Subgroups — C₁₃₇×#13622 #46863008

Order 35072.

294 conjugacy classes of subgroups.

The subgroup lattice has 226 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#508×C₁₃₇	C₂	1	1	yes	maximal, characteristic
2	C₂×M₆₄×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
3	#2158×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#1418×C₁₃₇	C₂	1	4	yes	maximal, complemented
5	C₂×#135×C₁₃₇	—	2	1	no	—
6	#508×C₁₃₇	—	2	1	no	—
7	#508×C₁₃₇	—	2	1	no	—
8	#508×C₁₃₇	C₄	1	2	yes	complemented
9	#508×C₁₃₇	C₂²	1	2	yes	—
10	C₂×Q₃₂×C₁₃₇	C₄	1	1	yes	characteristic, complemented
11	C₂×SD₃₂×C₁₃₇	—	2	1	no	—
12	C₂×D₁₆×C₁₃₇	C₄	1	1	yes	characteristic, complemented
13	C₂²×C₁₆×C₁₃₇	C₂²	1	1	yes	characteristic
14	#427×C₁₃₇	—	2	1	no	—
15	M₆₄×C₁₃₇	—	2	1	no	—
16	M₆₄×C₁₃₇	C₂²	1	2	yes	complemented
17	C₂×C₃₂×C₁₃₇	—	2	1	no	—
18	#367×C₁₃₇	C₂²	1	2	yes	complemented
19	#366×C₁₃₇	—	2	1	no	—
20	C₂×#41×C₁₃₇	—	4	1	no	—
21	#135×C₁₃₇	—	2	2	no	—
22	#135×C₁₃₇	—	2	2	no	—
23	#135×C₁₃₇	—	4	1	no	—
24	#135×C₁₃₇	—	4	1	no	—
25	C₂×Q₁₆×C₁₃₇	—	2	1	no	—
26	C₂×SD₁₆×C₁₃₇	—	4	1	no	—
27	C₂×D₈×C₁₃₇	—	2	1	no	—
28	C₂²×C₈×C₁₃₇	D₄	1	1	yes	characteristic
29	#117×C₁₃₇	—	4	1	no	—
30	Q₃₂×C₁₃₇	—	2	1	no	—
31	Q₃₂×C₁₃₇	C₂×C₄	1	2	yes	—
32	SD₃₂×C₁₃₇	—	2	2	no	—
33	SD₃₂×C₁₃₇	—	2	2	no	—
34	D₁₆×C₁₃₇	—	2	1	no	—
35	D₁₆×C₁₃₇	C₂×C₄	1	2	yes	—
36	C₂×C₁₆×C₁₃₇	C₂×C₄	1	1	yes	characteristic
37	C₂×C₁₆×C₁₃₇	C₂×C₄	1	1	yes	characteristic
38	C₂×C₁₆×C₁₃₇	D₄	1	1	yes	characteristic
39	C₂×C₁₆×C₁₃₇	D₄	1	1	yes	characteristic
40	C₂×C₁₆×C₁₃₇	D₄	1	1	yes	characteristic
41	C₂×C₁₆×C₁₃₇	C₂³	1	1	yes	characteristic
42	#107×C₁₃₇	—	2	2	no	—
43	#106×C₁₃₇	—	4	1	no	—
44	C₃₂×C₁₃₇	—	2	2	no	—
45	#41×C₁₃₇	—	4	2	no	—
46	#41×C₁₃₇	—	4	2	no	—
47	#41×C₁₃₇	—	8	1	no	—
48	#41×C₁₃₇	—	8	1	no	—
49	C₂×Q₈×C₁₃₇	—	4	1	no	—
50	C₂×D₄×C₁₃₇	—	4	1	no	—
51	C₂×D₄×C₁₃₇	—	8	1	no	—
52	C₂²×C₄×C₁₃₇	—	8	1	no	—
53	C₂²×C₄×C₁₃₇	D₈	1	1	yes	characteristic
54	Q₁₆×C₁₃₇	—	2	2	no	—
55	Q₁₆×C₁₃₇	—	4	1	no	—
56	SD₁₆×C₁₃₇	—	4	2	no	—
57	SD₁₆×C₁₃₇	—	4	2	no	—
58	D₈×C₁₃₇	—	2	2	no	—
59	D₈×C₁₃₇	—	4	1	no	—
60	C₂×C₈×C₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
61	C₂×C₈×C₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
