Subgroups — #3175×C₁₃₇ #31778186

Order 17536.

362 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#542×C₁₃₇	C₂	1	2	yes	maximal, complemented
2	#500×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
3	#495×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#489×C₁₃₇	C₂	1	2	yes	maximal, complemented
5	#468×C₁₃₇	C₂	1	2	yes	maximal, complemented
6	#467×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
7	#466×C₁₃₇	C₂	1	1	yes	maximal, characteristic
8	#440×C₁₃₇	C₂	1	2	yes	maximal, complemented
9	C₄×Q₁₆×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×D₈×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
11	#405×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₁₆×C₁₃₇	C₂²	1	1	yes	characteristic
13	C₂×SD₁₆×C₁₃₇	C₂²	1	2	yes	—
14	C₂×D₈×C₁₃₇	C₂²	1	1	yes	characteristic
15	C₂×M₁₆×C₁₃₇	C₂²	1	1	yes	characteristic
16	C₂²×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
17	C₄⋊Q₈×C₁₃₇	C₂²	1	2	yes	—
18	#126×C₁₃₇	—	2	2	no	—
19	#124×C₁₃₇	—	2	2	no	—
20	#124×C₁₃₇	C₂²	1	2	yes	—
21	#123×C₁₃₇	—	2	2	no	—
22	C₂²⋊Q₈×C₁₃₇	—	2	2	no	—
23	C₂²⋊Q₈×C₁₃₇	C₂²	1	2	yes	—
24	#121×C₁₃₇	C₂²	1	2	yes	—
25	C₄×Q₈×C₁₃₇	C₂²	1	2	yes	—
26	C₄×D₄×C₁₃₇	C₂²	1	2	yes	—
27	#117×C₁₃₇	C₂²	1	1	yes	characteristic
28	#107×C₁₃₇	C₂²	1	1	yes	characteristic
29	#107×C₁₃₇	C₂²	1	1	yes	characteristic
30	#107×C₁₃₇	C₂²	1	2	yes	—
31	#103×C₁₃₇	C₂²	1	2	yes	—
32	#103×C₁₃₇	C₂²	1	2	yes	—
33	#102×C₁₃₇	C₂²	1	2	yes	—
34	#102×C₁₃₇	C₂²	1	2	yes	—
35	#97×C₁₃₇	C₂²	1	1	yes	characteristic
36	#97×C₁₃₇	C₂²	1	1	yes	characteristic
37	C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
38	C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
39	C₂×Q₈×C₁₃₇	—	2	2	no	—
40	C₂×Q₈×C₁₃₇	C₂³	1	2	yes	—
41	C₂×D₄×C₁₃₇	—	2	2	no	—
42	C₂×D₄×C₁₃₇	C₂³	1	2	yes	—
43	C₂²×C₄×C₁₃₇	—	2	2	no	—
44	C₂²×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
45	Q₁₆×C₁₃₇	—	2	2	no	—
46	SD₁₆×C₁₃₇	—	2	4	no	—
47	D₈×C₁₃₇	—	2	2	no	—
48	M₁₆×C₁₃₇	—	2	2	no	—
49	C₂×C₈×C₁₃₇	—	2	2	no	—
50	C₂×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
51	C₂×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
52	C₂×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
53	C₂×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
54	C₄⋊C₄×C₁₃₇	—	2	2	no	—
