Finite Group Databasebeta

Subgroups — #16819157

Order 3712.

242 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 242 conjugacy classes of subgroups — too many to draw. Only its 61 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	#14271539	C₂	1	1	yes	maximal, characteristic, complemented
2	#14271539	C₂	1	1	yes	maximal, characteristic, complemented
3	#14271535	C₂	1	1	yes	maximal, characteristic, complemented
4	#14271510	C₂	1	1	yes	maximal, characteristic, complemented
5	#14271509	C₂	1	1	yes	maximal, characteristic, complemented
6	#14270885	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₉×#493	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×D₈×C₂₉	—	2	2	no	—
9	C₂₉×#127	C₂²	1	1	yes	characteristic
10	C₂₉×#127	C₂²	1	1	yes	characteristic
11	C₂×#90783	—	2	2	no	—
12	C₂×#90782	—	2	2	no	—
13	C₂×#90782	—	2	2	no	—
14	#11745236	C₂²	1	1	yes	characteristic
15	#11745231	C₂²	1	1	yes	characteristic
16	C₄×C₈×C₂₉	C₂²	1	1	yes	characteristic
17	#11745180	—	2	2	no	—
18	#11745169	—	2	2	no	—
19	C₄×#6597	C₂²	1	1	yes	characteristic
20	C₄×#6597	C₂²	1	1	yes	characteristic
21	C₂×D₄×C₂₉	—	2	1	no	—
22	C₂×D₄×C₂₉	—	2	1	no	—
23	C₂×D₄×C₂₉	—	2	1	no	—
24	C₂×D₄×C₂₉	—	2	1	no	—
25	C₂×D₄×C₂₉	—	2	1	no	—
26	C₂×D₄×C₂₉	—	2	1	no	—
27	C₂×D₁₁₆	—	2	1	no	—
28	C₂×D₁₁₆	—	2	1	no	—
29	C₂×D₁₁₆	—	2	1	no	—
30	C₂×Dic₅₈	—	2	1	no	—
31	D₈×C₂₉	—	4	4	no	—
32	C₂×C₈×C₂₉	D₄	1	2	yes	—
33	C₄²×C₂₉	C₂³	1	1	yes	characteristic
34	#90783	—	4	4	no	—
35	#90782	—	4	4	no	—
36	#90782	—	4	4	no	—
37	C₄⋊Dic₂₉	—	2	1	no	—
38	C₄⋊Dic₂₉	—	2	1	no	—
39	C₂×#6597	D₄	1	2	yes	—
40	C₂×#6597	D₄	1	2	yes	—
41	C₂³×C₂₉	—	4	1	no	—
42	C₂³×C₂₉	—	4	1	no	—
43	C₂²×D₂₉	—	4	1	no	—
44	D₄×C₂₉	—	4	2	no	—
45	D₄×C₂₉	—	4	2	no	—
46	D₄×C₂₉	—	4	2	no	—
47	D₄×C₂₉	—	4	2	no	—
48	D₄×C₂₉	—	4	2	no	—
49	D₄×C₂₉	—	4	2	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	C₂×D₄	1	1	yes	characteristic
53	C₂×Dic₂₉	—	4	1	no	—
54	D₁₁₆	—	4	2	no	—
55	D₁₁₆	—	4	2	no	—
56	D₁₁₆	—	4	2	no	—
57	Dic₅₈	—	4	2	no	—
58	C₈×C₂₉	—	2	2	no	—
59	#6597	—	2	2	no	—
60	#6597	D₈	1	4	yes	complemented
61	#1721	—	29	1	no	maximal, Sylow
62	C₂²×C₂₉	—	8	1	no	—
63	C₂²×C₂₉	—	8	1	no	—
64	C₂²×C₂₉	—	8	1	no	—
65	C₂²×C₂₉	—	8	1	no	—
66	C₂²×C₂₉	—	8	1	no	—
67	C₂²×C₂₉	—	8	1	no	—
68	C₂²×C₂₉	#127	1	1	yes	characteristic
69	C₂×D₂₉	—	8	1	no	—
70	C₂×D₂₉	—	8	1	no	—
71	C₂×D₂₉	—	8	1	no	—
72	C₄×C₂₉	C₂×D₈	1	2	yes	—
73	C₄×C₂₉	AGL(1,ℤ/8)	1	2	yes	—
74	C₄×C₂₉	AGL(1,ℤ/8)	1	2	yes	—
75	Dic₂₉	—	8	1	no	—
76	#493	—	29	1	no	—
77	#493	—	29	1	no	—
78	#493	D₂₉	1	1	yes	characteristic, complemented
79	#492	—	29	1	no	—
80	#486	—	29	1	no	—
81	#486	—	29	1	no	—
82	#322	—	29	1	no	—
83	C₂×C₂₉	—	16	1	no	—
84	C₂×C₂₉	—	16	1	no	—
85	C₂×C₂₉	#493	1	1	yes	characteristic
