Subgroups — C₁₃₇×#20683 #46870069

Order 35072.

1446 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#117×C₁₃₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#407×C₁₃₇	C₂	1	2	yes	maximal
3	#1818×C₁₃₇	C₂	1	8	yes	maximal, complemented
4	#1728×C₁₃₇	C₂	1	4	yes	maximal, complemented
5	C₂⁴×C₄×C₁₃₇	C₂²	1	1	yes	characteristic
6	C₂²×M₁₆×C₁₃₇	C₂²	1	4	yes	—
7	C₂×#117×C₁₃₇	—	2	8	no	—
8	C₂×#117×C₁₃₇	C₄	1	4	yes	—
9	C₂×#117×C₁₃₇	C₂²	1	4	yes	—
10	C₂²×C₄⋊C₄×C₁₃₇	C₄	1	2	yes	—
11	C₂²×C₂²⋊C₄×C₁₃₇	C₄	1	2	yes	—
12	C₂²×C₄²×C₁₃₇	C₂²	1	2	yes	—
13	#407×C₁₃₇	—	2	8	no	—
14	C₂×C₂²⋊C₈×C₁₃₇	C₂²	1	8	yes	—
15	#336×C₁₃₇	C₂²	1	16	yes	complemented
16	C₂⁵×C₁₃₇	C₂×C₄	1	1	yes	characteristic
17	C₂³×C₄×C₁₃₇	—	2	8	no	—
18	C₂³×C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
19	C₂³×C₄×C₁₃₇	C₂×C₄	1	2	yes	—
20	C₂³×C₄×C₁₃₇	C₂×C₄	1	4	yes	—
21	C₂³×C₄×C₁₃₇	D₄	1	8	yes	—
22	C₂³×C₄×C₁₃₇	C₂³	1	2	yes	—
23	C₂³×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
24	C₂×M₁₆×C₁₃₇	—	2	8	no	—
25	C₂×M₁₆×C₁₃₇	—	2	16	no	—
26	C₂²×C₈×C₁₃₇	C₂³	1	8	yes	—
27	#117×C₁₃₇	—	2	16	no	—
28	#117×C₁₃₇	—	2	16	no	—
29	C₂×C₄⋊C₄×C₁₃₇	—	2	8	no	—
30	C₂×C₄⋊C₄×C₁₃₇	C₂×C₄	1	8	yes	—
31	C₂×C₂²⋊C₄×C₁₃₇	—	2	8	no	—
32	C₂×C₂²⋊C₄×C₁₃₇	C₂×C₄	1	8	yes	—
33	C₂×C₄²×C₁₃₇	—	2	8	no	—
34	C₂×C₄²×C₁₃₇	C₂×C₄	1	4	yes	—
35	C₂×C₄²×C₁₃₇	C₂³	1	4	yes	—
36	C₂²⋊C₈×C₁₃₇	—	2	16	no	—
37	C₂⁴×C₁₃₇	—	2	4	no	—
38	C₂⁴×C₁₃₇	—	4	4	no	—
39	C₂⁴×C₁₃₇	C₂²⋊C₄	1	4	yes	—
40	C₂⁴×C₁₃₇	C₂²×C₄	1	2	yes	—
41	C₂⁴×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
42	C₂²×C₄×C₁₃₇	—	2	4	no	—
43	C₂²×C₄×C₁₃₇	—	2	8	no	—
44	C₂²×C₄×C₁₃₇	—	2	32	no	—
45	C₂²×C₄×C₁₃₇	—	2	8	no	—
46	C₂²×C₄×C₁₃₇	—	2	8	no	—
47	C₂²×C₄×C₁₃₇	—	4	8	no	—
48	C₂²×C₄×C₁₃₇	C₂²⋊C₄	1	4	yes	—
49	C₂²×C₄×C₁₃₇	C₂²⋊C₄	1	8	yes	—
50	C₂²×C₄×C₁₃₇	C₂²×C₄	1	8	yes	—
51	C₂²×C₄×C₁₃₇	C₂²×C₄	1	2	yes	—