55	C₄⋊C₄×C₁₃₇	—	2	2	no	—
56	C₄⋊C₄×C₁₃₇	—	2	2	no	—
57	C₄⋊C₄×C₁₃₇	D₄	1	1	yes	characteristic
58	C₄⋊C₄×C₁₃₇	D₄	1	1	yes	characteristic
59	C₄⋊C₄×C₁₃₇	C₂³	1	2	yes	—
60	C₄⋊C₄×C₁₃₇	C₂³	1	2	yes	—
61	C₂²⋊C₄×C₁₃₇	—	2	2	no	—
62	C₂²⋊C₄×C₁₃₇	—	2	2	no	—
63	C₂²⋊C₄×C₁₃₇	—	2	2	no	—
64	C₂²⋊C₄×C₁₃₇	—	2	2	no	—
65	C₂²⋊C₄×C₁₃₇	—	2	2	no	—
66	C₂²⋊C₄×C₁₃₇	D₄	1	1	yes	characteristic
67	C₂²⋊C₄×C₁₃₇	D₄	1	1	yes	characteristic
68	C₄²×C₁₃₇	—	2	2	no	—
69	C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
70	C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
71	C₂³×C₁₃₇	—	2	2	no	—
72	C₂³×C₁₃₇	C₂×D₄	1	1	yes	characteristic
73	Q₈×C₁₃₇	—	2	2	no	—
74	Q₈×C₁₃₇	—	2	2	no	—
75	Q₈×C₁₃₇	—	8	2	no	—
76	D₄×C₁₃₇	—	2	2	no	—
77	D₄×C₁₃₇	—	2	2	no	—
78	D₄×C₁₃₇	—	8	2	no	—
79	C₂×C₄×C₁₃₇	—	2	1	no	—
80	C₂×C₄×C₁₃₇	—	2	1	no	—
81	C₂×C₄×C₁₃₇	—	2	2	no	—
82	C₂×C₄×C₁₃₇	—	2	2	no	—
83	C₂×C₄×C₁₃₇	—	2	2	no	—
84	C₂×C₄×C₁₃₇	—	8	2	no	—
85	C₂×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
86	C₂×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
87	C₂×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
88	C₂×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
89	C₂×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
90	C₂×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
91	C₈×C₁₃₇	—	2	1	no	—
92	C₈×C₁₃₇	—	2	1	no	—
93	C₈×C₁₃₇	#41	1	2	yes	—
94	C₈×C₁₃₇	#41	1	2	yes	—
95	C₂²×C₁₃₇	—	2	1	no	—
96	C₂²×C₁₃₇	—	2	1	no	—
97	C₂²×C₁₃₇	—	2	1	no	—
98	C₂²×C₁₃₇	—	4	2	no	—
99	C₂²×C₁₃₇	—	4	2	no	—
100	C₂²×C₁₃₇	—	4	2	no	—
101	C₂²×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
102	C₄×C₁₃₇	—	2	1	no	—
103	C₄×C₁₃₇	—	2	1	no	—
104	C₄×C₁₃₇	—	2	1	no	—
105	C₄×C₁₃₇	—	2	1	no	—
106	C₄×C₁₃₇	—	2	1	no	—
107	C₄×C₁₃₇	—	4	2	no	—
108	C₄×C₁₃₇	—	4	2	no	—
109	C₄×C₁₃₇	—	4	2	no	—
110	C₄×C₁₃₇	C₂×#41	1	1	yes	characteristic
111	C₄×C₁₃₇	C₂×#41	1	1	yes	characteristic
112	C₂×C₁₃₇	—	4	1	no	—
113	C₂×C₁₃₇	—	8	2	no	—
114	C₂×C₁₃₇	#532	1	1	yes	characteristic
115	C₂×C₁₃₇	#576	1	1	yes	characteristic
116	C₂×C₁₃₇	#578	1	1	yes	characteristic
117	C₁₃₇	#3175	1	1	yes	characteristic, direct product factor, Sylow
118	#3175	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
119	#542	C₂×C₁₃₇	1	2	yes	complemented
120	#500	C₂×C₁₃₇	1	1	yes	characteristic, complemented
