Finite Group Databasebeta

Subgroups — #16836397

Order 3712.

322 conjugacy classes of subgroups; only the 148 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 148 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	#14272215	C₂	1	4	yes	maximal, complemented
2	#14272214	C₂	1	2	yes	maximal, complemented
3	Q₈×#6599	C₂	1	2	yes	maximal, complemented
4	#14272188	C₂	1	4	yes	maximal
5	#14272185	C₂	1	1	yes	maximal, characteristic
6	#14272181	C₂	1	1	yes	maximal, characteristic
7	#14271718	C₂	1	1	yes	maximal, characteristic
8	C₂₉×C₄⋊Q₈	C₄	1	1	yes	characteristic
9	#11745308	C₄	1	2	yes	—
10	Q₈×Dic₂₉	C₂²	1	2	yes	—
11	#11745259	C₄	1	2	yes	—
12	#11745254	C₂²	1	2	yes	—
13	#11745252	C₄	1	2	yes	—
14	#11745249	C₂²	1	2	yes	—
15	C₄²×D₂₉	C₂²	1	1	yes	characteristic
16	#11745231	C₄	1	1	yes	characteristic
17	#11745220	C₂²	1	2	yes	—
18	#11745220	C₂²	1	4	yes	—
19	#11745219	C₂²	1	2	yes	—
20	#11745219	C₂²	1	2	yes	—
21	#11745218	C₂²	1	2	yes	—
22	#11745218	C₂²	1	4	yes	—
23	C₄⋊#6599	C₂²	1	2	yes	—
24	C₄⋊#6599	C₂²	1	4	yes	—
25	C₄×#6599	C₂²	1	1	yes	characteristic
26	C₄×#6599	C₂²	1	1	yes	characteristic
27	C₄×#6599	C₂²	1	4	yes	—
28	C₂×Q₈×C₂₉	C₂×C₄	1	2	yes	—
29	C₂×C₄×D₂₉	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄×D₂₉	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄×D₂₉	C₂³	1	1	yes	characteristic
32	C₂×Dic₅₈	C₂×C₄	1	2	yes	—
33	C₂×#6599	C₂³	1	2	yes	—
34	C₂×#6599	C₂³	1	4	yes	—
35	C₂×#6599	C₂³	1	2	yes	—
36	C₄⋊C₄×C₂₉	C₂×C₄	1	2	yes	—
37	C₄⋊C₄×C₂₉	C₂×C₄	1	2	yes	—
38	C₄²×C₂₉	C₂×C₄	1	1	yes	characteristic
39	#90781	C₂×C₄	1	4	yes	—
40	C₄⋊Dic₂₉	C₂×C₄	1	2	yes	—
41	C₄⋊Dic₂₉	C₂×C₄	1	4	yes	—
42	C₄⋊Dic₂₉	C₂×C₄	1	2	yes	—
43	#90779	C₂×C₄	1	4	yes	—
44	C₄×Dic₂₉	C₂×C₄	1	1	yes	characteristic
45	C₄×Dic₂₉	C₂³	1	1	yes	characteristic
46	C₄×Dic₂₉	C₂³	1	4	yes	—
47	C₄×Dic₂₉	C₂³	1	1	yes	characteristic
48	C₂²×D₂₉	C₂²×C₄	1	1	yes	characteristic
49	Q₈×C₂₉	—	2	4	no	—
50	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₄×C₂₉	C₂²×C₄	1	4	yes	—
52	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
54	C₂×Dic₂₉	C₂²×C₄	1	1	yes	characteristic
55	C₂×Dic₂₉	C₂²×C₄	1	1	yes	characteristic
56	C₂×Dic₂₉	C₂²×C₄	1	4	yes	—
57	C₂×Dic₂₉	C₂⁴	1	1	yes	characteristic
58	C₄×D₂₉	—	2	2	no	—
59	C₄×D₂₉	—	2	2	no	—
60	C₄×D₂₉	—	4	1	no	—
61	Dic₅₈	—	4	2	no	—
62	#6599	—	2	2	no	—
63	#6599	—	2	4	no	—
64	#6599	—	2	2	no	—
65	#2985	—	29	1	no	maximal, Sylow
66	C₂²×C₂₉	C₂³×C₄	1	1	yes	characteristic
67	C₂×D₂₉	—	2	1	no	—
68	C₂×D₂₉	—	2	1	no	—
69	C₂×D₂₉	—	2	1	no	—
70	C₄×C₂₉	—	2	1	no	—
71	C₄×C₂₉	—	2	4	no	—
72	C₄×C₂₉	—	2	1	no	—
73	C₄×C₂₉	#131	1	2	yes	—
74	Dic₂₉	—	2	1	no	—
75	Dic₂₉	—	2	4	no	—
76	Dic₂₉	—	2	1	no	—
77	Dic₂₉	2₋¹⁺⁴	1	2	yes	—
78	#533	—	29	1	no	—
