Subgroups — #135×#3866 #18011346

Order 4736.

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×#3866	C₂	1	2	yes	maximal, complemented
2	Q₁₆×#3866	C₂	1	2	yes	maximal, complemented
3	SD₁₆×#3866	C₂	1	4	yes	maximal, complemented
4	D₈×#3866	C₂	1	2	yes	maximal, complemented
5	#14558543	C₂	1	1	yes	maximal, characteristic, complemented
6	#14558523	C₂	1	2	yes	maximal, complemented
7	C₂×C₈×#3866	C₂	1	1	yes	maximal, characteristic, complemented
8	#135×D₃₇	C₂	1	1	yes	maximal, characteristic
9	C₂×D₃₇⋊C₈	C₂²	1	1	yes	characteristic
10	Q₈×#3866	C₂²	1	4	yes	—
11	D₄×#3866	—	2	4	no	—
12	D₄×#3866	C₂²	1	4	yes	—
13	#11924615	—	2	2	no	—
14	#11924615	C₂²	1	2	yes	—
15	C₂×C₄×#3866	—	2	2	no	—
16	C₂×C₄×#3866	C₂²	1	1	yes	characteristic
17	#11924605	C₂²	1	2	yes	—
18	#11924604	C₂²	1	2	yes	—
19	#11924599	C₂²	1	4	yes	—
20	#11924598	C₂²	1	4	yes	—
21	C₈×#3866	C₂²	1	2	yes	—
22	C₈×#3866	C₂²	1	2	yes	—
23	#41×D₃₇	C₂²	1	2	yes	—
24	#11924559	C₄	1	1	yes	characteristic, complemented
25	#11924558	C₄	1	1	yes	characteristic, complemented
26	#11924557	C₄	1	2	yes	complemented
27	#11924556	C₄	1	2	yes	complemented
28	#11924555	C₄	1	1	yes	characteristic, complemented
29	Q₁₆×D₃₇	C₂²	1	1	yes	characteristic
30	SD₁₆×D₃₇	C₂²	1	2	yes	—
31	D₈×D₃₇	C₂²	1	1	yes	characteristic
32	C₂×C₈×D₃₇	C₂²	1	1	yes	characteristic
33	#135×C₃₇	C₄	1	1	yes	characteristic, complemented
34	C₂²×#3866	—	2	4	no	—
35	C₂²×#3866	D₄	1	2	yes	—
36	#41×C₃₇	C₂×C₄	1	2	yes	complemented
37	#10596966	C₂×C₄	1	2	yes	complemented
38	Q₈×D₃₇	C₂³	1	2	yes	—
39	#10596964	—	2	2	no	—
40	#10596964	C₂×C₄	1	2	yes	complemented
41	D₄×D₃₇	—	2	2	no	—
42	D₄×D₃₇	C₂³	1	2	yes	—
43	#10596962	—	2	2	no	—
44	#10596962	C₂×C₄	1	2	yes	complemented
45	C₂×C₄×D₃₇	—	2	2	no	—
46	C₂×C₄×D₃₇	C₂³	1	1	yes	characteristic
47	C₂²⋊#3866	—	2	2	no	—
48	C₂²⋊#3866	—	2	4	no	—
49	C₂×#64438	—	2	1	no	—
50	C₄⋊#3866	—	2	4	no	—
51	C₄⋊#3866	C₂³	1	4	yes	—
52	C₄×#3866	—	2	4	no	—
53	C₄×#3866	—	2	2	no	—
54	C₄×#3866	D₄	1	1	yes	characteristic
55	C₄×#3866	D₄	1	1	yes	characteristic
56	C₄×#3866	C₂³	1	2	yes	—
57	D₃₇⋊C₈	—	2	1	no	—
58	D₃₇⋊C₈	C₂³	1	2	yes	—
59	Q₁₆×C₃₇	C₂×C₄	1	1	yes	characteristic, complemented
60	SD₁₆×C₃₇	C₂×C₄	1	2	yes	complemented
61	D₈×C₃₇	C₂×C₄	1	1	yes	characteristic, complemented
62	C₂×C₈×C₃₇	C₂×C₄	1	1	yes	characteristic, complemented
63	#10596942	C₂×C₄	1	2	yes	complemented
64	#10596941	C₂×C₄	1	2	yes	complemented
65	#10596940	C₂×C₄	1	2	yes	complemented
66	#10596939	C₂×C₄	1	2	yes	complemented
67	C₂×#64436	C₂×C₄	1	1	yes	characteristic, complemented
68	Dic₁₄₈	C₂×C₄	1	1	yes	characteristic, complemented
