Subgroups — C₁₃₇×#33947 #46883333

Order 35072.

476 conjugacy classes of subgroups.

The subgroup lattice has 220 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#3380×C₁₃₇	C₂	1	2	yes	maximal, complemented
2	#2253×C₁₃₇	C₂	1	2	yes	maximal, complemented
3	#2252×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#2228×C₁₃₇	C₂	1	4	yes	maximal, complemented
5	#2226×C₁₃₇	C₂	1	2	yes	maximal, complemented
6	Q₈×C₁₆×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×D₁₆×C₁₃₇	C₂	1	2	yes	maximal
8	C₄×D₁₆×C₁₃₇	C₂	1	1	yes	maximal, characteristic
9	#554×C₁₃₇	—	2	2	no	—
10	C₂×D₁₆×C₁₃₇	C₂²	1	1	yes	characteristic
11	#500×C₁₃₇	C₂²	1	2	yes	—
12	#499×C₁₃₇	C₂²	1	4	yes	—
13	#478×C₁₃₇	—	2	4	no	—
14	#476×C₁₃₇	—	2	2	no	—
15	C₈×Q₈×C₁₃₇	C₂²	1	1	yes	characteristic
16	C₄×D₈×C₁₃₇	C₂²	1	2	yes	—
17	C₄×D₈×C₁₃₇	C₂²	1	4	yes	—
18	#367×C₁₃₇	C₂²	1	4	yes	—
19	#366×C₁₃₇	C₂²	1	2	yes	—
20	#366×C₁₃₇	C₂²	1	1	yes	characteristic
21	#366×C₁₃₇	C₂²	1	2	yes	—
22	C₄⋊C₁₆×C₁₃₇	C₂²	1	1	yes	characteristic
23	C₄⋊C₁₆×C₁₃₇	C₂²	1	2	yes	—
24	#357×C₁₃₇	C₂²	1	4	yes	—
25	#357×C₁₃₇	C₂²	1	2	yes	—
26	C₄×C₁₆×C₁₃₇	C₂²	1	1	yes	characteristic
27	C₄×C₁₆×C₁₃₇	C₂²	1	2	yes	—
28	C₂×D₈×C₁₃₇	C₂³	1	2	yes	—
29	C₄⋊Q₈×C₁₃₇	—	2	2	no	—
30	#125×C₁₃₇	—	2	4	no	—
31	C₂²⋊Q₈×C₁₃₇	—	4	2	no	—
32	C₂²⋊Q₈×C₁₃₇	—	4	4	no	—
33	C₄×Q₈×C₁₃₇	D₄	1	1	yes	characteristic
34	C₄×D₄×C₁₃₇	—	2	2	no	—
35	C₄×D₄×C₁₃₇	—	2	4	no	—
36	C₂×C₄⋊C₄×C₁₃₇	—	4	2	no	—
37	D₁₆×C₁₃₇	Q₈	1	4	yes	complemented
38	C₂×C₁₆×C₁₃₇	C₂³	1	1	yes	characteristic
39	C₂×C₁₆×C₁₃₇	C₂³	1	2	yes	—
40	C₂×C₁₆×C₁₃₇	C₂³	1	1	yes	characteristic
41	#107×C₁₃₇	—	2	2	no	—
42	#107×C₁₃₇	C₂³	1	4	yes	—
43	#107×C₁₃₇	C₂³	1	2	yes	—
44	#106×C₁₃₇	—	2	4	no	—
45	C₄⋊C₈×C₁₃₇	D₄	1	1	yes	characteristic
46	C₄⋊C₈×C₁₃₇	D₄	1	2	yes	—
47	#102×C₁₃₇	—	2	4	no	—
48	#102×C₁₃₇	—	2	2	no	—
49	C₄×C₈×C₁₃₇	C₂³	1	2	yes	—
50	C₄×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
51	C₂×Q₈×C₁₃₇	—	4	2	no	—
52	C₂×Q₈×C₁₃₇	D₈	1	1	yes	characteristic
53	C₂×D₄×C₁₃₇	—	2	2	no	—
54	C₂²×C₄×C₁₃₇	—	4	2	no	—
