Subgroups — C₁₃₇×#33909 #46883295

Order 35072.

468 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	#3342×C₁₃₇	C₂	1	2	yes	maximal
2	#2252×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
3	#2237×C₁₃₇	C₂	1	4	yes	maximal, complemented
4	#2225×C₁₃₇	C₂	1	2	yes	maximal, complemented
5	#2216×C₁₃₇	C₂	1	2	yes	maximal, complemented
6	C₄×Q₃₂×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
7	D₄×C₁₆×C₁₃₇	C₂	1	1	yes	maximal, characteristic
8	#2158×C₁₃₇	C₂	1	2	yes	maximal, complemented
9	#556×C₁₃₇	—	2	2	no	—
10	C₂×Q₃₂×C₁₃₇	C₂²	1	1	yes	characteristic
11	C₂²×C₁₆×C₁₃₇	C₂²	1	2	yes	—
12	#500×C₁₃₇	C₂²	1	2	yes	—
13	#485×C₁₃₇	—	2	4	no	—
14	#475×C₁₃₇	—	2	2	no	—
15	#467×C₁₃₇	C₂²	1	4	yes	—
16	C₄×Q₁₆×C₁₃₇	C₂²	1	2	yes	—
17	C₈×D₄×C₁₃₇	C₂²	1	1	yes	characteristic
18	#427×C₁₃₇	C₂²	1	4	yes	—
19	#367×C₁₃₇	C₂²	1	4	yes	—
20	#366×C₁₃₇	C₂²	1	1	yes	characteristic
21	#366×C₁₃₇	C₂²	1	2	yes	—
22	#366×C₁₃₇	C₂²	1	2	yes	—
23	C₄⋊C₁₆×C₁₃₇	C₂²	1	1	yes	characteristic
24	#358×C₁₃₇	C₂²	1	2	yes	—
25	#358×C₁₃₇	C₂²	1	4	yes	—
26	C₂²⋊C₁₆×C₁₃₇	C₂²	1	2	yes	—
27	C₄×C₁₆×C₁₃₇	C₂²	1	1	yes	characteristic
28	C₂×Q₁₆×C₁₃₇	C₂³	1	2	yes	—
29	C₂²×C₈×C₁₃₇	C₂³	1	2	yes	—
30	C₄⋊Q₈×C₁₃₇	—	2	2	no	—
31	#126×C₁₃₇	—	4	4	no	—
32	#124×C₁₃₇	—	4	2	no	—
33	C₂²⋊Q₈×C₁₃₇	—	2	4	no	—
34	C₄×Q₈×C₁₃₇	—	2	2	no	—
35	C₄×Q₈×C₁₃₇	—	4	2	no	—
36	C₄×D₄×C₁₃₇	D₄	1	1	yes	characteristic
37	#117×C₁₃₇	—	2	4	no	—
38	Q₃₂×C₁₃₇	—	2	2	no	—
39	C₂×C₁₆×C₁₃₇	—	2	4	no	—
40	C₂×C₁₆×C₁₃₇	C₂³	1	2	yes	—
41	C₂×C₁₆×C₁₃₇	C₂³	1	1	yes	characteristic
42	C₂×C₁₆×C₁₃₇	C₂³	1	1	yes	characteristic
43	#107×C₁₃₇	—	2	2	no	—
44	#107×C₁₃₇	C₂³	1	4	yes	—
45	#107×C₁₃₇	C₂³	1	2	yes	—
46	#106×C₁₃₇	—	2	4	no	—
47	C₄⋊C₈×C₁₃₇	D₄	1	1	yes	characteristic
48	#103×C₁₃₇	—	2	2	no	—
49	#103×C₁₃₇	—	2	4	no	—
50	C₂²⋊C₈×C₁₃₇	D₄	1	2	yes	—
51	C₄×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
52	C₂×Q₈×C₁₃₇	—	2	2	no	—
53	C₂×Q₈×C₁₃₇	—	4	2	no	—
54	C₂×D₄×C₁₃₇	D₈	1	1	yes	characteristic
55	C₂²×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
56	Q₁₆×C₁₃₇	—	2	2	no	—
