Subgroups — #16836297

Order 3712.

456 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14272212	C₂	1	2	yes	maximal, complemented
2	#14272190	C₂	1	4	yes	maximal, complemented
3	D₄×#6599	C₂	1	2	yes	maximal, complemented
4	#14272182	C₂	1	1	yes	maximal, characteristic, complemented
5	#14272180	C₂	1	2	yes	maximal, complemented
6	C₄×D₂₉⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₄×#6599	C₂	1	2	yes	maximal, complemented
8	C₄×#90807	C₂	1	1	yes	maximal, characteristic
9	C₂×#90807	C₄	1	1	yes	characteristic
10	C₂²×#6599	C₂²	1	4	yes	—
11	C₂×D₂₉⋊C₈	C₂²	1	2	yes	—
12	C₄×D₄×C₂₉	C₄	1	1	yes	characteristic
13	D₄×Dic₂₉	C₂²	1	1	yes	characteristic
14	C₄×C₂²⋊D₂₉	C₄	1	2	yes	—
15	C₂×C₄×Dic₂₉	C₂²	1	2	yes	—
16	#11745254	C₄	1	1	yes	characteristic
17	#11745249	C₂²	1	1	yes	characteristic
18	#11745243	C₂²	1	2	yes	—
19	#11745239	C₄	1	2	yes	—
20	C₄²×D₂₉	C₂²	1	1	yes	characteristic
21	C₄×Dic₅₈	C₄	1	1	yes	characteristic
22	C₂²⋊#6599	C₂²	1	4	yes	—
23	#11745220	C₂²	1	4	yes	—
24	#11745219	C₂²	1	2	yes	—
25	C₄⋊#6599	C₂²	1	2	yes	—
26	C₄×#6599	C₂²	1	1	yes	characteristic
27	C₄×#6599	C₂²	1	2	yes	—
28	C₄×#6599	C₂²	1	4	yes	—
29	C₄×#6599	C₂²	1	1	yes	characteristic
30	C₄×#6599	C₂²	1	2	yes	—
31	C₂×D₄×C₂₉	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₄×C₂₉	C₂×C₄	1	2	yes	—
33	C₂×C₂²⋊D₂₉	C₂×C₄	1	2	yes	—
34	C₂²×Dic₂₉	C₂×C₄	1	2	yes	—
35	C₂²×Dic₂₉	C₂³	1	2	yes	—
36	#90807	C₈	1	8	yes	complemented
37	C₂×C₄×D₂₉	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄×D₂₉	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄×D₂₉	C₂³	1	1	yes	characteristic
40	C₂×Dic₅₈	C₂×C₄	1	1	yes	characteristic
41	C₂×#6599	—	2	8	no	—
42	C₂×#6599	C₂³	1	2	yes	—
43	C₂×#6599	C₂³	1	4	yes	—
44	C₂×#6599	C₂³	1	2	yes	—
45	D₂₉⋊C₈	—	2	4	no	—
46	C₄⋊C₄×C₂₉	C₂×C₄	1	1	yes	characteristic
47	C₂²⋊C₄×C₂₉	C₂×C₄	1	2	yes	—
48	C₄²×C₂₉	C₂×C₄	1	1	yes	characteristic
49	C₂²⋊Dic₂₉	C₂×C₄	1	2	yes	—
50	#90781	C₂×C₄	1	2	yes	—
51	C₄⋊Dic₂₉	C₂×C₄	1	1	yes	characteristic
52	#90779	C₂×C₄	1	2	yes	—
53	#90779	C₂×C₄	1	2	yes	—
54	C₄×Dic₂₉	C₂×C₄	1	2	yes	—
55	C₄×Dic₂₉	C₂×C₄	1	2	yes	—
56	C₄×Dic₂₉	C₂×C₄	1	1	yes	characteristic
57	C₄×Dic₂₉	C₂³	1	1	yes	characteristic
58	C₄×Dic₂₉	C₂³	1	2	yes	—
59	C₄×Dic₂₉	C₂³	1	1	yes	characteristic
60	C₂³×C₂₉	C₂²×C₄	1	2	yes	—
61	C₂²×D₂₉	C₂²×C₄	1	1	yes	characteristic
62	D₄×C₂₉	C₂×C₈	1	4	yes	complemented
63	C₂×C₄×C₂₉	—	2	4	no	—
64	C₂×C₄×C₂₉	C₂²×C₄	1	2	yes	—
65	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
