Finite Group Databasebeta

Subgroups — #16818903

Order 3712.

176 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 176 conjugacy classes of subgroups — too many to draw. Only its 55 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	#14271511	C₂	1	1	yes	maximal, characteristic, complemented
2	#14271358	C₂	1	1	yes	maximal, characteristic
3	D₄×#6597	C₂	1	1	yes	maximal, characteristic, complemented
4	#14270913	C₂	1	1	yes	maximal, characteristic, complemented
5	#14270912	C₂	1	1	yes	maximal, characteristic, complemented
6	#14270906	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₉×#476	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₉×C₄⋊Q₈	C₂²	1	1	yes	characteristic
9	C₄×D₄×C₂₉	C₂²	1	1	yes	characteristic
10	C₂×#90782	—	2	1	no	—
11	C₂²×#6597	—	2	1	no	—
12	#11745236	C₂²	1	1	yes	characteristic
13	C₂₉×#106	—	2	1	no	—
14	C₂₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
15	C₂₉×#102	—	2	1	no	—
16	C₂²⋊#6597	—	2	1	no	—
17	#11745157	—	2	1	no	—
18	#11745157	—	2	1	no	—
19	#11745157	—	2	1	no	—
20	C₄⋊#6597	C₂²	1	1	yes	characteristic
21	C₄⋊#6597	C₂²	1	1	yes	characteristic
22	C₄×#6597	C₂²	1	1	yes	characteristic
23	C₂×Q₈×C₂₉	—	2	1	no	—
24	C₂×D₄×C₂₉	D₄	1	1	yes	characteristic
25	C₂²×C₄×C₂₉	—	2	1	no	—
26	C₂×D₁₁₆	—	2	1	no	—
27	C₂×D₁₁₆	—	2	1	no	—
28	C₂×D₁₁₆	—	2	1	no	—
29	C₂×C₈×C₂₉	—	2	1	no	—
30	C₄⋊C₄×C₂₉	—	2	1	no	—
31	C₄⋊C₄×C₂₉	—	2	1	no	—
32	C₄⋊C₄×C₂₉	D₄	1	1	yes	characteristic
33	C₂²⋊C₄×C₂₉	—	2	1	no	—
34	C₄²×C₂₉	C₂³	1	1	yes	characteristic
35	#90782	—	4	2	no	—
36	C₂×#6597	—	2	1	no	—
37	C₂×#6597	—	2	1	no	—
38	C₂×#6597	—	4	2	no	—
39	C₂×#6597	D₄	1	1	yes	characteristic
40	C₂×#6597	D₄	1	1	yes	characteristic
41	C₂³×C₂₉	—	2	1	no	—
42	C₂²×D₂₉	—	4	1	no	—
43	Q₈×C₂₉	—	8	1	no	—
44	D₄×C₂₉	—	2	1	no	—
45	D₄×C₂₉	SD₁₆	1	2	yes	—
46	C₂×C₄×C₂₉	—	2	1	no	—
47	C₂×C₄×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×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₂₉	#41	1	1	yes	characteristic
53	D₁₁₆	—	4	1	no	—
54	D₁₁₆	—	4	1	no	—
55	D₁₁₆	—	4	1	no	—
56	D₁₁₆	—	4	1	no	—
57	D₁₁₆	—	8	1	no	—
58	C₈×C₂₉	—	4	1	no	—
59	#6597	—	2	1	no	—
60	#6597	—	2	1	no	—
61	#6597	—	4	1	no	—
62	#6597	—	4	1	no	—
63	#1677	—	29	1	no	maximal, Sylow
64	C₂²×C₂₉	—	2	2	no	—
65	C₂²×C₂₉	—	4	1	no	—
66	C₂²×C₂₉	—	4	1	no	—
67	C₂²×C₂₉	#121	1	1	yes	characteristic
68	C₂×D₂₉	—	8	1	no	—
69	C₂×D₂₉	—	8	1	no	—
70	C₂×D₂₉	—	8	1	no	—
71	C₄×C₂₉	—	2	1	no	—
72	C₄×C₂₉	—	4	1	no	—
73	C₄×C₂₉	—	8	1	no	—
74	C₄×C₂₉	C₂×SD₁₆	1	1	yes	characteristic
75	C₄×C₂₉	#135	1	1	yes	characteristic
76	C₄×C₂₉	#135	1	1	yes	characteristic
77	C₄×C₂₉	AGL(1,ℤ/8)	1	1	yes	characteristic
78	#486	—	29	1	no	—
79	#476	D₂₉	1	1	yes	characteristic, complemented
80	#459	—	29	1	no	—
81	C₈×D₄	—	29	1	no	—
82	#334	—	29	1	no	—
83	#333	—	29	1	no	—
84	#331	—	29	1	no	—
85	C₂×C₂₉	—	4	2	no	—
86	C₂×C₂₉	#463	1	1	yes	characteristic
