Finite Group Databasebeta

Subgroups — #16835375

Order 3712.

260 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 72 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	#14272173	C₂	1	2	yes	maximal, complemented
2	#14272142	C₂	1	2	yes	maximal
3	#14272031	C₂	1	2	yes	maximal, complemented
4	C₂×#11745269	C₂	1	1	yes	maximal, characteristic
5	C₂×#90805	C₂²	1	1	yes	characteristic
6	#11745345	C₂²	1	4	yes	—
7	C₂×M₁₆×C₂₉	C₄	1	1	yes	characteristic, complemented
8	C₂²×#6597	C₄	1	1	yes	characteristic, complemented
9	#11745269	—	2	2	no	—
10	#11745269	—	2	2	no	—
11	C₂×#90772	C₄	1	1	yes	characteristic, complemented
12	C₂×#90772	C₂²	1	1	yes	characteristic
13	C₂×C₈×D₂₉	C₄	1	1	yes	characteristic, complemented
14	C₂×C₈×D₂₉	C₂²	1	1	yes	characteristic
15	#11745227	—	2	2	no	—
16	#11745215	—	2	2	no	—
17	#11745210	—	2	2	no	—
18	#11745209	—	2	2	no	—
19	C₂²×C₄×C₂₉	C₂×C₄	1	1	yes	characteristic
20	C₂×C₂²⋊D₂₉	D₄	1	1	yes	characteristic
21	C₂×C₂²⋊D₂₉	Q₈	1	1	yes	characteristic
22	#90805	—	2	2	no	—
23	#90805	—	2	2	no	—
24	C₂×D₁₁₆	D₄	1	1	yes	characteristic
25	C₂×C₄×D₂₉	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄×D₂₉	C₂³	1	1	yes	characteristic
27	C₂×Dic₅₈	Q₈	1	1	yes	characteristic
28	C₂²⋊#1181	—	2	4	no	—
29	C₄⋊#1181	—	2	4	no	—
30	C₄×#1181	—	2	4	no	—
31	M₁₆×C₂₉	—	2	2	no	—
32	C₂×C₈×C₂₉	C₂×C₄	1	1	yes	characteristic
33	C₂×C₈×C₂₉	C₂×C₄	1	1	yes	characteristic
34	C₂×#6597	—	2	1	no	—
35	C₂×#6597	—	2	1	no	—
36	C₂×#6597	C₂×C₄	1	1	yes	characteristic
37	C₂×#6597	C₂×C₄	1	1	yes	characteristic
38	#90772	—	2	2	no	—
39	#90772	—	2	2	no	—
40	C₈×D₂₉	—	2	1	no	—
41	C₈×D₂₉	—	2	1	no	—
42	C₈×D₂₉	D₄	1	2	yes	—
43	C₈×D₂₉	D₄	1	2	yes	—
44	C₂³×C₂₉	C₄⋊C₄	1	1	yes	characteristic
45	C₂²×D₂₉	C₄⋊C₄	1	1	yes	characteristic
46	C₂²×D₂₉	C₂×D₄	1	1	yes	characteristic
47	C₂×#1181	—	4	4	no	—
48	C₂×C₄×C₂₉	—	2	1	no	—
49	C₂×C₄×C₂₉	—	2	1	no	—
50	C₂×C₄×C₂₉	C₄⋊C₄	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	C₄⋊C₄	1	1	yes	characteristic
55	C₂×Dic₂₉	C₂×Q₈	1	1	yes	characteristic
56	D₁₁₆	—	2	2	no	—
57	C₄×D₂₉	—	2	1	no	—
58	C₄×D₂₉	—	2	1	no	—
59	C₄×D₂₉	C₂×D₄	1	2	yes	—
60	C₄×D₂₉	#41	1	2	yes	—
61	Dic₅₈	—	2	2	no	—
62	C₈×C₂₉	—	2	1	no	—
63	C₈×C₂₉	C₂²⋊C₄	1	2	yes	—
64	#6597	—	2	1	no	—
65	#6597	C₂²⋊C₄	1	1	yes	characteristic
66	#6597	C₂²⋊C₄	1	1	yes	characteristic
67	#1812	—	29	1	no	maximal, Sylow
68	C₂²×C₂₉	—	2	1	no	—
69	C₂²×C₂₉	—	4	1	no	—
70	C₂²×C₂₉	C₂×C₄⋊C₄	1	1	yes	characteristic
71	C₂×D₂₉	—	2	1	no	—
72	C₂×D₂₉	—	4	1	no	—
73	C₂×D₂₉	—	4	1	no	—
74	C₂×D₂₉	#121	1	2	yes	—
75	#1181	—	8	4	no	—
76	C₄×C₂₉	—	2	1	no	—
77	C₄×C₂₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
78	C₄×C₂₉	#117	1	1	yes	characteristic
79	Dic₂₉	—	2	1	no	—
80	Dic₂₉	C₂²⋊Q₈	1	2	yes	—
81	C₂×#131	—	29	1	no	—
82	#427	—	29	2	no	—
83	#410	—	29	2	no	—
84	H(3,ℤ/4)	—	29	2	no	—
85	C₂×C₂₉	—	2	1	no	—
86	C₂×C₂₉	—	4	1	no	—
87	C₂×C₂₉	#380	1	1	yes	characteristic
