Finite Group Databasebeta

Subgroups — #16821203

Order 3712.

232 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 232 conjugacy classes of subgroups — too many to draw. Only its 69 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×#11745281	C₂	1	1	yes	maximal, characteristic
2	C₂×#11745146	C₂	1	1	yes	maximal, characteristic, complemented
3	#107×D₂₉	C₂	1	1	yes	maximal, characteristic, complemented
4	#14270967	C₂	1	1	yes	maximal, characteristic, complemented
5	#14270965	C₂	1	1	yes	maximal, characteristic, complemented
6	#14270963	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₉×#358	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#90809	—	2	1	no	—
9	C₂×Q₁₆×C₂₉	C₂²	1	1	yes	characteristic
10	C₂×#90784	—	2	1	no	—
11	#11745281	—	2	2	no	—
12	#11745281	C₄	1	4	yes	complemented
13	C₂×D₂₃₂	C₂²	1	1	yes	characteristic
14	C₂×C₈×D₂₉	C₂²	1	1	yes	characteristic
15	C₄⋊C₄×D₂₉	—	2	1	no	—
16	C₂×C₁₆×C₂₉	C₂²	1	1	yes	characteristic
17	C₂₉×#107	C₂²	1	1	yes	characteristic
18	#11745166	C₂²	1	1	yes	characteristic
19	C₂×#90768	C₂²	1	1	yes	characteristic
20	#11745155	—	2	1	no	—
21	#11745146	—	2	2	no	—
22	C₂×Q₈×C₂₉	—	2	1	no	—
23	#90809	—	2	4	no	—
24	#90809	—	4	2	no	—
25	C₂×D₁₁₆	—	2	1	no	—
26	C₂×D₁₁₆	—	4	1	no	—
27	C₂×C₄×D₂₉	—	4	1	no	—
28	C₂×C₄×D₂₉	—	4	1	no	—
29	C₂×C₄×D₂₉	D₄	1	1	yes	characteristic
30	Q₁₆×C₂₉	—	2	1	no	—
31	Q₁₆×C₂₉	C₂×C₄	1	2	yes	complemented
32	C₂×C₈×C₂₉	C₂³	1	1	yes	characteristic
33	C₄⋊C₄×C₂₉	—	2	1	no	—
34	#90784	—	2	4	no	—
35	C₄⋊Dic₂₉	—	2	1	no	—
36	#90779	—	4	1	no	—
37	C₂×#6597	D₄	1	1	yes	characteristic
38	D₂₃₂	—	2	1	no	—
39	D₂₃₂	C₂×C₄	1	2	yes	complemented
40	C₈×D₂₉	C₂×C₄	1	2	yes	—
41	C₈×D₂₉	D₄	1	2	yes	—
42	C₁₆×C₂₉	—	2	1	no	—
43	#90768	—	2	1	no	—
44	C₂²×D₂₉	—	4	1	no	—
45	C₂²×D₂₉	D₈	1	1	yes	characteristic
46	Q₈×C₂₉	—	2	2	no	—
47	Q₈×C₂₉	—	4	1	no	—
48	C₂×C₄×C₂₉	—	4	1	no	—
49	C₂×C₄×C₂₉	—	4	1	no	—
50	C₂×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
51	C₂×Dic₂₉	—	4	1	no	—
52	C₂×Dic₂₉	D₈	1	1	yes	characteristic
53	D₁₁₆	—	2	2	no	—
54	D₁₁₆	—	4	4	no	—
55	D₁₁₆	—	4	1	no	—
56	C₄×D₂₉	—	4	4	no	—
57	C₄×D₂₉	—	8	2	no	—
58	C₄×D₂₉	C₂²⋊C₄	1	2	yes	—
59	C₄×D₂₉	SD₁₆	1	2	yes	—
60	C₈×C₂₉	C₂²×C₄	1	1	yes	characteristic
61	C₈×C₂₉	C₂×D₄	1	1	yes	characteristic
62	#6597	C₂²⋊C₄	1	2	yes	—
63	#2139	—	29	1	no	maximal, Sylow
64	C₂²×C₂₉	C₂×D₈	1	1	yes	characteristic
65	C₂×D₂₉	—	2	1	no	—
66	C₂×D₂₉	—	2	1	no	—
67	C₂×D₂₉	—	4	2	no	—
68	C₂×D₂₉	—	8	1	no	—
69	C₂×D₂₉	—	8	1	no	—
70	C₂×D₂₉	#102	1	2	yes	—
71	C₄×C₂₉	—	4	2	no	—
72	C₄×C₂₉	—	8	1	no	—
73	C₄×C₂₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
74	C₄×C₂₉	C₂×SD₁₆	1	1	yes	characteristic
75	Dic₂₉	—	8	1	no	—
76	Dic₂₉	#102	1	2	yes	—
77	C₂×#135	—	29	1	no	—
78	C₂×M₃₂	—	29	1	no	—
79	C₂×#107	—	29	1	no	—
80	#358	—	29	1	no	—
81	#358	D₂₉	1	1	yes	characteristic, complemented
82	#357	—	29	1	no	—
83	#357	—	29	1	no	—
84	C₂×C₂₉	C₂×#102	1	1	yes	characteristic