57	Q₁₆×C₁₃₇	—	2	2	no	—
58	C₂×C₈×C₁₃₇	—	2	2	no	—
59	C₂×C₈×C₁₃₇	—	2	2	no	—
60	C₂×C₈×C₁₃₇	C₂×D₄	1	1	yes	characteristic
61	C₂×C₈×C₁₃₇	C₂×D₄	1	2	yes	—
62	C₂×C₈×C₁₃₇	C₂⁴	1	1	yes	characteristic
63	C₄⋊C₄×C₁₃₇	—	2	2	no	—
64	C₄⋊C₄×C₁₃₇	—	2	4	no	—
65	C₄⋊C₄×C₁₃₇	—	4	2	no	—
66	C₄⋊C₄×C₁₃₇	—	4	4	no	—
67	C₄⋊C₄×C₁₃₇	—	4	2	no	—
68	C₄⋊C₄×C₁₃₇	D₈	1	1	yes	characteristic
69	C₂²⋊C₄×C₁₃₇	—	4	4	no	—
70	C₂²⋊C₄×C₁₃₇	—	4	4	no	—
71	C₂²⋊C₄×C₁₃₇	D₈	1	2	yes	—
72	C₄²×C₁₃₇	—	4	4	no	—
73	C₄²×C₁₃₇	—	4	2	no	—
74	C₄²×C₁₃₇	C₂×D₄	1	1	yes	characteristic
75	C₁₆×C₁₃₇	—	2	2	no	—
76	C₁₆×C₁₃₇	—	2	1	no	—
77	C₁₆×C₁₃₇	#41	1	2	yes	—
78	C₂³×C₁₃₇	C₂×D₈	1	2	yes	—
79	Q₈×C₁₃₇	—	4	2	no	—
80	Q₈×C₁₃₇	—	4	2	no	—
81	Q₈×C₁₃₇	—	8	4	no	—
82	D₄×C₁₃₇	—	2	2	no	—
83	C₂×C₄×C₁₃₇	—	2	2	no	—
84	C₂×C₄×C₁₃₇	—	2	2	no	—
85	C₂×C₄×C₁₃₇	—	4	2	no	—
86	C₂×C₄×C₁₃₇	—	4	4	no	—
87	C₂×C₄×C₁₃₇	—	4	2	no	—
88	C₂×C₄×C₁₃₇	C₂×D₈	1	2	yes	—
89	C₂×C₄×C₁₃₇	C₂×D₈	1	1	yes	characteristic
90	C₂×C₄×C₁₃₇	C₂×D₈	1	1	yes	characteristic
91	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
92	C₈×C₁₃₇	—	2	2	no	—
93	C₈×C₁₃₇	—	2	1	no	—
94	C₈×C₁₃₇	C₂×#41	1	1	yes	characteristic
95	C₈×C₁₃₇	2₋¹⁺⁴	1	1	yes	characteristic
96	C₂²×C₁₃₇	—	2	2	no	—
97	C₂²×C₁₃₇	—	2	2	no	—
98	C₂²×C₁₃₇	—	2	2	no	—
99	C₂²×C₁₃₇	C₂²×D₈	1	1	yes	characteristic
100	C₄×C₁₃₇	—	2	2	no	—
101	C₄×C₁₃₇	—	2	1	no	—
102	C₄×C₁₃₇	—	8	2	no	—
103	C₄×C₁₃₇	—	8	4	no	—
104	C₄×C₁₃₇	—	8	2	no	—
105	C₄×C₁₃₇	#508	1	2	yes	—
106	C₄×C₁₃₇	#547	1	1	yes	characteristic
107	C₄×C₁₃₇	#578	1	1	yes	characteristic
108	C₂×C₁₃₇	—	4	2	no	—
109	C₂×C₁₃₇	#3334	1	1	yes	characteristic
110	C₂×C₁₃₇	C₂×#508	1	1	yes	characteristic
111	C₂×C₁₃₇	#3417	1	1	yes	characteristic
112	#33909	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
113	C₁₃₇	#33909	1	1	yes	characteristic, direct product factor, Sylow
114	#3342	C₂×C₁₃₇	1	2	yes	—
115	#2252	C₂×C₁₃₇	1	1	yes	characteristic, complemented
116	#2237	C₂×C₁₃₇	1	4	yes	complemented
117	#2225	C₂×C₁₃₇	1	2	yes	complemented
118	#2216	C₂×C₁₃₇	1	2	yes	complemented
