Finite Group Databasebeta

Subgroups — #17996576

Order 4736.

206 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 49 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	#14558350	C₂	1	1	yes	maximal, characteristic
2	#14557976	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×Dic₂₉₆	C₂	1	1	yes	maximal, characteristic, complemented
4	#14557808	C₂	1	1	yes	maximal, characteristic, complemented
5	#14557350	C₂	1	1	yes	maximal, characteristic, complemented
6	#14557341	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₇×#350	C₂	1	1	yes	maximal, characteristic, complemented
8	#11924583	—	2	1	no	—
9	#11924569	—	2	1	no	—
10	#11924569	—	2	2	no	—
11	#11924566	—	2	1	no	—
12	#11924558	—	2	1	no	—
13	#11924558	C₂²	1	1	yes	characteristic
14	C₂×Dic₁₄₈	—	2	1	no	—
15	C₂×Dic₁₄₈	C₂²	1	1	yes	characteristic
16	C₂×Dic₁₄₈	C₂²	1	1	yes	characteristic
17	Dic₂₉₆	—	2	2	no	—
18	Dic₂₉₆	—	2	1	no	—
19	#11924481	—	2	1	no	—
20	#11924472	C₂²	1	1	yes	characteristic
21	#11924454	—	2	1	no	—
22	#131×C₃₇	C₂²	1	1	yes	characteristic
23	M₃₂×C₃₇	C₂²	1	1	yes	characteristic
24	C₂×C₁₆×C₃₇	C₂²	1	1	yes	characteristic
25	#41×C₃₇	D₄	1	1	yes	characteristic
26	Q₈×D₃₇	—	4	1	no	—
27	#10596964	—	4	2	no	—
28	#10596964	—	4	1	no	—
29	#10596962	—	2	2	no	—
30	#10596962	—	2	1	no	—
31	C₂×Dic₇₄	—	2	2	no	—
32	C₂×Dic₇₄	—	2	1	no	—
33	C₂×Dic₇₄	—	2	1	no	—
34	M₁₆×C₃₇	—	2	1	no	—
35	M₁₆×C₃₇	D₄	1	1	yes	characteristic
36	C₂×C₈×C₃₇	—	2	1	no	—
37	C₂×C₈×C₃₇	C₂³	1	1	yes	characteristic
38	#10596942	—	4	1	no	—
39	#10596940	—	4	1	no	—
40	#10596934	—	2	1	no	—
41	Dic₁₄₈	—	2	2	no	—
42	Dic₁₄₈	—	2	1	no	—
43	Dic₁₄₈	—	2	1	no	—
44	Dic₁₄₈	—	2	1	no	—
45	Dic₁₄₈	D₄	1	1	yes	characteristic
46	Dic₁₄₈	D₄	1	2	yes	—
47	D₂₉₆	D₄	1	1	yes	characteristic
48	#10596930	—	2	2	no	—
49	#10596930	—	2	1	no	—
50	C₁₆×C₃₇	—	2	1	no	—
51	C₁₆×C₃₇	—	2	1	no	—
52	Q₈×C₃₇	D₈	1	1	yes	characteristic
53	D₄×C₃₇	—	2	1	no	—
54	D₄×C₃₇	D₈	1	1	yes	characteristic
55	C₂×C₄×C₃₇	—	2	1	no	—
56	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
57	C₂²⋊D₃₇	—	4	2	no	—
58	C₂²⋊D₃₇	—	4	1	no	—
59	C₂×Dic₃₇	—	4	2	no	—
60	C₂×Dic₃₇	—	4	1	no	—
61	C₂×Dic₃₇	—	4	1	no	—
62	D₁₄₈	—	2	1	no	—
63	C₄×D₃₇	—	4	2	no	—
64	C₄×D₃₇	—	4	1	no	—
65	Dic₇₄	—	2	2	no	—
66	Dic₇₄	—	2	1	no	—
67	Dic₇₄	—	4	2	no	—
68	Dic₇₄	—	4	1	no	—
69	Dic₇₄	—	4	1	no	—
70	Dic₇₄	—	4	1	no	—
71	C₈×C₃₇	—	2	1	no	—
72	C₈×C₃₇	C₂×D₄	1	1	yes	characteristic
73	C₈×C₃₇	C₂×D₄	1	1	yes	characteristic
74	#64436	—	4	1	no	—
75	C₂²×C₃₇	—	2	1	no	—
76	C₂²×C₃₇	AGL(1,ℤ/8)	1	1	yes	characteristic
77	C₂×D₃₇	—	4	1	no	—
78	C₄×C₃₇	—	2	1	no	—
79	C₄×C₃₇	#120	1	1	yes	characteristic
80	C₄×C₃₇	C₂×D₈	1	1	yes	characteristic
81	Dic₃₇	—	4	1	no	—
82	Dic₃₇	—	4	2	no	—
83	Dic₃₇	—	8	1	no	—
84	#2195	—	37	1	no	maximal, Sylow
85	C₂×C₃₇	—	2	1	no	—
86	C₂×C₃₇	—	4	1	no	—
87	C₂×C₃₇	#447	1	1	yes	characteristic
88	D₃₇	—	8	1	no	—
89	#578	—	37	1	no	—
90	#510	—	37	1	no	—
91	C₂×Q₃₂	—	37	1	no	—
92	#473	—	37	1	no	—
