Subgroups — C₁₃₇×#17353 #46866739

Order 35072.

1238 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	C₂×#520×C₁₃₇	C₂	1	1	yes	maximal, characteristic
2	#2982×C₁₃₇	C₂	1	2	yes	maximal, complemented
3	#1460×C₁₃₇	C₂	1	6	yes	maximal, complemented
4	#1458×C₁₃₇	C₂	1	6	yes	maximal, complemented
5	C₂²×#41×C₁₃₇	C₄	1	1	yes	characteristic
6	#520×C₁₃₇	—	2	8	no	—
7	C₂×C₄×Q₈×C₁₃₇	C₄	1	1	yes	characteristic
8	C₂×C₄×Q₈×C₁₃₇	C₂²	1	1	yes	characteristic
9	C₂×C₄×D₄×C₁₃₇	C₄	1	3	yes	—
10	C₂×C₄×D₄×C₁₃₇	C₂²	1	3	yes	—
11	C₂×#117×C₁₃₇	C₂²	1	3	yes	—
12	C₂²×C₄⋊C₄×C₁₃₇	C₄	1	3	yes	—
13	C₈×Q₈×C₁₃₇	—	2	8	no	—
14	#431×C₁₃₇	C₂²	1	12	yes	—
15	#324×C₁₃₇	C₂²	1	4	yes	—
16	#323×C₁₃₇	C₂²	1	12	yes	—
17	C₂×#41×C₁₃₇	—	2	4	no	—
18	C₂×#41×C₁₃₇	—	2	4	no	—
19	C₂×#41×C₁₃₇	C₈	1	8	yes	complemented
20	C₂²×Q₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
21	C₂²×D₄×C₁₃₇	C₂×C₄	1	3	yes	—
22	C₂³×C₄×C₁₃₇	C₂×C₄	1	3	yes	—
23	C₂³×C₄×C₁₃₇	C₂×C₄	1	6	yes	—
24	C₄×Q₈×C₁₃₇	—	2	4	no	—
25	C₄×Q₈×C₁₃₇	D₄	1	8	yes	—
26	C₄×D₄×C₁₃₇	—	2	12	no	—
27	C₄×D₄×C₁₃₇	—	2	12	no	—
28	#117×C₁₃₇	—	2	12	no	—
29	C₂×C₄⋊C₄×C₁₃₇	—	2	12	no	—
30	C₂×C₄⋊C₄×C₁₃₇	C₂×C₄	1	3	yes	—
31	C₂×C₄⋊C₄×C₁₃₇	C₂×C₄	1	3	yes	—
32	C₂×C₄⋊C₄×C₁₃₇	C₂×C₄	1	3	yes	—
33	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	1	yes	characteristic
34	C₂×C₂²⋊C₄×C₁₃₇	C₂×C₄	1	3	yes	—
35	C₂×C₂²⋊C₄×C₁₃₇	C₂³	1	3	yes	—
36	C₂×C₄²×C₁₃₇	C₂×C₄	1	3	yes	—
37	C₂×C₄²×C₁₃₇	C₂³	1	3	yes	—
38	C₄⋊C₈×C₁₃₇	—	2	12	no	—
39	C₂²⋊C₈×C₁₃₇	C₂³	1	8	yes	—
40	C₄×C₈×C₁₃₇	—	2	12	no	—
41	C₂⁴×C₁₃₇	C₂²×C₄	1	3	yes	—
42	#41×C₁₃₇	—	2	32	no	—
43	C₂×Q₈×C₁₃₇	C₂²⋊C₄	1	4	yes	—
44	C₂×Q₈×C₁₃₇	C₂×C₈	1	4	yes	—
45	C₂×Q₈×C₁₃₇	M₁₆	1	4	yes	—
46	C₂×D₄×C₁₃₇	—	2	6	no	—
47	C₂×D₄×C₁₃₇	—	2	6	no	—
48	C₂×D₄×C₁₃₇	C₂×C₈	1	12	yes	—
49	C₂²×C₄×C₁₃₇	—	2	6	no	—
50	C₂²×C₄×C₁₃₇	—	2	6	no	—
51	C₂²×C₄×C₁₃₇	—	2	12	no	—
52	C₂²×C₄×C₁₃₇	—	8	6	no	—
53	C₂²×C₄×C₁₃₇	C₂×C₈	1	12	yes	—
54	C₂²×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
56	C₂²×C₄×C₁₃₇	C₂²×C₄	1	3	yes	—
57	C₂²×C₄×C₁₃₇	C₂²×C₄	1	3	yes	—
