Finite Group Databasebeta

Subgroups — #16835686

Order 3712.

228 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 70 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	#14272166	C₂	1	1	yes	maximal, characteristic
2	C₄×#90800	C₂	1	1	yes	maximal, characteristic
3	#14272043	C₂	1	2	yes	maximal
4	#14272043	C₂	1	2	yes	maximal
5	C₂×#11745251	C₂	1	1	yes	maximal, characteristic
6	C₂×#90800	C₂²	1	2	yes	—
7	C₂×#90800	C₂²	1	2	yes	—
8	C₂×C₄⋊C₄×C₂₉	C₄	1	1	yes	characteristic
9	C₂×C₄⋊Dic₂₉	C₄	1	1	yes	characteristic
10	C₂×#90779	C₄	1	1	yes	characteristic
11	C₂×#90779	C₄	1	1	yes	characteristic
12	C₂×#90779	C₂²	1	1	yes	characteristic
13	C₂×#90779	C₂²	1	1	yes	characteristic
14	C₂×C₄×Dic₂₉	C₂²	1	1	yes	characteristic
15	#11745251	—	2	1	no	—
16	#11745251	—	2	1	no	—
17	#11745251	—	2	1	no	—
18	#11745251	—	2	1	no	—
19	#11745220	—	2	1	no	—
20	#11745218	—	2	1	no	—
21	C₄⋊#6599	—	2	1	no	—
22	C₄×#6599	—	2	1	no	—
23	C₂²×C₄×C₂₉	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₄×C₂₉	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₄×C₂₉	C₂×C₄	1	1	yes	characteristic
26	C₂²×Dic₂₉	C₂×C₄	1	1	yes	characteristic
27	C₂²×Dic₂₉	C₂×C₄	1	1	yes	characteristic
28	C₂²×Dic₂₉	C₂×C₄	1	1	yes	characteristic
29	C₂²×Dic₂₉	C₂³	1	1	yes	characteristic
30	#90800	—	2	2	no	—
31	#90800	—	2	4	no	—
32	#90800	Q₈	1	4	yes	—
33	C₂×#6599	—	2	2	no	—
34	C₂×#6599	—	2	2	no	—
35	C₄⋊C₄×C₂₉	—	2	1	no	—
36	C₄⋊C₄×C₂₉	—	2	1	no	—
37	C₄⋊Dic₂₉	—	2	1	no	—
38	C₄⋊Dic₂₉	—	2	1	no	—
39	#90779	—	2	1	no	—
40	#90779	—	2	1	no	—
41	#90779	—	2	1	no	—
42	#90779	—	2	1	no	—
43	#90779	—	2	1	no	—
44	#90779	—	2	1	no	—
45	#90779	—	2	1	no	—
46	#90779	—	2	1	no	—
47	C₄×Dic₂₉	D₄	1	1	yes	characteristic
48	C₄×Dic₂₉	D₄	1	1	yes	characteristic
49	C₄×Dic₂₉	D₄	1	1	yes	characteristic
50	C₄×Dic₂₉	D₄	1	1	yes	characteristic
51	C₂³×C₂₉	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₄×C₂₉	—	2	1	no	—
53	C₂×C₄×C₂₉	—	2	1	no	—
54	C₂×C₄×C₂₉	—	4	1	no	—
55	C₂×C₄×C₂₉	—	4	1	no	—
56	C₂×C₄×C₂₉	—	4	1	no	—
57	C₂×C₄×C₂₉	C₂²⋊C₄	1	1	yes	characteristic
58	C₂×C₄×C₂₉	C₂²⋊C₄	1	1	yes	characteristic
59	C₂×Dic₂₉	—	2	1	no	—
60	C₂×Dic₂₉	—	2	1	no	—
61	C₂×Dic₂₉	—	4	1	no	—
62	C₂×Dic₂₉	—	4	1	no	—
63	C₂×Dic₂₉	—	4	1	no	—
64	C₂×Dic₂₉	C₂²⋊C₄	1	1	yes	characteristic
65	C₂×Dic₂₉	C₂²⋊C₄	1	1	yes	characteristic
66	C₂×Dic₂₉	C₂×D₄	1	1	yes	characteristic
67	C₂×Dic₂₉	C₂×D₄	1	1	yes	characteristic
68	C₂×Dic₂₉	C₂×Q₈	1	2	yes	—
69	C₂×Dic₂₉	#41	1	2	yes	—
70	#6599	—	2	4	no	—
71	#6599	—	4	2	no	—
72	#1998	—	29	1	no	maximal, Sylow
73	C₂²×C₂₉	—	2	1	no	—
74	C₂²×C₂₉	—	2	1	no	—
75	C₂²×C₂₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
76	C₂²×C₂₉	C₄×Q₈	1	2	yes	—
77	C₄×C₂₉	—	2	1	no	—
78	C₄×C₂₉	—	2	1	no	—
79	C₄×C₂₉	—	4	1	no	—
80	C₄×C₂₉	—	4	1	no	—
81	Dic₂₉	—	2	1	no	—
82	Dic₂₉	—	2	1	no	—
83	Dic₂₉	—	4	1	no	—
84	Dic₂₉	—	4	1	no	—
85	Dic₂₉	C₂²⋊Q₈	1	2	yes	—
86	Dic₂₉	#124	1	1	yes	characteristic
87	Dic₂₉	C₄⋊Q₈	1	1	yes	characteristic