79	#446	—	29	2	no	—
80	#446	—	29	4	no	—
81	C₈×Q₈	—	29	2	no	—
82	#433	—	29	4	no	—
83	#432	—	29	1	no	—
84	#431	—	29	1	no	—
85	C₂×C₂₉	#519	1	1	yes	characteristic
86	C₂×C₂₉	C₂×#131	1	1	yes	characteristic
87	C₂×C₂₉	#568	1	1	yes	characteristic
88	D₂₉	—	4	1	no	—
89	C₄⋊Q₈	—	29	1	no	—
90	C₄⋊Q₈	#1181	1	1	yes	characteristic
91	#125	—	29	2	no	—
92	C₂²⋊Q₈	—	29	2	no	—
93	C₂²⋊Q₈	—	29	2	no	—
94	C₄×Q₈	—	29	2	no	—
95	C₄×Q₈	—	29	2	no	—
96	#117	—	29	2	no	—
97	C₂×C₄²	—	29	1	no	—
98	C₄⋊C₈	—	29	2	no	—
99	C₄⋊C₈	—	29	2	no	—
100	C₄⋊C₈	—	29	4	no	—
101	C₄⋊C₈	—	29	4	no	—
102	C₂²⋊C₈	—	29	2	no	—
103	C₂²⋊C₈	—	29	2	no	—
104	#97	—	29	2	no	—
105	#97	—	29	4	no	—
106	C₄×C₈	—	29	1	no	—
107	C₄×C₈	—	29	1	no	—
108	C₄×C₈	—	29	4	no	—
109	C₂₉	#2985	1	1	yes	characteristic, complemented, Sylow
110	C₂×Q₈	—	29	2	no	—
111	C₂×Q₈	C₂×#1181	1	2	yes	—
112	C₂²×C₄	—	29	1	no	—
113	C₂²×C₄	—	29	1	no	—
114	C₂²×C₄	—	29	1	no	—
115	C₂×C₈	—	29	2	no	—
116	C₂×C₈	—	29	2	no	—
117	C₂×C₈	—	29	4	no	—
118	C₄⋊C₄	—	29	2	no	—
119	C₄⋊C₄	—	29	4	no	—
120	C₄⋊C₄	—	29	4	no	—
121	C₄⋊C₄	—	29	2	no	—
122	C₄⋊C₄	C₂×#1181	1	2	yes	—
123	C₄⋊C₄	C₂×#1181	1	2	yes	—
124	C₂²⋊C₄	—	29	4	no	—
125	C₄²	—	29	1	no	—
126	C₄²	—	29	4	no	—
127	C₄²	—	29	1	no	—
128	C₄²	—	29	1	no	—
129	C₄²	C₂×#1181	1	1	yes	characteristic
130	C₂³	—	29	1	no	—
131	Q₈	—	2	4	no	—
132	Q₈	—	116	2	no	—
133	C₂×C₄	—	29	1	no	—
134	C₂×C₄	—	29	1	no	—
135	C₂×C₄	—	29	1	no	—
136	C₂×C₄	—	29	4	no	—
137	C₂×C₄	—	58	2	no	—
138	C₂×C₄	—	58	2	no	—
139	C₂×C₄	—	116	1	no	—
140	C₂×C₄	C₂²×#1181	1	1	yes	characteristic
141	C₂×C₄	C₂²×#1181	1	1	yes	characteristic
142	C₂×C₄	C₂²×#1181	1	4	yes	—
143	C₂×C₄	C₂²×#1181	1	1	yes	characteristic
144	C₈	—	58	2	no	—
145	C₈	—	58	2	no	—
146	C₈	—	58	4	no	—
147	C₂²	—	58	1	no	—
148	C₂²	—	58	1	no	—
149	C₂²	—	58	1	no	—
150	C₂²	C₂³×#1181	1	1	yes	characteristic
151	C₄	—	2	1	no	—
152	C₄	—	2	4	no	—
153	C₄	—	2	1	no	—
154	C₄	—	29	2	no	—
155	C₄	—	58	1	no	—
156	C₄	—	58	1	no	—
157	C₄	—	58	4	no	—
158	C₄	#11745346	1	2	yes	—
159	C₂	—	116	1	no	—
160	C₂	#14272231	1	1	yes	characteristic
161	C₂	C₂×#11745346	1	1	yes	characteristic
162	C₂	#14272243	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×#1181
commutator, socle	C₂²×C₂₉	C₂³×C₄
Fitting	C₂₉×C₄⋊Q₈	C₄
prime radical	C₂²×D₂₉	C₂²×C₄
cosocle	C₂×Dic₂₉	C₂⁴
hypercenter	C₄⋊Q₈	#1181
hypocenter	C₂₉	#2985
LCS	G ▷ C₂²×C₂₉ ▷ C₂₉
UCS	C₁ ◁ C₂² ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
1. #14272215	C₂	semidirect
2. #14272214	C₂	semidirect
3. Q₈×#6599	C₂	semidirect
109. C₂₉	#2985	semidirect

Maximal subgroups

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

subgroup	index
#14272215	2
#14272214	2
Q₈×#6599	2
#14272188	2
#14272185	2
#14272181	2
#14271718	2
#2985	29