119	C₄×Q₃₂	C₂×C₁₃₇	1	1	yes	characteristic, complemented
120	D₄×C₁₆	C₂×C₁₃₇	1	1	yes	characteristic
121	#2158	C₂×C₁₃₇	1	2	yes	complemented
122	#556	—	2	2	no	—
123	C₂×Q₃₂	C₂²×C₁₃₇	1	1	yes	characteristic
124	C₂²×C₁₆	C₂²×C₁₃₇	1	2	yes	—
125	#500	C₂²×C₁₃₇	1	2	yes	—
126	#485	—	2	4	no	—
127	#475	—	2	2	no	—
128	#467	C₂²×C₁₃₇	1	4	yes	—
129	C₄×Q₁₆	C₂²×C₁₃₇	1	2	yes	—
130	C₈×D₄	C₂²×C₁₃₇	1	1	yes	characteristic
131	#427	C₂²×C₁₃₇	1	4	yes	—
132	#367	C₂²×C₁₃₇	1	4	yes	—
133	#366	C₂²×C₁₃₇	1	1	yes	characteristic
134	#366	C₂²×C₁₃₇	1	2	yes	—
135	#366	C₂²×C₁₃₇	1	2	yes	—
136	C₄⋊C₁₆	C₂²×C₁₃₇	1	1	yes	characteristic
137	#358	C₂²×C₁₃₇	1	2	yes	—
138	#358	C₂²×C₁₃₇	1	4	yes	—
139	C₂²⋊C₁₆	C₂²×C₁₃₇	1	2	yes	—
140	C₄×C₁₆	C₂²×C₁₃₇	1	1	yes	characteristic
141	C₂×Q₁₆	C₂³×C₁₃₇	1	2	yes	—
142	C₂²×C₈	C₂³×C₁₃₇	1	2	yes	—
143	C₄⋊Q₈	—	2	2	no	—
144	#126	—	4	4	no	—
145	#124	—	4	2	no	—
146	C₂²⋊Q₈	—	2	4	no	—
147	C₄×Q₈	—	2	2	no	—
148	C₄×Q₈	—	4	2	no	—
149	C₄×D₄	D₄×C₁₃₇	1	1	yes	characteristic
150	#117	—	2	4	no	—
151	Q₃₂	—	2	2	no	—
152	C₂×C₁₆	—	2	4	no	—
153	C₂×C₁₆	C₂³×C₁₃₇	1	1	yes	characteristic
154	C₂×C₁₆	C₂³×C₁₃₇	1	2	yes	—
155	C₂×C₁₆	C₂³×C₁₃₇	1	1	yes	characteristic
156	#107	—	2	2	no	—
157	#107	C₂³×C₁₃₇	1	4	yes	—
158	#107	C₂³×C₁₃₇	1	2	yes	—
159	#106	—	2	4	no	—
160	C₄⋊C₈	D₄×C₁₃₇	1	1	yes	characteristic
161	#103	—	2	2	no	—
162	#103	—	2	4	no	—
163	C₂²⋊C₈	D₄×C₁₃₇	1	2	yes	—
164	C₄×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
165	C₂×Q₈	—	2	2	no	—
166	C₂×Q₈	—	4	2	no	—
167	C₂×D₄	D₈×C₁₃₇	1	1	yes	characteristic
168	C₂²×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
169	Q₁₆	—	2	2	no	—
170	Q₁₆	—	2	2	no	—
171	C₂×C₈	—	2	2	no	—
172	C₂×C₈	—	2	2	no	—
173	C₂×C₈	C₂×D₄×C₁₃₇	1	1	yes	characteristic
174	C₂×C₈	C₂×D₄×C₁₃₇	1	2	yes	—
175	C₂×C₈	C₂⁴×C₁₃₇	1	1	yes	characteristic
176	C₄⋊C₄	—	2	2	no	—
177	C₄⋊C₄	—	2	4	no	—
178	C₄⋊C₄	—	4	2	no	—
179	C₄⋊C₄	—	4	4	no	—
180	C₄⋊C₄	—	4	2	no	—
181	C₄⋊C₄	D₈×C₁₃₇	1	1	yes	characteristic
182	C₂²⋊C₄	—	4	4	no	—
183	C₂²⋊C₄	—	4	4	no	—
184	C₂²⋊C₄	D₈×C₁₃₇	1	2	yes	—
185	C₄²	—	4	4	no	—
186	C₄²	—	4	2	no	—
