Finite Group Databasebeta

Subgroups — #16821119

Order 3712.

220 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 89 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×#11745263	C₂	1	1	yes	maximal, characteristic
2	C₂×#11745160	C₂	1	1	yes	maximal, characteristic, complemented
3	#14270946	C₂	1	4	yes	maximal, complemented
4	C₂×C₂₉×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#90805	C₄	1	1	yes	characteristic
6	C₂²×C₈×C₂₉	C₂²	1	1	yes	characteristic
7	C₂×#90777	C₄	1	1	yes	characteristic
8	#11745263	—	2	1	no	—
9	#11745263	—	2	1	no	—
10	#11745263	C₄	1	2	yes	—
11	#11745263	C₂²	1	2	yes	—
12	C₂×#90772	—	2	1	no	—
13	C₂×C₈×D₂₉	—	2	1	no	—
14	C₂₉×M₃₂	—	2	1	no	—
15	C₂₉×M₃₂	—	2	1	no	—
16	C₂×C₁₆×C₂₉	C₂²	1	2	yes	complemented
17	#11745160	—	2	1	no	—
18	#11745160	C₂²	1	2	yes	complemented
19	C₂×#90768	—	2	1	no	—
20	C₂²×C₄×C₂₉	C₂×C₄	1	1	yes	characteristic
21	C₂×C₂²⋊D₂₉	—	2	1	no	—
22	#90805	—	2	1	no	—
23	#90805	—	2	1	no	—
24	#90805	C₈	1	2	yes	—
25	#90805	C₂×C₄	1	2	yes	—
26	C₂×D₁₁₆	C₈	1	1	yes	characteristic
27	C₂×C₄×D₂₉	—	2	1	no	—
28	C₂×Dic₅₈	C₈	1	1	yes	characteristic
29	C₂×C₈×C₂₉	C₂×C₄	1	1	yes	characteristic
30	C₂×C₈×C₂₉	D₄	1	1	yes	characteristic
31	C₂×C₈×C₂₉	D₄	1	1	yes	characteristic
32	C₂×C₈×C₂₉	D₄	1	1	yes	characteristic
33	C₂×C₈×C₂₉	D₄	1	1	yes	characteristic
34	C₂×C₈×C₂₉	C₂³	1	1	yes	characteristic
35	#90777	—	2	1	no	—
36	#90777	C₂×C₄	1	2	yes	—
37	C₂×#6597	—	2	1	no	—
38	#90772	—	2	2	no	—
39	#90772	—	2	2	no	—
40	C₈×D₂₉	—	2	2	no	—
41	C₈×D₂₉	—	2	2	no	—
42	C₁₆×C₂₉	—	2	2	no	—
43	#90768	—	2	2	no	—
44	C₂³×C₂₉	M₁₆	1	1	yes	characteristic
45	C₂²×D₂₉	—	2	1	no	—
46	C₂×C₄×C₂₉	C₂²⋊C₄	1	1	yes	characteristic
47	C₂×C₄×C₂₉	C₂²⋊C₄	1	1	yes	characteristic
48	C₂×C₄×C₂₉	C₂×C₈	1	1	yes	characteristic
49	C₂×C₄×C₂₉	C₂×C₈	1	1	yes	characteristic
50	C₂×C₄×C₂₉	M₁₆	1	1	yes	characteristic
51	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
52	C₂²⋊D₂₉	—	2	2	no	—
53	C₂²⋊D₂₉	—	2	2	no	—
54	C₂×Dic₂₉	—	2	1	no	—
55	D₁₁₆	—	2	1	no	—
56	D₁₁₆	C₂×C₈	1	2	yes	—
57	C₄×D₂₉	—	2	2	no	—
58	C₄×D₂₉	—	2	2	no	—
59	Dic₅₈	—	2	1	no	—
60	Dic₅₈	C₂×C₈	1	2	yes	—
61	C₈×C₂₉	C₂²⋊C₄	1	1	yes	characteristic
62	C₈×C₂₉	C₂²⋊C₄	1	1	yes	characteristic
63	C₈×C₂₉	C₂×D₄	1	1	yes	characteristic
64	C₈×C₂₉	C₂×D₄	1	1	yes	characteristic
65	#6597	—	2	2	no	—
66	#2117	—	29	1	no	maximal, Sylow
67	C₂²×C₂₉	—	4	1	no	—
68	C₂²×C₂₉	C₂²⋊C₈	1	1	yes	characteristic
69	C₂²×C₂₉	C₂²⋊C₈	1	1	yes	characteristic
70	C₂²×C₂₉	C₂×M₁₆	1	1	yes	characteristic
71	C₂×D₂₉	—	2	2	no	—
72	C₂×D₂₉	—	4	2	no	—
73	C₄×C₂₉	C₂²⋊C₈	1	1	yes	characteristic
74	C₄×C₂₉	C₂²⋊C₈	1	1	yes	characteristic
75	C₄×C₂₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
76	C₄×C₂₉	C₂²×C₈	1	1	yes	characteristic
77	Dic₂₉	—	2	2	no	—
78	C₂×#131	—	29	1	no	—
79	C₂×M₃₂	—	29	1	no	—
80	C₂×M₃₂	D₂₉	1	1	yes	characteristic, complemented
81	#350	—	29	4	no	—
82	C₂×C₂₉	—	2	1	no	—
83	C₂×C₂₉	—	2	1	no	—
84	C₂×C₂₉	—	2	1	no	—
85	C₂×C₂₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
86	D₂₉	—	4	2	no	—
