Finite Group Databasebeta

Subgroups — #49175192

Order 37504.

142 conjugacy classes of subgroups; only the 44 normal subgroups are shown.

normal subgroups only

This group has 142 conjugacy classes of subgroups — too many to draw. Only its 44 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	#34037175	C₂	1	1	yes	maximal, characteristic
2	#34037169	C₂	1	1	yes	maximal, characteristic, complemented
3	#34037053	C₂	1	1	yes	maximal, characteristic, complemented
4	#34037048	C₂	1	1	yes	maximal, characteristic, complemented
5	#34037046	C₂	1	1	yes	maximal, characteristic
6	#34037019	C₂	1	1	yes	maximal, characteristic, complemented
7	#34036571	C₂	1	1	yes	maximal, characteristic
8	#22627712	—	2	1	no	—
9	Q₈×#11924078	—	2	1	no	—
10	#22627699	C₂²	1	1	yes	characteristic
11	#22627694	C₂²	1	1	yes	characteristic
12	#22627692	C₂²	1	1	yes	characteristic
13	#22627689	—	2	1	no	—
14	#22627687	—	2	1	no	—
15	#22627678	C₂²	1	1	yes	characteristic
16	#22627647	C₂²	1	1	yes	characteristic
17	Q₁₆×D₂₉₃	C₂²	1	1	yes	characteristic
18	#22627570	C₄	1	1	yes	characteristic, complemented
19	#22627569	C₄	1	1	yes	characteristic, complemented
20	#22627567	C₄	1	1	yes	characteristic, complemented
21	#22627563	C₂²	1	1	yes	characteristic
22	Q₃₂×C₂₉₃	C₄	1	1	yes	characteristic, complemented
23	#17989812	—	2	1	no	—
24	#17989812	—	4	1	no	—
25	D₂₉₃⋊C₈	—	4	1	no	—
26	D₂₉₃⋊C₈	D₄	1	1	yes	characteristic
27	C₄⋊#11924078	—	2	1	no	—
28	C₄⋊#11924078	—	4	1	no	—
29	C₄×#11924078	—	4	1	no	—
30	C₄×#11924078	D₄	1	1	yes	characteristic
31	#17989805	—	2	1	no	—
32	#17989803	—	2	1	no	—
33	Q₈×D₂₉₃	—	2	1	no	—
34	Dic₁₁₇₂	C₂×C₄	1	1	yes	characteristic
35	#17989792	—	2	1	no	—
36	#17989789	—	2	1	no	—
37	D₂₃₄₄	C₂×C₄	1	1	yes	characteristic
38	C₈×D₂₉₃	C₂³	1	1	yes	characteristic
39	#17989777	C₂×C₄	1	1	yes	characteristic
40	Q₁₆×C₂₉₃	C₂×C₄	1	1	yes	characteristic
41	Q₁₆×C₂₉₃	C₂×C₄	1	1	yes	characteristic
42	C₁₆×C₂₉₃	C₂×C₄	1	1	yes	characteristic
43	C₂×#11924078	—	2	1	no	—
44	C₂×#11924078	—	4	1	no	—
45	#14556880	—	2	1	no	—
46	#14556880	—	4	1	no	—
47	Dic₅₈₆	—	2	1	no	—
48	Dic₅₈₆	—	4	1	no	—
49	D₁₁₇₂	—	2	1	no	—
50	D₁₁₇₂	—	4	1	no	—
51	C₄×D₂₉₃	—	4	1	no	—
52	C₄×D₂₉₃	—	4	1	no	—
53	C₄×D₂₉₃	C₂×D₄	1	1	yes	characteristic
54	#14556873	#41	1	1	yes	characteristic
55	Q₈×C₂₉₃	—	2	1	no	—
56	Q₈×C₂₉₃	—	2	1	no	—
57	C₈×C₂₉₃	C₂²×C₄	1	1	yes	characteristic
58	C₂×D₂₉₃	—	4	1	no	—
59	C₂×D₂₉₃	AGL(1,ℤ/8)	1	1	yes	characteristic
60	#11924078	—	4	1	no	—
61	#11924078	—	8	1	no	—
62	C₄×C₂₉₃	—	4	1	no	—
63	C₄×C₂₉₃	—	4	1	no	—
64	C₄×C₂₉₃	C₄×D₄	1	1	yes	characteristic
65	Dic₂₉₃	—	4	1	no	—
66	Dic₂₉₃	AGL(1,ℤ/8)	1	1	yes	characteristic
67	C₂×C₂₉₃	#442	1	1	yes	characteristic
68	D₂₉₃	—	2	1	no	—