55	C₂²×C₄×C₁₃₇	—	4	4	no	—
56	D₈×C₁₃₇	—	2	2	no	—
57	D₈×C₁₃₇	C₂×Q₈	1	4	yes	—
58	C₂×C₈×C₁₃₇	C₂×D₄	1	2	yes	—
59	C₂×C₈×C₁₃₇	C₂×D₄	1	1	yes	characteristic
60	C₂×C₈×C₁₃₇	C₂⁴	1	1	yes	characteristic
61	C₄⋊C₄×C₁₃₇	—	2	2	no	—
62	C₄⋊C₄×C₁₃₇	—	2	4	no	—
63	C₄⋊C₄×C₁₃₇	—	4	4	no	—
64	C₄⋊C₄×C₁₃₇	—	4	2	no	—
65	C₄⋊C₄×C₁₃₇	—	4	4	no	—
66	C₄⋊C₄×C₁₃₇	D₈	1	1	yes	characteristic
67	C₄⋊C₄×C₁₃₇	D₈	1	2	yes	—
68	C₂²⋊C₄×C₁₃₇	—	4	2	no	—
69	C₂²⋊C₄×C₁₃₇	—	4	4	no	—
70	C₄²×C₁₃₇	C₂×D₄	1	2	yes	—
71	C₄²×C₁₃₇	C₂×D₄	1	1	yes	characteristic
72	C₁₆×C₁₃₇	—	2	1	no	—
73	C₁₆×C₁₃₇	—	2	2	no	—
74	C₁₆×C₁₃₇	C₂×Q₈	1	2	yes	—
75	C₂³×C₁₃₇	—	4	2	no	—
76	Q₈×C₁₃₇	—	2	2	no	—
77	Q₈×C₁₃₇	—	16	2	no	—
78	D₄×C₁₃₇	—	2	4	no	—
79	D₄×C₁₃₇	—	4	2	no	—
80	C₂×C₄×C₁₃₇	—	4	2	no	—
81	C₂×C₄×C₁₃₇	—	4	4	no	—
82	C₂×C₄×C₁₃₇	—	8	4	no	—
83	C₂×C₄×C₁₃₇	—	16	4	no	—
84	C₂×C₄×C₁₃₇	C₂×D₈	1	2	yes	—
85	C₂×C₄×C₁₃₇	C₂×D₈	1	1	yes	characteristic
86	C₂×C₄×C₁₃₇	C₂×D₈	1	2	yes	—
87	C₂×C₄×C₁₃₇	C₂×D₈	1	1	yes	characteristic
88	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
89	C₈×C₁₃₇	—	2	2	no	—
90	C₈×C₁₃₇	—	2	1	no	—
91	C₈×C₁₃₇	C₂²×Q₈	1	1	yes	characteristic
92	C₈×C₁₃₇	2₋¹⁺⁴	1	1	yes	characteristic
93	C₂²×C₁₃₇	—	4	4	no	—
94	C₂²×C₁₃₇	—	8	2	no	—
95	C₂²×C₁₃₇	—	8	2	no	—
96	C₂²×C₁₃₇	C₂²×D₈	1	1	yes	characteristic
97	C₄×C₁₃₇	—	2	2	no	—
98	C₄×C₁₃₇	—	2	2	no	—
99	C₄×C₁₃₇	—	2	1	no	—
100	C₄×C₁₃₇	—	8	2	no	—
101	C₄×C₁₃₇	—	8	4	no	—
102	C₄×C₁₃₇	#508	1	2	yes	—
103	C₄×C₁₃₇	D₄×Q₈	1	1	yes	characteristic
104	C₄×C₁₃₇	#578	1	1	yes	characteristic
105	C₂×C₁₃₇	—	8	4	no	—
106	C₂×C₁₃₇	Q₈×D₈	1	1	yes	characteristic
107	C₂×C₁₃₇	C₂×#508	1	1	yes	characteristic
108	C₂×C₁₃₇	#3415	1	1	yes	characteristic
109	#33947	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
110	C₁₃₇	#33947	1	1	yes	characteristic, direct product factor, Sylow
111	#3380	C₂×C₁₃₇	1	2	yes	complemented
112	#2253	C₂×C₁₃₇	1	2	yes	complemented
113	#2252	C₂×C₁₃₇	1	1	yes	characteristic, complemented
114	#2228	C₂×C₁₃₇	1	4	yes	complemented
115	#2226	C₂×C₁₃₇	1	2	yes	complemented
