Finite Group Databasebeta

Subgroups — #16820089

Order 3712.

186 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 186 conjugacy classes of subgroups — too many to draw. Only its 73 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	#14271199	C₂	1	1	yes	maximal, characteristic
2	#14271191	C₂	1	1	yes	maximal, characteristic
3	#14271119	C₂	1	1	yes	maximal, characteristic
4	#14270927	C₂	1	1	yes	maximal, characteristic
5	#14270923	C₂	1	1	yes	maximal, characteristic
6	#14270918	C₂	1	1	yes	maximal, characteristic
7	C₂₉×#339	C₂	1	1	yes	maximal, characteristic
8	C₂×M₁₆×C₂₉	C₂²	1	1	yes	characteristic
9	C₂₉×#117	C₂²	1	1	yes	characteristic
10	C₂×C₄²×C₂₉	C₂²	1	1	yes	characteristic
11	#11745288	C₂²	1	1	yes	characteristic
12	C₂×#90777	C₂²	1	1	yes	characteristic
13	C₂²×#6597	C₂²	1	1	yes	characteristic
14	C₂²×#6597	C₂²	1	1	yes	characteristic
15	#11745162	—	2	1	no	—
16	#11745162	—	2	1	no	—
17	#11745156	C₄	1	1	yes	characteristic, complemented
18	#11745156	C₄	1	1	yes	characteristic, complemented
19	#11745155	C₄	1	1	yes	characteristic, complemented
20	#11745155	C₄	1	1	yes	characteristic, complemented
21	#11745151	—	2	1	no	—
22	C₄×#6597	—	2	1	no	—
23	C₂²×C₄×C₂₉	—	2	1	no	—
24	C₂²×C₄×C₂₉	C₂³	1	1	yes	characteristic
25	M₁₆×C₂₉	—	2	1	no	—
26	M₁₆×C₂₉	—	2	1	no	—
27	C₂×C₈×C₂₉	—	2	1	no	—
28	C₄⋊C₄×C₂₉	C₂×C₄	1	1	yes	characteristic
29	C₄⋊C₄×C₂₉	C₂×C₄	1	1	yes	characteristic
30	C₂²⋊C₄×C₂₉	—	2	1	no	—
31	C₄²×C₂₉	—	2	1	no	—
32	C₄²×C₂₉	—	2	1	no	—
33	C₄²×C₂₉	—	2	1	no	—
34	C₂²⋊Dic₂₉	—	2	1	no	—
35	C₄⋊Dic₂₉	C₂×C₄	1	1	yes	characteristic
36	C₄⋊Dic₂₉	C₂×C₄	1	1	yes	characteristic
37	C₄×Dic₂₉	—	2	1	no	—
38	#90777	—	2	1	no	—
39	#90777	Q₈	1	2	yes	—
40	C₂×#6597	—	2	1	no	—
41	C₂×#6597	—	2	1	no	—
42	C₂×#6597	—	2	1	no	—
43	C₂×#6597	—	2	1	no	—
44	C₂×#6597	—	2	1	no	—
45	C₂×#6597	C₂×C₄	1	1	yes	characteristic
46	C₂×#6597	C₂×C₄	1	1	yes	characteristic
47	C₂×#6597	D₄	1	1	yes	characteristic
48	C₂×#6597	D₄	1	1	yes	characteristic
49	C₂³×C₂₉	#41	1	1	yes	characteristic
50	C₂×C₄×C₂₉	—	2	2	no	—
51	C₂×C₄×C₂₉	—	2	1	no	—
52	C₂×C₄×C₂₉	—	2	1	no	—
53	C₂×C₄×C₂₉	—	4	1	no	—
54	C₂×C₄×C₂₉	—	4	1	no	—
55	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
56	C₂×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
57	C₂×C₄×C₂₉	C₂×Q₈	1	1	yes	characteristic
58	C₂×C₄×C₂₉	#41	1	1	yes	characteristic
59	C₂×C₄×C₂₉	#41	1	1	yes	characteristic
60	C₂×C₄×C₂₉	#41	1	1	yes	characteristic
61	C₂×Dic₂₉	—	2	1	no	—
62	C₂×Dic₂₉	—	2	1	no	—
63	C₈×C₂₉	—	4	1	no	—
64	#6597	—	2	1	no	—
65	#6597	—	2	1	no	—
66	#6597	—	2	2	no	—
67	#6597	—	2	1	no	—
68	#6597	—	2	1	no	—
69	#1922	—	29	1	no	maximal, Sylow
70	C₂²×C₂₉	—	2	1	no	—
71	C₂²×C₂₉	—	2	1	no	—
72	C₂²×C₂₉	C₄×D₄	1	1	yes	characteristic
73	C₂²×C₂₉	#125	1	1	yes	characteristic
74	C₂²×C₂₉	#126	1	1	yes	characteristic
75	C₄×C₂₉	—	4	2	no	—
76	C₄×C₂₉	—	4	1	no	—
77	C₄×C₂₉	—	4	1	no	—
78	C₄×C₂₉	#117	1	1	yes	characteristic
79	C₄×C₂₉	C₄×Q₈	1	1	yes	characteristic
80	C₄×C₂₉	C₂²⋊Q₈	1	1	yes	characteristic
81	C₄×C₂₉	#123	1	1	yes	characteristic
82	Dic₂₉	—	4	1	no	—
83	Dic₂₉	—	4	1	no	—
84	#427	—	29	1	no	—
85	#424	—	29	1	no	—
86	#405	—	29	1	no	—
87	#341	—	29	1	no	—
88	#339	—	29	1	no	—
