Subgroups — #16817376

Order 3712.

188 conjugacy classes of subgroups.

The subgroup lattice has 162 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊#6597	C₂	1	1	yes	maximal, characteristic
2	#14271033	C₂	1	1	yes	maximal, characteristic
3	C₂×C₂₉×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₈×C₂₉	—	2	1	no	—
5	C₂×C₄²×C₂₉	C₂²	1	1	yes	characteristic
6	C₂×C₄⋊Dic₂₉	—	2	1	no	—
7	C₂×C₄⋊Dic₂₉	—	2	1	no	—
8	C₂×C₄⋊Dic₂₉	C₄	1	2	yes	—
9	C₂²×#6597	—	2	1	no	—
10	C₂₉×C₄⋊C₈	—	2	1	no	—
11	C₂₉×C₄⋊C₈	C₄	1	2	yes	complemented
12	C₄⋊#6597	—	2	1	no	—
13	C₄⋊#6597	C₄	1	2	yes	complemented
14	C₂²×C₄×C₂₉	C₂×C₄	1	1	yes	characteristic
15	C₂²×C₄×C₂₉	D₄	1	1	yes	characteristic
16	C₂²×C₄×C₂₉	D₄	1	1	yes	characteristic
17	C₂²×Dic₂₉	—	2	2	no	—
18	C₂×C₈×C₂₉	—	2	2	no	—
19	C₂×C₈×C₂₉	—	4	1	no	—
20	C₂×C₈×C₂₉	—	4	1	no	—
21	C₄²×C₂₉	C₂×C₄	1	1	yes	characteristic
22	C₄²×C₂₉	C₂×C₄	1	1	yes	characteristic
23	C₄²×C₂₉	D₄	1	1	yes	characteristic
24	C₄²×C₂₉	Q₈	1	1	yes	characteristic
25	C₄⋊Dic₂₉	—	4	1	no	—
26	C₄⋊Dic₂₉	—	4	1	no	—
27	C₄⋊Dic₂₉	—	4	1	no	—
28	C₄⋊Dic₂₉	—	4	1	no	—
29	C₄⋊Dic₂₉	—	4	2	no	—
30	C₂×#6597	—	2	2	no	—
31	C₂×#6597	—	4	1	no	—
32	C₂×#6597	—	4	1	no	—
33	C₂³×C₂₉	C₂²⋊C₄	1	1	yes	characteristic
34	C₂×C₄×C₂₉	—	2	1	no	—
35	C₂×C₄×C₂₉	—	2	1	no	—
36	C₂×C₄×C₂₉	C₄²	1	1	yes	characteristic
37	C₂×C₄×C₂₉	C₂²⋊C₄	1	1	yes	characteristic
38	C₂×C₄×C₂₉	C₂²⋊C₄	1	1	yes	characteristic
39	C₂×C₄×C₂₉	C₄⋊C₄	1	1	yes	characteristic
40	C₂×C₄×C₂₉	C₄⋊C₄	1	1	yes	characteristic
41	C₂×C₄×C₂₉	C₄⋊C₄	1	1	yes	characteristic
42	C₂×C₄×C₂₉	D₈	1	1	yes	characteristic
43	C₂×C₄×C₂₉	D₈	1	1	yes	characteristic
44	C₂×C₄×C₂₉	SD₁₆	1	1	yes	characteristic
45	C₂×C₄×C₂₉	SD₁₆	1	1	yes	characteristic
46	C₂×C₄×C₂₉	SD₁₆	1	1	yes	characteristic
47	C₂×C₄×C₂₉	SD₁₆	1	1	yes	characteristic
48	C₂×C₄×C₂₉	Q₁₆	1	1	yes	characteristic
49	C₂×C₄×C₂₉	Q₁₆	1	1	yes	characteristic
50	C₂×Dic₂₉	—	4	2	no	—
51	C₂×Dic₂₉	—	4	2	no	—
52	C₂×Dic₂₉	—	4	2	no	—
53	C₈×C₂₉	—	4	2	no	—
54	#6597	—	4	2	no	—
55	#1296	—	29	1	no	maximal, Sylow
56	C₂²×C₂₉	#95	1	1	yes	characteristic
57	C₂²×C₂₉	#100	1	1	yes	characteristic
58	C₂²×C₂₉	#101	1	1	yes	characteristic
59	C₂²×C₂₉	#102	1	1	yes	characteristic
60	C₂²×C₂₉	#102	1	1	yes	characteristic
61	C₂²×C₂₉	#103	1	1	yes	characteristic
62	C₂²×C₂₉	#103	1	1	yes	characteristic
63	C₄×C₂₉	—	2	1	no	—
64	C₄×C₂₉	—	2	1	no	—
65	C₄×C₂₉	#102	1	1	yes	characteristic
66	C₄×C₂₉	#102	1	1	yes	characteristic
67	C₄×C₂₉	#103	1	1	yes	characteristic
68	C₄×C₂₉	#103	1	1	yes	characteristic
69	C₄×C₂₉	#106	1	1	yes	characteristic
70	C₄×C₂₉	#106	1	1	yes	characteristic
71	C₄×C₂₉	#107	1	1	yes	characteristic
72	C₄×C₂₉	#107	1	1	yes	characteristic
73	Dic₂₉	—	8	2	no	—
74	C₂×C₄⋊C₈	—	29	1	no	—
75	C₂×C₄⋊C₈	D₂₉	1	1	yes	characteristic
76	#384	—	29	1	no	—
77	C₂×C₂₉	#331	1	1	yes	characteristic
78	C₂×C₂₉	#332	1	1	yes	characteristic
79	C₂×C₂₉	#332	1	1	yes	characteristic
80	C₂×C₂₉	#333	1	1	yes	characteristic
81	C₂×C₂₉	#340	1	1	yes	characteristic
82	C₂×C₂₉	#340	1	1	yes	characteristic
83	C₂×C₂₉	#343	1	1	yes	characteristic
84	C₂²×C₈	—	2	1	no	—
85	C₂²×C₈	—	58	1	no	—
