Subgroups — #135×#1181 #16836242

Order 3712.

500 conjugacy classes of subgroups.

The subgroup lattice has 262 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	Q₁₆×#1181	C₂	1	2	yes	maximal, complemented
3	SD₁₆×#1181	C₂	1	4	yes	maximal, complemented
4	D₈×#1181	C₂	1	2	yes	maximal, complemented
5	#14272173	C₂	1	1	yes	maximal, characteristic, complemented
6	#14272153	C₂	1	2	yes	maximal, complemented
7	C₂×C₈×#1181	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	C₈×#1181	C₂²	1	2	yes	—
33	C₈×#1181	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	1	yes	characteristic
55	C₄×#1181	D₄	1	1	yes	characteristic
56	C₄×#1181	C₂³	1	2	yes	—
57	D₂₉⋊C₈	—	2	1	no	—
58	D₂₉⋊C₈	C₂³	1	2	yes	—
59	Q₁₆×C₂₉	C₂×C₄	1	1	yes	characteristic, complemented
60	SD₁₆×C₂₉	C₂×C₄	1	2	yes	complemented
61	D₈×C₂₉	C₂×C₄	1	1	yes	characteristic, complemented
62	C₂×C₈×C₂₉	C₂×C₄	1	1	yes	characteristic, complemented
63	#90785	C₂×C₄	1	2	yes	complemented
64	#90784	C₂×C₄	1	2	yes	complemented
65	#90783	C₂×C₄	1	2	yes	complemented
66	#90782	C₂×C₄	1	2	yes	complemented
67	C₂×#6597	C₂×C₄	1	1	yes	characteristic, complemented
68	Dic₁₁₆	C₂×C₄	1	1	yes	characteristic, complemented
69	D₂₃₂	C₂×C₄	1	1	yes	characteristic, complemented
70	#90773	C₂×C₄	1	2	yes	complemented
71	C₈×D₂₉	C₂×C₄	1	1	yes	characteristic, complemented
72	C₈×D₂₉	C₂×C₄	1	1	yes	characteristic, complemented
73	C₈×D₂₉	C₂³	1	1	yes	characteristic
74	C₈×D₂₉	C₂³	1	1	yes	characteristic
75	C₂²×D₂₉	—	2	2	no	—
76	C₂²×D₂₉	C₂×D₄	1	1	yes	characteristic
77	C₂×#1181	—	2	4	no	—
78	C₂×#1181	—	2	4	no	—
79	C₂×#1181	—	4	8	no	—
80	C₂×#1181	C₂×D₄	1	2	yes	—
81	C₂×#1181	C₂×D₄	1	2	yes	—
82	Q₈×C₂₉	C₂²×C₄	1	2	yes	—
83	D₄×C₂₉	—	2	2	no	—
84	D₄×C₂₉	C₂²×C₄	1	2	yes	—
85	C₂×C₄×C₂₉	—	2	2	no	—
86	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
87	C₂²⋊D₂₉	—	2	2	no	—
88	C₂²⋊D₂₉	—	2	2	no	—
89	C₂²⋊D₂₉	—	2	2	no	—
90	C₂×Dic₂₉	—	2	2	no	—
91	C₂×Dic₂₉	#41	1	1	yes	characteristic
92	D₁₁₆	—	2	2	no	—
93	D₁₁₆	C₂²×C₄	1	2	yes	—
94	C₄×D₂₉	—	2	2	no	—
95	C₄×D₂₉	—	2	2	no	—
96	C₄×D₂₉	—	2	2	no	—
97	C₄×D₂₉	C₂²×C₄	1	1	yes	characteristic
98	C₄×D₂₉	C₂×D₄	1	1	yes	characteristic
99	C₄×D₂₉	#41	1	1	yes	characteristic
100	C₄×D₂₉	C₂⁴	1	1	yes	characteristic
101	Dic₅₈	—	2	2	no	—
102	Dic₅₈	C₂²×C₄	1	2	yes	—
103	#6599	—	2	2	no	—
104	C₈×C₂₉	C₂²×C₄	1	1	yes	characteristic
105	C₈×C₂₉	C₂²×C₄	1	1	yes	characteristic
106	#6597	C₂²×C₄	1	1	yes	characteristic
107	#6597	C₂²×C₄	1	1	yes	characteristic
108	C₄×#135	—	29	1	no	maximal, Sylow
