Finite Group Databasebeta

Subgroups — #16821440

Order 3712.

226 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 226 conjugacy classes of subgroups — too many to draw. Only its 49 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	#14271963	C₂	1	1	yes	maximal, characteristic, complemented
2	#14271601	C₂	1	2	yes	maximal, complemented
3	#14271416	C₂	1	1	yes	maximal, characteristic, complemented
4	#14270974	C₂	1	2	yes	maximal, complemented
5	C₂₉×#349	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#90805	—	2	1	no	—
7	C₂×M₁₆×C₂₉	C₂²	1	1	yes	characteristic
8	#11745292	—	2	1	no	—
9	#11745271	—	2	2	no	—
10	#11745270	—	2	2	no	—
11	#11745266	—	2	1	no	—
12	#11745266	C₂²	1	2	yes	—
13	C₂×D₂₃₂	C₂²	1	1	yes	characteristic, complemented
14	C₂₉×M₃₂	C₂²	1	2	yes	complemented
15	#11745169	—	2	1	no	—
16	#11745165	C₂²	1	1	yes	characteristic, complemented
17	#11745148	—	2	2	no	—
18	D₄₆₄	—	2	2	no	—
19	C₂²×C₄×C₂₉	D₄	1	1	yes	characteristic
20	C₂×C₂²⋊D₂₉	—	4	1	no	—
21	C₂×C₂²⋊D₂₉	—	4	1	no	—
22	#90805	—	2	2	no	—
23	#90805	—	2	2	no	—
24	#90805	—	4	1	no	—
25	#90805	—	4	1	no	—
26	C₂×D₁₁₆	—	2	1	no	—
27	C₂×D₁₁₆	—	2	1	no	—
28	C₂×C₄×D₂₉	—	4	1	no	—
29	C₂×Dic₅₈	—	2	1	no	—
30	M₁₆×C₂₉	—	2	2	no	—
31	C₂×C₈×C₂₉	D₄	1	1	yes	characteristic
32	C₂×C₈×C₂₉	C₂³	1	1	yes	characteristic
33	#90781	—	4	1	no	—
34	C₄⋊Dic₂₉	—	2	1	no	—
35	Dic₁₁₆	—	2	1	no	—
36	Dic₁₁₆	D₄	1	2	yes	complemented
37	D₂₃₂	—	2	1	no	—
38	D₂₃₂	—	2	2	no	—
39	D₂₃₂	D₄	1	2	yes	complemented
40	#90773	—	2	2	no	—
41	#90773	—	2	2	no	—
42	C₁₆×C₂₉	—	2	2	no	—
43	C₂³×C₂₉	D₈	1	1	yes	characteristic
44	C₂²×D₂₉	—	4	1	no	—
45	C₂²×D₂₉	—	4	1	no	—
46	C₂×C₄×C₂₉	—	2	1	no	—
47	C₂×C₄×C₂₉	—	2	1	no	—
48	C₂×C₄×C₂₉	D₈	1	1	yes	characteristic
49	C₂×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
50	C₂²⋊D₂₉	—	4	2	no	—
51	C₂²⋊D₂₉	—	4	2	no	—
52	C₂²⋊D₂₉	—	8	2	no	—
53	C₂×Dic₂₉	—	4	1	no	—
54	C₂×Dic₂₉	—	4	1	no	—
55	D₁₁₆	—	2	2	no	—
56	D₁₁₆	—	4	1	no	—
57	D₁₁₆	—	4	1	no	—
58	D₁₁₆	—	4	1	no	—
59	C₄×D₂₉	—	4	2	no	—
60	C₄×D₂₉	—	4	2	no	—
61	Dic₅₈	—	2	2	no	—
62	Dic₅₈	—	4	1	no	—
63	C₈×C₂₉	—	2	1	no	—
64	C₈×C₂₉	C₂×D₄	1	2	yes	—
65	#2190	—	29	1	no	maximal, Sylow
66	C₂²×C₂₉	—	2	1	no	—
67	C₂²×C₂₉	—	4	1	no	—
68	C₂²×C₂₉	C₂×D₈	1	1	yes	characteristic
69	C₂×D₂₉	—	4	2	no	—
70	C₂×D₂₉	—	8	2	no	—
71	C₂×D₂₉	—	8	1	no	—
72	C₂×D₂₉	—	8	1	no	—
73	C₂×D₂₉	—	8	1	no	—
74	C₄×C₂₉	—	2	1	no	—
75	C₄×C₂₉	#120	1	1	yes	characteristic
76	C₄×C₂₉	AGL(1,ℤ/8)	1	1	yes	characteristic
77	Dic₂₉	—	4	2	no	—
78	Dic₂₉	—	8	1	no	—
79	#575	—	29	1	no	—
80	#509	—	29	2	no	—
81	#469	—	29	1	no	—
82	#360	—	29	2	no	—
83	#349	D₂₉	1	1	yes	characteristic, complemented
84	C₂×C₂₉	—	2	1	no	—
85	C₂×C₂₉	—	4	1	no	—
86	C₂×C₂₉	#447	1	1	yes	characteristic
87	D₂₉	—	8	2	no	—
88	D₂₉	—	16	1	no	—
89	C₂×#41	—	58	1	no	—
90	#137	—	58	2	no	—