88	D₂₉	—	2	2	no	—
89	D₂₉	—	4	1	no	—
90	C₂×#41	—	29	1	no	—
91	#131	—	58	2	no	—
92	#131	—	58	2	no	—
93	C₂×M₁₆	—	29	1	no	—
94	C₂×M₁₆	—	29	1	no	—
95	C₂×M₁₆	#1181	1	1	yes	characteristic, complemented
96	C₂²×C₈	—	29	1	no	—
97	C₂²×C₈	—	29	1	no	—
98	C₂²×C₈	—	29	1	no	—
99	#117	—	29	4	no	—
100	#107	—	58	2	no	—
101	#106	—	58	2	no	—
102	#99	—	58	2	no	—
103	#99	—	58	2	no	—
104	C₂₉	#1812	1	1	yes	characteristic, complemented, Sylow
105	#41	—	58	2	no	—
106	#41	—	58	2	no	—
107	C₂×Q₈	—	29	1	no	—
108	C₂×D₄	—	29	1	no	—
109	C₂×D₄	—	29	1	no	—
110	C₂×D₄	—	29	1	no	—
111	C₂²×C₄	—	29	1	no	—
112	C₂²×C₄	—	29	1	no	—
113	C₂²×C₄	C₂×#1181	1	1	yes	characteristic
114	M₁₆	—	2	2	no	—
115	M₁₆	—	58	2	no	—
116	M₁₆	—	58	2	no	—
117	C₂×C₈	—	29	1	no	—
118	C₂×C₈	—	29	2	no	—
119	C₂×C₈	—	29	2	no	—
120	C₂×C₈	—	29	1	no	—
121	C₂×C₈	—	58	1	no	—
122	C₂×C₈	—	58	1	no	—
123	C₂×C₈	—	58	1	no	—
124	C₂×C₈	—	58	1	no	—
125	C₂×C₈	C₂×#1181	1	1	yes	characteristic
126	C₂×C₈	C₂×#1181	1	1	yes	characteristic
127	C₄⋊C₄	—	58	4	no	—
128	C₂²⋊C₄	—	58	4	no	—
129	C₄²	—	58	4	no	—
130	C₂³	—	29	1	no	—
131	C₂³	—	29	1	no	—
132	C₂³	C₄⋊#1181	1	1	yes	characteristic
133	Q₈	—	58	2	no	—
134	D₄	—	58	2	no	—
135	D₄	—	58	2	no	—
136	D₄	—	58	2	no	—
137	C₂×C₄	—	2	1	no	—
138	C₂×C₄	—	2	1	no	—
139	C₂×C₄	—	29	1	no	—
140	C₂×C₄	—	29	2	no	—
141	C₂×C₄	—	29	2	no	—
142	C₂×C₄	—	29	1	no	—
143	C₂×C₄	—	58	1	no	—
144	C₂×C₄	—	58	1	no	—
145	C₂×C₄	—	116	4	no	—
146	C₂×C₄	C₄⋊#1181	1	1	yes	characteristic
147	C₂×C₄	C₂²×#1181	1	1	yes	characteristic
148	C₈	—	2	1	no	—
149	C₈	—	29	1	no	—
150	C₈	—	29	1	no	—
151	C₈	—	58	1	no	—
152	C₈	C₂²⋊#1181	1	2	yes	—
153	C₂²	—	2	1	no	—
154	C₂²	—	4	1	no	—
155	C₂²	—	29	2	no	—
156	C₂²	—	58	1	no	—
157	C₂²	—	116	1	no	—
158	C₂²	—	116	1	no	—
159	C₂²	C₂×C₄⋊#1181	1	1	yes	characteristic
160	C₄	—	2	1	no	—
161	C₄	—	29	2	no	—
162	C₄	—	58	1	no	—
163	C₄	—	232	4	no	—
164	C₄	#11745345	1	1	yes	characteristic
165	C₄	C₂×C₂²⋊#1181	1	1	yes	characteristic
166	C₂	—	2	1	no	—
167	C₂	—	4	1	no	—
168	C₂	—	58	2	no	—
169	C₂	—	116	1	no	—
170	C₂	#14272105	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#11745345
commutator	C₂×C₄×C₂₉	C₂²×C₄
Frattini	C₂×C₄	C₂²×#1181
Fitting	C₂×M₁₆×C₂₉	C₄
socle	C₂×C₂₉	#380
prime radical	C₂×#90805	C₂²
cosocle	C₂×C₄×D₂₉	C₂³
hypercenter	C₂×M₁₆	#1181
hypocenter	C₂₉	#1812
LCS	G ▷ C₂×C₄×C₂₉ ▷ C₂×C₂₉ ▷ C₂₉
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ C₂×M₁₆

Complements

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

N	H ≅ G/N	type
1. #14272173	C₂	semidirect
3. #14272031	C₂	semidirect
7. C₂×M₁₆×C₂₉	C₄	semidirect
8. C₂²×#6597	C₄	semidirect
11. C₂×#90772	C₄	semidirect
13. C₂×C₈×D₂₉	C₄	semidirect
95. C₂×M₁₆	#1181	semidirect
104. C₂₉	#1812	semidirect

Maximal subgroups

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

subgroup	index
#14272173	2
#14272142	2
#14272031	2
C₂×#11745269	2
#1812	29