Subgroups — C₁₃₇×#33960 #46883346

Order 35072.

508 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#3393×C₁₃₇	C₂	1	2	yes	maximal, complemented
2	#2249×C₁₃₇	C₂	1	2	yes	maximal
3	#2247×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#2211×C₁₃₇	C₂	1	4	yes	maximal, complemented
5	#2208×C₁₃₇	C₂	1	2	yes	maximal, complemented
6	Q₈×C₁₆×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×Q₃₂×C₁₃₇	C₂	1	2	yes	maximal, complemented
8	C₄×D₁₆×C₁₃₇	C₂	1	1	yes	maximal, characteristic
9	#556×C₁₃₇	—	2	2	no	—
10	C₂×Q₃₂×C₁₃₇	C₂²	1	2	yes	—
11	C₂×Q₃₂×C₁₃₇	C₂²	1	2	yes	—
12	C₂×SD₃₂×C₁₃₇	C₂²	1	4	yes	—
13	C₂×D₁₆×C₁₃₇	C₂²	1	1	yes	characteristic
14	#495×C₁₃₇	C₂²	1	4	yes	—
15	#494×C₁₃₇	C₂²	1	2	yes	—
16	#464×C₁₃₇	—	2	4	no	—
17	#461×C₁₃₇	—	2	2	no	—
18	C₈×Q₈×C₁₃₇	C₂²	1	1	yes	characteristic
19	C₄×Q₁₆×C₁₃₇	C₂²	1	4	yes	—
20	C₄×D₈×C₁₃₇	C₂²	1	2	yes	—
21	#366×C₁₃₇	C₂²	1	1	yes	characteristic
22	C₄⋊C₁₆×C₁₃₇	C₂²	1	1	yes	characteristic
23	C₄⋊C₁₆×C₁₃₇	C₂²	1	2	yes	—
24	#358×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×Q₁₆×C₁₃₇	—	2	2	no	—
29	C₂×Q₁₆×C₁₃₇	C₂³	1	4	yes	—
30	C₂×SD₁₆×C₁₃₇	—	2	4	no	—
31	C₂×D₈×C₁₃₇	C₂³	1	2	yes	—
32	C₄⋊Q₈×C₁₃₇	—	2	2	no	—
33	#126×C₁₃₇	—	4	4	no	—
34	#124×C₁₃₇	—	2	4	no	—
35	C₂²⋊Q₈×C₁₃₇	—	4	2	no	—
36	C₄×Q₈×C₁₃₇	—	2	4	no	—
37	C₄×Q₈×C₁₃₇	D₄	1	1	yes	characteristic
38	C₄×D₄×C₁₃₇	—	2	2	no	—
39	#117×C₁₃₇	—	4	2	no	—
40	Q₃₂×C₁₃₇	—	2	4	no	—
41	Q₃₂×C₁₃₇	—	4	2	no	—
42	SD₃₂×C₁₃₇	—	4	4	no	—
43	D₁₆×C₁₃₇	—	2	2	no	—
44	C₂×C₁₆×C₁₃₇	C₂³	1	2	yes	—
45	C₂×C₁₆×C₁₃₇	C₂³	1	1	yes	characteristic
46	C₂×C₁₆×C₁₃₇	C₂³	1	1	yes	characteristic
47	#107×C₁₃₇	C₂³	1	2	yes	—
48	C₄⋊C₈×C₁₃₇	D₄	1	1	yes	characteristic
49	C₄⋊C₈×C₁₃₇	D₄	1	2	yes	—
50	#103×C₁₃₇	—	2	4	no	—
51	#102×C₁₃₇	—	2	2	no	—
52	C₄×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
53	C₄×C₈×C₁₃₇	C₂³	1	2	yes	—
54	C₂×Q₈×C₁₃₇	—	2	4	no	—
55	C₂×Q₈×C₁₃₇	D₈	1	1	yes	characteristic
56	C₂×D₄×C₁₃₇	—	2	2	no	—
57	C₂²×C₄×C₁₃₇	—	4	2	no	—
58	Q₁₆×C₁₃₇	—	2	4	no	—
59	Q₁₆×C₁₃₇	—	2	4	no	—