89	#339	D₂₉	1	1	yes	characteristic
90	#336	—	29	1	no	—
91	C₂×C₂₉	—	2	1	no	—
92	C₂×C₂₉	—	2	1	no	—
93	C₂×C₂₉	#387	1	1	yes	characteristic
94	C₂×C₂₉	#443	1	1	yes	characteristic
95	C₂×C₂₉	#444	1	1	yes	characteristic
96	C₂×M₁₆	—	29	1	no	—
97	C₂×M₁₆	C₂×D₂₉	1	1	yes	characteristic
98	C₂²×C₈	—	29	1	no	—
99	C₂²×C₈	—	29	1	no	—
100	#117	—	29	1	no	—
101	#117	C₂×D₂₉	1	1	yes	characteristic
102	C₂×C₄²	C₂×D₂₉	1	1	yes	characteristic
103	#107	—	29	1	no	—
104	#107	—	29	1	no	—
105	#106	—	29	1	no	—
106	#106	—	29	1	no	—
107	C₄⋊C₈	—	58	1	no	—
108	C₄⋊C₈	—	58	1	no	—
109	#97	—	58	1	no	—
110	C₄×C₈	—	58	1	no	—
111	C₂₉	#1922	1	1	yes	characteristic, complemented, Sylow
112	C₂²×C₄	—	2	1	no	—
113	C₂²×C₄	C₂²×D₂₉	1	1	yes	characteristic
114	M₁₆	—	2	1	no	—
115	M₁₆	—	2	1	no	—
116	M₁₆	—	29	2	no	—
117	M₁₆	—	58	1	no	—
118	C₂×C₈	—	2	1	no	—
119	C₂×C₈	—	29	1	no	—
120	C₂×C₈	—	29	1	no	—
121	C₂×C₈	—	29	1	no	—
122	C₂×C₈	—	29	1	no	—
123	C₂×C₈	—	58	1	no	—
124	C₂×C₈	—	58	1	no	—
125	C₂×C₈	—	58	1	no	—
126	C₂×C₈	—	58	1	no	—
127	C₂×C₈	—	58	1	no	—
128	C₄⋊C₄	—	29	1	no	—
129	C₄⋊C₄	—	29	1	no	—
130	C₄⋊C₄	C₄×D₂₉	1	1	yes	characteristic
131	C₄⋊C₄	C₄×D₂₉	1	1	yes	characteristic
132	C₂²⋊C₄	—	2	1	no	—
133	C₂²⋊C₄	—	58	1	no	—
134	C₄²	—	2	1	no	—
135	C₄²	—	2	1	no	—
136	C₄²	—	2	1	no	—
137	C₄²	—	58	1	no	—
138	C₂³	#90807	1	1	yes	characteristic
139	C₂×C₄	—	2	2	no	—
140	C₂×C₄	—	2	1	no	—
141	C₂×C₄	—	2	1	no	—
142	C₂×C₄	—	4	1	no	—
143	C₂×C₄	—	4	1	no	—
144	C₂×C₄	—	58	1	no	—
145	C₂×C₄	—	58	1	no	—
146	C₂×C₄	C₂×C₄×D₂₉	1	1	yes	characteristic
147	C₂×C₄	#90805	1	1	yes	characteristic
148	C₂×C₄	#90805	1	1	yes	characteristic
149	C₂×C₄	D₄×D₂₉	1	1	yes	characteristic
150	C₂×C₄	#90807	1	1	yes	characteristic
151	C₂×C₄	Q₈×D₂₉	1	1	yes	characteristic
152	C₈	—	4	1	no	—
153	C₈	—	58	1	no	—
154	C₈	—	58	1	no	—
155	C₈	—	58	1	no	—
156	C₈	—	58	1	no	—
157	C₈	—	58	2	no	—
158	C₂²	—	2	1	no	—
159	C₂²	—	2	1	no	—
160	C₂²	#11745241	1	1	yes	characteristic
161	C₂²	#11745243	1	1	yes	characteristic
162	C₂²	#11745251	1	1	yes	characteristic
163	C₄	—	4	2	no	—
164	C₄	—	4	1	no	—
165	C₄	—	4	1	no	—
166	C₄	—	116	1	no	—
167	C₄	—	116	1	no	—
168	C₄	#11745234	1	1	yes	characteristic
169	C₄	#11745245	1	1	yes	characteristic
170	C₄	#11745249	1	1	yes	characteristic
171	C₄	#11745258	1	1	yes	characteristic
172	C₂	—	2	1	no	—
173	C₂	—	2	1	no	—
174	C₂	#14271045	1	1	yes	characteristic
175	C₂	#14271277	1	1	yes	characteristic
176	C₂	#14271282	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₄×D₂₉
commutator	C₂×C₄×C₂₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₉
Fitting	C₂₉×#339	C₂
socle	C₂²×C₂₉	C₄×D₄
prime radical	C₂³×C₂₉	#41
cosocle	C₂²×C₄×C₂₉	C₂³
hypercenter	#339	D₂₉
hypocenter	C₂₉	#1922
LCS	G ▷ C₂×C₄×C₂₉ ▷ C₂×C₂₉ ▷ C₂₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #339

Complements

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

N	H ≅ G/N	type
17. #11745156	C₄	semidirect
18. #11745156	C₄	semidirect
19. #11745155	C₄	semidirect
20. #11745155	C₄	semidirect
111. C₂₉	#1922	semidirect

Maximal subgroups

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

subgroup	index
#14271199	2
#14271191	2
#14271119	2
#14270927	2
#14270923	2
#14270918	2
C₂₉×#339	2
#1922	29