₈	—	137	1	no	—
131	C₂²	—	2	1	no	—
132	C₂²	—	274	1	no	—
133	C₂²	#17984175	1	1	yes	characteristic
134	C₄	—	2	1	no	—
135	C₄	—	4	1	no	—
136	C₄	—	274	1	no	—
137	C₄	D₄×#10587739	1	1	yes	characteristic
138	C₄	C₂×#14555518	1	1	yes	characteristic
139	C₂	—	2	1	no	—
140	C₂	—	4	1	no	—
141	C₂	—	548	1	no	—
142	C₂	#22578430	1	1	yes	characteristic

	subgroup	G/H
center	C₄	D₄×#10587739
commutator	C₄×C₁₃₇	C₂²×C₈
Frattini	C₂×C₄	C₂²×#10587739
Fitting	#104×C₁₃₇	C₄
socle	C₂×C₁₃₇	C₈×D₄
prime radical	#17984108	C₄
cosocle	C₂×#11764040	C₂³
hypercenter	#104	#10587739
hypocenter	C₁₃₇	#2171
LCS	G ▷ C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ #104

Complements

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

N	H ≅ G/N	type
1. #22578462	C₂	semidirect
2. #22578461	C₂	semidirect
3. #22578326	C₂	semidirect
4. #22578302	C₂	semidirect
5. #22578301	C₂	semidirect
6. #22578291	C₂	semidirect
71. C₁₃₇	#2171	semidirect

Maximal subgroups

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

subgroup	index
#22578462	2
#22578461	2
#22578326	2
#22578302	2
#22578301	2
#22578291	2
#22577517	2
#2171	137