116	Q₈×C₁₆	C₂×C₁₃₇	1	1	yes	characteristic, complemented
117	C₄×D₁₆	C₂×C₁₃₇	1	1	yes	characteristic
118	C₄×D₁₆	C₂×C₁₃₇	1	2	yes	—
119	#554	—	2	2	no	—
120	C₂×D₁₆	C₂²×C₁₃₇	1	1	yes	characteristic
121	#500	C₂²×C₁₃₇	1	2	yes	—
122	#499	C₂²×C₁₃₇	1	4	yes	—
123	#478	—	2	4	no	—
124	#476	—	2	2	no	—
125	C₈×Q₈	C₂²×C₁₃₇	1	1	yes	characteristic
126	C₄×D₈	C₂²×C₁₃₇	1	2	yes	—
127	C₄×D₈	C₂²×C₁₃₇	1	4	yes	—
128	#367	C₂²×C₁₃₇	1	4	yes	—
129	#366	C₂²×C₁₃₇	1	2	yes	—
130	#366	C₂²×C₁₃₇	1	1	yes	characteristic
131	#366	C₂²×C₁₃₇	1	2	yes	—
132	C₄⋊C₁₆	C₂²×C₁₃₇	1	1	yes	characteristic
133	C₄⋊C₁₆	C₂²×C₁₃₇	1	2	yes	—
134	#357	C₂²×C₁₃₇	1	4	yes	—
135	#357	C₂²×C₁₃₇	1	2	yes	—
136	C₄×C₁₆	C₂²×C₁₃₇	1	1	yes	characteristic
137	C₄×C₁₆	C₂²×C₁₃₇	1	2	yes	—
138	C₂×D₈	C₂³×C₁₃₇	1	2	yes	—
139	C₄⋊Q₈	—	2	2	no	—
140	#125	—	2	4	no	—
141	C₂²⋊Q₈	—	4	2	no	—
142	C₂²⋊Q₈	—	4	4	no	—
143	C₄×Q₈	D₄×C₁₃₇	1	1	yes	characteristic
144	C₄×D₄	—	2	2	no	—
145	C₄×D₄	—	2	4	no	—
146	C₂×C₄⋊C₄	—	4	2	no	—
147	D₁₆	Q₈×C₁₃₇	1	4	yes	complemented
148	C₂×C₁₆	C₂³×C₁₃₇	1	1	yes	characteristic
149	C₂×C₁₆	C₂³×C₁₃₇	1	2	yes	—
150	C₂×C₁₆	C₂³×C₁₃₇	1	1	yes	characteristic
151	#107	—	2	2	no	—
152	#107	C₂³×C₁₃₇	1	4	yes	—
153	#107	C₂³×C₁₃₇	1	2	yes	—
154	#106	—	2	4	no	—
155	C₄⋊C₈	D₄×C₁₃₇	1	1	yes	characteristic
156	C₄⋊C₈	D₄×C₁₃₇	1	2	yes	—
157	#102	—	2	4	no	—
158	#102	—	2	2	no	—
159	C₄×C₈	C₂³×C₁₃₇	1	2	yes	—
160	C₄×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
161	C₂×Q₈	—	4	2	no	—
162	C₂×Q₈	D₈×C₁₃₇	1	1	yes	characteristic
163	C₂×D₄	—	2	2	no	—
164	C₂²×C₄	—	4	2	no	—
165	C₂²×C₄	—	4	4	no	—
166	D₈	—	2	2	no	—
167	D₈	C₂×Q₈×C₁₃₇	1	4	yes	—
168	C₂×C₈	C₂×D₄×C₁₃₇	1	2	yes	—
169	C₂×C₈	C₂×D₄×C₁₃₇	1	1	yes	characteristic
170	C₂×C₈	C₂⁴×C₁₃₇	1	1	yes	characteristic
171	C₄⋊C₄	—	2	2	no	—
172	C₄⋊C₄	—	2	4	no	—
173	C₄⋊C₄	—	4	4	no	—
174	C₄⋊C₄	—	4	2	no	—
175	C₄⋊C₄	—	4	4	no	—
176	C₄⋊C₄	D₈×C₁₃₇	1	1	yes	characteristic
177	C₄⋊C₄	D₈×C₁₃₇	1	2	yes	—
178	C₂²⋊C₄	—	4	2	no	—
179	C₂²⋊C₄	—	4	4	no	—
180	C₄²	C₂×D₄×C₁₃₇	1	2	yes	—
181	C₄²	C₂×D₄×C₁₃₇	1	1	yes	characteristic
182	C₁₆	—	2	1	no	—
