Finite Group Databasebeta

Subgroups — #16821221

Order 3712.

220 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 81 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×#11745266	C₂	1	1	yes	maximal, characteristic
2	C₂×#11745167	C₂	1	1	yes	maximal, characteristic, complemented
3	#14270971	C₂	1	4	yes	maximal, complemented
4	C₂×C₂₉×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#90805	—	2	1	no	—
6	C₂²×C₈×C₂₉	C₂²	1	1	yes	characteristic
7	C₂×#90777	—	2	1	no	—
8	C₂×Dic₁₁₆	C₄	1	1	yes	characteristic
9	#11745266	—	2	1	no	—
10	#11745266	—	2	1	no	—
11	#11745266	C₄	1	2	yes	—
12	#11745266	C₂²	1	2	yes	—
13	C₂×D₂₃₂	C₄	1	1	yes	characteristic
14	C₂×#90773	—	2	1	no	—
15	C₂₉×M₃₂	—	2	1	no	—
16	C₂₉×M₃₂	—	2	1	no	—
17	C₂×C₁₆×C₂₉	C₂²	1	2	yes	complemented
18	#11745167	—	2	1	no	—
19	#11745167	C₂²	1	2	yes	complemented
20	C₂²×C₄×C₂₉	D₄	1	1	yes	characteristic
21	C₂×C₂²⋊D₂₉	—	4	1	no	—
22	#90805	—	2	2	no	—
23	#90805	—	2	2	no	—
24	#90805	—	4	1	no	—
25	#90805	—	4	1	no	—
26	C₂×D₁₁₆	—	2	1	no	—
27	C₂×C₄×D₂₉	—	4	1	no	—
28	C₂×Dic₅₈	—	2	1	no	—
29	C₂×C₈×C₂₉	C₂×C₄	1	1	yes	characteristic
30	C₂×C₈×C₂₉	C₂×C₄	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	2	no	—
36	#90777	—	4	1	no	—
37	C₂×#6597	—	4	1	no	—
38	Dic₁₁₆	—	2	1	no	—
39	Dic₁₁₆	C₂×C₄	1	2	yes	—
40	D₂₃₂	—	2	1	no	—
41	D₂₃₂	C₂×C₄	1	2	yes	—
42	#90773	—	2	2	no	—
43	#90773	—	2	2	no	—
44	C₁₆×C₂₉	—	2	2	no	—
45	C₂³×C₂₉	SD₁₆	1	1	yes	characteristic
46	C₂²×D₂₉	—	4	1	no	—
47	C₂×C₄×C₂₉	C₂²⋊C₄	1	1	yes	characteristic
48	C₂×C₄×C₂₉	C₂²⋊C₄	1	1	yes	characteristic
49	C₂×C₄×C₂₉	D₈	1	1	yes	characteristic
50	C₂×C₄×C₂₉	D₈	1	1	yes	characteristic
51	C₂×C₄×C₂₉	SD₁₆	1	1	yes	characteristic
52	C₂×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
53	C₂²⋊D₂₉	—	4	2	no	—
54	C₂²⋊D₂₉	—	4	2	no	—
55	C₂×Dic₂₉	—	4	1	no	—
56	D₁₁₆	—	2	2	no	—
57	D₁₁₆	—	4	1	no	—
58	C₄×D₂₉	—	4	2	no	—
59	C₄×D₂₉	—	4	2	no	—
60	Dic₅₈	—	2	2	no	—
61	Dic₅₈	—	4	1	no	—
62	C₈×C₂₉	C₂²⋊C₄	1	1	yes	characteristic
63	C₈×C₂₉	C₂²⋊C₄	1	1	yes	characteristic
64	C₈×C₂₉	C₂²×C₄	1	1	yes	characteristic
65	C₈×C₂₉	C₂×D₄	1	1	yes	characteristic
66	#6597	—	4	2	no	—
67	#2143	—	29	1	no	maximal, Sylow
68	C₂²×C₂₉	—	4	1	no	—
69	C₂²×C₂₉	#102	1	1	yes	characteristic
70	C₂²×C₂₉	#102	1	1	yes	characteristic
71	C₂²×C₂₉	C₂×SD₁₆	1	1	yes	characteristic
72	C₂×D₂₉	—	4	2	no	—
73	C₂×D₂₉	—	8	2	no	—
74	C₄×C₂₉	#102	1	1	yes	characteristic
75	C₄×C₂₉	#102	1	1	yes	characteristic
76	C₄×C₂₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
77	C₄×C₂₉	C₂×D₈	1	1	yes	characteristic
78	Dic₂₉	—	4	2	no	—
79	C₂×#135	—	29	1	no	—
80	C₂×M₃₂	D₂₉	1	1	yes	characteristic, complemented
81	C₂×#108	—	29	1	no	—
82	#359	—	29	4	no	—
83	C₂×C₂₉	—	2	1	no	—
84	C₂×C₂₉	—	2	1	no	—
85	C₂×C₂₉	—	2	1	no	—
86	C₂×C₂₉	C₂×#102	1	1	yes	characteristic
87	D₂₉	—	8	2	no	—
88	C₂×#41	—	58	1	no	—
89	#135	—	29	2	no	—
90	#135	—	29	2	no	—
91	#135	—	58	1	no	—
92	#135	—	58	1	no	—
93	C₂×Q₁₆	—	29	1	no	—
