Finite Group Databasebeta

Subgroups — #16819028

Order 3712.

170 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 170 conjugacy classes of subgroups — too many to draw. Only its 49 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	#14271450	C₂	1	1	yes	maximal, characteristic, complemented
3	#14271359	C₂	1	1	yes	maximal, characteristic
4	#14271240	C₂	1	1	yes	maximal, characteristic, complemented
5	#14270915	C₂	1	1	yes	maximal, characteristic, complemented
6	#14270906	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₉×#333	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₉×C₄⋊Q₈	C₂²	1	1	yes	characteristic
9	C₂×D₂₃₂	—	2	1	no	—
10	C₂×#90772	—	2	1	no	—
11	#11745236	C₂²	1	1	yes	characteristic
12	C₄×D₁₁₆	C₂²	1	1	yes	characteristic
13	C₂₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
14	C₂₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
15	#11745169	—	2	1	no	—
16	#11745168	—	2	1	no	—
17	#11745157	—	2	1	no	—
18	#11745157	—	2	1	no	—
19	#11745157	—	2	1	no	—
20	#11745156	—	2	1	no	—
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×D₁₁₆	—	2	1	no	—
25	C₂×D₁₁₆	—	2	1	no	—
26	C₂×D₁₁₆	—	2	1	no	—
27	C₂×D₁₁₆	D₄	1	1	yes	characteristic
28	C₂×C₄×D₂₉	—	2	1	no	—
29	C₂×C₈×C₂₉	—	2	1	no	—
30	C₂×C₈×C₂₉	—	2	1	no	—
31	C₄⋊C₄×C₂₉	—	2	1	no	—
32	C₄⋊C₄×C₂₉	—	2	1	no	—
33	C₄²×C₂₉	C₂³	1	1	yes	characteristic
34	#90781	—	2	1	no	—
35	C₄⋊Dic₂₉	D₄	1	1	yes	characteristic
36	C₂×#6597	—	2	1	no	—
37	C₂×#6597	D₄	1	1	yes	characteristic
38	C₂×#6597	D₄	1	1	yes	characteristic
39	D₂₃₂	—	4	2	no	—
40	#90772	—	4	2	no	—
41	C₂²×D₂₉	—	2	1	no	—
42	C₂²×D₂₉	—	4	1	no	—
43	Q₈×C₂₉	—	8	1	no	—
44	C₂×C₄×C₂₉	—	4	1	no	—
45	C₂×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₂₉	#41	1	1	yes	characteristic
48	C₂×Dic₂₉	—	2	1	no	—
49	D₁₁₆	—	2	1	no	—
50	D₁₁₆	—	2	1	no	—
51	D₁₁₆	—	4	1	no	—
52	D₁₁₆	—	4	1	no	—
53	D₁₁₆	—	4	1	no	—
54	D₁₁₆	—	4	1	no	—
55	D₁₁₆	—	8	1	no	—
56	C₄×D₂₉	—	4	1	no	—
57	C₄×D₂₉	—	4	1	no	—
58	C₈×C₂₉	—	4	1	no	—
59	C₈×C₂₉	—	4	1	no	—
60	#6597	—	2	1	no	—
61	#6597	—	2	1	no	—
62	#6597	—	4	1	no	—
63	#1697	—	29	1	no	maximal, Sylow
64	C₂²×C₂₉	#121	1	1	yes	characteristic
65	C₂×D₂₉	—	4	1	no	—
66	C₂×D₂₉	—	4	1	no	—
67	C₂×D₂₉	—	4	1	no	—
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	—	8	1	no	—
73	C₄×C₂₉	#135	1	1	yes	characteristic
74	C₄×C₂₉	AGL(1,ℤ/8)	1	1	yes	characteristic
75	C₄×C₂₉	AGL(1,ℤ/8)	1	1	yes	characteristic
76	C₄×C₂₉	#137	1	1	yes	characteristic
77	Dic₂₉	—	4	1	no	—
78	#486	—	29	1	no	—
79	#476	—	29	1	no	—
80	#459	—	29	1	no	—
81	#436	—	29	1	no	—
82	#335	—	29	1	no	—
83	#333	D₂₉	1	1	yes	characteristic, complemented
84	#331	—	29	1	no	—
85	C₂×C₂₉	#463	1	1	yes	characteristic
86	C₂×C₂₉	#468	1	1	yes	characteristic