52	C₂²×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
53	C₂²×C₄×C₁₃₇	C₂×D₄	1	8	yes	—
54	C₂²×C₄×C₁₃₇	C₂×D₄	1	4	yes	—
55	C₂²×C₄×C₁₃₇	#41	1	8	yes	—
56	C₂²×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
57	M₁₆×C₁₃₇	—	4	16	no	—
58	C₂×C₈×C₁₃₇	—	2	8	no	—
59	C₂×C₈×C₁₃₇	—	2	16	no	—
60	C₄⋊C₄×C₁₃₇	—	2	16	no	—
61	C₂²⋊C₄×C₁₃₇	—	2	16	no	—
62	C₄²×C₁₃₇	—	2	8	no	—
63	C₄²×C₁₃₇	—	2	8	no	—
64	C₂³×C₁₃₇	—	2	4	no	—
65	C₂³×C₁₃₇	—	2	4	no	—
66	C₂³×C₁₃₇	—	2	8	no	—
67	C₂³×C₁₃₇	—	2	8	no	—
68	C₂³×C₁₃₇	—	4	8	no	—
69	C₂³×C₁₃₇	—	4	16	no	—
70	C₂³×C₁₃₇	C₂×C₂²⋊C₄	1	4	yes	—
71	C₂³×C₁₃₇	C₂×C₂²⋊C₄	1	2	yes	—
72	C₂³×C₁₃₇	#117	1	4	yes	—
73	C₂³×C₁₃₇	C₂³×C₄	1	1	yes	characteristic
74	C₂×C₄×C₁₃₇	—	2	4	no	—
75	C₂×C₄×C₁₃₇	—	2	8	no	—
76	C₂×C₄×C₁₃₇	—	2	16	no	—
77	C₂×C₄×C₁₃₇	—	2	16	no	—
78	C₂×C₄×C₁₃₇	—	2	8	no	—
79	C₂×C₄×C₁₃₇	—	2	16	no	—
80	C₂×C₄×C₁₃₇	—	4	16	no	—
81	C₂×C₄×C₁₃₇	C₂×C₂²⋊C₄	1	4	yes	—
82	C₂×C₄×C₁₃₇	C₂×C₂²⋊C₄	1	2	yes	—
83	C₂×C₄×C₁₃₇	#117	1	4	yes	—
84	C₂×C₄×C₁₃₇	#123	1	16	yes	—
85	C₂×C₄×C₁₃₇	C₂²×D₄	1	2	yes	—
86	C₂×C₄×C₁₃₇	C₂×#41	1	4	yes	—
87	C₈×C₁₃₇	—	4	8	no	—
88	C₂²×C₁₃₇	—	2	4	no	—
89	C₂²×C₁₃₇	—	2	8	no	—
90	C₂²×C₁₃₇	—	2	8	no	—
91	C₂²×C₁₃₇	—	2	4	no	—
92	C₂²×C₁₃₇	—	4	16	no	—
93	C₂²×C₁₃₇	—	4	8	no	—
94	C₂²×C₁₃₇	#381	1	4	yes	—
95	C₂²×C₁₃₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
96	C₂²×C₁₃₇	C₂×#117	1	2	yes	—
97	C₂²×C₁₃₇	#568	1	4	yes	—
98	C₄×C₁₃₇	—	2	4	no	—
99	C₄×C₁₃₇	—	4	8	no	—
100	C₄×C₁₃₇	#381	1	4	yes	—
101	C₄×C₁₃₇	C₂×#123	1	4	yes	—
102	C₂×C₁₃₇	—	2	4	no	—
103	C₂×C₁₃₇	—	4	4	no	—
104	C₂×C₁₃₇	C₂×#381	1	1	yes	characteristic
105	C₂×C₁₃₇	#2879	1	2	yes	—
106	C₂×C₁₃₇	#2921	1	4	yes	—
107	#20683	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
108	C₁₃₇	#20683	1	1	yes	characteristic, direct product factor, Sylow
109	C₂²×#117	C₂×C₁₃₇	1	1	yes	characteristic
110	C₂×#407	C₂×C₁₃₇	1	2	yes	—