183	C₁₆	—	2	2	no	—
184	C₁₆	C₂×Q₈×C₁₃₇	1	2	yes	—
185	C₂³	—	4	2	no	—
186	Q₈	—	2	2	no	—
187	Q₈	—	16	2	no	—
188	D₄	—	2	4	no	—
189	D₄	—	4	2	no	—
190	C₂×C₄	—	4	2	no	—
191	C₂×C₄	—	4	4	no	—
192	C₂×C₄	—	8	4	no	—
193	C₂×C₄	—	16	4	no	—
194	C₂×C₄	C₂×D₈×C₁₃₇	1	2	yes	—
195	C₂×C₄	C₂×D₈×C₁₃₇	1	1	yes	characteristic
196	C₂×C₄	C₂×D₈×C₁₃₇	1	2	yes	—
197	C₂×C₄	C₂×D₈×C₁₃₇	1	1	yes	characteristic
198	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
199	C₈	—	2	2	no	—
200	C₈	—	2	1	no	—
201	C₈	C₂²×Q₈×C₁₃₇	1	1	yes	characteristic
202	C₈	2₋¹⁺⁴×C₁₃₇	1	1	yes	characteristic
203	C₂²	—	4	4	no	—
204	C₂²	—	8	2	no	—
205	C₂²	—	8	2	no	—
206	C₂²	C₂²×D₈×C₁₃₇	1	1	yes	characteristic
207	C₄	—	2	2	no	—
208	C₄	—	2	2	no	—
209	C₄	—	2	1	no	—
210	C₄	—	8	2	no	—
211	C₄	—	8	4	no	—
212	C₄	#508×C₁₃₇	1	2	yes	—
213	C₄	D₄×Q₈×C₁₃₇	1	1	yes	characteristic
214	C₄	#578×C₁₃₇	1	1	yes	characteristic
215	C₂	—	8	4	no	—
216	C₂	Q₈×D₈×C₁₃₇	1	1	yes	characteristic
217	C₂	C₂×#508×C₁₃₇	1	1	yes	characteristic
218	C₂	#3415×C₁₃₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃₇	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₁₃₇
prime radical	C₂×D₁₆×C₁₃₇	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃₇ ◁ C₄×Q₈×C₁₃₇ ◁ C₈×Q₈×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. #3380×C₁₃₇	C₂	semidirect
2. #2253×C₁₃₇	C₂	semidirect
3. #2252×C₁₃₇	C₂	semidirect
4. #2228×C₁₃₇	C₂	semidirect
5. #2226×C₁₃₇	C₂	semidirect
6. Q₈×C₁₆×C₁₃₇	C₂	semidirect
37. D₁₆×C₁₃₇	Q₈	semidirect
109. #33947	C₁₃₇	direct (N × H)
110. C₁₃₇	#33947	direct (N × H)
111. #3380	C₂×C₁₃₇	semidirect
112. #2253	C₂×C₁₃₇	semidirect
113. #2252	C₂×C₁₃₇	semidirect
114. #2228	C₂×C₁₃₇	semidirect
115. #2226	C₂×C₁₃₇	semidirect
116. Q₈×C₁₆	C₂×C₁₃₇	semidirect
147. D₁₆	Q₈×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
#3380×C₁₃₇	2
#2253×C₁₃₇	2
#2252×C₁₃₇	2
#2228×C₁₃₇	2
#2226×C₁₃₇	2
Q₈×C₁₆×C₁₃₇	2
C₄×D₁₆×C₁₃₇	2
#33947	137

Subgroups — C137×#33947 #46883333

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#33947 #46883333