86	C₂×C₄⋊C₄	—	29	2	no	—
87	C₂×C₄⋊C₄	—	58	1	no	—
88	C₂×C₄⋊C₄	—	58	1	no	—
89	C₂×C₄²	C₂×D₂₉	1	1	yes	characteristic
90	C₄⋊C₈	—	2	1	no	—
91	C₄⋊C₈	—	29	2	no	—
92	C₄⋊C₈	—	58	1	no	—
93	C₄⋊C₈	Dic₂₉	1	2	yes	complemented
94	C₂₉	#1296	1	1	yes	characteristic, complemented, Sylow
95	C₂²×C₄	—	58	2	no	—
96	C₂²×C₄	C₄×D₂₉	1	1	yes	characteristic
97	C₂²×C₄	D₁₁₆	1	1	yes	characteristic
98	C₂²×C₄	C₂²⋊D₂₉	1	1	yes	characteristic
99	C₂×C₈	—	2	2	no	—
100	C₂×C₈	—	4	1	no	—
101	C₂×C₈	—	4	1	no	—
102	C₂×C₈	—	58	2	no	—
103	C₂×C₈	—	116	1	no	—
104	C₂×C₈	—	116	1	no	—
105	C₄⋊C₄	—	116	2	no	—
106	C₄⋊C₄	—	116	1	no	—
107	C₄⋊C₄	—	116	1	no	—
108	C₄⋊C₄	—	116	1	no	—
109	C₄⋊C₄	—	116	1	no	—
110	C₄²	Dic₅₈	1	1	yes	characteristic
111	C₄²	C₄×D₂₉	1	1	yes	characteristic
112	C₄²	C₂×Dic₂₉	1	1	yes	characteristic
113	C₄²	C₂²⋊D₂₉	1	1	yes	characteristic
114	C₂³	#90781	1	1	yes	characteristic
115	C₂×C₄	—	2	1	no	—
116	C₂×C₄	—	2	1	no	—
117	C₂×C₄	—	116	2	no	—
118	C₂×C₄	—	116	2	no	—
119	C₂×C₄	—	116	2	no	—
120	C₂×C₄	#90773	1	1	yes	characteristic
121	C₂×C₄	#90773	1	1	yes	characteristic
122	C₂×C₄	D₂₃₂	1	1	yes	characteristic
123	C₂×C₄	Dic₁₁₆	1	1	yes	characteristic
124	C₂×C₄	C₄×Dic₂₉	1	1	yes	characteristic
125	C₂×C₄	#90779	1	1	yes	characteristic
126	C₂×C₄	#90779	1	1	yes	characteristic
127	C₂×C₄	C₄⋊Dic₂₉	1	1	yes	characteristic
128	C₂×C₄	#90781	1	1	yes	characteristic
129	C₂×C₄	#90782	1	1	yes	characteristic
130	C₂×C₄	#90783	1	1	yes	characteristic
131	C₂×C₄	#90784	1	1	yes	characteristic
132	C₂×C₄	#90785	1	1	yes	characteristic
133	C₂×C₄	C₂²⋊Dic₂₉	1	1	yes	characteristic
134	C₈	—	4	2	no	—
135	C₈	—	116	2	no	—
136	C₂²	#11745157	1	1	yes	characteristic
137	C₂²	#11745158	1	1	yes	characteristic
138	C₂²	#11745164	1	1	yes	characteristic
139	C₂²	#11745169	1	1	yes	characteristic
140	C₂²	#11745171	1	1	yes	characteristic
141	C₂²	#11745172	1	1	yes	characteristic
142	C₂²	#11745179	1	1	yes	characteristic
143	C₄	—	2	1	no	—
144	C₄	—	2	1	no	—
145	C₄	—	232	2	no	—
146	C₄	#11745155	1	1	yes	characteristic
147	C₄	#11745156	1	1	yes	characteristic
148	C₄	#11745164	1	1	yes	characteristic
149	C₄	#11745165	1	1	yes	characteristic
150	C₄	#11745166	1	1	yes	characteristic
151	C₄	#11745169	1	1	yes	characteristic
152	C₄	#11745180	1	1	yes	characteristic
153	C₄	#11745183	1	1	yes	characteristic
154	C₂	#14270906	1	1	yes	characteristic
155	C₂	#14270908	1	1	yes	characteristic
156	C₂	#14270910	1	1	yes	characteristic
157	C₂	#14270911	1	1	yes	characteristic
158	C₂	#14270925	1	1	yes	characteristic
159	C₂	#14270926	1	1	yes	characteristic
160	C₂	#14270931	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#90781
commutator	C₂×C₄×C₂₉	C₄²
Frattini	C₂×C₄²	C₂×D₂₉
Fitting	C₂×C₂₉×C₄⋊C₈	C₂
socle, prime radical	C₂³×C₂₉	C₂²⋊C₄
cosocle	C₂×C₄²×C₂₉	C₂²
hypercenter	C₂×C₄⋊C₈	D₂₉
hypocenter	C₂₉	#1296
LCS	G ▷ C₂×C₄×C₂₉ ▷ C₂²×C₂₉ ▷ C₂₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
11. C₂₉×C₄⋊C₈	C₄	semidirect
13. C₄⋊#6597	C₄	semidirect
93. C₄⋊C₈	Dic₂₉	semidirect
94. C₂₉	#1296	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊#6597	2
#14271033	2
C₂×C₂₉×C₄⋊C₈	2
#1296	29