111	#1818	C₂×C₁₃₇	1	8	yes	complemented
112	#1728	C₂×C₁₃₇	1	4	yes	complemented
113	C₂⁴×C₄	C₂²×C₁₃₇	1	1	yes	characteristic
114	C₂²×M₁₆	C₂²×C₁₃₇	1	4	yes	—
115	C₂×#117	—	2	8	no	—
116	C₂×#117	C₄×C₁₃₇	1	4	yes	—
117	C₂×#117	C₂²×C₁₃₇	1	4	yes	—
118	C₂²×C₄⋊C₄	C₄×C₁₃₇	1	2	yes	—
119	C₂²×C₂²⋊C₄	C₄×C₁₃₇	1	2	yes	—
120	C₂²×C₄²	C₂²×C₁₃₇	1	2	yes	—
121	#407	—	2	8	no	—
122	C₂×C₂²⋊C₈	C₂²×C₁₃₇	1	8	yes	—
123	#336	C₂²×C₁₃₇	1	16	yes	complemented
124	C₂⁵	C₂×C₄×C₁₃₇	1	1	yes	characteristic
125	C₂³×C₄	—	2	8	no	—
126	C₂³×C₄	C₂×C₄×C₁₃₇	1	1	yes	characteristic
127	C₂³×C₄	C₂×C₄×C₁₃₇	1	2	yes	—
128	C₂³×C₄	C₂×C₄×C₁₃₇	1	4	yes	—
129	C₂³×C₄	D₄×C₁₃₇	1	8	yes	—
130	C₂³×C₄	C₂³×C₁₃₇	1	2	yes	—
131	C₂³×C₄	C₂³×C₁₃₇	1	1	yes	characteristic
132	C₂×M₁₆	—	2	8	no	—
133	C₂×M₁₆	—	2	16	no	—
134	C₂²×C₈	C₂³×C₁₃₇	1	8	yes	—
135	#117	—	2	16	no	—
136	#117	—	2	16	no	—
137	C₂×C₄⋊C₄	—	2	8	no	—
138	C₂×C₄⋊C₄	C₂×C₄×C₁₃₇	1	8	yes	—
139	C₂×C₂²⋊C₄	—	2	8	no	—
140	C₂×C₂²⋊C₄	C₂×C₄×C₁₃₇	1	8	yes	—
141	C₂×C₄²	—	2	8	no	—
142	C₂×C₄²	C₂×C₄×C₁₃₇	1	4	yes	—
143	C₂×C₄²	C₂³×C₁₃₇	1	4	yes	—
144	C₂²⋊C₈	—	2	16	no	—
145	C₂⁴	—	2	4	no	—
146	C₂⁴	—	4	4	no	—
147	C₂⁴	C₂²⋊C₄×C₁₃₇	1	4	yes	—
148	C₂⁴	C₂²×C₄×C₁₃₇	1	2	yes	—
149	C₂⁴	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
150	C₂²×C₄	—	2	4	no	—
151	C₂²×C₄	—	2	8	no	—
152	C₂²×C₄	—	2	32	no	—
153	C₂²×C₄	—	2	8	no	—
154	C₂²×C₄	—	2	8	no	—
155	C₂²×C₄	—	4	8	no	—
156	C₂²×C₄	C₂²⋊C₄×C₁₃₇	1	4	yes	—
157	C₂²×C₄	C₂²⋊C₄×C₁₃₇	1	8	yes	—
158	C₂²×C₄	C₂²×C₄×C₁₃₇	1	8	yes	—
159	C₂²×C₄	C₂²×C₄×C₁₃₇	1	2	yes	—
160	C₂²×C₄	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
161	C₂²×C₄	C₂×D₄×C₁₃₇	1	8	yes	—
162	C₂²×C₄	C₂×D₄×C₁₃₇	1	4	yes	—
163	C₂²×C₄	#41×C₁₃₇	1	8	yes	—
164	C₂²×C₄	C₂⁴×C₁₃₇	1	1	yes	characteristic
165	M₁₆	—	4	16	no	—
166	C₂×C₈	—	2	8	no	—
167	C₂×C₈	—	2	16	no	—
168	C₄⋊C₄	—	2	16	no	—
169	C₂²⋊C₄	—	2	16	no	—
170	C₄²	—	2	8	no	—
