Subgroups — #16820211

Order 3712.

260 conjugacy classes of subgroups.

The subgroup lattice has 164 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	C₂×C₄⋊#6597	C₂	1	2	yes	maximal
3	C₂×C₄×#6597	C₂	1	1	yes	maximal, characteristic
4	#14270918	C₂	1	2	yes	maximal
5	C₄×C₄⋊C₄×C₂₉	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊C₄×C₂₉	C₄	1	1	yes	characteristic
7	C₂×C₄⋊C₄×C₂₉	C₄	1	1	yes	characteristic
8	C₂×C₄²×C₂₉	C₂²	1	2	yes	—
9	C₂×C₄²×C₂₉	C₂²	1	1	yes	characteristic
10	C₂²×#6597	C₂²	1	2	yes	—
11	C₂²×#6597	C₂²	1	1	yes	characteristic
12	C₂²×#6597	C₂²	1	1	yes	characteristic
13	C₂₉×#95	C₄	1	2	yes	—
14	C₄⋊#6597	—	2	2	no	—
15	C₄⋊#6597	—	2	4	no	—
16	C₄×#6597	—	2	2	no	—
17	C₂²×C₄×C₂₉	C₂×C₄	1	2	yes	—
18	C₂²×C₄×C₂₉	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₄×C₂₉	C₂×C₄	1	2	yes	—
20	C₂²×C₄×C₂₉	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₄×C₂₉	C₂³	1	1	yes	characteristic
22	C₄⋊C₄×C₂₉	—	2	2	no	—
23	C₄⋊C₄×C₂₉	C₈	1	4	yes	complemented
24	C₄²×C₂₉	—	2	2	no	—
25	C₄²×C₂₉	—	2	2	no	—
26	C₄²×C₂₉	D₄	1	2	yes	—
27	C₄²×C₂₉	Q₈	1	2	yes	—
28	C₂×#6597	—	2	1	no	—
29	C₂×#6597	—	2	1	no	—
30	C₂×#6597	—	2	1	no	—
31	C₂×#6597	—	2	1	no	—
32	C₂×#6597	—	2	2	no	—
33	C₂×#6597	—	2	2	no	—
34	C₂×#6597	—	2	2	no	—
35	C₂×#6597	D₄	1	2	yes	—
36	C₂×#6597	Q₈	1	2	yes	—
37	C₂³×C₂₉	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₄×C₂₉	—	2	2	no	—
39	C₂×C₄×C₂₉	—	2	1	no	—
40	C₂×C₄×C₂₉	—	2	1	no	—
41	C₂×C₄×C₂₉	—	2	2	no	—
42	C₂×C₄×C₂₉	—	2	2	no	—
43	C₂×C₄×C₂₉	—	2	2	no	—
44	C₂×C₄×C₂₉	—	2	2	no	—
45	C₂×C₄×C₂₉	C₄⋊C₄	1	2	yes	—
46	C₂×C₄×C₂₉	C₄⋊C₄	1	2	yes	—
47	C₂×C₄×C₂₉	C₂×C₈	1	4	yes	—
48	C₂×C₄×C₂₉	C₂×C₈	1	2	yes	—
49	C₂×C₄×C₂₉	M₁₆	1	2	yes	—
50	C₂×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₂₉	C₂×Q₈	1	1	yes	characteristic
53	C₂×C₄×C₂₉	C₂×Q₈	1	1	yes	characteristic
54	C₂×C₄×C₂₉	#41	1	1	yes	characteristic
55	C₂×C₄×C₂₉	#41	1	1	yes	characteristic
56	#6597	—	2	2	no	—
57	#6597	—	2	2	no	—
58	#6597	—	4	2	no	—
59	#1939	—	29	1	no	maximal, Sylow
60	C₂²×C₂₉	C₂×C₄⋊C₄	1	1	yes	characteristic
61	C₂²×C₂₉	C₄×D₄	1	1	yes	characteristic
62	C₂²×C₂₉	C₄×Q₈	1	1	yes	characteristic
63	C₂²×C₂₉	C₂²×C₈	1	1	yes	characteristic
64	C₂²×C₂₉	C₂×M₁₆	1	1	yes	characteristic
65	C₂²×C₂₉	#131	1	1	yes	characteristic
66	C₂²×C₂₉	#131	1	1	yes	characteristic
67	C₄×C₂₉	—	2	2	no	—
68	C₄×C₂₉	—	2	4	no	—
69	C₄×C₂₉	—	4	2	no	—
70	C₄×C₂₉	C₄⋊C₈	1	4	yes	complemented
71	C₄×C₂₉	#121	1	1	yes	characteristic
72	C₄×C₂₉	C₂²⋊Q₈	1	1	yes	characteristic
73	C₄×C₂₉	#125	1	1	yes	characteristic
74	C₄×C₂₉	C₄⋊Q₈	1	1	yes	characteristic
75	C₂×C₄⋊C₈	—	29	2	no	—
76	C₂×C₄⋊C₈	—	29	1	no	—
77	C₂×C₄×C₈	—	29	1	no	—
78	C₄×C₄⋊C₄	D₂₉	1	1	yes	characteristic
79	#336	—	29	2	no	—
80	C₂×C₂₉	#389	1	1	yes	characteristic
81	C₂×C₂₉	C₂×C₄⋊C₈	1	1	yes	characteristic
82	C₂×C₂₉	#424	1	1	yes	characteristic
83	C₂×C₂₉	C₈×D₄	1	1	yes	characteristic
84	C₂×C₂₉	#436	1	1	yes	characteristic
85	C₂×C₂₉	C₈×Q₈	1	1	yes	characteristic
