Subgroups — #16836246

Order 3712.

492 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#41×#1181	C₂	1	2	yes	maximal, complemented
2	#14272204	C₂	1	2	yes	maximal, complemented
3	#14272203	C₂	1	2	yes	maximal, complemented
4	#14272202	C₂	1	4	yes	maximal, complemented
5	#14272173	C₂	1	1	yes	maximal, characteristic, complemented
6	#14272153	C₂	1	2	yes	maximal, complemented
7	C₂×#11745208	C₂	1	1	yes	maximal, characteristic, complemented
8	#135×D₂₉	C₂	1	1	yes	maximal, characteristic
9	#41×D₂₉	C₂²	1	2	yes	—
10	Q₈×#1181	C₂²	1	4	yes	—
11	D₄×#1181	—	2	4	no	—
12	D₄×#1181	C₂²	1	4	yes	—
13	#11745345	—	2	2	no	—
14	#11745345	C₂²	1	2	yes	—
15	C₂×C₄×#1181	—	2	2	no	—
16	C₂×C₄×#1181	C₂²	1	1	yes	characteristic
17	C₂×D₂₉⋊C₈	C₂²	1	1	yes	characteristic
18	C₂₉×#135	C₄	1	1	yes	characteristic, complemented
19	#11745312	C₄	1	2	yes	complemented
20	#11745281	C₄	1	1	yes	characteristic, complemented
21	Q₁₆×D₂₉	C₂²	1	1	yes	characteristic
22	#11745278	C₄	1	2	yes	complemented
23	SD₁₆×D₂₉	C₂²	1	2	yes	—
24	#11745274	C₄	1	1	yes	characteristic, complemented
25	D₈×D₂₉	C₂²	1	1	yes	characteristic
26	#11745266	C₄	1	1	yes	characteristic, complemented
27	C₂×C₈×D₂₉	C₂²	1	1	yes	characteristic
28	#11745225	C₂²	1	4	yes	—
29	#11745223	C₂²	1	4	yes	—
30	#11745210	C₂²	1	2	yes	—
31	#11745209	C₂²	1	2	yes	—
32	#11745208	C₂²	1	2	yes	—
33	#11745208	C₂²	1	2	yes	—
34	C₂²×#1181	—	2	4	no	—
35	C₂²×#1181	D₄	1	2	yes	—
36	#41×C₂₉	C₂×C₄	1	2	yes	complemented
37	#90809	C₂×C₄	1	2	yes	complemented
38	Q₈×D₂₉	C₂³	1	2	yes	—
39	#90807	—	2	2	no	—
40	#90807	C₂×C₄	1	2	yes	complemented
41	D₄×D₂₉	—	2	2	no	—
42	D₄×D₂₉	C₂³	1	2	yes	—
43	#90805	—	2	2	no	—
44	#90805	C₂×C₄	1	2	yes	complemented
45	C₂×C₄×D₂₉	—	2	2	no	—
46	C₂×C₄×D₂₉	C₂³	1	1	yes	characteristic
47	C₂²⋊#1181	—	2	2	no	—
48	C₂²⋊#1181	—	2	4	no	—
49	C₂×#6599	—	2	1	no	—
50	C₄⋊#1181	—	2	4	no	—
51	C₄⋊#1181	C₂³	1	4	yes	—
52	C₄×#1181	—	2	4	no	—
53	C₄×#1181	—	2	2	no	—
54	C₄×#1181	D₄	1	2	yes	—
55	C₄×#1181	C₂³	1	2	yes	—
56	D₂₉⋊C₈	—	2	1	no	—
57	D₂₉⋊C₈	C₂³	1	2	yes	—
58	Q₁₆×C₂₉	C₂×C₄	1	1	yes	characteristic, complemented
59	SD₁₆×C₂₉	C₂×C₄	1	2	yes	complemented
60	D₈×C₂₉	C₂×C₄	1	1	yes	characteristic, complemented
61	C₂×C₈×C₂₉	C₂×C₄	1	1	yes	characteristic, complemented
62	#90785	C₂×C₄	1	2	yes	complemented
63	#90784	C₂×C₄	1	2	yes	complemented
64	#90783	C₂×C₄	1	2	yes	complemented
65	#90782	C₂×C₄	1	2	yes	complemented
66	C₂×#6597	C₂×C₄	1	1	yes	characteristic, complemented
67	Dic₁₁₆	C₂×C₄	1	1	yes	characteristic, complemented