86	C₂×C₂₉	#496	1	1	yes	characteristic
87	C₂×C₂₉	#496	1	1	yes	characteristic
88	D₂₉	—	16	1	no	—
89	C₂×SD₁₆	—	58	2	no	—
90	C₂×D₈	—	2	2	no	—
91	C₂×D₈	—	58	2	no	—
92	C₂×D₈	—	58	2	no	—
93	C₄⋊Q₈	—	29	1	no	—
94	#127	—	29	1	no	—
95	#127	C₂×D₂₉	1	1	yes	characteristic
96	#127	C₂×D₂₉	1	1	yes	characteristic
97	#102	—	58	2	no	—
98	#102	—	58	2	no	—
99	C₄×C₈	—	29	1	no	—
100	C₄×C₈	—	29	1	no	—
101	C₄×C₈	C₂×D₂₉	1	1	yes	characteristic
102	C₂₉	#1721	1	1	yes	characteristic, complemented, Sylow
103	C₂×Q₈	—	58	1	no	—
104	C₂×D₄	—	2	1	no	—
105	C₂×D₄	—	2	1	no	—
106	C₂×D₄	—	2	1	no	—
107	C₂×D₄	—	2	1	no	—
108	C₂×D₄	—	2	1	no	—
109	C₂×D₄	—	2	1	no	—
110	C₂×D₄	—	58	1	no	—
111	C₂×D₄	—	58	1	no	—
112	C₂×D₄	—	58	1	no	—
113	SD₁₆	—	116	4	no	—
114	D₈	—	4	4	no	—
115	D₈	—	116	4	no	—
116	D₈	—	116	4	no	—
117	C₂×C₈	—	29	2	no	—
118	C₂×C₈	—	29	2	no	—
119	C₂×C₈	C₂²⋊D₂₉	1	2	yes	—
120	C₄⋊C₄	—	58	1	no	—
121	C₄⋊C₄	—	58	1	no	—
122	C₄²	C₂²×D₂₉	1	1	yes	characteristic
123	C₂³	—	4	1	no	—
124	C₂³	—	4	1	no	—
125	C₂³	—	116	1	no	—
126	Q₈	—	116	2	no	—
127	D₄	—	4	2	no	—
128	D₄	—	4	2	no	—
129	D₄	—	4	2	no	—
130	D₄	—	4	2	no	—
131	D₄	—	4	2	no	—
132	D₄	—	4	2	no	—
133	D₄	—	116	2	no	—
134	D₄	—	116	2	no	—
135	D₄	—	116	2	no	—
136	C₂×C₄	—	116	1	no	—
137	C₂×C₄	D₄×D₂₉	1	1	yes	characteristic
138	C₂×C₄	D₄×D₂₉	1	1	yes	characteristic
139	C₂×C₄	C₂×C₂²⋊D₂₉	1	1	yes	characteristic
140	C₈	—	2	2	no	—
141	C₈	—	29	4	no	—
142	C₈	—	58	2	no	—
143	C₂²	—	8	1	no	—
144	C₂²	—	8	1	no	—
145	C₂²	—	8	1	no	—
146	C₂²	—	8	1	no	—
147	C₂²	—	8	1	no	—
148	C₂²	—	8	1	no	—
149	C₂²	—	232	1	no	—
150	C₂²	—	232	1	no	—
151	C₂²	—	232	1	no	—
152	C₂²	#11745302	1	1	yes	characteristic
153	C₄	—	232	1	no	—
154	C₄	D₈×D₂₉	1	2	yes	—
155	C₄	#11745273	1	2	yes	—
156	C₄	#11745294	1	2	yes	—
157	C₂	—	16	1	no	—
158	C₂	—	16	1	no	—
159	C₂	—	464	1	no	—
160	C₂	#14271538	1	1	yes	characteristic
161	C₂	#14271550	1	1	yes	characteristic
162	C₂	#14271551	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#11745302
commutator, cosocle	C₄²×C₂₉	C₂³
Frattini	C₄²	C₂²×D₂₉
Fitting	C₂₉×#493	C₂
socle	C₂²×C₂₉	#127
hypercenter	#493	D₂₉
hypocenter	C₂₉	#1721
LCS	G ▷ C₄²×C₂₉ ▷ C₂²×C₂₉ ▷ C₂₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #493

Complements

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

N	H ≅ G/N	type
1. #14271539	C₂	semidirect
2. #14271539	C₂	semidirect
3. #14271535	C₂	semidirect
4. #14271510	C₂	semidirect
5. #14271509	C₂	semidirect
6. #14270885	C₂	semidirect
7. C₂₉×#493	C₂	semidirect
60. #6597	D₈	semidirect
78. #493	D₂₉	semidirect
102. C₂₉	#1721	semidirect

Maximal subgroups

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

subgroup	index
#14271539	2
#14271535	2
#14271510	2
#14271509	2
#14270885	2
C₂₉×#493	2
#1721	29