Finite Group Databasebeta

Subgroups — #16819370

Order 3712.

232 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 232 conjugacy classes of subgroups — too many to draw. Only its 105 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×#11745170	C₂	1	1	yes	maximal, characteristic
2	#14271199	C₂	1	1	yes	maximal, characteristic
3	#14271121	C₂	1	1	yes	maximal, characteristic
4	#14271119	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄×#6597	C₂	1	1	yes	maximal, characteristic
6	#14270923	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	#11745170	C₄	1	4	yes	complemented
16	#11745163	—	2	1	no	—
17	C₈×Dic₂₉	—	2	1	no	—
18	#11745156	C₄	1	2	yes	complemented
19	#11745155	C₄	1	2	yes	complemented
20	#11745151	—	2	1	no	—
21	C₄×#6597	—	2	1	no	—
22	C₄×#6597	C₄	1	4	yes	complemented
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	C₂×C₄	1	2	yes	—
27	C₂×C₈×C₂₉	—	2	1	no	—
28	C₄⋊C₄×C₂₉	C₂×C₄	1	2	yes	—
29	C₂²⋊C₄×C₂₉	—	2	1	no	—
30	C₄²×C₂₉	—	2	1	no	—
31	C₄²×C₂₉	—	2	1	no	—
32	C₄²×C₂₉	C₂×C₄	1	2	yes	—
33	C₂²⋊Dic₂₉	—	2	1	no	—
34	C₄⋊Dic₂₉	C₂×C₄	1	2	yes	—
35	C₄×Dic₂₉	—	2	1	no	—
36	#90777	—	2	1	no	—
37	#90777	C₂×C₄	1	2	yes	—
38	C₂×#6597	—	2	1	no	—
39	C₂×#6597	C₂×C₄	1	2	yes	—
40	C₂×#6597	C₂×C₄	1	1	yes	characteristic
41	C₂×#6597	C₂×C₄	1	1	yes	characteristic
42	C₂×#6597	C₂×C₄	1	2	yes	—
43	C₂×#6597	C₂×C₄	1	2	yes	—
44	C₂×#6597	D₄	1	1	yes	characteristic
45	C₂×#6597	D₄	1	1	yes	characteristic
46	C₂×#6597	Q₈	1	2	yes	—
47	C₂³×C₂₉	#41	1	1	yes	characteristic
48	C₂×C₄×C₂₉	—	2	4	no	—
49	C₂×C₄×C₂₉	—	2	2	no	—
50	C₂×C₄×C₂₉	—	2	2	no	—
51	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
54	C₂×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
55	C₂×C₄×C₂₉	C₂×Q₈	1	1	yes	characteristic
56	C₂×C₄×C₂₉	#41	1	1	yes	characteristic
57	C₂×Dic₂₉	—	2	2	no	—
58	C₈×C₂₉	—	2	2	no	—
59	#6597	—	2	2	no	—
60	#6597	C₄²	1	2	yes	—
61	#6597	C₄²	1	2	yes	—
62	#6597	C₄⋊C₄	1	2	yes	—
63	#6597	C₄⋊C₄	1	2	yes	—
64	#1772	—	29	1	no	maximal, Sylow
65	C₂²×C₂₉	—	2	2	no	—
66	C₂²×C₂₉	#117	1	1	yes	characteristic
67	C₂²×C₂₉	C₄×D₄	1	1	yes	characteristic
68	C₂²×C₂₉	C₄×Q₈	1	1	yes	characteristic
69	C₄×C₂₉	—	2	4	no	—
70	C₄×C₂₉	—	4	2	no	—
71	C₄×C₂₉	C₂×C₄²	1	1	yes	characteristic
72	C₄×C₂₉	C₂×C₄⋊C₄	1	1	yes	characteristic
73	C₄×C₂₉	C₄×D₄	1	1	yes	characteristic
74	C₄×C₂₉	C₄×Q₈	1	1	yes	characteristic
75	Dic₂₉	—	4	2	no	—
76	C₂×#108	—	29	1	no	—
77	#427	—	29	1	no	—
78	#405	—	29	1	no	—
79	#405	—	29	1	no	—
80	C₂×C₄×C₈	—	29	1	no	—
81	#339	—	29	1	no	—
82	#339	D₂₉	1	1	yes	characteristic
83	C₂×C₂₉	—	2	1	no	—