68	D₂₃₂	C₂×C₄	1	1	yes	characteristic, complemented
69	#90773	C₂×C₄	1	2	yes	complemented
70	C₈×D₂₉	C₂×C₄	1	1	yes	characteristic, complemented
71	C₈×D₂₉	C₂×C₄	1	1	yes	characteristic, complemented
72	C₈×D₂₉	C₂³	1	1	yes	characteristic
73	C₈×D₂₉	C₂³	1	1	yes	characteristic
74	C₂²×D₂₉	—	2	2	no	—
75	C₂²×D₂₉	C₂×D₄	1	1	yes	characteristic
76	C₂×#1181	—	2	4	no	—
77	C₂×#1181	—	4	2	no	—
78	C₂×#1181	—	4	8	no	—
79	C₂×#1181	C₂×D₄	1	4	yes	—
80	Q₈×C₂₉	C₂²×C₄	1	2	yes	—
81	D₄×C₂₉	—	2	2	no	—
82	D₄×C₂₉	C₂²×C₄	1	2	yes	—
83	C₂×C₄×C₂₉	—	2	2	no	—
84	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
85	C₂²⋊D₂₉	—	2	2	no	—
86	C₂²⋊D₂₉	—	2	2	no	—
87	C₂²⋊D₂₉	—	2	2	no	—
88	C₂×Dic₂₉	—	2	2	no	—
89	C₂×Dic₂₉	#41	1	1	yes	characteristic
90	D₁₁₆	—	2	2	no	—
91	D₁₁₆	C₂²×C₄	1	2	yes	—
92	C₄×D₂₉	—	2	2	no	—
93	C₄×D₂₉	—	2	2	no	—
94	C₄×D₂₉	—	2	2	no	—
95	C₄×D₂₉	C₂²×C₄	1	1	yes	characteristic
96	C₄×D₂₉	C₂×D₄	1	1	yes	characteristic
97	C₄×D₂₉	#41	1	1	yes	characteristic
98	C₄×D₂₉	C₂⁴	1	1	yes	characteristic
99	Dic₅₈	—	2	2	no	—
100	Dic₅₈	C₂²×C₄	1	2	yes	—
101	#6599	—	2	2	no	—
102	C₈×C₂₉	C₂²×C₄	1	1	yes	characteristic
103	C₈×C₂₉	C₂²×C₄	1	1	yes	characteristic
104	#6597	C₂²×C₄	1	1	yes	characteristic
105	#6597	C₂²×C₄	1	1	yes	characteristic
106	#2943	—	29	1	no	maximal, Sylow
107	C₂²×C₂₉	—	2	2	no	—
108	C₂²×C₂₉	C₄×D₄	1	1	yes	characteristic
109	C₂×D₂₉	—	2	2	no	—
110	C₂×D₂₉	—	2	2	no	—
111	C₂×D₂₉	—	4	4	no	—
112	C₂×D₂₉	C₄×D₄	1	1	yes	characteristic
113	C₂×D₂₉	C₂²×D₄	1	1	yes	characteristic
114	#1181	—	2	4	no	—
115	#1181	—	4	4	no	—
116	C₄×C₂₉	—	2	2	no	—
117	C₄×C₂₉	C₄×D₄	1	1	yes	characteristic
118	C₄×C₂₉	C₂³×C₄	1	1	yes	characteristic
119	Dic₂₉	—	2	2	no	—
120	Dic₂₉	C₄×D₄	1	1	yes	characteristic
121	Dic₂₉	C₂×#41	1	1	yes	characteristic
122	C₂×#135	—	29	1	no	—
123	C₄×#41	—	29	2	no	—
124	#442	—	29	2	no	—
125	#441	—	29	2	no	—
126	#440	—	29	4	no	—
127	#427	—	29	1	no	—
128	#416	—	29	2	no	—
129	C₂×#97	—	29	1	no	—
130	C₂×C₂₉	—	2	1	no	—
131	C₂×C₂₉	—	4	2	no	—
132	C₂×C₂₉	C₂×C₄×D₄	1	1	yes	characteristic
133	D₂₉	—	2	1	no	—
134	D₂₉	—	4	2	no	—
135	D₂₉	#575	1	2	yes	—
136	C₂×#41	—	29	2	no	—
137	#135	—	29	2	no	—
138	#135	—	29	1	no	—
139	#135	—	29	2	no	—
140	#135	—	29	1	no	—
141	#135	—	29	1	no	—
142	#135	#1181	1	1	yes	characteristic, complemented
143	C₂×Q₁₆	—	29	1	no	—
144	C₂×SD₁₆	—	29	2	no	—
145	C₂×D₈	—	29	1	no	—
146	C₂²×C₈	—	29	1	no	—
147	C₂²×C₈	—	29	1	no	—
148	C₄×Q₈	—	29	4	no	—
149	C₄×D₄	—	29	4	no	—
150	C₄×D₄	—	58	4	no	—
151	#117	—	29	2	no	—
152	#117	—	58	2	no	—
153	C₂×C₄²	—	29	1	no	—
