Finite Group Databasebeta

Subgroups — #16835766

Order 3712.

258 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 52 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	#14272144	C₂	1	2	yes	maximal, complemented
2	#14272069	C₂	1	2	yes	maximal, complemented
3	#14272068	C₂	1	2	yes	maximal, complemented
4	#14271714	C₂	1	1	yes	maximal, characteristic
5	C₂×#90809	C₄	1	1	yes	characteristic, complemented
6	C₂×#90809	C₂²	1	1	yes	characteristic
7	#11745345	—	2	2	no	—
8	C₂₉×C₄⋊Q₈	C₄	1	1	yes	characteristic, complemented
9	#11745309	C₄	1	1	yes	characteristic, complemented
10	#11745309	C₂²	1	1	yes	characteristic
11	#11745257	—	2	2	no	—
12	#11745255	—	2	2	no	—
13	#11745236	C₂²	1	1	yes	characteristic
14	C₄²×D₂₉	C₄	1	1	yes	characteristic, complemented
15	#11745215	—	2	2	no	—
16	#11745215	C₂²	1	4	yes	—
17	C₂×Q₈×C₂₉	C₂×C₄	1	1	yes	characteristic
18	C₂×Q₈×C₂₉	C₂×C₄	1	1	yes	characteristic
19	#90809	—	4	1	no	—
20	#90809	—	4	1	no	—
21	#90809	—	4	1	no	—
22	#90809	—	4	1	no	—
23	C₂×D₁₁₆	—	2	1	no	—
24	C₂×D₁₁₆	—	2	2	no	—
25	C₂×D₁₁₆	C₂×C₄	1	1	yes	characteristic
26	C₂×D₁₁₆	D₄	1	2	yes	—
27	C₂×D₁₁₆	C₂³	1	1	yes	characteristic
28	C₂×C₄×D₂₉	—	2	1	no	—
29	C₂×C₄×D₂₉	—	2	1	no	—
30	C₂×C₄×D₂₉	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄×D₂₉	D₄	1	2	yes	—
32	C₂²⋊#1181	—	2	4	no	—
33	C₄⋊#1181	—	4	2	no	—
34	C₄×#1181	—	4	2	no	—
35	C₄⋊C₄×C₂₉	—	2	2	no	—
36	C₄²×C₂₉	C₂×C₄	1	1	yes	characteristic
37	#90781	—	2	2	no	—
38	#90781	—	2	2	no	—
39	C₄×Dic₂₉	—	2	1	no	—
40	C₄×Dic₂₉	C₂×C₄	1	1	yes	characteristic
41	C₂²×D₂₉	—	2	1	no	—
42	C₂²×D₂₉	C₂²⋊C₄	1	1	yes	characteristic
43	C₂²×D₂₉	C₂×D₄	1	2	yes	—
44	C₂×#1181	—	4	4	no	—
45	Q₈×C₂₉	—	4	1	no	—
46	Q₈×C₂₉	—	4	1	no	—
47	C₂×C₄×C₂₉	—	2	1	no	—
48	C₂×C₄×C₂₉	—	2	1	no	—
49	C₂×C₄×C₂₉	C₂²⋊C₄	1	2	yes	—
50	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
51	C₂×Dic₂₉	—	2	1	no	—
52	C₂×Dic₂₉	C₂²⋊C₄	1	1	yes	characteristic
53	D₁₁₆	—	4	2	no	—
54	D₁₁₆	—	4	1	no	—
55	D₁₁₆	—	4	1	no	—
56	D₁₁₆	—	4	4	no	—
57	D₁₁₆	—	4	2	no	—
58	C₄×D₂₉	—	2	2	no	—
59	C₄×D₂₉	—	2	2	no	—
60	C₄×D₂₉	—	2	1	no	—
61	C₄×D₂₉	—	2	1	no	—
62	C₄×D₂₉	—	4	2	no	—
63	C₄×D₂₉	—	4	1	no	—
64	C₄×D₂₉	—	4	1	no	—
65	#2126	—	29	1	no	maximal, Sylow
66	C₂²×C₂₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
67	C₂×D₂₉	—	2	1	no	—
68	C₂×D₂₉	—	2	2	no	—
69	C₂×D₂₉	—	4	1	no	—
70	C₂×D₂₉	—	4	2	no	—
71	C₂×D₂₉	—	4	2	no	—
72	C₂×D₂₉	—	4	1	no	—
73	#1181	—	8	4	no	—
74	C₄×C₂₉	—	2	2	no	—
75	C₄×C₂₉	—	2	1	no	—
76	C₄×C₂₉	—	2	2	no	—
77	C₄×C₂₉	—	4	1	no	—
78	Dic₂₉	—	4	1	no	—
79	Dic₂₉	#99	1	1	yes	characteristic
80	Dic₂₉	#99	1	1	yes	characteristic
81	#532	—	29	1	no	—
82	#410	—	29	2	no	—
83	#354	—	29	2	no	—
84	#353	—	29	2	no	—
85	C₂×C₂₉	—	2	1	no	—
86	C₂×C₂₉	C₂×#99	1	1	yes	characteristic
87	D₂₉	—	4	2	no	—
88	D₂₉	—	4	1	no	—
