Finite Group Databasebeta

Subgroups — #16835534

Order 3712.

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

normal subgroups only

This group has 312 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	#14272243	C₂	1	1	yes	maximal, characteristic
2	#14272156	C₂	1	2	yes	maximal, complemented
3	#14272144	C₂	1	1	yes	maximal, characteristic
4	#14272030	C₂	1	2	yes	maximal, complemented
5	#14271738	C₂	1	1	yes	maximal, characteristic
6	C₂×#90809	C₂²	1	1	yes	characteristic
7	C₂×Q₈×D₂₉	C₂²	1	1	yes	characteristic
8	#11745348	—	2	4	no	—
9	#11745345	C₂²	1	2	yes	—
10	D₂₉⋊M₁₆	—	2	2	no	—
11	D₂₉⋊M₁₆	C₂²	1	2	yes	—
12	C₂₉×C₄⋊Q₈	C₄	1	1	yes	characteristic, complemented
13	#11745309	C₄	1	1	yes	characteristic, complemented
14	#11745306	C₄	1	1	yes	characteristic, complemented
15	#11745258	—	2	2	no	—
16	#11745256	—	2	2	no	—
17	#11745237	C₄	1	1	yes	characteristic, complemented
18	#11745234	C₂²	1	1	yes	characteristic
19	#11745225	—	2	4	no	—
20	#11745215	—	2	2	no	—
21	C₂×Q₈×C₂₉	C₂×C₄	1	1	yes	characteristic
22	C₂×Q₈×C₂₉	C₂×C₄	1	1	yes	characteristic
23	#90809	—	2	2	no	—
24	#90809	—	2	2	no	—
25	Q₈×D₂₉	—	2	2	no	—
26	Q₈×D₂₉	—	2	2	no	—
27	C₂×D₁₁₆	C₂×C₄	1	1	yes	characteristic
28	C₂×D₁₁₆	D₄	1	2	yes	—
29	C₂×C₄×D₂₉	—	2	1	no	—
30	C₂×C₄×D₂₉	D₄	1	2	yes	—
31	C₂×C₄×D₂₉	C₂³	1	1	yes	characteristic
32	C₂×Dic₅₈	—	2	1	no	—
33	C₂×Dic₅₈	C₂×C₄	1	1	yes	characteristic
34	C₂²⋊#1181	—	2	2	no	—
35	#90800	—	2	2	no	—
36	C₄⋊#1181	—	2	2	no	—
37	C₄×#1181	—	2	2	no	—
38	#90796	—	2	4	no	—
39	#90796	—	2	2	no	—
40	D₂₉⋊C₈	—	2	4	no	—
41	D₂₉⋊C₈	D₄	1	4	yes	—
42	C₄⋊C₄×C₂₉	—	2	2	no	—
43	C₄²×C₂₉	C₂×C₄	1	1	yes	characteristic
44	#90781	—	2	2	no	—
45	#90781	—	2	2	no	—
46	#90781	—	2	1	no	—
47	#90779	—	2	2	no	—
48	#90779	—	2	1	no	—
49	C₄×Dic₂₉	C₂×C₄	1	1	yes	characteristic
50	C₂²×D₂₉	C₂²⋊C₄	1	2	yes	—
51	C₂²×D₂₉	C₂×D₄	1	1	yes	characteristic
52	C₂×#1181	—	4	2	no	—
53	Q₈×C₂₉	—	2	2	no	—
54	Q₈×C₂₉	—	2	2	no	—
55	C₂×C₄×C₂₉	—	2	1	no	—
56	C₂×C₄×C₂₉	—	2	1	no	—
57	C₂×C₄×C₂₉	C₂²⋊C₄	1	2	yes	—
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	C₂×D₄	1	1	yes	characteristic
62	D₁₁₆	—	2	4	no	—
63	D₁₁₆	—	2	2	no	—
64	C₄×D₂₉	—	2	2	no	—
65	C₄×D₂₉	—	2	2	no	—
66	C₄×D₂₉	—	4	2	no	—
67	C₄×D₂₉	C₂×D₄	1	2	yes	—
68	C₄×D₂₉	#41	1	2	yes	—
69	Dic₅₈	—	2	2	no	—
70	Dic₅₈	—	4	1	no	—
71	Dic₅₈	—	4	1	no	—
72	#6599	—	2	4	no	—
73	#1909	—	29	1	no	maximal, Sylow
74	C₂²×C₂₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
75	C₂×D₂₉	—	2	2	no	—
76	C₂×D₂₉	—	4	1	no	—
77	C₂×D₂₉	—	4	2	no	—
78	C₂×D₂₉	#121	1	2	yes	—
79	#1181	—	8	2	no	—
80	C₄×C₂₉	—	2	2	no	—
81	C₄×C₂₉	—	4	1	no	—
82	C₄×C₂₉	—	4	1	no	—
83	C₄×C₂₉	C₄×D₄	1	2	yes	—
84	Dic₂₉	—	4	1	no	—
85	Dic₂₉	—	4	1	no	—
86	Dic₂₉	#120	1	1	yes	characteristic
87	Dic₂₉	#123	1	1	yes	characteristic
88	#568	—	29	1	no	—
89	#536	—	29	1	no	—
90	#419	—	29	2	no	—
91	#410	—	29	1	no	—
92	H(3,ℤ/4)	—	29	2	no	—