66	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
67	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
68	C₂²⋊D₂₉	C₂×C₈	1	8	yes	complemented
69	C₂×Dic₂₉	—	2	4	no	—
70	C₂×Dic₂₉	C₂×C₈	1	8	yes	complemented
71	C₂×Dic₂₉	C₂²×C₄	1	2	yes	—
72	C₂×Dic₂₉	C₂²×C₄	1	1	yes	characteristic
73	C₂×Dic₂₉	C₂²×C₄	1	1	yes	characteristic
74	C₂×Dic₂₉	C₂²×C₄	1	2	yes	—
75	C₂×Dic₂₉	C₂⁴	1	1	yes	characteristic
76	C₄×D₂₉	—	2	2	no	—
77	C₄×D₂₉	—	2	2	no	—
78	C₄×D₂₉	C₂×C₈	1	4	yes	complemented
79	Dic₅₈	C₂×C₈	1	4	yes	complemented
80	#6599	—	2	2	no	—
81	#6599	—	2	4	no	—
82	#6599	#41	1	4	yes	complemented
83	C₈×#41	—	29	1	no	maximal, Sylow
84	C₂²×C₂₉	—	2	2	no	—
85	C₂²×C₂₉	—	2	2	no	—
86	C₂²×C₂₉	C₂²×C₈	1	4	yes	—
87	C₂²×C₂₉	C₂³×C₄	1	1	yes	characteristic
88	C₂×D₂₉	—	2	1	no	—
89	C₂×D₂₉	—	2	1	no	—
90	C₂×D₂₉	C₂²×C₈	1	2	yes	—
91	C₄×C₂₉	—	2	1	no	—
92	C₄×C₂₉	—	2	2	no	—
93	C₄×C₂₉	C₂²×C₈	1	2	yes	—
94	C₄×C₂₉	#131	1	2	yes	—
95	Dic₂₉	—	2	2	no	—
96	Dic₂₉	—	2	1	no	—
97	Dic₂₉	C₂²×C₈	1	4	yes	—
98	Dic₂₉	C₂²×C₈	1	2	yes	—
99	Dic₂₉	C₂×#41	1	2	yes	—
100	C₄×#41	—	29	1	no	—
101	C₈×Q₈	—	29	2	no	—
102	C₈×D₄	—	29	2	no	—
103	C₈×D₄	—	29	4	no	—
104	#431	—	29	2	no	—
105	#431	—	29	1	no	—
106	C₂×C₄×C₈	—	29	2	no	—
107	C₂×C₄×C₈	—	29	1	no	—
108	C₂×C₂₉	—	2	4	no	—
109	C₂×C₂₉	C₄×#41	1	1	yes	characteristic
110	C₂×C₂₉	C₂³×C₈	1	1	yes	characteristic
111	C₂×C₂₉	C₂×#131	1	1	yes	characteristic
112	D₂₉	—	2	2	no	—
113	C₂×#41	—	29	1	no	—
114	C₂²×C₈	—	29	2	no	—
115	C₂²×C₈	—	29	4	no	—
116	C₄×Q₈	—	29	1	no	—
117	C₄×Q₈	—	29	1	no	—
118	C₄×D₄	—	29	1	no	—
119	C₄×D₄	—	29	2	no	—
120	C₄×D₄	—	29	2	no	—
121	C₄×D₄	#1181	1	1	yes	characteristic
122	#117	—	29	1	no	—
123	#117	—	29	2	no	—
124	C₂×C₄²	—	29	2	no	—
125	C₂×C₄²	—	29	1	no	—
126	C₄⋊C₈	—	29	2	no	—
127	C₄⋊C₈	—	29	4	no	—
128	C₂²⋊C₈	—	29	2	no	—
129	C₂²⋊C₈	—	29	4	no	—
130	C₄×C₈	—	29	1	no	—
131	C₄×C₈	—	29	2	no	—
132	C₄×C₈	—	29	1	no	—
133	C₄×C₈	—	29	4	no	—
134	C₄×C₈	—	29	2	no	—
135	C₂₉	C₈×#41	1	1	yes	characteristic, complemented, Sylow
136	#41	—	29	8	no	—
137	C₂×Q₈	—	29	1	no	—
138	C₂×D₄	—	29	2	no	—
139	C₂×D₄	C₂×#1181	1	1	yes	characteristic
140	C₂²×C₄	—	29	2	no	—
141	C₂²×C₄	—	29	1	no	—
142	C₂²×C₄	—	29	2	no	—
143	C₂²×C₄	—	29	1	no	—
144	C₂²×C₄	—	29	1	no	—
145	C₂²×C₄	C₂×#1181	1	2	yes	—
146	C₂×C₈	—	29	2	no	—
147	C₂×C₈	—	29	4	no	—
148	C₂×C₈	—	29	2	no	—
149	C₂×C₈	—	58	4	no	—
150	C₂×C₈	—	58	8	no	—
151	C₄⋊C₄	—	29	2	no	—
152	C₄⋊C₄	—	29	2	no	—