93	#362	—	37	1	no	—
94	#359	—	37	1	no	—
95	#350	D₃₇	1	1	yes	characteristic, complemented
96	C₃₇	#2195	1	1	yes	characteristic, complemented, Sylow
97	2₋¹⁺⁴	—	74	1	no	—
98	#137	—	74	2	no	—
99	#137	—	74	1	no	—
100	#137	—	74	1	no	—
101	#135	—	37	1	no	—
102	#135	—	74	1	no	—
103	C₂×Q₁₆	—	37	1	no	—
104	C₂×Q₁₆	—	37	1	no	—
105	C₂×Q₁₆	—	74	1	no	—
106	#131	C₂×D₃₇	1	1	yes	characteristic
107	Q₃₂	—	74	2	no	—
108	Q₃₂	—	74	1	no	—
109	SD₃₂	—	74	1	no	—
110	M₃₂	C₂×D₃₇	1	1	yes	characteristic
111	C₂×C₁₆	C₂×D₃₇	1	1	yes	characteristic
112	#108	—	37	1	no	—
113	#101	—	74	1	no	—
114	#41	—	74	2	no	—
115	#41	—	74	1	no	—
116	#41	—	148	2	no	—
117	#41	—	148	1	no	—
118	#41	D₁₄₈	1	1	yes	characteristic
119	C₂×Q₈	—	74	2	no	—
120	C₂×Q₈	—	74	1	no	—
121	C₂×Q₈	—	74	1	no	—
122	C₂×Q₈	—	148	1	no	—
123	Q₁₆	—	37	2	no	—
124	Q₁₆	—	37	1	no	—
125	Q₁₆	—	74	2	no	—
126	Q₁₆	—	74	1	no	—
127	Q₁₆	—	74	1	no	—
128	Q₁₆	—	74	1	no	—
129	Q₁₆	—	148	1	no	—
130	SD₁₆	—	74	2	no	—
131	SD₁₆	—	74	1	no	—
132	SD₁₆	—	148	1	no	—
133	D₈	—	37	1	no	—
134	M₁₆	—	2	1	no	—
135	M₁₆	—	74	1	no	—
136	M₁₆	D₁₄₈	1	1	yes	characteristic
137	C₂×C₈	—	2	1	no	—
138	C₂×C₈	C₂²×D₃₇	1	1	yes	characteristic
139	C₁₆	—	2	1	no	—
140	C₁₆	—	2	1	no	—
141	Q₈	—	74	2	no	—
142	Q₈	—	74	1	no	—
143	Q₈	—	148	2	no	—
144	Q₈	—	148	1	no	—
145	Q₈	—	148	1	no	—
146	Q₈	—	148	1	no	—
147	Q₈	D₂₉₆	1	1	yes	characteristic
148	D₄	—	2	1	no	—
149	D₄	—	74	1	no	—
150	D₄	—	148	2	no	—
151	D₄	—	148	1	no	—
152	D₄	D₂₉₆	1	1	yes	characteristic
153	C₂×C₄	—	2	1	no	—
154	C₂×C₄	—	148	2	no	—
155	C₂×C₄	—	148	1	no	—
156	C₂×C₄	—	148	2	no	—
157	C₂×C₄	—	148	1	no	—
158	C₂×C₄	—	148	1	no	—
159	C₂×C₄	C₂×D₁₄₈	1	1	yes	characteristic
160	C₈	—	2	1	no	—
161	C₈	—	148	1	no	—
162	C₈	D₄×D₃₇	1	1	yes	characteristic
163	C₈	D₄×D₃₇	1	1	yes	characteristic
164	C₂²	—	2	1	no	—
165	C₂²	—	148	1	no	—
166	C₂²	#11924564	1	1	yes	characteristic
167	C₄	—	2	1	no	—
168	C₄	—	296	1	no	—
169	C₄	—	148	2	no	—
170	C₄	—	148	1	no	—
171	C₄	#11924506	1	1	yes	characteristic
172	C₄	C₂×D₂₉₆	1	1	yes	characteristic
173	C₂	—	2	1	no	—
174	C₂	—	4	1	no	—
175	C₂	—	296	1	no	—
176	C₂	#14557669	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#14557669
commutator, cosocle	C₂×C₈×C₃₇	C₂³
Frattini	C₂×C₈	C₂²×D₃₇
Fitting	C₃₇×#350	C₂
socle	C₂×C₃₇	#447
prime radical	#14558350	C₂
hypercenter	#350	D₃₇
hypocenter	C₃₇	#2195
LCS	G ▷ C₂×C₈×C₃₇ ▷ C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #131 ◁ #350

Complements

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

N	H ≅ G/N	type
2. #14557976	C₂	semidirect
3. C₂×Dic₂₉₆	C₂	semidirect
4. #14557808	C₂	semidirect
5. #14557350	C₂	semidirect
6. #14557341	C₂	semidirect
7. C₃₇×#350	C₂	semidirect
95. #350	D₃₇	semidirect
96. C₃₇	#2195	semidirect

Maximal subgroups

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

subgroup	index
#14558350	2
#14557976	2
C₂×Dic₂₉₆	2
#14557808	2
#14557350	2
#14557341	2
C₃₇×#350	2
#2195	37