154	C₂×C₄²	—	58	2	no	—
155	#107	—	29	2	no	—
156	#106	—	29	2	no	—
157	#103	—	29	4	no	—
158	#102	—	29	4	no	—
159	#97	—	29	2	no	—
160	#97	—	29	2	no	—
161	C₂₉	#2943	1	1	yes	characteristic, complemented, Sylow
162	#41	—	29	2	no	—
163	#41	—	29	2	no	—
164	#41	—	29	2	no	—
165	#41	—	58	2	no	—
166	#41	—	58	2	no	—
167	#41	C₂×#1181	1	2	yes	complemented
168	C₂×Q₈	—	29	2	no	—
169	C₂×D₄	—	29	2	no	—
170	C₂×D₄	—	58	2	no	—
171	C₂²×C₄	—	29	1	no	—
172	C₂²×C₄	—	29	2	no	—
173	C₂²×C₄	—	58	2	no	—
174	C₂²×C₄	—	58	4	no	—
175	Q₁₆	—	29	2	no	—
176	Q₁₆	—	29	1	no	—
177	Q₁₆	C₂×#1181	1	1	yes	characteristic, complemented
178	SD₁₆	—	29	2	no	—
179	SD₁₆	—	29	2	no	—
180	SD₁₆	—	29	2	no	—
181	SD₁₆	C₂×#1181	1	2	yes	complemented
182	D₈	—	29	2	no	—
183	D₈	—	29	1	no	—
184	D₈	C₂×#1181	1	1	yes	characteristic, complemented
185	C₂×C₈	—	29	1	no	—
186	C₂×C₈	—	29	1	no	—
187	C₂×C₈	—	29	1	no	—
188	C₂×C₈	—	29	1	no	—
189	C₂×C₈	—	29	1	no	—
190	C₂×C₈	—	29	2	no	—
191	C₂×C₈	—	58	1	no	—
192	C₂×C₈	—	58	1	no	—
193	C₂×C₈	C₂×#1181	1	1	yes	characteristic, complemented
194	C₄⋊C₄	—	29	4	no	—
195	C₄⋊C₄	—	58	4	no	—
196	C₂²⋊C₄	—	58	4	no	—
197	C₂²⋊C₄	—	58	2	no	—
198	C₄²	—	29	2	no	—
199	C₄²	—	29	2	no	—
200	C₄²	—	58	2	no	—
201	C₄²	—	58	4	no	—
202	C₂³	—	29	1	no	—
203	C₂³	—	58	2	no	—
204	Q₈	—	29	2	no	—
205	Q₈	—	58	2	no	—
206	Q₈	C₂²×#1181	1	2	yes	—
207	D₄	—	2	2	no	—
208	D₄	—	29	2	no	—
209	D₄	—	58	2	no	—
210	D₄	—	58	2	no	—
211	D₄	—	58	2	no	—
212	D₄	—	58	2	no	—
213	D₄	C₂²×#1181	1	2	yes	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	29	1	no	—
216	C₂×C₄	—	29	1	no	—
217	C₂×C₄	—	29	1	no	—
218	C₂×C₄	—	29	1	no	—
219	C₂×C₄	—	29	1	no	—
220	C₂×C₄	—	29	4	no	—
221	C₂×C₄	—	58	2	no	—
222	C₂×C₄	—	58	2	no	—
223	C₂×C₄	—	58	2	no	—
224	C₂×C₄	—	58	2	no	—
225	C₂×C₄	—	58	4	no	—
226	C₂×C₄	—	116	2	no	—
227	C₂×C₄	—	116	8	no	—
228	C₂×C₄	C₂²×#1181	1	1	yes	characteristic
229	C₈	—	29	1	no	—
230	C₈	—	29	1	no	—
231	C₈	—	58	2	no	—
232	C₈	C₂²×#1181	1	1	yes	characteristic
233	C₈	C₂²×#1181	1	1	yes	characteristic
234	C₂²	—	2	2	no	—
235	C₂²	—	29	1	no	—
236	C₂²	—	29	1	no	—
237	C₂²	—	58	2	no	—
238	C₂²	—	58	2	no	—
239	C₂²	—	116	4	no	—
240	C₂²	D₄×#1181	1	1	yes	characteristic
241	C₄	—	2	2	no	—
242	C₄	—	29	1	no	—
243	C₄	—	29	1	no	—
244	C₄	—	58	4	no	—
245	C₄	—	58	2	no	—
246	C₄	—	116	4	no	—
247	C₄	D₄×#1181	1	1	yes	characteristic
248	C₄	C₂³×#1181	1	1	yes	characteristic
249	C₂	—	2	1	no	—
250	C₂	—	4	2	no	—
251	C₂	—	29	2	no	—
252	C₂	—	58	1	no	—
253	C₂	—	116	2	no	—
254	C₂	C₂×D₄×#1181	1	1	yes	characteristic