86	C₂×C₂₉	#446	1	1	yes	characteristic
87	C₂²×C₈	—	29	1	no	—
88	C₂²×C₈	—	29	1	no	—
89	C₂²×C₈	—	29	2	no	—
90	C₂×C₄⋊C₄	Dic₂₉	1	1	yes	characteristic
91	C₂×C₄⋊C₄	Dic₂₉	1	1	yes	characteristic
92	C₂×C₄²	C₂×D₂₉	1	2	yes	—
93	C₂×C₄²	C₂×D₂₉	1	1	yes	characteristic
94	C₄⋊C₈	—	58	4	no	—
95	C₄⋊C₈	—	58	2	no	—
96	C₄×C₈	—	58	2	no	—
97	#95	Dic₂₉	1	2	yes	—
98	C₂₉	#1939	1	1	yes	characteristic, complemented, Sylow
99	C₂²×C₄	C₂×Dic₂₉	1	2	yes	—
100	C₂²×C₄	C₂×Dic₂₉	1	2	yes	—
101	C₂²×C₄	C₂×Dic₂₉	1	1	yes	characteristic
102	C₂²×C₄	C₂×Dic₂₉	1	1	yes	characteristic
103	C₂²×C₄	C₂²×D₂₉	1	1	yes	characteristic
104	C₂×C₈	—	29	2	no	—
105	C₂×C₈	—	29	2	no	—
106	C₂×C₈	—	58	1	no	—
107	C₂×C₈	—	58	1	no	—
108	C₂×C₈	—	58	1	no	—
109	C₂×C₈	—	58	1	no	—
110	C₂×C₈	—	58	2	no	—
111	C₂×C₈	—	58	2	no	—
112	C₂×C₈	—	58	2	no	—
113	C₄⋊C₄	—	2	2	no	—
114	C₄⋊C₄	#6597	1	4	yes	complemented
115	C₄²	—	2	2	no	—
116	C₄²	—	2	2	no	—
117	C₄²	Dic₅₈	1	2	yes	—
118	C₄²	D₁₁₆	1	2	yes	—
119	C₂³	C₂²×Dic₂₉	1	1	yes	characteristic
120	C₂×C₄	—	2	2	no	—
121	C₂×C₄	—	2	1	no	—
122	C₂×C₄	—	2	1	no	—
123	C₂×C₄	—	2	2	no	—
124	C₂×C₄	—	2	2	no	—
125	C₂×C₄	—	2	2	no	—
126	C₂×C₄	—	2	2	no	—
127	C₂×C₄	C₂×#6597	1	4	yes	—
128	C₂×C₄	C₂×#6597	1	2	yes	—
129	C₂×C₄	#90777	1	2	yes	—
130	C₂×C₄	C₄⋊Dic₂₉	1	2	yes	—
131	C₂×C₄	C₄⋊Dic₂₉	1	2	yes	—
132	C₂×C₄	C₂×Dic₅₈	1	1	yes	characteristic
133	C₂×C₄	C₂×D₁₁₆	1	1	yes	characteristic
134	C₂×C₄	D₄×D₂₉	1	1	yes	characteristic
135	C₂×C₄	#90807	1	1	yes	characteristic
136	C₂×C₄	Q₈×D₂₉	1	1	yes	characteristic
137	C₂×C₄	#90809	1	1	yes	characteristic
138	C₈	—	58	2	no	—
139	C₈	—	58	2	no	—
140	C₈	—	116	2	no	—
141	C₂²	C₂²×#6597	1	1	yes	characteristic
142	C₂²	C₂×#90777	1	1	yes	characteristic
143	C₂²	C₂×C₄⋊Dic₂₉	1	1	yes	characteristic
144	C₂²	D₄×Dic₂₉	1	1	yes	characteristic
145	C₂²	Q₈×Dic₂₉	1	1	yes	characteristic
146	C₂²	#11745310	1	1	yes	characteristic
147	C₂²	#11745310	1	1	yes	characteristic
148	C₄	—	2	2	no	—
149	C₄	—	2	4	no	—
150	C₄	—	4	2	no	—
151	C₄	C₄⋊#6597	1	4	yes	complemented
152	C₄	#11745250	1	1	yes	characteristic
153	C₄	#11745252	1	1	yes	characteristic
154	C₄	#11745257	1	1	yes	characteristic
155	C₄	#11745259	1	1	yes	characteristic
156	C₂	#14271062	1	1	yes	characteristic
157	C₂	C₂×C₄⋊#6597	1	1	yes	characteristic
158	C₂	#14271190	1	1	yes	characteristic
159	C₂	D₄×#6597	1	1	yes	characteristic
160	C₂	#14271241	1	1	yes	characteristic
161	C₂	Q₈×#6597	1	1	yes	characteristic
162	C₂	#14271294	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₂₉
commutator	C₂²×C₂₉	C₂²×C₈
Fitting	C₄×C₄⋊C₄×C₂₉	C₂
socle, prime radical	C₂³×C₂₉	C₂²×C₄
cosocle	C₂²×C₄×C₂₉	C₂³
hypercenter	C₄×C₄⋊C₄	D₂₉
hypocenter	C₂₉	#1939
LCS	G ▷ C₂²×C₂₉ ▷ C₂₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
23. C₄⋊C₄×C₂₉	C₈	semidirect
70. C₄×C₂₉	C₄⋊C₈	semidirect
98. C₂₉	#1939	semidirect
114. C₄⋊C₄	#6597	semidirect
151. C₄	C₄⋊#6597	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
C₂×C₄×#6597	2
#14270918	2
C₄×C₄⋊C₄×C₂₉	2
#1939	29