62	C₂×C₈×C₁₃₇	D₈	1	1	yes	characteristic
63	C₂×C₈×C₁₃₇	SD₁₆	1	1	yes	characteristic
64	C₂×C₈×C₁₃₇	SD₁₆	1	1	yes	characteristic
65	C₂×C₈×C₁₃₇	C₂×D₄	1	1	yes	characteristic
66	C₄⋊C₄×C₁₃₇	—	4	2	no	—
67	C₂²⋊C₄×C₁₃₇	—	8	1	no	—
68	C₄²×C₁₃₇	—	8	1	no	—
69	C₁₆×C₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
70	C₁₆×C₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
71	C₁₆×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
72	C₁₆×C₁₃₇	C₂×D₄	1	1	yes	characteristic
73	C₂³×C₁₃₇	—	8	1	no	—
74	C₂³×C₁₃₇	D₁₆	1	1	yes	characteristic
75	Q₈×C₁₃₇	—	4	2	no	—
76	Q₈×C₁₃₇	—	8	1	no	—
77	D₄×C₁₃₇	—	4	2	no	—
78	D₄×C₁₃₇	—	8	2	no	—
79	D₄×C₁₃₇	—	8	2	no	—
80	D₄×C₁₃₇	—	8	1	no	—
81	C₂×C₄×C₁₃₇	—	8	1	no	—
82	C₂×C₄×C₁₃₇	—	8	2	no	—
83	C₂×C₄×C₁₃₇	—	8	2	no	—
84	C₂×C₄×C₁₃₇	—	8	2	no	—
85	C₂×C₄×C₁₃₇	#102	1	1	yes	characteristic
86	C₂×C₄×C₁₃₇	#102	1	1	yes	characteristic
87	C₂×C₄×C₁₃₇	D₁₆	1	1	yes	characteristic
88	C₂×C₄×C₁₃₇	SD₃₂	1	1	yes	characteristic
89	C₂×C₄×C₁₃₇	SD₃₂	1	1	yes	characteristic
90	C₂×C₄×C₁₃₇	C₂×D₈	1	1	yes	characteristic
91	C₈×C₁₃₇	#102	1	1	yes	characteristic
92	C₈×C₁₃₇	#102	1	1	yes	characteristic
93	C₈×C₁₃₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
94	C₈×C₁₃₇	C₂×SD₁₆	1	1	yes	characteristic
95	C₂²×C₁₃₇	—	4	1	no	—
96	C₂²×C₁₃₇	—	8	2	no	—
97	C₂²×C₁₃₇	—	16	2	no	—
98	C₂²×C₁₃₇	#357	1	1	yes	characteristic
99	C₂²×C₁₃₇	#357	1	1	yes	characteristic
100	C₂²×C₁₃₇	C₂×D₁₆	1	1	yes	characteristic
101	C₄×C₁₃₇	—	8	2	no	—
102	C₄×C₁₃₇	—	16	2	no	—
103	C₄×C₁₃₇	#357	1	1	yes	characteristic
104	C₄×C₁₃₇	#357	1	1	yes	characteristic
105	C₄×C₁₃₇	C₂×#102	1	1	yes	characteristic
106	C₄×C₁₃₇	C₂×SD₃₂	1	1	yes	characteristic
107	C₂×C₁₃₇	—	2	1	no	—
108	C₂×C₁₃₇	—	2	1	no	—
109	C₂×C₁₃₇	—	2	1	no	—
110	C₂×C₁₃₇	—	16	2	no	—
111	C₂×C₁₃₇	C₂×#357	1	1	yes	characteristic
112	#13622	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
113	C₁₃₇	#13622	1	1	yes	characteristic, direct product factor, Sylow
114	C₂×#508	C₂×C₁₃₇	1	1	yes	characteristic
115	C₂×M₆₄	C₂×C₁₃₇	1	1	yes	characteristic, complemented
116	#2158	C₂×C₁₃₇	1	1	yes	characteristic, complemented
117	#1418	C₂×C₁₃₇	1	4	yes	complemented