88	C₂×#125	—	29	1	no	—
89	#423	—	29	1	no	—
90	C₄×M₁₆	—	29	1	no	—
91	#343	—	29	2	no	—
92	#343	—	29	2	no	—
93	C₂×C₂₉	—	2	2	no	—
94	C₂×C₂₉	#391	1	1	yes	characteristic
95	C₂×C₂₉	#413	1	1	yes	characteristic
96	C₂×C₂₉	#413	1	1	yes	characteristic
97	C₂×M₁₆	—	29	2	no	—
98	C₂×M₁₆	—	29	2	no	—
99	#125	—	58	1	no	—
100	#125	—	58	1	no	—
101	#125	—	58	1	no	—
102	#125	—	58	1	no	—
103	C₂×C₄⋊C₄	—	29	1	no	—
104	C₂×C₄⋊C₄	—	29	1	no	—
105	C₂×C₄⋊C₄	—	29	1	no	—
106	C₂×C₄⋊C₄	—	29	1	no	—
107	C₂×C₄⋊C₄	—	29	1	no	—
108	C₂×C₄⋊C₄	#1181	1	1	yes	characteristic
109	C₂×C₄²	—	29	1	no	—
110	C₄⋊C₈	—	58	1	no	—
111	C₄⋊C₈	—	58	1	no	—
112	#97	—	58	1	no	—
113	C₄×C₈	—	58	1	no	—
114	C₂₉	#1998	1	1	yes	characteristic, complemented, Sylow
115	C₂²×C₄	—	29	1	no	—
116	C₂²×C₄	—	29	1	no	—
117	C₂²×C₄	—	29	1	no	—
118	C₂²×C₄	—	29	1	no	—
119	C₂²×C₄	C₂×#1181	1	1	yes	characteristic
120	C₂²×C₄	C₂×#1181	1	1	yes	characteristic
121	C₂²×C₄	C₂×#1181	1	1	yes	characteristic
122	M₁₆	—	29	4	no	—
123	M₁₆	—	58	2	no	—
124	M₁₆	—	58	4	no	—
125	C₂×C₈	—	58	2	no	—
126	C₂×C₈	—	58	2	no	—
127	C₄⋊C₄	—	2	1	no	—
128	C₄⋊C₄	—	2	1	no	—
129	C₄⋊C₄	—	58	1	no	—
130	C₄⋊C₄	—	58	1	no	—
131	C₄⋊C₄	—	58	1	no	—
132	C₄⋊C₄	—	58	1	no	—
133	C₄⋊C₄	—	58	1	no	—
134	C₄⋊C₄	—	58	1	no	—
135	C₄⋊C₄	—	58	1	no	—
136	C₄⋊C₄	—	58	1	no	—
137	C₄⋊C₄	—	58	1	no	—
138	C₄⋊C₄	—	58	1	no	—
139	C₄²	—	29	1	no	—
140	C₄²	—	29	1	no	—
141	C₄²	—	29	1	no	—
142	C₄²	—	29	1	no	—
143	C₂³	C₂²×#1181	1	1	yes	characteristic
144	C₂×C₄	—	2	1	no	—
145	C₂×C₄	—	2	1	no	—
146	C₂×C₄	—	4	1	no	—
147	C₂×C₄	—	4	1	no	—
148	C₂×C₄	—	4	1	no	—
149	C₂×C₄	—	29	2	no	—
150	C₂×C₄	—	29	2	no	—
151	C₂×C₄	—	29	1	no	—
152	C₂×C₄	—	29	1	no	—
153	C₂×C₄	—	29	1	no	—
154	C₂×C₄	—	29	1	no	—
155	C₂×C₄	—	58	1	no	—
156	C₂×C₄	—	58	1	no	—
157	C₂×C₄	—	116	1	no	—
158	C₂×C₄	—	116	1	no	—
159	C₂×C₄	—	116	1	no	—
160	C₂×C₄	C₂²⋊#1181	1	1	yes	characteristic
161	C₂×C₄	C₂²⋊#1181	1	1	yes	characteristic
162	C₈	—	58	4	no	—
163	C₈	—	116	2	no	—
164	C₂²	—	2	1	no	—
165	C₂²	—	2	1	no	—
166	C₂²	Q₈×#1181	1	2	yes	—
167	C₂²	C₂×C₂²⋊#1181	1	1	yes	characteristic
168	C₄	—	2	1	no	—
169	C₄	—	2	1	no	—
170	C₄	—	4	1	no	—
171	C₄	—	4	1	no	—
172	C₄	—	29	2	no	—
173	C₄	—	29	1	no	—
174	C₄	—	29	1	no	—
175	C₄	—	58	1	no	—
176	C₄	—	58	1	no	—
177	C₄	—	116	1	no	—
178	C₄	—	116	1	no	—
179	C₂	—	2	2	no	—
180	C₂	(C₂×#1181)⋊Q₈	1	1	yes	characteristic
181	C₂	#14272149	1	1	yes	characteristic
182	C₂	#14272150	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊#1181
commutator, prime radical	C₂³×C₂₉	C₂²×C₄
Frattini	C₂³	C₂²×#1181
Fitting	C₂×C₄⋊C₄×C₂₉	C₄
socle	C₂²×C₂₉	C₂×C₂²⋊C₄
cosocle	C₂²×Dic₂₉	C₂³
hypercenter	C₂×C₄⋊C₄	#1181
hypocenter	C₂₉	#1998
LCS	G ▷ C₂³×C₂₉ ▷ C₂×C₂₉ ▷ C₂₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
114. C₂₉	#1998	semidirect

Maximal subgroups

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

subgroup	index
#14272166	2
C₄×#90800	2
#14272043	2
C₂×#11745251	2
#1998	29