187	C₄²	C₂×D₄×C₁₃₇	1	1	yes	characteristic
188	C₁₆	—	2	2	no	—
189	C₁₆	—	2	1	no	—
190	C₁₆	#41×C₁₃₇	1	2	yes	—
191	C₂³	C₂×D₈×C₁₃₇	1	2	yes	—
192	Q₈	—	4	2	no	—
193	Q₈	—	4	2	no	—
194	Q₈	—	8	4	no	—
195	D₄	—	2	2	no	—
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	4	2	no	—
199	C₂×C₄	—	4	4	no	—
200	C₂×C₄	—	4	2	no	—
201	C₂×C₄	C₂×D₈×C₁₃₇	1	2	yes	—
202	C₂×C₄	C₂×D₈×C₁₃₇	1	1	yes	characteristic
203	C₂×C₄	C₂×D₈×C₁₃₇	1	1	yes	characteristic
204	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
205	C₈	—	2	2	no	—
206	C₈	—	2	1	no	—
207	C₈	C₂×#41×C₁₃₇	1	1	yes	characteristic
208	C₈	2₋¹⁺⁴×C₁₃₇	1	1	yes	characteristic
209	C₂²	—	2	2	no	—
210	C₂²	—	2	2	no	—
211	C₂²	—	2	2	no	—
212	C₂²	C₂²×D₈×C₁₃₇	1	1	yes	characteristic
213	C₄	—	2	2	no	—
214	C₄	—	2	1	no	—
215	C₄	—	8	2	no	—
216	C₄	—	8	4	no	—
217	C₄	—	8	2	no	—
218	C₄	#508×C₁₃₇	1	2	yes	—
219	C₄	#547×C₁₃₇	1	1	yes	characteristic
220	C₄	#578×C₁₃₇	1	1	yes	characteristic
221	C₂	—	4	2	no	—
222	C₂	#3334×C₁₃₇	1	1	yes	characteristic
223	C₂	C₂×#508×C₁₃₇	1	1	yes	characteristic
224	C₂	#3417×C₁₃₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃₇	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₁₃₇
prime radical	C₂×D₄×C₁₃₇	D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃₇ ◁ C₄×D₄×C₁₃₇ ◁ C₈×D₄×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. #2252×C₁₃₇	C₂	semidirect
3. #2237×C₁₃₇	C₂	semidirect
4. #2225×C₁₃₇	C₂	semidirect
5. #2216×C₁₃₇	C₂	semidirect
6. C₄×Q₃₂×C₁₃₇	C₂	semidirect
8. #2158×C₁₃₇	C₂	semidirect
112. #33909	C₁₃₇	direct (N × H)
113. C₁₃₇	#33909	direct (N × H)
115. #2252	C₂×C₁₃₇	semidirect
116. #2237	C₂×C₁₃₇	semidirect
117. #2225	C₂×C₁₃₇	semidirect
118. #2216	C₂×C₁₃₇	semidirect
119. C₄×Q₃₂	C₂×C₁₃₇	semidirect
121. #2158	C₂×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
#3342×C₁₃₇	2
#2252×C₁₃₇	2
#2237×C₁₃₇	2
#2225×C₁₃₇	2
#2216×C₁₃₇	2
C₄×Q₃₂×C₁₃₇	2
D₄×C₁₆×C₁₃₇	2
#2158×C₁₃₇	2
#33909	137

Subgroups — C137×#33909 #46883295

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#33909 #46883295