69	D₂₉₆	C₂×C₄	1	1	yes	characteristic, complemented
70	#10596930	C₂×C₄	1	2	yes	complemented
71	C₈×D₃₇	C₂×C₄	1	1	yes	characteristic, complemented
72	C₈×D₃₇	C₂×C₄	1	1	yes	characteristic, complemented
73	C₈×D₃₇	C₂³	1	1	yes	characteristic
74	C₈×D₃₇	C₂³	1	1	yes	characteristic
75	C₂²×D₃₇	—	2	2	no	—
76	C₂²×D₃₇	C₂×D₄	1	1	yes	characteristic
77	C₂×#3866	—	2	4	no	—
78	C₂×#3866	—	2	4	no	—
79	C₂×#3866	—	4	8	no	—
80	C₂×#3866	C₂×D₄	1	2	yes	—
81	C₂×#3866	C₂×D₄	1	2	yes	—
82	Q₈×C₃₇	C₂²×C₄	1	2	yes	—
83	D₄×C₃₇	—	2	2	no	—
84	D₄×C₃₇	C₂²×C₄	1	2	yes	—
85	C₂×C₄×C₃₇	—	2	2	no	—
86	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
87	C₂²⋊D₃₇	—	2	2	no	—
88	C₂²⋊D₃₇	—	2	2	no	—
89	C₂²⋊D₃₇	—	2	2	no	—
90	C₂×Dic₃₇	—	2	2	no	—
91	C₂×Dic₃₇	#41	1	1	yes	characteristic
92	D₁₄₈	—	2	2	no	—
93	D₁₄₈	C₂²×C₄	1	2	yes	—
94	C₄×D₃₇	—	2	2	no	—
95	C₄×D₃₇	—	2	2	no	—
96	C₄×D₃₇	—	2	2	no	—
97	C₄×D₃₇	C₂²×C₄	1	1	yes	characteristic
98	C₄×D₃₇	C₂×D₄	1	1	yes	characteristic
99	C₄×D₃₇	#41	1	1	yes	characteristic
100	C₄×D₃₇	C₂⁴	1	1	yes	characteristic
101	Dic₇₄	—	2	2	no	—
102	Dic₇₄	C₂²×C₄	1	2	yes	—
103	#64438	—	2	2	no	—
104	C₈×C₃₇	C₂²×C₄	1	1	yes	characteristic
105	C₈×C₃₇	C₂²×C₄	1	1	yes	characteristic
106	#64436	C₂²×C₄	1	1	yes	characteristic
107	#64436	C₂²×C₄	1	1	yes	characteristic
108	C₂²×C₃₇	—	2	2	no	—
109	C₂²×C₃₇	C₄×D₄	1	1	yes	characteristic
110	C₂×D₃₇	—	2	2	no	—
111	C₂×D₃₇	—	2	2	no	—
112	C₂×D₃₇	—	4	4	no	—
113	C₂×D₃₇	C₄×D₄	1	1	yes	characteristic
114	C₂×D₃₇	C₂²×D₄	1	1	yes	characteristic
115	#3866	—	2	2	no	—
116	#3866	—	4	4	no	—
117	#3866	#135	1	4	yes	direct product factor
118	C₄×C₃₇	—	2	2	no	—
119	C₄×C₃₇	C₄×D₄	1	1	yes	characteristic
120	C₄×C₃₇	C₂³×C₄	1	1	yes	characteristic
121	Dic₃₇	—	2	2	no	—
122	Dic₃₇	C₄×D₄	1	1	yes	characteristic
123	Dic₃₇	C₂×#41	1	1	yes	characteristic
124	C₄×#135	—	37	1	no	maximal, Sylow
125	C₂×C₃₇	—	2	1	no	—
126	C₂×C₃₇	—	4	2	no	—
127	C₂×C₃₇	C₂×C₄×D₄	1	1	yes	characteristic
128	D₃₇	—	2	1	no	—
129	D₃₇	—	4	2	no	—
130	D₃₇	C₂×#135	1	2	yes	—
131	C₂×#135	—	37	1	no	—
132	C₄×#41	—	37	2	no	—
133	C₄×Q₁₆	—	37	2	no	—
134	C₄×SD₁₆	—	37	4	no	—
135	C₄×D₈	—	37	2	no	—
136	#427	—	37	1	no	—
137	#416	—	37	2	no	—
138	C₂×C₄×C₈	—	37	1	no	—
139	C₃₇	C₄×#135	1	1	yes	characteristic, complemented, Sylow
140	C₂×#41	—	37	2	no	—
141	#135	—	37	2	no	—
142	#135	—	37	1	no	—
143	#135	—	37	2	no	—
144	#135	—	37	1	no	—
145	#135	—	37	1	no	—
146	#135	#3866	1	1	yes	characteristic, direct product factor
147	C₂×Q₁₆	—	37	1	no	—
148	C₂×SD₁₆	—	37	2	no	—
149	C₂×D₈	—	37	1	no	—
150	C₂²×C₈	—	37	1	no	—
151	C₂²×C₈	—	37	1	no	—
152	C₄×Q₈	—	37	4	no	—
153	C₄×D₄	—	37	4	no	—
