Finite Group Databasebeta

Subgroups — #16835649

Order 3712.

284 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 66 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	#14272173	C₂	1	1	yes	maximal, characteristic, complemented
2	#14272161	C₂	1	2	yes	maximal, complemented
3	#14272143	C₂	1	2	yes	maximal, complemented
4	#14272126	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#11745274	C₂	1	1	yes	maximal, characteristic
6	C₂×#90807	C₂²	1	2	yes	—
7	#11745345	C₂²	1	2	yes	—
8	#11745345	C₂²	1	1	yes	characteristic
9	D₂₉⋊M₁₆	C₂²	1	1	yes	characteristic
10	C₂×D₈×C₂₉	C₄	1	1	yes	characteristic, complemented
11	C₂×#90783	C₄	1	2	yes	complemented
12	#11745274	—	2	1	no	—
13	#11745274	—	2	1	no	—
14	#11745274	—	2	1	no	—
15	#11745274	—	2	1	no	—
16	C₂×Dic₁₁₆	C₄	1	1	yes	characteristic, complemented
17	C₂×C₈×D₂₉	C₂²	1	1	yes	characteristic
18	#11745227	—	2	4	no	—
19	#11745224	—	2	4	no	—
20	#11745210	—	2	1	no	—
21	#11745209	—	2	1	no	—
22	#11745208	—	2	1	no	—
23	C₈×#1181	—	2	1	no	—
24	C₂×D₄×C₂₉	C₂×C₄	1	2	yes	—
25	C₂×C₂²⋊D₂₉	—	2	2	no	—
26	C₂²×Dic₂₉	—	2	2	no	—
27	#90807	—	2	2	no	—
28	#90807	—	2	2	no	—
29	#90807	—	2	2	no	—
30	#90807	—	2	2	no	—
31	C₂×C₄×D₂₉	C₂³	1	1	yes	characteristic
32	C₂×Dic₅₈	C₂×C₄	1	2	yes	—
33	C₂²⋊#1181	—	2	2	no	—
34	C₂²⋊#1181	D₄	1	2	yes	—
35	#90800	—	2	1	no	—
36	C₄⋊#1181	—	2	2	no	—
37	C₄⋊#1181	—	2	1	no	—
38	C₄×#1181	—	2	1	no	—
39	C₄×#1181	—	2	2	no	—
40	#90796	D₄	1	2	yes	—
41	D₂₉⋊C₈	—	2	1	no	—
42	D₈×C₂₉	—	2	1	no	—
43	D₈×C₂₉	—	2	1	no	—
44	C₂×C₈×C₂₉	C₂×C₄	1	1	yes	characteristic
45	#90783	—	2	2	no	—
46	#90783	—	2	2	no	—
47	C₂×#6597	C₂×C₄	1	1	yes	characteristic
48	Dic₁₁₆	—	2	1	no	—
49	Dic₁₁₆	—	2	1	no	—
50	C₈×D₂₉	D₄	1	1	yes	characteristic
51	C₈×D₂₉	D₄	1	1	yes	characteristic
52	C₈×D₂₉	D₄	1	1	yes	characteristic
53	C₈×D₂₉	D₄	1	1	yes	characteristic
54	C₂³×C₂₉	—	2	2	no	—
55	C₂²×D₂₉	C₂×D₄	1	1	yes	characteristic
56	C₂×#1181	—	2	2	no	—
57	C₂×#1181	—	4	2	no	—
58	D₄×C₂₉	—	2	2	no	—
59	D₄×C₂₉	—	2	2	no	—
60	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
61	C₂²⋊D₂₉	—	4	2	no	—
62	C₂²⋊D₂₉	—	4	2	no	—
63	C₂×Dic₂₉	—	2	2	no	—
64	C₂×Dic₂₉	—	4	2	no	—
65	C₂×Dic₂₉	—	4	2	no	—
66	C₂×Dic₂₉	#41	1	1	yes	characteristic
67	C₄×D₂₉	C₂×D₄	1	1	yes	characteristic
68	C₄×D₂₉	C₂×D₄	1	1	yes	characteristic
69	C₄×D₂₉	C₂×D₄	1	1	yes	characteristic
70	C₄×D₂₉	#41	1	1	yes	characteristic
71	Dic₅₈	—	2	2	no	—
72	Dic₅₈	—	2	2	no	—
73	#6599	—	2	2	no	—
74	C₈×C₂₉	C₂²⋊C₄	1	1	yes	characteristic
75	C₈×C₂₉	C₂²⋊C₄	1	1	yes	characteristic
76	#6597	C₂²⋊C₄	1	1	yes	characteristic
77	#6597	C₂²⋊C₄	1	1	yes	characteristic
78	#1976	—	29	1	no	maximal, Sylow
79	C₂²×C₂₉	—	4	2	no	—
80	C₂²×C₂₉	—	8	2	no	—
81	C₂²×C₂₉	C₄×D₄	1	1	yes	characteristic
82	C₂×D₂₉	—	4	1	no	—
83	C₂×D₂₉	#121	1	1	yes	characteristic
84	C₂×D₂₉	#127	1	1	yes	characteristic
85	#1181	—	4	2	no	—
86	#1181	—	8	2	no	—
87	C₄×C₂₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
88	C₄×C₂₉	C₄×D₄	1	1	yes	characteristic
89	Dic₂₉	—	4	2	no	—
90	Dic₂₉	#121	1	1	yes	characteristic
91	Dic₂₉	#124	1	1	yes	characteristic
92	C₂×#135	—	29	1	no	—
93	#427	—	29	1	no	—
94	#421	—	29	2	no	—
95	#410	—	29	2	no	—
96	#405	—	29	1	no	—