87	C₂×C₂₉	#472	1	1	yes	characteristic
88	D₂₉	—	8	1	no	—
89	D₂₉	—	16	1	no	—
90	C₂×D₈	—	58	1	no	—
91	C₂×M₁₆	—	58	1	no	—
92	C₄⋊Q₈	C₂×D₂₉	1	1	yes	characteristic
93	#127	—	29	1	no	—
94	C₄×D₄	—	29	1	no	—
95	#106	—	58	1	no	—
96	C₄⋊C₈	—	29	1	no	—
97	C₄⋊C₈	C₂×D₂₉	1	1	yes	characteristic
98	C₄⋊C₈	C₂×D₂₉	1	1	yes	characteristic
99	#102	—	58	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	#1697	1	1	yes	characteristic, complemented, Sylow
106	C₂×Q₈	—	2	1	no	—
107	C₂×D₄	—	29	1	no	—
108	C₂×D₄	—	58	1	no	—
109	C₂×D₄	—	58	1	no	—
110	C₂×D₄	—	58	1	no	—
111	C₂²×C₄	—	58	1	no	—
112	D₈	—	116	2	no	—
113	M₁₆	—	116	2	no	—
114	C₂×C₈	—	2	1	no	—
115	C₂×C₈	—	2	1	no	—
116	C₂×C₈	—	29	1	no	—
117	C₂×C₈	—	29	1	no	—
118	C₂×C₈	—	58	1	no	—
119	C₄⋊C₄	—	2	1	no	—
120	C₄⋊C₄	—	2	1	no	—
121	C₄⋊C₄	—	29	1	no	—
122	C₂²⋊C₄	—	58	1	no	—
123	C₄²	C₂²×D₂₉	1	1	yes	characteristic
124	C₂³	—	58	1	no	—
125	C₂³	—	116	1	no	—
126	Q₈	—	8	1	no	—
127	D₄	—	58	1	no	—
128	D₄	—	58	1	no	—
129	D₄	—	116	1	no	—
130	D₄	—	116	1	no	—
131	D₄	—	116	1	no	—
132	D₄	—	116	1	no	—
133	D₄	—	232	1	no	—
134	C₂×C₄	—	4	1	no	—
135	C₂×C₄	—	58	1	no	—
136	C₂×C₄	—	116	1	no	—
137	C₂×C₄	—	116	1	no	—
138	C₂×C₄	#90805	1	1	yes	characteristic
139	C₂×C₄	D₄×D₂₉	1	1	yes	characteristic
140	C₂×C₄	D₄×D₂₉	1	1	yes	characteristic
141	C₈	—	4	1	no	—
142	C₈	—	4	1	no	—
143	C₈	—	58	1	no	—
144	C₈	—	58	1	no	—
145	C₈	—	116	1	no	—
146	C₂²	—	116	1	no	—
147	C₂²	—	116	1	no	—
148	C₂²	—	116	1	no	—
149	C₂²	—	232	1	no	—
150	C₂²	—	232	1	no	—
151	C₂²	—	232	1	no	—
152	C₂²	#11745246	1	1	yes	characteristic
153	C₄	—	2	1	no	—
154	C₄	—	8	1	no	—
155	C₄	—	116	1	no	—
156	C₄	#11745276	1	1	yes	characteristic
157	C₄	#11745276	1	1	yes	characteristic
158	C₄	#11745280	1	1	yes	characteristic
159	C₄	#11745281	1	1	yes	characteristic
160	C₂	—	464	1	no	—
161	C₂	—	232	1	no	—
162	C₂	#14271381	1	1	yes	characteristic
163	C₂	#14271412	1	1	yes	characteristic
164	C₂	#14271432	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#11745246
commutator, cosocle	C₄²×C₂₉	C₂³
Frattini	C₄²	C₂²×D₂₉
Fitting	C₂₉×#333	C₂
socle	C₂²×C₂₉	#121
prime radical	#14271359	C₂
hypercenter	#333	D₂₉
hypocenter	C₂₉	#1697
LCS	G ▷ C₄²×C₂₉ ▷ C₂²×C₂₉ ▷ C₂₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #333

Complements

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

N	H ≅ G/N	type
1. #14271511	C₂	semidirect
2. #14271450	C₂	semidirect
4. #14271240	C₂	semidirect
5. #14270915	C₂	semidirect
6. #14270906	C₂	semidirect
7. C₂₉×#333	C₂	semidirect
83. #333	D₂₉	semidirect
105. C₂₉	#1697	semidirect

Maximal subgroups

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

subgroup	index
#14271511	2
#14271450	2
#14271359	2
#14271240	2
#14270915	2
#14270906	2
C₂₉×#333	2
#1697	29