154	C₄×D₄	—	74	4	no	—
155	#117	—	37	2	no	—
156	#117	—	74	2	no	—
157	C₂×C₄²	—	37	1	no	—
158	C₂×C₄²	—	74	2	no	—
159	#107	—	37	2	no	—
160	#106	—	37	2	no	—
161	#103	—	37	4	no	—
162	#102	—	37	4	no	—
163	C₄×C₈	—	37	2	no	—
164	C₄×C₈	—	37	2	no	—
165	#41	—	37	2	no	—
166	#41	—	37	2	no	—
167	#41	—	37	2	no	—
168	#41	—	74	2	no	—
169	#41	—	74	2	no	—
170	#41	C₂×#3866	1	2	yes	complemented
171	C₂×Q₈	—	37	2	no	—
172	C₂×D₄	—	37	2	no	—
173	C₂×D₄	—	74	2	no	—
174	C₂²×C₄	—	37	1	no	—
175	C₂²×C₄	—	37	2	no	—
176	C₂²×C₄	—	74	2	no	—
177	C₂²×C₄	—	74	4	no	—
178	Q₁₆	—	37	2	no	—
179	Q₁₆	—	37	1	no	—
180	Q₁₆	C₂×#3866	1	1	yes	characteristic, complemented
181	SD₁₆	—	37	2	no	—
182	SD₁₆	—	37	2	no	—
183	SD₁₆	—	37	2	no	—
184	SD₁₆	C₂×#3866	1	2	yes	complemented
185	D₈	—	37	2	no	—
186	D₈	—	37	1	no	—
187	D₈	C₂×#3866	1	1	yes	characteristic, complemented
188	C₂×C₈	—	37	1	no	—
189	C₂×C₈	—	37	1	no	—
190	C₂×C₈	—	37	1	no	—
191	C₂×C₈	—	37	1	no	—
192	C₂×C₈	—	37	1	no	—
193	C₂×C₈	—	37	2	no	—
194	C₂×C₈	—	74	1	no	—
195	C₂×C₈	—	74	1	no	—
196	C₂×C₈	C₂×#3866	1	1	yes	characteristic, complemented
197	C₄⋊C₄	—	37	4	no	—
198	C₄⋊C₄	—	74	4	no	—
199	C₂²⋊C₄	—	74	4	no	—
200	C₂²⋊C₄	—	74	2	no	—
201	C₄²	—	37	1	no	—
202	C₄²	—	37	1	no	—
203	C₄²	—	37	2	no	—
204	C₄²	—	74	2	no	—
205	C₄²	—	74	4	no	—
206	C₂³	—	37	1	no	—
207	C₂³	—	74	2	no	—
208	Q₈	—	37	2	no	—
209	Q₈	—	74	2	no	—
210	Q₈	C₂²×#3866	1	2	yes	—
211	D₄	—	2	2	no	—
212	D₄	—	37	2	no	—
213	D₄	—	74	2	no	—
214	D₄	—	74	2	no	—
215	D₄	—	74	2	no	—
216	D₄	—	74	2	no	—
217	D₄	C₂²×#3866	1	2	yes	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	37	2	no	—
220	C₂×C₄	—	37	1	no	—
221	C₂×C₄	—	37	1	no	—
222	C₂×C₄	—	37	1	no	—
223	C₂×C₄	—	37	1	no	—
224	C₂×C₄	—	37	1	no	—
225	C₂×C₄	—	37	2	no	—
226	C₂×C₄	—	74	2	no	—
227	C₂×C₄	—	74	2	no	—
228	C₂×C₄	—	74	2	no	—
229	C₂×C₄	—	74	4	no	—
230	C₂×C₄	—	74	2	no	—
231	C₂×C₄	—	74	4	no	—
232	C₂×C₄	—	148	8	no	—
233	C₂×C₄	C₂²×#3866	1	1	yes	characteristic
234	C₈	—	37	1	no	—
235	C₈	—	37	1	no	—
236	C₈	—	74	2	no	—
237	C₈	C₂²×#3866	1	1	yes	characteristic
238	C₈	C₂²×#3866	1	1	yes	characteristic
239	C₂²	—	2	2	no	—
240	C₂²	—	37	1	no	—
241	C₂²	—	37	1	no	—
242	C₂²	—	74	2	no	—
243	C₂²	—	74	2	no	—
244	C₂²	—	148	4	no	—
245	C₂²	D₄×#3866	1	1	yes	characteristic
246	C₄	—	2	2	no	—
247	C₄	—	37	4	no	—
248	C₄	—	37	1	no	—
249	C₄	—	37	1	no	—
250	C₄	—	74	2	no	—
251	C₄	—	74	2	no	—
252	C₄	—	148	4	no	—
253	C₄	D₄×#3866	1	1	yes	characteristic
254	C₄	C₂³×#3866	1	1	yes	characteristic
255	C₂	—	2	1	no	—
256	C₂	—	4	2	no	—
257	C₂	—	37	2	no	—
258	C₂	—	74	1	no	—
259	C₂	—	148	2	no	—
260	C₂	C₂×D₄×#3866	1	1	yes	characteristic