109	C₂²×C₂₉	—	2	2	no	—
110	C₂²×C₂₉	C₄×D₄	1	1	yes	characteristic
111	C₂×D₂₉	—	2	2	no	—
112	C₂×D₂₉	—	2	2	no	—
113	C₂×D₂₉	—	4	4	no	—
114	C₂×D₂₉	C₄×D₄	1	1	yes	characteristic
115	C₂×D₂₉	C₂²×D₄	1	1	yes	characteristic
116	#1181	—	2	2	no	—
117	#1181	—	4	4	no	—
118	#1181	#135	1	4	yes	direct product factor
119	C₄×C₂₉	—	2	2	no	—
120	C₄×C₂₉	C₄×D₄	1	1	yes	characteristic
121	C₄×C₂₉	C₂³×C₄	1	1	yes	characteristic
122	Dic₂₉	—	2	2	no	—
123	Dic₂₉	C₄×D₄	1	1	yes	characteristic
124	Dic₂₉	C₂×#41	1	1	yes	characteristic
125	C₂×#135	—	29	1	no	—
126	C₄×#41	—	29	2	no	—
127	C₄×Q₁₆	—	29	2	no	—
128	C₄×SD₁₆	—	29	4	no	—
129	C₄×D₈	—	29	2	no	—
130	#427	—	29	1	no	—
131	#416	—	29	2	no	—
132	C₂×C₄×C₈	—	29	1	no	—
133	C₂×C₂₉	—	2	1	no	—
134	C₂×C₂₉	—	4	2	no	—
135	C₂×C₂₉	C₂×C₄×D₄	1	1	yes	characteristic
136	D₂₉	—	2	1	no	—
137	D₂₉	—	4	2	no	—
138	D₂₉	C₂×#135	1	2	yes	—
139	C₂×#41	—	29	2	no	—
140	#135	—	29	2	no	—
141	#135	—	29	1	no	—
142	#135	—	29	2	no	—
143	#135	—	29	1	no	—
144	#135	—	29	1	no	—
145	#135	#1181	1	1	yes	characteristic, direct product factor
146	C₂×Q₁₆	—	29	1	no	—
147	C₂×SD₁₆	—	29	2	no	—
148	C₂×D₈	—	29	1	no	—
149	C₂²×C₈	—	29	1	no	—
150	C₂²×C₈	—	29	1	no	—
151	C₄×Q₈	—	29	4	no	—
152	C₄×D₄	—	29	4	no	—
153	C₄×D₄	—	58	4	no	—
154	#117	—	29	2	no	—
155	#117	—	58	2	no	—
156	C₂×C₄²	—	29	1	no	—
157	C₂×C₄²	—	58	2	no	—
158	#107	—	29	2	no	—
159	#106	—	29	2	no	—
160	#103	—	29	4	no	—
161	#102	—	29	4	no	—
162	C₄×C₈	—	29	2	no	—
163	C₄×C₈	—	29	2	no	—
164	C₂₉	C₄×#135	1	1	yes	characteristic, complemented, Sylow
165	#41	—	29	2	no	—
166	#41	—	29	2	no	—
167	#41	—	29	2	no	—
168	#41	—	58	2	no	—
169	#41	—	58	2	no	—
170	#41	C₂×#1181	1	2	yes	complemented
171	C₂×Q₈	—	29	2	no	—
172	C₂×D₄	—	29	2	no	—
173	C₂×D₄	—	58	2	no	—
174	C₂²×C₄	—	29	1	no	—
175	C₂²×C₄	—	29	2	no	—
176	C₂²×C₄	—	58	2	no	—
177	C₂²×C₄	—	58	4	no	—
178	Q₁₆	—	29	2	no	—
179	Q₁₆	—	29	1	no	—
180	Q₁₆	C₂×#1181	1	1	yes	characteristic, complemented
181	SD₁₆	—	29	2	no	—
182	SD₁₆	—	29	2	no	—
183	SD₁₆	—	29	2	no	—
184	SD₁₆	C₂×#1181	1	2	yes	complemented
185	D₈	—	29	2	no	—
186	D₈	—	29	1	no	—
187	D₈	C₂×#1181	1	1	yes	characteristic, complemented
188	C₂×C₈	—	29	1	no	—
189	C₂×C₈	—	29	1	no	—
190	C₂×C₈	—	29	1	no	—
191	C₂×C₈	—	29	1	no	—
192	C₂×C₈	—	29	1	no	—
193	C₂×C₈	—	29	2	no	—
194	C₂×C₈	—	58	1	no	—
195	C₂×C₈	—	58	1	no	—
196	C₂×C₈	C₂×#1181	1	1	yes	characteristic, complemented
197	C₄⋊C₄	—	29	4	no	—
198	C₄⋊C₄	—	58	4	no	—
199	C₂²⋊C₄	—	58	4	no	—
200	C₂²⋊C₄	—	58	2	no	—
201	C₄²	—	29	2	no	—