58	C₂²×C₄×C₁₃₇	C₂²×C₄	1	3	yes	—
59	C₂²×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
60	C₂×C₈×C₁₃₇	—	2	8	no	—
61	C₄⋊C₄×C₁₃₇	—	2	6	no	—
62	C₄⋊C₄×C₁₃₇	—	2	6	no	—
63	C₄⋊C₄×C₁₃₇	—	2	2	no	—
64	C₄⋊C₄×C₁₃₇	C₂²⋊C₄	1	12	yes	—
65	C₂²⋊C₄×C₁₃₇	—	2	6	no	—
66	C₂²⋊C₄×C₁₃₇	—	2	6	no	—
67	C₄²×C₁₃₇	—	2	6	no	—
68	C₄²×C₁₃₇	C₂×D₄	1	12	yes	—
69	C₂³×C₁₃₇	—	2	3	no	—
70	C₂³×C₁₃₇	—	2	3	no	—
71	C₂³×C₁₃₇	—	4	3	no	—
72	C₂³×C₁₃₇	—	4	3	no	—
73	C₂³×C₁₃₇	C₂²×C₈	1	6	yes	—
74	C₂³×C₁₃₇	C₂³×C₄	1	1	yes	characteristic
75	Q₈×C₁₃₇	C₂²⋊C₈	1	16	yes	—
76	D₄×C₁₃₇	—	2	24	no	—
77	C₂×C₄×C₁₃₇	—	2	1	no	—
78	C₂×C₄×C₁₃₇	—	2	1	no	—
79	C₂×C₄×C₁₃₇	—	2	24	no	—
80	C₂×C₄×C₁₃₇	—	2	1	no	—
81	C₂×C₄×C₁₃₇	—	2	3	no	—
82	C₂×C₄×C₁₃₇	—	4	3	no	—
83	C₂×C₄×C₁₃₇	—	4	1	no	—
84	C₂×C₄×C₁₃₇	—	4	3	no	—
85	C₂×C₄×C₁₃₇	—	4	1	no	—
86	C₂×C₄×C₁₃₇	—	8	12	no	—
87	C₂×C₄×C₁₃₇	C₂×C₂²⋊C₄	1	6	yes	—
88	C₂×C₄×C₁₃₇	C₂×C₂²⋊C₄	1	6	yes	—
89	C₂×C₄×C₁₃₇	C₂²×C₈	1	2	yes	—
90	C₂×C₄×C₁₃₇	C₂²×C₈	1	6	yes	—
91	C₂×C₄×C₁₃₇	C₂×M₁₆	1	6	yes	—
92	C₂×C₄×C₁₃₇	C₂²×D₄	1	2	yes	—
93	C₈×C₁₃₇	—	4	8	no	—
94	C₂²×C₁₃₇	—	2	1	no	—
95	C₂²×C₁₃₇	—	2	1	no	—
96	C₂²×C₁₃₇	—	2	6	no	—
97	C₂²×C₁₃₇	—	4	6	no	—
98	C₂²×C₁₃₇	—	4	6	no	—
99	C₂²×C₁₃₇	—	4	3	no	—
100	C₂²×C₁₃₇	—	4	3	no	—
101	C₂²×C₁₃₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
102	C₂²×C₁₃₇	C₂³×C₈	1	1	yes	characteristic
103	C₂²×C₁₃₇	C₂²×M₁₆	1	1	yes	characteristic
104	C₄×C₁₃₇	—	2	2	no	—
105	C₄×C₁₃₇	—	2	2	no	—
106	C₄×C₁₃₇	—	2	6	no	—
107	C₄×C₁₃₇	—	4	1	no	—
108	C₄×C₁₃₇	—	4	3	no	—
109	C₄×C₁₃₇	C₂×C₂²⋊C₈	1	12	yes	—
110	C₂×C₁₃₇	—	2	1	no	—
111	C₂×C₁₃₇	—	2	1	no	—
112	C₂×C₁₃₇	—	4	6	no	—
113	C₂×C₁₃₇	C₂²×C₂²⋊C₈	1	1	yes	characteristic
114	C₂×C₁₃₇	#2884	1	1	yes	characteristic
115	C₂×C₁₃₇	#2888	1	1	yes	characteristic
116	#17353	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
117	C₁₃₇	#17353	1	1	yes	characteristic, direct product factor, Sylow
118	C₂×#520	C₂×C₁₃₇	1	1	yes	characteristic
119	#2982	C₂×C₁₃₇	1	2	yes	complemented
120	#1460	C₂×C₁₃₇	1	6	yes	complemented
121	#1458	C₂×C₁₃₇	1	6	yes	complemented