85	C₂×C₂₉	#509	1	1	yes	characteristic
86	C₂×C₂₉	#510	1	1	yes	characteristic
87	D₂₉	—	2	2	no	—
88	D₂₉	—	8	2	no	—
89	C₂×#41	—	58	1	no	—
90	#135	—	29	4	no	—
91	#135	—	58	2	no	—
92	C₂×Q₁₆	C₂×D₂₉	1	1	yes	characteristic
93	C₂×SD₁₆	—	58	1	no	—
94	C₂×D₈	—	29	1	no	—
95	C₂²×C₈	—	29	1	no	—
96	C₂×C₄⋊C₄	—	58	1	no	—
97	M₃₂	—	58	2	no	—
98	C₂×C₁₆	—	29	1	no	—
99	C₂×C₁₆	C₂×D₂₉	1	1	yes	characteristic
100	#107	—	29	1	no	—
101	#107	—	58	1	no	—
102	#107	C₂×D₂₉	1	1	yes	characteristic
103	C₂₉	#2139	1	1	yes	characteristic, complemented, Sylow
104	#41	—	58	4	no	—
105	#41	—	116	2	no	—
106	C₂×Q₈	—	2	1	no	—
107	C₂×D₄	—	58	1	no	—
108	C₂×D₄	—	116	1	no	—
109	C₂²×C₄	—	29	1	no	—
110	C₂²×C₄	—	116	1	no	—
111	C₂²×C₄	—	116	1	no	—
112	Q₁₆	—	2	1	no	—
113	Q₁₆	C₄×D₂₉	1	2	yes	complemented
114	SD₁₆	—	58	4	no	—
115	D₈	—	29	2	no	—
116	D₈	—	58	1	no	—
117	C₂×C₈	—	29	1	no	—
118	C₂×C₈	—	29	2	no	—
119	C₂×C₈	—	29	2	no	—
120	C₂×C₈	C₂²×D₂₉	1	1	yes	characteristic
121	C₄⋊C₄	—	2	1	no	—
122	C₄⋊C₄	—	58	1	no	—
123	C₄⋊C₄	—	116	1	no	—
124	C₁₆	—	2	1	no	—
125	C₁₆	—	58	1	no	—
126	C₂³	—	29	1	no	—
127	C₂³	—	116	1	no	—
128	Q₈	—	2	2	no	—
129	Q₈	—	4	1	no	—
130	D₄	—	58	2	no	—
131	D₄	—	116	4	no	—
132	D₄	—	116	1	no	—
133	C₂×C₄	—	4	1	no	—
134	C₂×C₄	—	4	1	no	—
135	C₂×C₄	—	29	1	no	—
136	C₂×C₄	—	29	2	no	—
137	C₂×C₄	—	29	2	no	—
138	C₂×C₄	—	116	4	no	—
139	C₂×C₄	—	116	1	no	—
140	C₂×C₄	—	232	2	no	—
141	C₂×C₄	D₄×D₂₉	1	1	yes	characteristic
142	C₈	—	29	2	no	—
143	C₈	C₂×C₄×D₂₉	1	1	yes	characteristic
144	C₈	D₄×D₂₉	1	1	yes	characteristic
145	C₂²	—	29	2	no	—
146	C₂²	—	58	1	no	—
147	C₂²	—	58	1	no	—
148	C₂²	—	116	2	no	—
149	C₂²	—	232	1	no	—
150	C₂²	—	232	1	no	—
151	C₂²	D₈×D₂₉	1	1	yes	characteristic
152	C₄	—	4	2	no	—
153	C₄	—	8	1	no	—
154	C₄	—	29	2	no	—
155	C₄	—	232	1	no	—
156	C₄	C₂²⋊C₄×D₂₉	1	1	yes	characteristic
157	C₄	SD₁₆×D₂₉	1	1	yes	characteristic
158	C₂	—	58	2	no	—
159	C₂	—	232	2	no	—
160	C₂	#102×D₂₉	1	1	yes	characteristic
161	C₂	#14271600	1	1	yes	characteristic
162	C₂	#14271605	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₈×D₂₉
commutator	C₈×C₂₉	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₂₉
Fitting	C₂₉×#358	C₂
socle	C₂²×C₂₉	C₂×D₈
prime radical	C₂×#11745281	C₂
cosocle	C₂×C₈×C₂₉	C₂³
hypercenter	#358	D₂₉
hypocenter	C₂₉	#2139
LCS	G ▷ C₈×C₂₉ ▷ C₄×C₂₉ ▷ C₂×C₂₉ ▷ C₂₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #358

Complements

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

N	H ≅ G/N	type
2. C₂×#11745146	C₂	semidirect
3. #107×D₂₉	C₂	semidirect
4. #14270967	C₂	semidirect
5. #14270965	C₂	semidirect
6. #14270963	C₂	semidirect
7. C₂₉×#358	C₂	semidirect
12. #11745281	C₄	semidirect
31. Q₁₆×C₂₉	C₂×C₄	semidirect
39. D₂₃₂	C₂×C₄	semidirect
81. #358	D₂₉	semidirect
103. C₂₉	#2139	semidirect
113. Q₁₆	C₄×D₂₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11745281	2
C₂×#11745146	2
#107×D₂₉	2
#14270967	2
#14270965	2
#14270963	2
C₂₉×#358	2
#2139	29