97	C₂×C₂₉	—	2	1	no	—
98	C₂×C₂₉	—	8	2	no	—
99	C₂×C₂₉	#390	1	1	yes	characteristic
100	D₂₉	—	2	1	no	—
101	D₂₉	—	2	1	no	—
102	C₂×#41	—	29	2	no	—
103	#135	—	58	1	no	—
104	#135	—	58	1	no	—
105	#135	—	58	1	no	—
106	#135	—	58	1	no	—
107	C₂×Q₁₆	—	29	1	no	—
108	C₂×SD₁₆	—	29	2	no	—
109	C₂×D₈	#1181	1	1	yes	characteristic, complemented
110	C₂×M₁₆	—	29	1	no	—
111	C₂²×C₈	—	29	1	no	—
112	#117	—	29	1	no	—
113	#117	—	29	2	no	—
114	#107	—	58	1	no	—
115	#106	—	58	1	no	—
116	#104	—	58	4	no	—
117	#99	—	58	4	no	—
118	#97	—	58	1	no	—
119	C₄×C₈	—	58	1	no	—
120	C₂₉	#1976	1	1	yes	characteristic, complemented, Sylow
121	#41	—	58	2	no	—
122	#41	—	58	2	no	—
123	#41	—	58	2	no	—
124	#41	—	58	2	no	—
125	C₂×Q₈	—	29	2	no	—
126	C₂×D₄	—	58	2	no	—
127	C₂×D₄	C₂×#1181	1	2	yes	—
128	C₂²×C₄	—	29	1	no	—
129	C₂²×C₄	—	58	2	no	—
130	Q₁₆	—	58	1	no	—
131	Q₁₆	—	58	1	no	—
132	SD₁₆	—	58	2	no	—
133	SD₁₆	—	58	2	no	—
134	D₈	—	2	1	no	—
135	D₈	—	2	1	no	—
136	M₁₆	—	29	2	no	—
137	M₁₆	—	58	1	no	—
138	C₂×C₈	—	29	1	no	—
139	C₂×C₈	—	29	1	no	—
140	C₂×C₈	—	29	1	no	—
141	C₂×C₈	—	29	1	no	—
142	C₂×C₈	—	29	1	no	—
143	C₂×C₈	—	58	1	no	—
144	C₂×C₈	C₂×#1181	1	1	yes	characteristic
145	C₄⋊C₄	—	58	1	no	—
146	C₄⋊C₄	—	58	2	no	—
147	C₂²⋊C₄	—	29	2	no	—
148	C₂²⋊C₄	—	58	2	no	—
149	C₄²	—	58	1	no	—
150	C₄²	—	58	2	no	—
151	C₂³	—	2	2	no	—
152	C₂³	—	29	1	no	—
153	Q₈	—	58	2	no	—
154	Q₈	—	58	2	no	—
155	D₄	—	2	2	no	—
156	D₄	—	2	2	no	—
157	D₄	—	116	2	no	—
158	D₄	—	116	2	no	—
159	C₂×C₄	—	29	1	no	—
160	C₂×C₄	—	29	1	no	—
161	C₂×C₄	—	29	1	no	—
162	C₂×C₄	—	29	1	no	—
163	C₂×C₄	—	29	1	no	—
164	C₂×C₄	—	58	2	no	—
165	C₂×C₄	—	58	2	no	—
166	C₂×C₄	—	116	2	no	—
167	C₂×C₄	—	116	2	no	—
168	C₂×C₄	—	116	2	no	—
169	C₂×C₄	C₂²×#1181	1	1	yes	characteristic
170	C₈	—	29	1	no	—
171	C₈	—	29	1	no	—
172	C₈	—	58	2	no	—
173	C₈	C₂²⋊#1181	1	1	yes	characteristic
174	C₈	C₂²⋊#1181	1	1	yes	characteristic
175	C₂²	—	4	2	no	—
176	C₂²	—	8	2	no	—
177	C₂²	—	29	1	no	—
178	C₂²	—	29	1	no	—
179	C₂²	—	116	1	no	—
180	C₂²	D₄×#1181	1	1	yes	characteristic
181	C₄	—	29	1	no	—
182	C₄	—	29	1	no	—
183	C₄	—	116	2	no	—
184	C₄	—	116	2	no	—
185	C₄	—	232	2	no	—
186	C₄	D₄×#1181	1	1	yes	characteristic
187	C₄	C₂×C₂²⋊#1181	1	1	yes	characteristic
188	C₂	—	2	1	no	—
189	C₂	—	8	2	no	—
190	C₂	—	58	1	no	—
191	C₂	—	58	1	no	—
192	C₂	#14272115	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#14272115
commutator	C₂×C₄×C₂₉	C₂²×C₄
Frattini	C₂×C₄	C₂²×#1181
Fitting	C₂×D₈×C₂₉	C₄
socle	C₂×C₂₉	#390
prime radical	C₂×#11745274	C₂
cosocle	C₂×C₄×D₂₉	C₂³
hypercenter	C₂×D₈	#1181
hypocenter	C₂₉	#1976
LCS	G ▷ C₂×C₄×C₂₉ ▷ C₂×C₂₉ ▷ C₂₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×D₈

Complements

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

N	H ≅ G/N	type
1. #14272173	C₂	semidirect
2. #14272161	C₂	semidirect
3. #14272143	C₂	semidirect
4. #14272126	C₂	semidirect
10. C₂×D₈×C₂₉	C₄	semidirect
11. C₂×#90783	C₄	semidirect
16. C₂×Dic₁₁₆	C₄	semidirect
109. C₂×D₈	#1181	semidirect
120. C₂₉	#1976	semidirect

Maximal subgroups

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

subgroup	index
#14272173	2
#14272161	2
#14272143	2
#14272126	2
C₂×#11745274	2
#1976	29