87	C₂×C₂₉	#465	1	1	yes	characteristic
88	C₂×C₂₉	#472	1	1	yes	characteristic
89	D₂₉	—	16	1	no	—
90	C₂×D₈	—	58	1	no	—
91	C₂²×C₈	—	58	1	no	—
92	C₄⋊Q₈	C₂×D₂₉	1	1	yes	characteristic
93	#127	—	29	1	no	—
94	C₄×D₄	C₂×D₂₉	1	1	yes	characteristic
95	#106	—	2	1	no	—
96	C₄⋊C₈	—	29	1	no	—
97	C₄⋊C₈	—	29	1	no	—
98	C₄⋊C₈	C₂×D₂₉	1	1	yes	characteristic
99	#102	—	2	1	no	—
100	#102	—	58	1	no	—
101	#102	—	58	1	no	—
102	#102	—	58	1	no	—
103	C₂²⋊C₈	—	58	1	no	—
104	C₄×C₈	—	29	1	no	—
105	C₂₉	#1677	1	1	yes	characteristic, complemented, Sylow
106	C₂×Q₈	—	2	1	no	—
107	C₂×D₄	—	58	1	no	—
108	C₂×D₄	—	58	1	no	—
109	C₂×D₄	—	58	1	no	—
110	C₂×D₄	C₂²⋊D₂₉	1	1	yes	characteristic
111	C₂²×C₄	—	2	1	no	—
112	D₈	—	116	2	no	—
113	C₂×C₈	—	2	1	no	—
114	C₂×C₈	—	29	1	no	—
115	C₂×C₈	—	29	1	no	—
116	C₂×C₈	—	58	1	no	—
117	C₂×C₈	—	58	1	no	—
118	C₂×C₈	—	116	2	no	—
119	C₄⋊C₄	—	2	1	no	—
120	C₄⋊C₄	—	2	1	no	—
121	C₄⋊C₄	C₂²⋊D₂₉	1	1	yes	characteristic
122	C₂²⋊C₄	—	2	1	no	—
123	C₄²	C₂²×D₂₉	1	1	yes	characteristic
124	C₂³	—	2	1	no	—
125	C₂³	—	116	1	no	—
126	Q₈	—	8	1	no	—
127	D₄	—	2	1	no	—
128	D₄	—	116	1	no	—
129	D₄	—	116	1	no	—
130	D₄	—	116	1	no	—
131	D₄	—	116	1	no	—
132	D₄	—	232	1	no	—
133	D₄	#90784	1	2	yes	—
134	C₂×C₄	—	2	1	no	—
135	C₂×C₄	—	4	1	no	—
136	C₂×C₄	—	4	1	no	—
137	C₂×C₄	—	4	1	no	—
138	C₂×C₄	D₄×D₂₉	1	1	yes	characteristic
139	C₂×C₄	#90807	1	1	yes	characteristic
140	C₂×C₄	C₂×C₂²⋊D₂₉	1	1	yes	characteristic
141	C₈	—	4	1	no	—
142	C₈	—	58	1	no	—
143	C₈	—	58	1	no	—
144	C₈	—	116	1	no	—
145	C₈	—	116	1	no	—
146	C₂²	—	2	2	no	—
147	C₂²	—	4	1	no	—
148	C₂²	—	4	1	no	—
149	C₂²	—	232	1	no	—
150	C₂²	—	232	1	no	—
151	C₂²	—	232	1	no	—
152	C₂²	#11745301	1	1	yes	characteristic
153	C₄	—	2	1	no	—
154	C₄	—	4	1	no	—
155	C₄	—	8	1	no	—
156	C₄	#11745276	1	1	yes	characteristic
157	C₄	#11745278	1	1	yes	characteristic
158	C₄	C₂×#90784	1	1	yes	characteristic
159	C₄	#11745312	1	1	yes	characteristic
160	C₂	—	4	2	no	—
161	C₂	—	464	1	no	—
162	C₂	#14271380	1	1	yes	characteristic
163	C₂	#14271394	1	1	yes	characteristic
164	C₂	#14271431	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#11745301
commutator, cosocle	C₄²×C₂₉	C₂³
Frattini	C₄²	C₂²×D₂₉
Fitting	C₂₉×#476	C₂
socle	C₂²×C₂₉	#121
prime radical	#14271358	C₂
hypercenter	#476	D₂₉
hypocenter	C₂₉	#1677
LCS	G ▷ C₄²×C₂₉ ▷ C₂²×C₂₉ ▷ C₂₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #476

Complements

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

N	H ≅ G/N	type
1. #14271511	C₂	semidirect
3. D₄×#6597	C₂	semidirect
4. #14270913	C₂	semidirect
5. #14270912	C₂	semidirect
6. #14270906	C₂	semidirect
7. C₂₉×#476	C₂	semidirect
79. #476	D₂₉	semidirect
105. C₂₉	#1677	semidirect

Maximal subgroups

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

subgroup	index
#14271511	2
#14271358	2
D₄×#6597	2
#14270913	2
#14270912	2
#14270906	2
C₂₉×#476	2
#1677	29