122	C₂²×#41	C₄×C₁₃₇	1	1	yes	characteristic
123	#520	—	2	8	no	—
124	C₂×C₄×Q₈	C₄×C₁₃₇	1	1	yes	characteristic
125	C₂×C₄×Q₈	C₂²×C₁₃₇	1	1	yes	characteristic
126	C₂×C₄×D₄	C₄×C₁₃₇	1	3	yes	—
127	C₂×C₄×D₄	C₂²×C₁₃₇	1	3	yes	—
128	C₂×#117	C₂²×C₁₃₇	1	3	yes	—
129	C₂²×C₄⋊C₄	C₄×C₁₃₇	1	3	yes	—
130	C₈×Q₈	—	2	8	no	—
131	#431	C₂²×C₁₃₇	1	12	yes	—
132	#324	C₂²×C₁₃₇	1	4	yes	—
133	#323	C₂²×C₁₃₇	1	12	yes	—
134	C₂×#41	—	2	4	no	—
135	C₂×#41	—	2	4	no	—
136	C₂×#41	C₈×C₁₃₇	1	8	yes	complemented
137	C₂²×Q₈	C₂×C₄×C₁₃₇	1	1	yes	characteristic
138	C₂²×D₄	C₂×C₄×C₁₃₇	1	3	yes	—
139	C₂³×C₄	C₂×C₄×C₁₃₇	1	3	yes	—
140	C₂³×C₄	C₂×C₄×C₁₃₇	1	6	yes	—
141	C₄×Q₈	—	2	4	no	—
142	C₄×Q₈	D₄×C₁₃₇	1	8	yes	—
143	C₄×D₄	—	2	12	no	—
144	C₄×D₄	—	2	12	no	—
145	#117	—	2	12	no	—
146	C₂×C₄⋊C₄	—	2	12	no	—
147	C₂×C₄⋊C₄	C₂×C₄×C₁₃₇	1	3	yes	—
148	C₂×C₄⋊C₄	C₂×C₄×C₁₃₇	1	3	yes	—
149	C₂×C₄⋊C₄	C₂×C₄×C₁₃₇	1	3	yes	—
150	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	1	yes	characteristic
151	C₂×C₂²⋊C₄	C₂×C₄×C₁₃₇	1	3	yes	—
152	C₂×C₂²⋊C₄	C₂³×C₁₃₇	1	3	yes	—
153	C₂×C₄²	C₂×C₄×C₁₃₇	1	3	yes	—
154	C₂×C₄²	C₂³×C₁₃₇	1	3	yes	—
155	C₄⋊C₈	—	2	12	no	—
156	C₂²⋊C₈	C₂³×C₁₃₇	1	8	yes	—
157	C₄×C₈	—	2	12	no	—
158	C₂⁴	C₂²×C₄×C₁₃₇	1	3	yes	—
159	#41	—	2	32	no	—
160	C₂×Q₈	C₂²⋊C₄×C₁₃₇	1	4	yes	—
161	C₂×Q₈	C₂×C₈×C₁₃₇	1	4	yes	—
162	C₂×Q₈	M₁₆×C₁₃₇	1	4	yes	—
163	C₂×D₄	—	2	6	no	—
164	C₂×D₄	—	2	6	no	—
165	C₂×D₄	C₂×C₈×C₁₃₇	1	12	yes	—
166	C₂²×C₄	—	2	6	no	—
167	C₂²×C₄	—	2	6	no	—
168	C₂²×C₄	—	2	12	no	—
169	C₂²×C₄	—	8	6	no	—
170	C₂²×C₄	C₂×C₈×C₁₃₇	1	12	yes	—
171	C₂²×C₄	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
172	C₂²×C₄	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
173	C₂²×C₄	C₂²×C₄×C₁₃₇	1	3	yes	—
174	C₂²×C₄	C₂²×C₄×C₁₃₇	1	3	yes	—
175	C₂²×C₄	C₂²×C₄×C₁₃₇	1	3	yes	—
176	C₂²×C₄	C₂⁴×C₁₃₇	1	1	yes	characteristic
177	C₂×C₈	—	2	8	no	—
178	C₄⋊C₄	—	2	6	no	—
179	C₄⋊C₄	—	2	6	no	—
180	C₄⋊C₄	—	2	2	no	—
181	C₄⋊C₄	C₂²⋊C₄×C₁₃₇	1	12	yes	—
182	C₂²⋊C₄	—	2	6	no	—
183	C₂²⋊C₄	—	2	6	no	—
184	C₄²	—	2	6	no	—
185	C₄²	C₂×D₄×C₁₃₇	1	12	yes	—
186	C₂³	—	2	3	no	—