153	C₄⋊C₄	—	29	1	no	—
154	C₄⋊C₄	C₂×#1181	1	1	yes	characteristic
155	C₂²⋊C₄	—	29	2	no	—
156	C₂²⋊C₄	—	29	2	no	—
157	C₂²⋊C₄	C₂×#1181	1	2	yes	—
158	C₄²	—	29	2	no	—
159	C₄²	—	29	1	no	—
160	C₄²	—	29	2	no	—
161	C₄²	—	29	1	no	—
162	C₄²	—	29	1	no	—
163	C₄²	—	29	2	no	—
164	C₄²	C₂×#1181	1	1	yes	characteristic
165	C₂³	—	29	1	no	—
166	C₂³	C₂²×#1181	1	2	yes	—
167	Q₈	—	29	4	no	—
168	D₄	—	29	8	no	—
169	D₄	D₂₉⋊C₈	1	4	yes	complemented
170	C₂×C₄	—	2	4	no	—
171	C₂×C₄	—	29	8	no	—
172	C₂×C₄	—	29	1	no	—
173	C₂×C₄	—	29	1	no	—
174	C₂×C₄	—	29	2	no	—
175	C₂×C₄	—	29	4	no	—
176	C₂×C₄	—	29	2	no	—
177	C₂×C₄	—	29	1	no	—
178	C₂×C₄	—	58	2	no	—
179	C₂×C₄	—	58	4	no	—
180	C₂×C₄	—	58	2	no	—
181	C₂×C₄	C₂²×#1181	1	2	yes	—
182	C₂×C₄	C₂²×#1181	1	1	yes	characteristic
183	C₂×C₄	C₂²×#1181	1	1	yes	characteristic
184	C₂×C₄	C₂²×#1181	1	1	yes	characteristic
185	C₈	—	29	4	no	—
186	C₈	—	58	2	no	—
187	C₈	—	58	4	no	—
188	C₂²	—	2	2	no	—
189	C₂²	—	2	2	no	—
190	C₂²	—	29	2	no	—
191	C₂²	—	58	1	no	—
192	C₂²	—	58	1	no	—
193	C₂²	C₂×D₂₉⋊C₈	1	4	yes	—
194	C₂²	C₂³×#1181	1	1	yes	characteristic
195	C₄	—	2	1	no	—
196	C₄	—	2	2	no	—
197	C₄	—	29	2	no	—
198	C₄	—	29	2	no	—
199	C₄	—	29	4	no	—
200	C₄	—	58	2	no	—
201	C₄	—	58	1	no	—
202	C₄	C₂×D₂₉⋊C₈	1	2	yes	—
203	C₄	#11745346	1	2	yes	—
204	C₂	—	2	4	no	—
205	C₂	—	58	2	no	—
206	C₂	#41×#1181	1	1	yes	characteristic
207	C₂	C₂²×D₂₉⋊C₈	1	1	yes	characteristic
208	C₂	C₂×#11745346	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×#1181
commutator	C₂×C₂₉	C₂³×C₈
Frattini	C₂²	C₂³×#1181
Fitting	C₄×D₄×C₂₉	C₄
socle	C₂²×C₂₉	C₂³×C₄
prime radical	C₂×#90807	C₄
cosocle	C₂×Dic₂₉	C₂⁴
hypercenter	C₄×D₄	#1181
hypocenter	C₂₉	C₈×#41
LCS	G ▷ C₂×C₂₉ ▷ C₂₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄×D₄

Complements

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

N	H ≅ G/N	type
1. #14272212	C₂	semidirect
2. #14272190	C₂	semidirect
3. D₄×#6599	C₂	semidirect
4. #14272182	C₂	semidirect
5. #14272180	C₂	semidirect
6. C₄×D₂₉⋊C₈	C₂	semidirect
7. C₂×C₄×#6599	C₂	semidirect
36. #90807	C₈	semidirect
62. D₄×C₂₉	C₂×C₈	semidirect
68. C₂²⋊D₂₉	C₂×C₈	semidirect
70. C₂×Dic₂₉	C₂×C₈	semidirect
78. C₄×D₂₉	C₂×C₈	semidirect
79. Dic₅₈	C₂×C₈	semidirect
82. #6599	#41	semidirect
135. C₂₉	C₈×#41	semidirect
169. D₄	D₂₉⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
#14272212	2
#14272190	2
D₄×#6599	2
#14272182	2
#14272180	2
C₄×D₂₉⋊C₈	2
C₂×C₄×#6599	2
C₄×#90807	2
C₈×#41	29