60	Q₁₆×C₁₃₇	—	8	2	no	—
61	SD₁₆×C₁₃₇	—	8	4	no	—
62	D₈×C₁₃₇	—	2	2	no	—
63	D₈×C₁₃₇	—	2	2	no	—
64	C₂×C₈×C₁₃₇	C₂×D₄	1	1	yes	characteristic
65	C₂×C₈×C₁₃₇	C₂×D₄	1	2	yes	—
66	C₂×C₈×C₁₃₇	C₂⁴	1	1	yes	characteristic
67	C₄⋊C₄×C₁₃₇	—	2	2	no	—
68	C₄⋊C₄×C₁₃₇	—	4	2	no	—
69	C₄⋊C₄×C₁₃₇	—	4	4	no	—
70	C₄⋊C₄×C₁₃₇	—	4	2	no	—
71	C₄⋊C₄×C₁₃₇	D₈	1	1	yes	characteristic
72	C₄⋊C₄×C₁₃₇	D₈	1	2	yes	—
73	C₂²⋊C₄×C₁₃₇	—	4	4	no	—
74	C₂²⋊C₄×C₁₃₇	—	4	2	no	—
75	C₂²⋊C₄×C₁₃₇	—	4	4	no	—
76	C₄²×C₁₃₇	—	4	4	no	—
77	C₄²×C₁₃₇	C₂×D₄	1	1	yes	characteristic
78	C₄²×C₁₃₇	C₂×D₄	1	2	yes	—
79	C₁₆×C₁₃₇	—	2	2	no	—
80	C₁₆×C₁₃₇	—	2	1	no	—
81	C₁₆×C₁₃₇	#41	1	2	yes	—
82	C₂³×C₁₃₇	—	4	2	no	—
83	Q₈×C₁₃₇	—	2	2	no	—
84	Q₈×C₁₃₇	—	4	4	no	—
85	Q₈×C₁₃₇	—	4	4	no	—
86	D₄×C₁₃₇	—	4	2	no	—
87	D₄×C₁₃₇	—	4	2	no	—
88	C₂×C₄×C₁₃₇	—	4	2	no	—
89	C₂×C₄×C₁₃₇	—	4	4	no	—
90	C₂×C₄×C₁₃₇	—	8	4	no	—
91	C₂×C₄×C₁₃₇	C₂×D₈	1	2	yes	—
92	C₂×C₄×C₁₃₇	C₂×D₈	1	1	yes	characteristic
93	C₂×C₄×C₁₃₇	C₂×D₈	1	2	yes	—
94	C₂×C₄×C₁₃₇	C₂×D₈	1	1	yes	characteristic
95	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
96	C₈×C₁₃₇	—	2	2	no	—
97	C₈×C₁₃₇	—	2	1	no	—
98	C₈×C₁₃₇	C₂×#41	1	1	yes	characteristic
99	C₈×C₁₃₇	H(5,𝔽₂)	1	1	yes	characteristic
100	C₂²×C₁₃₇	—	8	2	no	—
101	C₂²×C₁₃₇	—	8	2	no	—
102	C₂²×C₁₃₇	—	8	2	no	—
103	C₂²×C₁₃₇	C₂²×D₈	1	1	yes	characteristic
104	C₄×C₁₃₇	—	2	2	no	—
105	C₄×C₁₃₇	—	2	2	no	—
106	C₄×C₁₃₇	—	2	1	no	—
107	C₄×C₁₃₇	—	8	4	no	—
108	C₄×C₁₃₇	—	8	2	no	—
109	C₄×C₁₃₇	#508	1	2	yes	—
110	C₄×C₁₃₇	#550	1	1	yes	characteristic
111	C₄×C₁₃₇	#576	1	1	yes	characteristic
112	C₂×C₁₃₇	—	16	2	no	—
113	C₂×C₁₃₇	#3391	1	1	yes	characteristic
114	C₂×C₁₃₇	C₂×#508	1	1	yes	characteristic
115	C₂×C₁₃₇	#3417	1	1	yes	characteristic
116	#33960	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
117	C₁₃₇	#33960	1	1	yes	characteristic, direct product factor, Sylow
118	#3393	C₂×C₁₃₇	1	2	yes	complemented
119	#2249	C₂×C₁₃₇	1	2	yes	—
120	#2247	C₂×C₁₃₇	1	1	yes	characteristic, complemented
121	#2211	C₂×C₁₃₇	1	4	yes	complemented