93	C₂×C₂₉	—	2	1	no	—
94	C₂×C₂₉	#386	1	1	yes	characteristic
95	D₂₉	—	2	2	no	—
96	D₂₉	—	4	2	no	—
97	C₂×#41	—	29	1	no	—
98	C₂²×Q₈	—	29	1	no	—
99	#131	—	58	4	no	—
100	C₂×M₁₆	—	29	2	no	—
101	C₂×M₁₆	—	58	2	no	—
102	C₄⋊Q₈	—	29	1	no	—
103	C₄⋊Q₈	#1181	1	1	yes	characteristic, complemented
104	#124	—	29	1	no	—
105	#124	—	29	1	no	—
106	#123	—	58	2	no	—
107	C₂²⋊Q₈	—	58	2	no	—
108	#117	—	29	1	no	—
109	#117	—	29	2	no	—
110	#103	—	58	4	no	—
111	#99	—	58	2	no	—
112	C₂₉	#1909	1	1	yes	characteristic, complemented, Sylow
113	#41	—	58	2	no	—
114	#41	—	58	2	no	—
115	C₂×Q₈	—	29	1	no	—
116	C₂×Q₈	—	58	1	no	—
117	C₂×Q₈	—	58	2	no	—
118	C₂×Q₈	—	58	2	no	—
119	C₂×Q₈	C₂×#1181	1	1	yes	characteristic
120	C₂×Q₈	C₂×#1181	1	1	yes	characteristic
121	C₂×D₄	—	29	2	no	—
122	C₂×D₄	—	29	1	no	—
123	C₂²×C₄	—	29	1	no	—
124	C₂²×C₄	—	29	2	no	—
125	C₂²×C₄	—	58	1	no	—
126	M₁₆	—	58	2	no	—
127	M₁₆	—	58	2	no	—
128	M₁₆	—	58	4	no	—
129	C₂×C₈	—	29	4	no	—
130	C₂×C₈	—	58	4	no	—
131	C₄⋊C₄	—	2	2	no	—
132	C₄⋊C₄	—	58	1	no	—
133	C₄⋊C₄	—	58	2	no	—
134	C₄⋊C₄	—	58	2	no	—
135	C₂²⋊C₄	—	58	1	no	—
136	C₂²⋊C₄	—	58	2	no	—
137	C₂²⋊C₄	—	58	2	no	—
138	C₂²⋊C₄	—	58	2	no	—
139	C₄²	—	29	1	no	—
140	C₄²	—	58	2	no	—
141	C₄²	C₂×#1181	1	1	yes	characteristic
142	C₂³	—	29	1	no	—
143	C₂³	—	29	2	no	—
144	Q₈	—	2	2	no	—
145	Q₈	—	2	2	no	—
146	Q₈	—	58	2	no	—
147	Q₈	—	116	1	no	—
148	Q₈	—	116	1	no	—
149	D₄	—	58	4	no	—
150	D₄	—	58	2	no	—
151	C₂×C₄	—	2	1	no	—
152	C₂×C₄	—	2	1	no	—
153	C₂×C₄	—	29	1	no	—
154	C₂×C₄	—	29	2	no	—
155	C₂×C₄	—	29	2	no	—
156	C₂×C₄	—	58	2	no	—
157	C₂×C₄	—	58	2	no	—
158	C₂×C₄	—	58	1	no	—
159	C₂×C₄	—	58	1	no	—
160	C₂×C₄	—	116	2	no	—
161	C₂×C₄	—	116	2	no	—
162	C₂×C₄	C₂²⋊#1181	1	2	yes	—
163	C₂×C₄	C₂²×#1181	1	1	yes	characteristic
164	C₈	—	58	4	no	—
165	C₂²	—	29	2	no	—
166	C₂²	—	58	2	no	—
167	C₂²	—	116	1	no	—
168	C₂²	—	116	2	no	—
169	C₂²	C₂×C₂²⋊#1181	1	1	yes	characteristic
170	C₄	—	2	2	no	—
171	C₄	—	4	1	no	—
172	C₄	—	4	1	no	—
173	C₄	—	29	1	no	—
174	C₄	—	29	1	no	—
175	C₄	—	116	1	no	—
176	C₄	—	116	1	no	—
177	C₄	—	232	2	no	—
178	C₄	D₄×#1181	1	2	yes	—
179	C₂	—	2	1	no	—
180	C₂	—	58	2	no	—
181	C₂	—	116	2	no	—
182	C₂	#14272111	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#14272111
commutator	C₂×C₄×C₂₉	C₂²×C₄
Frattini	C₂×C₄	C₂²×#1181
Fitting	C₂₉×C₄⋊Q₈	C₄
socle	C₂×C₂₉	#386
prime radical	C₂×#90809	C₂²
cosocle	C₂×C₄×D₂₉	C₂³
hypercenter	C₄⋊Q₈	#1181
hypocenter	C₂₉	#1909
LCS	G ▷ C₂×C₄×C₂₉ ▷ C₂×C₂₉ ▷ C₂₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
2. #14272156	C₂	semidirect
4. #14272030	C₂	semidirect
12. C₂₉×C₄⋊Q₈	C₄	semidirect
13. #11745309	C₄	semidirect
14. #11745306	C₄	semidirect
17. #11745237	C₄	semidirect
103. C₄⋊Q₈	#1181	semidirect
112. C₂₉	#1909	semidirect

Maximal subgroups

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

subgroup	index
#14272243	2
#14272156	2
#14272144	2
#14272030	2
#14271738	2
#1909	29