94	C₂×SD₁₆	—	58	1	no	—
95	C₂×D₈	—	29	1	no	—
96	C₂×M₁₆	—	58	1	no	—
97	C₂²×C₈	C₂×D₂₉	1	1	yes	characteristic
98	M₃₂	—	2	1	no	—
99	M₃₂	—	2	1	no	—
100	C₂×C₁₆	C₂×D₂₉	1	2	yes	complemented
101	#108	—	29	2	no	—
102	#108	—	58	1	no	—
103	C₂₉	#2143	1	1	yes	characteristic, complemented, Sylow
104	#41	—	58	2	no	—
105	#41	—	58	2	no	—
106	#41	—	116	1	no	—
107	#41	—	116	1	no	—
108	C₂×Q₈	—	58	1	no	—
109	C₂×D₄	—	58	1	no	—
110	C₂×D₄	—	116	1	no	—
111	C₂²×C₄	—	116	1	no	—
112	C₂²×C₄	D₁₁₆	1	1	yes	characteristic
113	Q₁₆	—	29	2	no	—
114	Q₁₆	—	58	1	no	—
115	SD₁₆	—	58	2	no	—
116	SD₁₆	—	58	2	no	—
117	D₈	—	29	2	no	—
118	D₈	—	58	1	no	—
119	M₁₆	—	58	2	no	—
120	M₁₆	—	116	1	no	—
121	C₂×C₈	—	116	1	no	—
122	C₂×C₈	C₄×D₂₉	1	1	yes	characteristic
123	C₂×C₈	C₄×D₂₉	1	1	yes	characteristic
124	C₂×C₈	D₁₁₆	1	1	yes	characteristic
125	C₂×C₈	C₂²⋊D₂₉	1	1	yes	characteristic
126	C₂×C₈	C₂²⋊D₂₉	1	1	yes	characteristic
127	C₂×C₈	C₂²×D₂₉	1	1	yes	characteristic
128	C₁₆	—	2	2	no	—
129	C₂³	—	116	1	no	—
130	C₂³	#90773	1	1	yes	characteristic
131	Q₈	—	58	2	no	—
132	Q₈	—	116	1	no	—
133	D₄	—	58	2	no	—
134	D₄	—	116	2	no	—
135	D₄	—	116	2	no	—
136	D₄	—	116	1	no	—
137	C₂×C₄	—	116	1	no	—
138	C₂×C₄	—	116	2	no	—
139	C₂×C₄	—	116	2	no	—
140	C₂×C₄	#90773	1	1	yes	characteristic
141	C₂×C₄	D₂₃₂	1	1	yes	characteristic
142	C₂×C₄	D₂₃₂	1	1	yes	characteristic
143	C₂×C₄	#90781	1	1	yes	characteristic
144	C₂×C₄	#90781	1	1	yes	characteristic
145	C₂×C₄	C₂×D₁₁₆	1	1	yes	characteristic
146	C₈	—	116	2	no	—
147	C₈	#90781	1	1	yes	characteristic
148	C₈	#90781	1	1	yes	characteristic
149	C₈	C₂×C₄×D₂₉	1	1	yes	characteristic
150	C₈	C₂×C₂²⋊D₂₉	1	1	yes	characteristic
151	C₂²	—	4	1	no	—
152	C₂²	—	116	2	no	—
153	C₂²	—	232	2	no	—
154	C₂²	#11745169	1	1	yes	characteristic
155	C₂²	#11745169	1	1	yes	characteristic
156	C₂²	C₂×#90773	1	1	yes	characteristic
157	C₄	—	116	2	no	—
158	C₄	#11745169	1	1	yes	characteristic
159	C₄	#11745169	1	1	yes	characteristic
160	C₄	C₂×D₂₃₂	1	1	yes	characteristic
161	C₄	C₂×#90781	1	1	yes	characteristic
162	C₂	—	2	1	no	—
163	C₂	—	2	1	no	—
164	C₂	—	2	1	no	—
165	C₂	—	232	2	no	—
166	C₂	C₂×#11745169	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×D₂₃₂
commutator	C₈×C₂₉	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₂₉
Fitting	C₂×C₂₉×M₃₂	C₂
socle	C₂×C₂₉	C₂×#102
prime radical	C₂×#11745266	C₂
cosocle	C₂×C₈×C₂₉	C₂³
hypercenter	C₂×M₃₂	D₂₉
hypocenter	C₂₉	#2143
LCS	G ▷ C₈×C₂₉ ▷ C₄×C₂₉ ▷ C₂×C₂₉ ▷ C₂₉
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×M₃₂

Complements

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

N	H ≅ G/N	type
2. C₂×#11745167	C₂	semidirect
3. #14270971	C₂	semidirect
4. C₂×C₂₉×M₃₂	C₂	semidirect
17. C₂×C₁₆×C₂₉	C₂²	semidirect
19. #11745167	C₂²	semidirect
80. C₂×M₃₂	D₂₉	semidirect
100. C₂×C₁₆	C₂×D₂₉	semidirect
103. C₂₉	#2143	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11745266	2
C₂×#11745167	2
#14270971	2
C₂×C₂₉×M₃₂	2
#2143	29