171	C₄²	—	2	8	no	—
172	C₂³	—	2	4	no	—
173	C₂³	—	2	8	no	—
174	C₂³	—	2	4	no	—
175	C₂³	—	2	8	no	—
176	C₂³	—	4	8	no	—
177	C₂³	—	4	16	no	—
178	C₂³	C₂×C₂²⋊C₄×C₁₃₇	1	4	yes	—
179	C₂³	C₂×C₂²⋊C₄×C₁₃₇	1	2	yes	—
180	C₂³	#117×C₁₃₇	1	4	yes	—
181	C₂³	C₂³×C₄×C₁₃₇	1	1	yes	characteristic
182	C₂×C₄	—	2	8	no	—
183	C₂×C₄	—	2	4	no	—
184	C₂×C₄	—	2	16	no	—
185	C₂×C₄	—	2	16	no	—
186	C₂×C₄	—	2	8	no	—
187	C₂×C₄	—	2	16	no	—
188	C₂×C₄	—	4	16	no	—
189	C₂×C₄	C₂×C₂²⋊C₄×C₁₃₇	1	4	yes	—
190	C₂×C₄	C₂×C₂²⋊C₄×C₁₃₇	1	2	yes	—
191	C₂×C₄	#117×C₁₃₇	1	4	yes	—
192	C₂×C₄	#123×C₁₃₇	1	16	yes	—
193	C₂×C₄	C₂²×D₄×C₁₃₇	1	2	yes	—
194	C₂×C₄	C₂×#41×C₁₃₇	1	4	yes	—
195	C₈	—	4	8	no	—
196	C₂²	—	2	4	no	—
197	C₂²	—	2	8	no	—
198	C₂²	—	2	4	no	—
199	C₂²	—	2	8	no	—
200	C₂²	—	4	16	no	—
201	C₂²	—	4	8	no	—
202	C₂²	#381×C₁₃₇	1	4	yes	—
203	C₂²	C₂²×C₂²⋊C₄×C₁₃₇	1	1	yes	characteristic
204	C₂²	C₂×#117×C₁₃₇	1	2	yes	—
205	C₂²	#568×C₁₃₇	1	4	yes	—
206	C₄	—	2	4	no	—
207	C₄	—	4	8	no	—
208	C₄	#381×C₁₃₇	1	4	yes	—
209	C₄	C₂×#123×C₁₃₇	1	4	yes	—
210	C₂	—	2	4	no	—
211	C₂	—	4	4	no	—
212	C₂	C₂×#381×C₁₃₇	1	1	yes	characteristic
213	C₂	#2879×C₁₃₇	1	2	yes	—
214	C₂	#2921×C₁₃₇	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄×C₁₃₇	C₂⁴
commutator	C₂³	C₂³×C₄×C₁₃₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃₇
socle	C₂³×C₁₃₇	C₂³×C₄
prime radical	C₂⁵×C₁₃₇	C₂×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. #1818×C₁₃₇	C₂	semidirect
4. #1728×C₁₃₇	C₂	semidirect
15. #336×C₁₃₇	C₂²	semidirect
107. #20683	C₁₃₇	direct (N × H)
108. C₁₃₇	#20683	direct (N × H)
111. #1818	C₂×C₁₃₇	semidirect
112. #1728	C₂×C₁₃₇	semidirect
123. #336	C₂²×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#117×C₁₃₇	2
C₂×#407×C₁₃₇	2
#1818×C₁₃₇	2
#1728×C₁₃₇	2
#20683	137

Subgroups — C137×#20683 #46870069

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#20683 #46870069