	subgroup	G/H
center	C₄	D₄×#1181
commutator	C₄×C₂₉	C₂³×C₄
Frattini	C₄	C₂³×#1181
Fitting	C₂₉×#135	C₄
socle	C₂×C₂₉	C₂×C₄×D₄
prime radical	#135×D₂₉	C₂
cosocle	C₄×D₂₉	C₂⁴
hypercenter	#135	#1181
hypocenter	C₂₉	#2943
LCS	G ▷ C₄×C₂₉ ▷ C₂×C₂₉ ▷ C₂₉
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ #135

Complements

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

N	H ≅ G/N	type
1. #41×#1181	C₂	semidirect
2. #14272204	C₂	semidirect
3. #14272203	C₂	semidirect
4. #14272202	C₂	semidirect
5. #14272173	C₂	semidirect
6. #14272153	C₂	semidirect
7. C₂×#11745208	C₂	semidirect
18. C₂₉×#135	C₄	semidirect
19. #11745312	C₄	semidirect
20. #11745281	C₄	semidirect
22. #11745278	C₄	semidirect
24. #11745274	C₄	semidirect
26. #11745266	C₄	semidirect
36. #41×C₂₉	C₂×C₄	semidirect
37. #90809	C₂×C₄	semidirect
40. #90807	C₂×C₄	semidirect
44. #90805	C₂×C₄	semidirect
58. Q₁₆×C₂₉	C₂×C₄	semidirect
59. SD₁₆×C₂₉	C₂×C₄	semidirect
60. D₈×C₂₉	C₂×C₄	semidirect
61. C₂×C₈×C₂₉	C₂×C₄	semidirect
62. #90785	C₂×C₄	semidirect
63. #90784	C₂×C₄	semidirect
64. #90783	C₂×C₄	semidirect
65. #90782	C₂×C₄	semidirect
66. C₂×#6597	C₂×C₄	semidirect
67. Dic₁₁₆	C₂×C₄	semidirect
68. D₂₃₂	C₂×C₄	semidirect
69. #90773	C₂×C₄	semidirect
70. C₈×D₂₉	C₂×C₄	semidirect
71. C₈×D₂₉	C₂×C₄	semidirect
142. #135	#1181	semidirect
161. C₂₉	#2943	semidirect
167. #41	C₂×#1181	semidirect
177. Q₁₆	C₂×#1181	semidirect
181. SD₁₆	C₂×#1181	semidirect
184. D₈	C₂×#1181	semidirect
193. C₂×C₈	C₂×#1181	semidirect

Maximal subgroups

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

subgroup	index
#41×#1181	2
#14272204	2
#14272203	2
#14272202	2
#14272173	2
#14272153	2
C₂×#11745208	2
#135×D₂₉	2
#2943	29