122	#2208	C₂×C₁₃₇	1	2	yes	complemented
123	Q₈×C₁₆	C₂×C₁₃₇	1	1	yes	characteristic, complemented
124	C₄×Q₃₂	C₂×C₁₃₇	1	2	yes	complemented
125	C₄×D₁₆	C₂×C₁₃₇	1	1	yes	characteristic
126	#556	—	2	2	no	—
127	C₂×Q₃₂	C₂²×C₁₃₇	1	2	yes	—
128	C₂×Q₃₂	C₂²×C₁₃₇	1	2	yes	—
129	C₂×SD₃₂	C₂²×C₁₃₇	1	4	yes	—
130	C₂×D₁₆	C₂²×C₁₃₇	1	1	yes	characteristic
131	#495	C₂²×C₁₃₇	1	4	yes	—
132	#494	C₂²×C₁₃₇	1	2	yes	—
133	#464	—	2	4	no	—
134	#461	—	2	2	no	—
135	C₈×Q₈	C₂²×C₁₃₇	1	1	yes	characteristic
136	C₄×Q₁₆	C₂²×C₁₃₇	1	4	yes	—
137	C₄×D₈	C₂²×C₁₃₇	1	2	yes	—
138	#366	C₂²×C₁₃₇	1	1	yes	characteristic
139	C₄⋊C₁₆	C₂²×C₁₃₇	1	1	yes	characteristic
140	C₄⋊C₁₆	C₂²×C₁₃₇	1	2	yes	—
141	#358	C₂²×C₁₃₇	1	4	yes	—
142	#357	C₂²×C₁₃₇	1	2	yes	—
143	C₄×C₁₆	C₂²×C₁₃₇	1	1	yes	characteristic
144	C₄×C₁₆	C₂²×C₁₃₇	1	2	yes	—
145	C₂×Q₁₆	—	2	2	no	—
146	C₂×Q₁₆	C₂³×C₁₃₇	1	4	yes	—
147	C₂×SD₁₆	—	2	4	no	—
148	C₂×D₈	C₂³×C₁₃₇	1	2	yes	—
149	C₄⋊Q₈	—	2	2	no	—
150	#126	—	4	4	no	—
151	#124	—	2	4	no	—
152	C₂²⋊Q₈	—	4	2	no	—
153	C₄×Q₈	—	2	4	no	—
154	C₄×Q₈	D₄×C₁₃₇	1	1	yes	characteristic
155	C₄×D₄	—	2	2	no	—
156	#117	—	4	2	no	—
157	Q₃₂	—	2	4	no	—
158	Q₃₂	—	4	2	no	—
159	SD₃₂	—	4	4	no	—
160	D₁₆	—	2	2	no	—
161	C₂×C₁₆	C₂³×C₁₃₇	1	1	yes	characteristic
162	C₂×C₁₆	C₂³×C₁₃₇	1	2	yes	—
163	C₂×C₁₆	C₂³×C₁₃₇	1	1	yes	characteristic
164	#107	C₂³×C₁₃₇	1	2	yes	—
165	C₄⋊C₈	D₄×C₁₃₇	1	1	yes	characteristic
166	C₄⋊C₈	D₄×C₁₃₇	1	2	yes	—
167	#103	—	2	4	no	—
168	#102	—	2	2	no	—
169	C₄×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
170	C₄×C₈	C₂³×C₁₃₇	1	2	yes	—
171	C₂×Q₈	—	2	4	no	—
172	C₂×Q₈	D₈×C₁₃₇	1	1	yes	characteristic
173	C₂×D₄	—	2	2	no	—
174	C₂²×C₄	—	4	2	no	—
175	Q₁₆	—	2	4	no	—
176	Q₁₆	—	2	4	no	—
177	Q₁₆	—	8	2	no	—
178	SD₁₆	—	8	4	no	—
179	D₈	—	2	2	no	—
180	D₈	—	2	2	no	—
181	C₂×C₈	C₂×D₄×C₁₃₇	1	1	yes	characteristic
182	C₂×C₈	C₂×D₄×C₁₃₇	1	2	yes	—
183	C₂×C₈	C₂⁴×C₁₃₇	1	1	yes	characteristic
184	C₄⋊C₄	—	2	2	no	—
185	C₄⋊C₄	—	4	2	no	—
186	C₄⋊C₄	—	4	4	no	—
187	C₄⋊C₄	—	4	2	no	—
188	C₄⋊C₄	D₈×C₁₃₇	1	1	yes	characteristic