187	C₂³	—	2	3	no	—
188	C₂³	—	4	3	no	—
189	C₂³	—	4	3	no	—
190	C₂³	C₂²×C₈×C₁₃₇	1	6	yes	—
191	C₂³	C₂³×C₄×C₁₃₇	1	1	yes	characteristic
192	Q₈	C₂²⋊C₈×C₁₃₇	1	16	yes	—
193	D₄	—	2	24	no	—
194	C₂×C₄	—	2	1	no	—
195	C₂×C₄	—	2	1	no	—
196	C₂×C₄	—	2	24	no	—
197	C₂×C₄	—	2	1	no	—
198	C₂×C₄	—	2	3	no	—
199	C₂×C₄	—	4	3	no	—
200	C₂×C₄	—	4	1	no	—
201	C₂×C₄	—	4	3	no	—
202	C₂×C₄	—	4	1	no	—
203	C₂×C₄	—	8	12	no	—
204	C₂×C₄	C₂×C₂²⋊C₄×C₁₃₇	1	6	yes	—
205	C₂×C₄	C₂×C₂²⋊C₄×C₁₃₇	1	6	yes	—
206	C₂×C₄	C₂²×C₈×C₁₃₇	1	2	yes	—
207	C₂×C₄	C₂²×C₈×C₁₃₇	1	6	yes	—
208	C₂×C₄	C₂×M₁₆×C₁₃₇	1	6	yes	—
209	C₂×C₄	C₂²×D₄×C₁₃₇	1	2	yes	—
210	C₈	—	4	8	no	—
211	C₂²	—	2	1	no	—
212	C₂²	—	2	1	no	—
213	C₂²	—	2	6	no	—
214	C₂²	—	4	6	no	—
215	C₂²	—	4	6	no	—
216	C₂²	—	4	3	no	—
217	C₂²	—	4	3	no	—
218	C₂²	C₂²×C₂²⋊C₄×C₁₃₇	1	1	yes	characteristic
219	C₂²	C₂³×C₈×C₁₃₇	1	1	yes	characteristic
220	C₂²	C₂²×M₁₆×C₁₃₇	1	1	yes	characteristic
221	C₄	—	2	2	no	—
222	C₄	—	2	2	no	—
223	C₄	—	2	6	no	—
224	C₄	—	4	1	no	—
225	C₄	—	4	3	no	—
226	C₄	C₂×C₂²⋊C₈×C₁₃₇	1	12	yes	—
227	C₂	—	2	1	no	—
228	C₂	—	2	1	no	—
229	C₂	—	4	6	no	—
230	C₂	C₂²×C₂²⋊C₈×C₁₃₇	1	1	yes	characteristic
231	C₂	#2884×C₁₃₇	1	1	yes	characteristic
232	C₂	#2888×C₁₃₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃₇	C₂²×C₂²⋊C₄
commutator	C₂²	C₂³×C₈×C₁₃₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃₇
prime radical	C₂²×#41×C₁₃₇	C₄
LCS	G ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃₇ ◁ C₂×C₄×Q₈×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. #2982×C₁₃₇	C₂	semidirect
3. #1460×C₁₃₇	C₂	semidirect
4. #1458×C₁₃₇	C₂	semidirect
19. C₂×#41×C₁₃₇	C₈	semidirect
116. #17353	C₁₃₇	direct (N × H)
117. C₁₃₇	#17353	direct (N × H)
119. #2982	C₂×C₁₃₇	semidirect
120. #1460	C₂×C₁₃₇	semidirect
121. #1458	C₂×C₁₃₇	semidirect
136. C₂×#41	C₈×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#520×C₁₃₇	2
#2982×C₁₃₇	2
#1460×C₁₃₇	2
#1458×C₁₃₇	2
#17353	137

Subgroups — C137×#17353 #46866739

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#17353 #46866739