	subgroup	G/H
center	C₄	D₄×#3866
commutator	C₄×C₃₇	C₂³×C₄
Frattini	C₄	C₂³×#3866
Fitting	#135×C₃₇	C₄
socle	C₂×C₃₇	C₂×C₄×D₄
prime radical	#135×D₃₇	C₂
cosocle	C₄×D₃₇	C₂⁴
hypercenter	#135	#3866
hypocenter	C₃₇	C₄×#135
LCS	G ▷ C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ #135

Complements

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

N	H ≅ G/N	type
1. #41×#3866	C₂	semidirect
2. Q₁₆×#3866	C₂	semidirect
3. SD₁₆×#3866	C₂	semidirect
4. D₈×#3866	C₂	semidirect
5. #14558543	C₂	semidirect
6. #14558523	C₂	semidirect
7. C₂×C₈×#3866	C₂	semidirect
24. #11924559	C₄	semidirect
25. #11924558	C₄	semidirect
26. #11924557	C₄	semidirect
27. #11924556	C₄	semidirect
28. #11924555	C₄	semidirect
33. #135×C₃₇	C₄	semidirect
36. #41×C₃₇	C₂×C₄	semidirect
37. #10596966	C₂×C₄	semidirect
40. #10596964	C₂×C₄	semidirect
44. #10596962	C₂×C₄	semidirect
59. Q₁₆×C₃₇	C₂×C₄	semidirect
60. SD₁₆×C₃₇	C₂×C₄	semidirect
61. D₈×C₃₇	C₂×C₄	semidirect
62. C₂×C₈×C₃₇	C₂×C₄	semidirect
63. #10596942	C₂×C₄	semidirect
64. #10596941	C₂×C₄	semidirect
65. #10596940	C₂×C₄	semidirect
66. #10596939	C₂×C₄	semidirect
67. C₂×#64436	C₂×C₄	semidirect
68. Dic₁₄₈	C₂×C₄	semidirect
69. D₂₉₆	C₂×C₄	semidirect
70. #10596930	C₂×C₄	semidirect
71. C₈×D₃₇	C₂×C₄	semidirect
72. C₈×D₃₇	C₂×C₄	semidirect
117. #3866	#135	direct (N × H)
139. C₃₇	C₄×#135	semidirect
146. #135	#3866	direct (N × H)
170. #41	C₂×#3866	semidirect
180. Q₁₆	C₂×#3866	semidirect
184. SD₁₆	C₂×#3866	semidirect
187. D₈	C₂×#3866	semidirect
196. C₂×C₈	C₂×#3866	semidirect

Maximal subgroups

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

subgroup	index
#41×#3866	2
Q₁₆×#3866	2
SD₁₆×#3866	2
D₈×#3866	2
#14558543	2
#14558523	2
C₂×C₈×#3866	2
#135×D₃₇	2
C₄×#135	37