189	C₄⋊C₄	D₈×C₁₃₇	1	2	yes	—
190	C₂²⋊C₄	—	4	4	no	—
191	C₂²⋊C₄	—	4	2	no	—
192	C₂²⋊C₄	—	4	4	no	—
193	C₄²	—	4	4	no	—
194	C₄²	C₂×D₄×C₁₃₇	1	1	yes	characteristic
195	C₄²	C₂×D₄×C₁₃₇	1	2	yes	—
196	C₁₆	—	2	2	no	—
197	C₁₆	—	2	1	no	—
198	C₁₆	#41×C₁₃₇	1	2	yes	—
199	C₂³	—	4	2	no	—
200	Q₈	—	2	2	no	—
201	Q₈	—	4	4	no	—
202	Q₈	—	4	4	no	—
203	D₄	—	4	2	no	—
204	D₄	—	4	2	no	—
205	C₂×C₄	—	4	2	no	—
206	C₂×C₄	—	4	4	no	—
207	C₂×C₄	—	8	4	no	—
208	C₂×C₄	C₂×D₈×C₁₃₇	1	2	yes	—
209	C₂×C₄	C₂×D₈×C₁₃₇	1	1	yes	characteristic
210	C₂×C₄	C₂×D₈×C₁₃₇	1	2	yes	—
211	C₂×C₄	C₂×D₈×C₁₃₇	1	1	yes	characteristic
212	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
213	C₈	—	2	2	no	—
214	C₈	—	2	1	no	—
215	C₈	C₂×#41×C₁₃₇	1	1	yes	characteristic
216	C₈	H(5,𝔽₂)×C₁₃₇	1	1	yes	characteristic
217	C₂²	—	8	2	no	—
218	C₂²	—	8	2	no	—
219	C₂²	—	8	2	no	—
220	C₂²	C₂²×D₈×C₁₃₇	1	1	yes	characteristic
221	C₄	—	2	2	no	—
222	C₄	—	2	2	no	—
223	C₄	—	2	1	no	—
224	C₄	—	8	4	no	—
225	C₄	—	8	2	no	—
226	C₄	#508×C₁₃₇	1	2	yes	—
227	C₄	#550×C₁₃₇	1	1	yes	characteristic
228	C₄	#576×C₁₃₇	1	1	yes	characteristic
229	C₂	—	16	2	no	—
230	C₂	#3391×C₁₃₇	1	1	yes	characteristic
231	C₂	C₂×#508×C₁₃₇	1	1	yes	characteristic
232	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×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. #3393×C₁₃₇	C₂	semidirect
3. #2247×C₁₃₇	C₂	semidirect
4. #2211×C₁₃₇	C₂	semidirect
5. #2208×C₁₃₇	C₂	semidirect
6. Q₈×C₁₆×C₁₃₇	C₂	semidirect
7. C₄×Q₃₂×C₁₃₇	C₂	semidirect
116. #33960	C₁₃₇	direct (N × H)
117. C₁₃₇	#33960	direct (N × H)
118. #3393	C₂×C₁₃₇	semidirect
120. #2247	C₂×C₁₃₇	semidirect
121. #2211	C₂×C₁₃₇	semidirect
122. #2208	C₂×C₁₃₇	semidirect
123. Q₈×C₁₆	C₂×C₁₃₇	semidirect
124. C₄×Q₃₂	C₂×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
#3393×C₁₃₇	2
#2249×C₁₃₇	2
#2247×C₁₃₇	2
#2211×C₁₃₇	2
#2208×C₁₃₇	2
Q₈×C₁₆×C₁₃₇	2
C₄×Q₃₂×C₁₃₇	2
C₄×D₁₆×C₁₃₇	2
#33960	137

Subgroups — C137×#33960 #46883346

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#33960 #46883346