Finite Group Databasebeta

Subgroups — #31796999

Order 17536.

184 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 102 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×#17984163	C₂	1	1	yes	maximal, characteristic
2	C₂×#17984161	C₂	1	1	yes	maximal, characteristic
3	#22578292	C₂	1	1	yes	maximal, characteristic, complemented
4	#22578291	C₂	1	1	yes	maximal, characteristic, complemented
5	#22578289	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×#14555513	C₂	1	1	yes	maximal, characteristic, complemented
7	#22577360	C₂	1	1	yes	maximal, characteristic
8	#17984163	C₄	1	4	yes	complemented
9	#17984161	C₄	1	4	yes	complemented
10	C₂×#14555513	C₂²	1	1	yes	characteristic
11	C₂×#14555513	C₂²	1	1	yes	characteristic
12	C₂×#14555513	C₂²	1	1	yes	characteristic
13	C₂×#14555513	C₂²	1	1	yes	characteristic
14	C₂×#14555493	C₄	1	1	yes	characteristic
15	C₂×C₈×D₁₃₇	C₂²	1	1	yes	characteristic
16	#17984045	C₄	1	1	yes	characteristic
17	#17983998	C₄	1	1	yes	characteristic
18	C₄×#11764040	C₂²	1	1	yes	characteristic
19	C₈×Dic₁₃₇	C₂²	1	1	yes	characteristic
20	#97×C₁₃₇	C₄	1	1	yes	characteristic
21	#14555513	C₂×C₄	1	2	yes	—
22	#14555513	C₂×C₄	1	2	yes	—
23	#14555513	C₂×C₄	1	2	yes	—
24	#14555513	C₂×C₄	1	2	yes	—
25	C₂×C₄×D₁₃₇	C₂×C₄	1	1	yes	characteristic
26	#14555493	C₈	1	4	yes	—
27	C₂×#11764040	C₂×C₄	1	1	yes	characteristic
28	C₂×#11764040	C₂³	1	1	yes	characteristic
29	C₈×D₁₃₇	C₂×C₄	1	2	yes	—
30	C₈×D₁₃₇	C₂×C₄	1	2	yes	—
31	#14555489	C₈	1	2	yes	—
32	#14555487	C₈	1	2	yes	—
33	C₄×Dic₁₃₇	C₂×C₄	1	1	yes	characteristic
34	C₂×C₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
35	C₂×C₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
36	C₄²×C₁₃₇	C₂×C₄	1	1	yes	characteristic
37	C₂²×D₁₃₇	C₂×C₈	1	1	yes	characteristic
38	C₂×Dic₁₃₇	C₂×C₈	1	2	yes	—
39	C₂×Dic₁₃₇	C₂×C₈	1	1	yes	characteristic
40	C₄×D₁₃₇	C₄²	1	2	yes	—
41	C₄×D₁₃₇	C₂×C₈	1	2	yes	—
42	#11764040	C₂×C₈	1	2	yes	—
43	#11764040	C₂²×C₄	1	1	yes	characteristic
44	#11764040	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₁₃₇	C₂×C₈	1	2	yes	—
46	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
47	C₈×C₁₃₇	C₄²	1	2	yes	—
48	C₈×C₁₃₇	C₂×C₈	1	2	yes	—
49	C₂²×C₁₃₇	C₂²×C₈	1	1	yes	characteristic
50	C₂×D₁₃₇	—	2	2	no	—
51	C₂×D₁₃₇	C₄×C₈	1	2	yes	—
52	C₄×C₁₃₇	—	2	2	no	—
53	C₄×C₁₃₇	C₂×C₄²	1	1	yes	characteristic
54	C₄×C₁₃₇	C₂²×C₈	1	1	yes	characteristic
55	Dic₁₃₇	—	2	2	no	—
56	Dic₁₃₇	C₄×C₈	1	2	yes	—