118	C₂×#135	—	2	1	no	—
119	#508	—	2	1	no	—
120	#508	—	2	1	no	—
121	#508	C₄×C₁₃₇	1	2	yes	complemented
122	#508	C₂²×C₁₃₇	1	2	yes	—
123	C₂×Q₃₂	C₄×C₁₃₇	1	1	yes	characteristic, complemented
124	C₂×SD₃₂	—	2	1	no	—
125	C₂×D₁₆	C₄×C₁₃₇	1	1	yes	characteristic, complemented
126	C₂²×C₁₆	C₂²×C₁₃₇	1	1	yes	characteristic
127	#427	—	2	1	no	—
128	M₆₄	—	2	1	no	—
129	M₆₄	C₂²×C₁₃₇	1	2	yes	complemented
130	C₂×C₃₂	—	2	1	no	—
131	#367	C₂²×C₁₃₇	1	2	yes	complemented
132	#366	—	2	1	no	—
133	C₂×#41	—	4	1	no	—
134	#135	—	2	2	no	—
135	#135	—	2	2	no	—
136	#135	—	4	1	no	—
137	#135	—	4	1	no	—
138	C₂×Q₁₆	—	2	1	no	—
139	C₂×SD₁₆	—	4	1	no	—
140	C₂×D₈	—	2	1	no	—
141	C₂²×C₈	D₄×C₁₃₇	1	1	yes	characteristic
142	#117	—	4	1	no	—
143	Q₃₂	—	2	1	no	—
144	Q₃₂	C₂×C₄×C₁₃₇	1	2	yes	—
145	SD₃₂	—	2	2	no	—
146	SD₃₂	—	2	2	no	—
147	D₁₆	—	2	1	no	—
148	D₁₆	C₂×C₄×C₁₃₇	1	2	yes	—
149	C₂×C₁₆	C₂×C₄×C₁₃₇	1	1	yes	characteristic
150	C₂×C₁₆	C₂×C₄×C₁₃₇	1	1	yes	characteristic
151	C₂×C₁₆	D₄×C₁₃₇	1	1	yes	characteristic
152	C₂×C₁₆	D₄×C₁₃₇	1	1	yes	characteristic
153	C₂×C₁₆	D₄×C₁₃₇	1	1	yes	characteristic
154	C₂×C₁₆	C₂³×C₁₃₇	1	1	yes	characteristic
155	#107	—	2	2	no	—
156	#106	—	4	1	no	—
157	C₃₂	—	2	2	no	—
158	#41	—	4	2	no	—
159	#41	—	4	2	no	—
160	#41	—	8	1	no	—
161	#41	—	8	1	no	—
162	C₂×Q₈	—	4	1	no	—
163	C₂×D₄	—	4	1	no	—
164	C₂×D₄	—	8	1	no	—
165	C₂²×C₄	—	8	1	no	—
166	C₂²×C₄	D₈×C₁₃₇	1	1	yes	characteristic
167	Q₁₆	—	2	2	no	—
168	Q₁₆	—	4	1	no	—
169	SD₁₆	—	4	2	no	—
170	SD₁₆	—	4	2	no	—
171	D₈	—	2	2	no	—
172	D₈	—	4	1	no	—
173	C₂×C₈	C₂²⋊C₄×C₁₃₇	1	1	yes	characteristic
174	C₂×C₈	C₂²⋊C₄×C₁₃₇	1	1	yes	characteristic
175	C₂×C₈	D₈×C₁₃₇	1	1	yes	characteristic
176	C₂×C₈	SD₁₆×C₁₃₇	1	1	yes	characteristic
177	C₂×C₈	SD₁₆×C₁₃₇	1	1	yes	characteristic
178	C₂×C₈	C₂×D₄×C₁₃₇	1	1	yes	characteristic
179	C₄⋊C₄	—	4	2	no	—
180	C₂²⋊C₄	—	8	1	no	—
181	C₄²	—	8	1	no	—
182	C₁₆	C₂²⋊C₄×C₁₃₇	1	1	yes	characteristic
183	C₁₆	C₂²⋊C₄×C₁₃₇	1	1	yes	characteristic
184	C₁₆	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
185	C₁₆	C₂×D₄×C₁₃₇	1	1	yes	characteristic
186	C₂³	—	8	1	no	—
187	C₂³	D₁₆×C₁₃₇	1	1	yes	characteristic
188	Q₈	—	4	2	no	—
189	Q₈	—	8	1	no	—
190	D₄	—	4	2	no	—