202	C₄²	—	29	1	no	—
203	C₄²	—	29	1	no	—
204	C₄²	—	58	2	no	—
205	C₄²	—	58	4	no	—
206	C₂³	—	29	1	no	—
207	C₂³	—	58	2	no	—
208	Q₈	—	29	2	no	—
209	Q₈	—	58	2	no	—
210	Q₈	C₂²×#1181	1	2	yes	—
211	D₄	—	2	2	no	—
212	D₄	—	29	2	no	—
213	D₄	—	58	2	no	—
214	D₄	—	58	2	no	—
215	D₄	—	58	2	no	—
216	D₄	—	58	2	no	—
217	D₄	C₂²×#1181	1	2	yes	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	29	2	no	—
220	C₂×C₄	—	29	1	no	—
221	C₂×C₄	—	29	1	no	—
222	C₂×C₄	—	29	1	no	—
223	C₂×C₄	—	29	1	no	—
224	C₂×C₄	—	29	1	no	—
225	C₂×C₄	—	29	2	no	—
226	C₂×C₄	—	58	2	no	—
227	C₂×C₄	—	58	2	no	—
228	C₂×C₄	—	58	2	no	—
229	C₂×C₄	—	58	4	no	—
230	C₂×C₄	—	58	2	no	—
231	C₂×C₄	—	58	4	no	—
232	C₂×C₄	—	116	8	no	—
233	C₂×C₄	C₂²×#1181	1	1	yes	characteristic
234	C₈	—	29	1	no	—
235	C₈	—	29	1	no	—
236	C₈	—	58	2	no	—
237	C₈	C₂²×#1181	1	1	yes	characteristic
238	C₈	C₂²×#1181	1	1	yes	characteristic
239	C₂²	—	2	2	no	—
240	C₂²	—	29	1	no	—
241	C₂²	—	29	1	no	—
242	C₂²	—	58	2	no	—
243	C₂²	—	58	2	no	—
244	C₂²	—	116	4	no	—
245	C₂²	D₄×#1181	1	1	yes	characteristic
246	C₄	—	2	2	no	—
247	C₄	—	29	4	no	—
248	C₄	—	29	1	no	—
249	C₄	—	29	1	no	—
250	C₄	—	58	2	no	—
251	C₄	—	58	2	no	—
252	C₄	—	116	4	no	—
253	C₄	D₄×#1181	1	1	yes	characteristic
254	C₄	C₂³×#1181	1	1	yes	characteristic
255	C₂	—	2	1	no	—
256	C₂	—	4	2	no	—
257	C₂	—	29	2	no	—
258	C₂	—	58	1	no	—
259	C₂	—	116	2	no	—
260	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	C₄×#135
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. Q₁₆×#1181	C₂	semidirect
3. SD₁₆×#1181	C₂	semidirect
4. D₈×#1181	C₂	semidirect
5. #14272173	C₂	semidirect
6. #14272153	C₂	semidirect
7. C₂×C₈×#1181	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
59. Q₁₆×C₂₉	C₂×C₄	semidirect
60. SD₁₆×C₂₉	C₂×C₄	semidirect
61. D₈×C₂₉	C₂×C₄	semidirect
62. C₂×C₈×C₂₉	C₂×C₄	semidirect
63. #90785	C₂×C₄	semidirect
64. #90784	C₂×C₄	semidirect
65. #90783	C₂×C₄	semidirect
66. #90782	C₂×C₄	semidirect
67. C₂×#6597	C₂×C₄	semidirect
68. Dic₁₁₆	C₂×C₄	semidirect
69. D₂₃₂	C₂×C₄	semidirect
70. #90773	C₂×C₄	semidirect
71. C₈×D₂₉	C₂×C₄	semidirect
72. C₈×D₂₉	C₂×C₄	semidirect
118. #1181	#135	direct (N × H)
145. #135	#1181	direct (N × H)
164. C₂₉	C₄×#135	semidirect
170. #41	C₂×#1181	semidirect
180. Q₁₆	C₂×#1181	semidirect
184. SD₁₆	C₂×#1181	semidirect
187. D₈	C₂×#1181	semidirect
196. 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
Q₁₆×#1181	2
SD₁₆×#1181	2
D₈×#1181	2
#14272173	2
#14272153	2
C₂×C₈×#1181	2
#135×D₂₉	2
C₄×#135	29