57	C₂×C₁₃₇	C₂×C₄×C₈	1	1	yes	characteristic
58	C₂×C₁₃₇	#504	1	2	yes	—
59	D₁₃₇	—	2	2	no	—
60	C₁₃₇	#2108	1	1	yes	characteristic, complemented, Sylow
61	#2108	—	137	1	no	maximal, Sylow
62	C₂×M₃₂	—	137	1	no	—
63	C₂²×C₁₆	—	137	1	no	—
64	#405	—	137	1	no	—
65	#346	—	137	1	no	—
66	#346	—	137	1	no	—
67	C₄×C₁₆	—	137	1	no	—
68	C₄×C₁₆	—	137	1	no	—
69	C₂×M₁₆	—	137	1	no	—
70	C₂²×C₈	—	137	1	no	—
71	#117	—	137	1	no	—
72	M₃₂	—	137	4	no	—
73	C₂×C₁₆	—	137	1	no	—
74	C₂×C₁₆	—	137	1	no	—
75	C₂×C₁₆	—	137	1	no	—
76	C₂×C₁₆	—	137	1	no	—
77	C₂×C₁₆	—	137	4	no	—
78	#97	—	137	1	no	—
79	#97	#10587739	1	1	yes	characteristic
80	C₄×C₈	—	137	1	no	—
81	C₄×C₈	—	137	1	no	—
82	C₂²×C₄	—	137	1	no	—
83	M₁₆	—	137	4	no	—
84	C₂×C₈	—	137	1	no	—
85	C₂×C₈	—	137	1	no	—
86	C₂×C₈	—	137	2	no	—
87	C₂×C₈	—	137	2	no	—
88	C₂×C₈	C₂×#10587739	1	1	yes	characteristic
89	C₂×C₈	C₂×#10587739	1	1	yes	characteristic
90	C₄⋊C₄	—	137	2	no	—
91	C₂²⋊C₄	—	137	2	no	—
92	C₄²	—	137	1	no	—
93	C₄²	C₂×#10587739	1	1	yes	characteristic
94	C₁₆	—	137	2	no	—
95	C₁₆	—	137	2	no	—
96	C₁₆	—	137	2	no	—
97	C₁₆	—	137	2	no	—
98	C₂³	—	137	1	no	—
99	C₂×C₄	—	137	2	no	—
100	C₂×C₄	—	137	1	no	—
101	C₂×C₄	—	137	2	no	—
102	C₂×C₄	—	137	2	no	—
103	C₂×C₄	#14555518	1	2	yes	—
104	C₂×C₄	C₂²×#10587739	1	1	yes	characteristic
105	C₈	—	137	1	no	—
106	C₈	—	137	1	no	—
107	C₈	—	137	2	no	—
108	C₈	C₄×#10587739	1	2	yes	—
109	C₈	#14555518	1	2	yes	—
110	C₂²	—	274	2	no	—
111	C₂²	—	137	2	no	—
112	C₂²	C₂×#14555518	1	1	yes	characteristic
113	C₄	—	2	2	no	—
114	C₄	—	274	2	no	—
115	C₄	—	137	2	no	—
116	C₄	C₂×C₄×#10587739	1	1	yes	characteristic
117	C₄	C₂×#14555518	1	1	yes	characteristic
118	C₂	—	274	2	no	—
119	C₂	C₄×#14555518	1	1	yes	characteristic
120	C₂	#22578462	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×#10587739
commutator	C₂×C₁₃₇	C₂×C₄×C₈
Fitting	#97×C₁₃₇	C₄
socle	C₂²×C₁₃₇	C₂²×C₈
prime radical	C₂²×D₁₃₇	C₂×C₈
cosocle	C₂×#11764040	C₂³
hypercenter	#97	#10587739
hypocenter	C₁₃₇	#2108
LCS	G ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂×C₄ ◁ #97

Complements

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

N	H ≅ G/N	type
3. #22578292	C₂	semidirect
4. #22578291	C₂	semidirect
5. #22578289	C₂	semidirect
6. C₄×#14555513	C₂	semidirect
8. #17984163	C₄	semidirect
9. #17984161	C₄	semidirect
60. C₁₃₇	#2108	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17984163	2
C₂×#17984161	2
#22578292	2
#22578291	2
#22578289	2
C₄×#14555513	2
#22577360	2
#2108	137