191	D₄	—	8	2	no	—
192	D₄	—	8	2	no	—
193	D₄	—	8	1	no	—
194	C₂×C₄	—	8	1	no	—
195	C₂×C₄	—	8	2	no	—
196	C₂×C₄	—	8	2	no	—
197	C₂×C₄	—	8	2	no	—
198	C₂×C₄	#102×C₁₃₇	1	1	yes	characteristic
199	C₂×C₄	#102×C₁₃₇	1	1	yes	characteristic
200	C₂×C₄	D₁₆×C₁₃₇	1	1	yes	characteristic
201	C₂×C₄	SD₃₂×C₁₃₇	1	1	yes	characteristic
202	C₂×C₄	SD₃₂×C₁₃₇	1	1	yes	characteristic
203	C₂×C₄	C₂×D₈×C₁₃₇	1	1	yes	characteristic
204	C₈	#102×C₁₃₇	1	1	yes	characteristic
205	C₈	#102×C₁₃₇	1	1	yes	characteristic
206	C₈	C₂×C₂²⋊C₄×C₁₃₇	1	1	yes	characteristic
207	C₈	C₂×SD₁₆×C₁₃₇	1	1	yes	characteristic
208	C₂²	—	4	1	no	—
209	C₂²	—	8	2	no	—
210	C₂²	—	16	2	no	—
211	C₂²	#357×C₁₃₇	1	1	yes	characteristic
212	C₂²	#357×C₁₃₇	1	1	yes	characteristic
213	C₂²	C₂×D₁₆×C₁₃₇	1	1	yes	characteristic
214	C₄	—	8	2	no	—
215	C₄	—	16	2	no	—
216	C₄	#357×C₁₃₇	1	1	yes	characteristic
217	C₄	#357×C₁₃₇	1	1	yes	characteristic
218	C₄	C₂×#102×C₁₃₇	1	1	yes	characteristic
219	C₄	C₂×SD₃₂×C₁₃₇	1	1	yes	characteristic
220	C₂	—	2	1	no	—
221	C₂	—	2	1	no	—
222	C₂	—	2	1	no	—
223	C₂	—	16	2	no	—
224	C₂	C₂×#357×C₁₃₇	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₁₃₇	#357
commutator	C₁₆	C₂²×C₄×C₁₃₇
Frattini, cosocle	C₂×C₁₆	C₂³×C₁₃₇
socle	C₂×C₁₃₇	C₂×#357
prime radical	C₂×#508×C₁₃₇	C₂
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₁₃₇ ◁ C₂²×C₄×C₁₃₇ ◁ C₂²×C₈×C₁₃₇ ◁ C₂²×C₁₆×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×M₆₄×C₁₃₇	C₂	semidirect
3. #2158×C₁₃₇	C₂	semidirect
4. #1418×C₁₃₇	C₂	semidirect
8. #508×C₁₃₇	C₄	semidirect
10. C₂×Q₃₂×C₁₃₇	C₄	semidirect
12. C₂×D₁₆×C₁₃₇	C₄	semidirect
16. M₆₄×C₁₃₇	C₂²	semidirect
18. #367×C₁₃₇	C₂²	semidirect
112. #13622	C₁₃₇	direct (N × H)
113. C₁₃₇	#13622	direct (N × H)
115. C₂×M₆₄	C₂×C₁₃₇	semidirect
116. #2158	C₂×C₁₃₇	semidirect
117. #1418	C₂×C₁₃₇	semidirect
121. #508	C₄×C₁₃₇	semidirect
123. C₂×Q₃₂	C₄×C₁₃₇	semidirect
125. C₂×D₁₆	C₄×C₁₃₇	semidirect
129. M₆₄	C₂²×C₁₃₇	semidirect
131. #367	C₂²×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#508×C₁₃₇	2
C₂×M₆₄×C₁₃₇	2
#2158×C₁₃₇	2
#1418×C₁₃₇	2
#13622	137

Subgroups — C137×#13622 #46863008

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#13622 #46863008