87	C₂×#41	—	29	1	no	—
88	#131	—	29	2	no	—
89	#131	—	29	2	no	—
90	#131	—	58	1	no	—
91	#131	—	58	1	no	—
92	C₂×M₁₆	—	29	1	no	—
93	C₂×M₁₆	—	58	1	no	—
94	C₂²×C₈	—	58	1	no	—
95	C₂²×C₈	C₂×D₂₉	1	1	yes	characteristic
96	M₃₂	—	2	1	no	—
97	M₃₂	—	2	1	no	—
98	M₃₂	—	29	2	no	—
99	M₃₂	—	58	1	no	—
100	C₂×C₁₆	—	58	1	no	—
101	C₂×C₁₆	C₂×D₂₉	1	2	yes	complemented
102	C₂₉	#2117	1	1	yes	characteristic, complemented, Sylow
103	#41	—	29	2	no	—
104	#41	—	29	2	no	—
105	#41	—	58	1	no	—
106	#41	—	58	1	no	—
107	C₂×Q₈	—	29	1	no	—
108	C₂×D₄	—	29	1	no	—
109	C₂×D₄	—	58	1	no	—
110	C₂²×C₄	—	58	1	no	—
111	C₂²×C₄	C₄×D₂₉	1	1	yes	characteristic
112	M₁₆	—	29	2	no	—
113	M₁₆	—	58	1	no	—
114	M₁₆	—	58	2	no	—
115	M₁₆	—	58	2	no	—
116	C₂×C₈	—	58	1	no	—
117	C₂×C₈	—	58	2	no	—
118	C₂×C₈	—	58	2	no	—
119	C₂×C₈	C₄×D₂₉	1	1	yes	characteristic
120	C₂×C₈	D₁₁₆	1	1	yes	characteristic
121	C₂×C₈	D₁₁₆	1	1	yes	characteristic
122	C₂×C₈	C₂²⋊D₂₉	1	1	yes	characteristic
123	C₂×C₈	C₂²⋊D₂₉	1	1	yes	characteristic
124	C₂×C₈	C₂²×D₂₉	1	1	yes	characteristic
125	C₁₆	—	2	2	no	—
126	C₁₆	—	58	2	no	—
127	C₂³	—	58	1	no	—
128	C₂³	#90772	1	1	yes	characteristic
129	Q₈	—	29	2	no	—
130	Q₈	—	58	1	no	—
131	D₄	—	29	2	no	—
132	D₄	—	58	2	no	—
133	D₄	—	58	2	no	—
134	D₄	—	58	1	no	—
135	C₂×C₄	—	58	1	no	—
136	C₂×C₄	—	58	2	no	—
137	C₂×C₄	—	58	2	no	—
138	C₂×C₄	C₈×D₂₉	1	1	yes	characteristic
139	C₂×C₄	C₈×D₂₉	1	1	yes	characteristic
140	C₂×C₄	#90772	1	1	yes	characteristic
141	C₂×C₄	#90781	1	1	yes	characteristic
142	C₂×C₄	#90781	1	1	yes	characteristic
143	C₂×C₄	C₂×C₄×D₂₉	1	1	yes	characteristic
144	C₈	—	58	2	no	—
145	C₈	#90781	1	1	yes	characteristic
146	C₈	#90781	1	1	yes	characteristic
147	C₈	C₂×D₁₁₆	1	1	yes	characteristic
148	C₈	C₂×C₂²⋊D₂₉	1	1	yes	characteristic
149	C₂²	—	4	1	no	—
150	C₂²	—	58	2	no	—
151	C₂²	—	116	2	no	—
152	C₂²	#11745168	1	1	yes	characteristic
153	C₂²	#11745168	1	1	yes	characteristic
154	C₂²	C₂×#90772	1	1	yes	characteristic
155	C₄	—	58	2	no	—
156	C₄	#11745168	1	1	yes	characteristic
157	C₄	#11745168	1	1	yes	characteristic
158	C₄	C₂×C₈×D₂₉	1	1	yes	characteristic
159	C₄	C₂×#90781	1	1	yes	characteristic
160	C₂	—	2	1	no	—
161	C₂	—	2	1	no	—
162	C₂	—	2	1	no	—
163	C₂	—	116	2	no	—
164	C₂	C₂×#11745168	1	1	yes	characteristic

	subgroup	G/H
center	C₈	C₂×D₁₁₆
commutator	C₄×C₂₉	C₂²×C₈
Frattini	C₂×C₈	C₂²×D₂₉
Fitting	C₂×C₂₉×M₃₂	C₂
socle	C₂×C₂₉	C₂×C₂²⋊C₈
prime radical	C₂×#90805	C₄
cosocle	C₂×C₈×C₂₉	C₂³
hypercenter	C₂×M₃₂	D₂₉
hypocenter	C₂₉	#2117
LCS	G ▷ C₄×C₂₉ ▷ C₂×C₂₉ ▷ C₂₉
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ C₂×M₃₂

Complements

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

N	H ≅ G/N	type
2. C₂×#11745160	C₂	semidirect
3. #14270946	C₂	semidirect
4. C₂×C₂₉×M₃₂	C₂	semidirect
16. C₂×C₁₆×C₂₉	C₂²	semidirect
18. #11745160	C₂²	semidirect
80. C₂×M₃₂	D₂₉	semidirect
101. C₂×C₁₆	C₂×D₂₉	semidirect
102. C₂₉	#2117	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11745263	2
C₂×#11745160	2
#14270946	2
C₂×C₂₉×M₃₂	2
#2117	29