69	D₂₉₃	—	8	1	no	—
70	C₂₉₃	#2180	1	1	yes	characteristic, complemented, Sylow
71	#2180	—	293	1	no	maximal, Sylow
72	#510	—	293	1	no	—
73	#444	—	293	1	no	—
74	#441	—	293	1	no	—
75	#365	—	293	1	no	—
76	#362	—	293	1	no	—
77	#360	—	293	1	no	—
78	#347	—	293	1	no	—
79	#135	—	293	1	no	—
80	C₂×Q₁₆	—	293	1	no	—
81	#131	—	586	1	no	—
82	C₄×Q₈	—	586	1	no	—
83	Q₃₂	—	293	1	no	—
84	Q₃₂	#11924078	1	1	yes	characteristic, complemented
85	SD₃₂	—	293	1	no	—
86	SD₃₂	—	293	1	no	—
87	M₃₂	—	293	1	no	—
88	M₃₂	—	293	1	no	—
89	#108	—	293	1	no	—
90	#106	—	293	1	no	—
91	#104	—	586	1	no	—
92	#103	—	586	1	no	—
93	#97	—	293	1	no	—
94	#41	—	586	1	no	—
95	C₂×Q₈	—	586	1	no	—
96	Q₁₆	—	293	1	no	—
97	Q₁₆	—	586	1	no	—
98	Q₁₆	C₂×#11924078	1	1	yes	characteristic
99	Q₁₆	C₂×#11924078	1	1	yes	characteristic
100	SD₁₆	—	586	1	no	—
101	D₈	—	293	1	no	—
102	M₁₆	—	586	1	no	—
103	M₁₆	—	1172	1	no	—
104	C₂×C₈	—	293	1	no	—
105	C₂×C₈	—	293	1	no	—
106	C₂×C₈	—	1172	1	no	—
107	C₄⋊C₄	—	586	1	no	—
108	C₄⋊C₄	—	1172	1	no	—
109	C₄²	—	293	1	no	—
110	C₄²	—	1172	1	no	—
111	C₁₆	—	293	1	no	—
112	C₁₆	—	586	1	no	—
113	C₁₆	C₂×#11924078	1	1	yes	characteristic
114	Q₈	—	2	1	no	—
115	Q₈	—	2	1	no	—
116	Q₈	—	586	1	no	—
117	Q₈	—	1172	1	no	—
118	D₄	—	586	1	no	—
119	D₄	—	1172	1	no	—
120	C₂×C₄	—	293	1	no	—
121	C₂×C₄	—	586	1	no	—
122	C₂×C₄	—	1172	1	no	—
123	C₂×C₄	—	1172	1	no	—
124	C₂×C₄	—	1172	1	no	—
125	C₈	—	293	1	no	—
126	C₈	—	586	1	no	—
127	C₈	—	1172	1	no	—
128	C₈	C₂²×#11924078	1	1	yes	characteristic
129	C₂²	—	293	1	no	—
130	C₂²	—	1172	1	no	—
131	C₄	—	4	1	no	—
132	C₄	—	4	1	no	—
133	C₄	—	293	1	no	—
134	C₄	—	2344	1	no	—
135	C₄	—	1172	1	no	—
136	C₄	—	1172	1	no	—
137	C₄	D₄×#11924078	1	1	yes	characteristic
138	C₂	—	2344	1	no	—
139	C₂	—	586	1	no	—
140	C₂	#34037170	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#34037170
commutator	C₈×C₂₉₃	C₂²×C₄
Frattini	C₈	C₂²×#11924078
Fitting	Q₃₂×C₂₉₃	C₄
socle	C₂×C₂₉₃	#442
prime radical	#22627647	C₂²
cosocle	C₈×D₂₉₃	C₂³
hypercenter	Q₃₂	#11924078
hypocenter	C₂₉₃	#2180
LCS	G ▷ C₈×C₂₉₃ ▷ C₄×C₂₉₃ ▷ C₂×C₂₉₃ ▷ C₂₉₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ Q₃₂

Complements

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

N	H ≅ G/N	type
2. #34037169	C₂	semidirect
3. #34037053	C₂	semidirect
4. #34037048	C₂	semidirect
6. #34037019	C₂	semidirect
18. #22627570	C₄	semidirect
19. #22627569	C₄	semidirect
20. #22627567	C₄	semidirect
22. Q₃₂×C₂₉₃	C₄	semidirect
70. C₂₉₃	#2180	semidirect
84. Q₃₂	#11924078	semidirect

Maximal subgroups

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

subgroup	index
#34037175	2
#34037169	2
#34037053	2
#34037048	2
#34037046	2
#34037019	2
#34036571	2
#2180	293