89	D₂₉	—	8	1	no	—
90	C₂×#41	—	29	1	no	—
91	C₂×#41	—	29	1	no	—
92	C₄⋊Q₈	#1181	1	1	yes	characteristic, complemented
93	#127	—	29	1	no	—
94	#124	—	29	1	no	—
95	#124	—	29	1	no	—
96	#121	—	58	2	no	—
97	C₄×D₄	—	58	2	no	—
98	#117	—	58	2	no	—
99	C₂×C₄²	—	29	1	no	—
100	#99	—	29	4	no	—
101	#99	—	58	2	no	—
102	C₂₉	#2126	1	1	yes	characteristic, complemented, Sylow
103	#41	—	116	1	no	—
104	#41	—	116	1	no	—
105	#41	—	116	1	no	—
106	#41	—	116	1	no	—
107	C₂×Q₈	C₂×#1181	1	1	yes	characteristic
108	C₂×Q₈	C₂×#1181	1	1	yes	characteristic
109	C₂×D₄	—	29	2	no	—
110	C₂×D₄	—	29	1	no	—
111	C₂×D₄	—	29	1	no	—
112	C₂×D₄	—	58	2	no	—
113	C₂×D₄	—	58	1	no	—
114	C₂²×C₄	—	29	2	no	—
115	C₂²×C₄	—	29	1	no	—
116	C₂²×C₄	—	58	1	no	—
117	C₂²×C₄	—	58	1	no	—
118	C₄⋊C₄	—	2	2	no	—
119	C₄⋊C₄	—	116	2	no	—
120	C₂²⋊C₄	—	58	2	no	—
121	C₂²⋊C₄	—	58	2	no	—
122	C₂²⋊C₄	—	58	4	no	—
123	C₄²	—	29	1	no	—
124	C₄²	—	58	1	no	—
125	C₄²	—	116	2	no	—
126	C₄²	C₂×#1181	1	1	yes	characteristic
127	C₂³	—	29	2	no	—
128	C₂³	—	29	1	no	—
129	C₂³	—	58	1	no	—
130	Q₈	—	4	1	no	—
131	Q₈	—	4	1	no	—
132	D₄	—	116	2	no	—
133	D₄	—	116	1	no	—
134	D₄	—	116	1	no	—
135	D₄	—	116	4	no	—
136	D₄	—	116	2	no	—
137	C₂×C₄	—	2	1	no	—
138	C₂×C₄	—	2	1	no	—
139	C₂×C₄	—	29	1	no	—
140	C₂×C₄	—	58	2	no	—
141	C₂×C₄	—	58	2	no	—
142	C₂×C₄	—	58	1	no	—
143	C₂×C₄	—	58	1	no	—
144	C₂×C₄	—	58	1	no	—
145	C₂×C₄	—	116	2	no	—
146	C₂×C₄	—	116	1	no	—
147	C₂×C₄	—	116	1	no	—
148	C₂×C₄	—	116	4	no	—
149	C₂×C₄	C₂²⋊#1181	1	2	yes	—
150	C₂×C₄	C₂²×#1181	1	1	yes	characteristic
151	C₂²	—	58	2	no	—
152	C₂²	—	58	1	no	—
153	C₂²	—	116	1	no	—
154	C₂²	—	116	2	no	—
155	C₂²	—	116	2	no	—
156	C₂²	—	116	1	no	—
157	C₂²	C₂×C₂²⋊#1181	1	1	yes	characteristic
158	C₄	—	2	2	no	—
159	C₄	—	2	2	no	—
160	C₄	—	2	1	no	—
161	C₄	—	4	1	no	—
162	C₄	—	29	1	no	—
163	C₄	—	29	1	no	—
164	C₄	—	116	1	no	—
165	C₄	—	232	4	no	—
166	C₂	—	2	1	no	—
167	C₂	—	116	2	no	—
168	C₂	—	116	1	no	—
169	C₂	—	232	1	no	—
170	C₂	D₂₉⋊#99	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₂₉⋊#99
commutator	C₂×C₄×C₂₉	C₂²×C₄
Frattini	C₂×C₄	C₂²×#1181
Fitting	C₂₉×C₄⋊Q₈	C₄
socle	C₂×C₂₉	C₂×#99
prime radical	#14271714	C₂
cosocle	C₂×D₁₁₆	C₂³
hypercenter	C₄⋊Q₈	#1181
hypocenter	C₂₉	#2126
LCS	G ▷ C₂×C₄×C₂₉ ▷ C₂²×C₂₉ ▷ C₂×C₂₉ ▷ C₂₉
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×Q₈ ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
1. #14272144	C₂	semidirect
2. #14272069	C₂	semidirect
3. #14272068	C₂	semidirect
5. C₂×#90809	C₄	semidirect
8. C₂₉×C₄⋊Q₈	C₄	semidirect
9. #11745309	C₄	semidirect
14. C₄²×D₂₉	C₄	semidirect
92. C₄⋊Q₈	#1181	semidirect
102. C₂₉	#2126	semidirect

Maximal subgroups

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

subgroup	index
#14272144	2
#14272069	2
#14272068	2
#14271714	2
#2126	29