91	AGL(1,ℤ/8)	—	58	2	no	—
92	#135	—	29	2	no	—
93	#135	—	58	1	no	—
94	C₂×D₈	—	29	1	no	—
95	C₂×M₁₆	C₂×D₂₉	1	1	yes	characteristic
96	#121	—	58	1	no	—
97	SD₃₂	—	58	2	no	—
98	D₁₆	—	58	2	no	—
99	M₃₂	C₂×D₂₉	1	2	yes	complemented
100	#106	—	29	1	no	—
101	#102	—	58	1	no	—
102	C₂₉	#2190	1	1	yes	characteristic, complemented, Sylow
103	#41	—	58	2	no	—
104	#41	—	58	2	no	—
105	#41	—	116	1	no	—
106	#41	—	116	1	no	—
107	C₂×Q₈	—	58	1	no	—
108	C₂×D₄	—	58	1	no	—
109	C₂×D₄	—	58	1	no	—
110	C₂×D₄	—	116	1	no	—
111	C₂×D₄	—	116	1	no	—
112	C₂²×C₄	—	116	1	no	—
113	C₂²×C₄	D₁₁₆	1	1	yes	characteristic
114	Q₁₆	—	29	2	no	—
115	Q₁₆	—	58	1	no	—
116	SD₁₆	—	58	2	no	—
117	SD₁₆	—	58	2	no	—
118	D₈	—	29	2	no	—
119	D₈	—	58	1	no	—
120	D₈	—	58	2	no	—
121	M₁₆	—	2	2	no	—
122	C₂×C₈	D₁₁₆	1	1	yes	characteristic
123	C₂×C₈	C₂²×D₂₉	1	1	yes	characteristic
124	C₄⋊C₄	—	58	1	no	—
125	C₂²⋊C₄	—	116	1	no	—
126	C₁₆	—	2	2	no	—
127	C₂³	—	116	1	no	—
128	C₂³	—	116	1	no	—
129	C₂³	D₂₃₂	1	1	yes	characteristic
130	Q₈	—	58	2	no	—
131	Q₈	—	116	1	no	—
132	D₄	—	58	2	no	—
133	D₄	—	116	1	no	—
134	D₄	—	116	2	no	—
135	D₄	—	116	2	no	—
136	D₄	—	116	1	no	—
137	D₄	—	116	1	no	—
138	D₄	—	232	2	no	—
139	C₂×C₄	—	2	1	no	—
140	C₂×C₄	—	2	1	no	—
141	C₂×C₄	—	116	2	no	—
142	C₂×C₄	—	116	1	no	—
143	C₂×C₄	—	116	2	no	—
144	C₂×C₄	—	116	1	no	—
145	C₂×C₄	D₂₃₂	1	1	yes	characteristic
146	C₂×C₄	C₂×D₁₁₆	1	1	yes	characteristic
147	C₈	—	2	1	no	—
148	C₈	D₄×D₂₉	1	2	yes	—
149	C₂²	—	2	1	no	—
150	C₂²	—	4	1	no	—
151	C₂²	—	116	2	no	—
152	C₂²	—	232	1	no	—
153	C₂²	—	232	1	no	—
154	C₂²	—	232	2	no	—
155	C₂²	—	232	1	no	—
156	C₂²	C₂×D₂₃₂	1	1	yes	characteristic
157	C₄	—	2	1	no	—
158	C₄	—	116	2	no	—
159	C₄	—	232	1	no	—
160	C₄	#11745244	1	1	yes	characteristic
161	C₄	#11745270	1	1	yes	characteristic
162	C₂	—	2	1	no	—
163	C₂	—	4	1	no	—
164	C₂	—	464	1	no	—
165	C₂	—	232	2	no	—
166	C₂	#14271299	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#14271299
commutator, cosocle	C₂×C₈×C₂₉	C₂³
Frattini	C₂×C₈	C₂²×D₂₉
Fitting	C₂₉×#349	C₂
socle	C₂×C₂₉	#447
hypercenter	#349	D₂₉
hypocenter	C₂₉	#2190
LCS	G ▷ C₂×C₈×C₂₉ ▷ C₄×C₂₉ ▷ C₂×C₂₉ ▷ C₂₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×M₁₆ ◁ #349

Complements

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

N	H ≅ G/N	type
1. #14271963	C₂	semidirect
2. #14271601	C₂	semidirect
3. #14271416	C₂	semidirect
4. #14270974	C₂	semidirect
5. C₂₉×#349	C₂	semidirect
13. C₂×D₂₃₂	C₂²	semidirect
14. C₂₉×M₃₂	C₂²	semidirect
16. #11745165	C₂²	semidirect
36. Dic₁₁₆	D₄	semidirect
39. D₂₃₂	D₄	semidirect
83. #349	D₂₉	semidirect
99. M₃₂	C₂×D₂₉	semidirect
102. C₂₉	#2190	semidirect

Maximal subgroups

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

subgroup	index
#14271963	2
#14271601	2
#14271416	2
#14270974	2
C₂₉×#349	2
#2190	29