84	C₂×C₂₉	—	2	1	no	—
85	C₂×C₂₉	C₄×C₄⋊C₄	1	1	yes	characteristic
86	C₂×C₂₉	#443	1	2	yes	—
87	C₂×M₁₆	—	29	1	no	—
88	C₂×M₁₆	C₂×D₂₉	1	1	yes	characteristic
89	C₂²×C₈	—	29	1	no	—
90	C₂²×C₈	—	29	1	no	—
91	#117	—	29	1	no	—
92	#117	C₂×D₂₉	1	1	yes	characteristic
93	C₂×C₄²	C₂×D₂₉	1	1	yes	characteristic
94	#108	—	29	4	no	—
95	#107	—	29	2	no	—
96	#106	—	29	2	no	—
97	#97	—	58	1	no	—
98	#97	—	58	1	no	—
99	C₄×C₈	—	29	4	no	—
100	C₄×C₈	—	58	1	no	—
101	C₄×C₈	—	58	1	no	—
102	C₂₉	#1772	1	1	yes	characteristic, complemented, Sylow
103	C₂²×C₄	—	2	1	no	—
104	C₂²×C₄	C₂²×D₂₉	1	1	yes	characteristic
105	M₁₆	—	2	1	no	—
106	M₁₆	—	29	2	no	—
107	M₁₆	—	58	1	no	—
108	M₁₆	C₄×D₂₉	1	2	yes	—
109	C₂×C₈	—	2	1	no	—
110	C₂×C₈	—	29	2	no	—
111	C₂×C₈	—	29	2	no	—
112	C₂×C₈	—	29	1	no	—
113	C₂×C₈	—	29	1	no	—
114	C₂×C₈	—	29	2	no	—
115	C₂×C₈	—	29	1	no	—
116	C₂×C₈	—	29	1	no	—
117	C₂×C₈	—	29	2	no	—
118	C₂×C₈	—	58	1	no	—
119	C₄⋊C₄	—	29	2	no	—
120	C₄⋊C₄	C₄×D₂₉	1	2	yes	—
121	C₂²⋊C₄	—	2	1	no	—
122	C₂²⋊C₄	—	58	1	no	—
123	C₄²	—	2	1	no	—
124	C₄²	—	2	1	no	—
125	C₄²	—	58	1	no	—
126	C₄²	C₄×D₂₉	1	2	yes	—
127	C₂³	#90807	1	1	yes	characteristic
128	C₂×C₄	—	2	4	no	—
129	C₂×C₄	—	2	2	no	—
130	C₂×C₄	—	2	2	no	—
131	C₂×C₄	—	58	2	no	—
132	C₂×C₄	C₂×C₄×D₂₉	1	1	yes	characteristic
133	C₂×C₄	C₂×C₄×D₂₉	1	1	yes	characteristic
134	C₂×C₄	C₂×C₄×D₂₉	1	1	yes	characteristic
135	C₂×C₄	D₄×D₂₉	1	1	yes	characteristic
136	C₂×C₄	#90807	1	1	yes	characteristic
137	C₂×C₄	Q₈×D₂₉	1	1	yes	characteristic
138	C₈	—	2	2	no	—
139	C₈	—	29	2	no	—
140	C₈	—	29	2	no	—
141	C₈	—	29	2	no	—
142	C₈	—	29	2	no	—
143	C₈	—	58	2	no	—
144	C₂²	—	2	2	no	—
145	C₂²	#11745239	1	1	yes	characteristic
146	C₂²	#11745243	1	1	yes	characteristic
147	C₂²	#11745249	1	1	yes	characteristic
148	C₄	—	2	4	no	—
149	C₄	—	4	2	no	—
150	C₄	—	116	2	no	—
151	C₄	C₄²×D₂₉	1	1	yes	characteristic
152	C₄	#11745243	1	1	yes	characteristic
153	C₄	#11745249	1	1	yes	characteristic
154	C₄	C₄⋊C₄×D₂₉	1	1	yes	characteristic
155	C₂	—	2	1	no	—
156	C₂	—	2	1	no	—
157	C₂	#14271014	1	1	yes	characteristic
158	C₂	#14271277	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	D₄×D₂₉
commutator	C₄×C₂₉	C₂×C₄²
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	#1772
LCS	G ▷ 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
15. #11745170	C₄	semidirect
18. #11745156	C₄	semidirect
19. #11745155	C₄	semidirect
22. C₄×#6597	C₄	semidirect
102. C₂₉	#1772	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11745170	2
#14271199	2
#14271121	2
#14271119	2
C₂×C₄×#6597	2
#14270923	2
C₂₉×#339	2
#1772	29