121	#495	C₂×C₁₃₇	1	1	yes	characteristic, complemented
122	#489	C₂×C₁₃₇	1	2	yes	complemented
123	#468	C₂×C₁₃₇	1	2	yes	complemented
124	#467	C₂×C₁₃₇	1	1	yes	characteristic, complemented
125	#466	C₂×C₁₃₇	1	1	yes	characteristic
126	#440	C₂×C₁₃₇	1	2	yes	complemented
127	C₄×Q₁₆	C₂×C₁₃₇	1	1	yes	characteristic, complemented
128	C₄×D₈	C₂×C₁₃₇	1	1	yes	characteristic, complemented
129	#405	C₂×C₁₃₇	1	1	yes	characteristic, complemented
130	C₂×Q₁₆	C₂²×C₁₃₇	1	1	yes	characteristic
131	C₂×SD₁₆	C₂²×C₁₃₇	1	2	yes	—
132	C₂×D₈	C₂²×C₁₃₇	1	1	yes	characteristic
133	C₂×M₁₆	C₂²×C₁₃₇	1	1	yes	characteristic
134	C₂²×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
135	C₄⋊Q₈	C₂²×C₁₃₇	1	2	yes	—
136	#126	—	2	2	no	—
137	#124	—	2	2	no	—
138	#124	C₂²×C₁₃₇	1	2	yes	—
139	#123	—	2	2	no	—
140	C₂²⋊Q₈	—	2	2	no	—
141	C₂²⋊Q₈	C₂²×C₁₃₇	1	2	yes	—
142	#121	C₂²×C₁₃₇	1	2	yes	—
143	C₄×Q₈	C₂²×C₁₃₇	1	2	yes	—
144	C₄×D₄	C₂²×C₁₃₇	1	2	yes	—
145	#117	C₂²×C₁₃₇	1	1	yes	characteristic
146	#107	C₂²×C₁₃₇	1	1	yes	characteristic
147	#107	C₂²×C₁₃₇	1	1	yes	characteristic
148	#107	C₂²×C₁₃₇	1	2	yes	—
149	#103	C₂²×C₁₃₇	1	2	yes	—
150	#103	C₂²×C₁₃₇	1	2	yes	—
151	#102	C₂²×C₁₃₇	1	2	yes	—
152	#102	C₂²×C₁₃₇	1	2	yes	—
153	#97	C₂²×C₁₃₇	1	1	yes	characteristic
154	#97	C₂²×C₁₃₇	1	1	yes	characteristic
155	C₄×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
156	C₄×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
157	C₂×Q₈	—	2	2	no	—
158	C₂×Q₈	C₂³×C₁₃₇	1	2	yes	—
159	C₂×D₄	—	2	2	no	—
160	C₂×D₄	C₂³×C₁₃₇	1	2	yes	—
161	C₂²×C₄	—	2	2	no	—
162	C₂²×C₄	C₂³×C₁₃₇	1	1	yes	characteristic
163	Q₁₆	—	2	2	no	—
164	SD₁₆	—	2	4	no	—
165	D₈	—	2	2	no	—
166	M₁₆	—	2	2	no	—
167	C₂×C₈	—	2	2	no	—
168	C₂×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
169	C₂×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
170	C₂×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
171	C₂×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
172	C₄⋊C₄	—	2	2	no	—
173	C₄⋊C₄	—	2	2	no	—
174	C₄⋊C₄	—	2	2	no	—
175	C₄⋊C₄	D₄×C₁₃₇	1	1	yes	characteristic
176	C₄⋊C₄	D₄×C₁₃₇	1	1	yes	characteristic
177	C₄⋊C₄	C₂³×C₁₃₇	1	2	yes	—
178	C₄⋊C₄	C₂³×C₁₃₇	1	2	yes	—
179	C₂²⋊C₄	—	2	2	no	—
180	C₂²⋊C₄	—	2	2	no	—
181	C₂²⋊C₄	—	2	2	no	—
182	C₂²⋊C₄	—	2	2	no	—
183	C₂²⋊C₄	—	2	2	no	—
184	C₂²⋊C₄	D₄×C₁₃₇	1	1	yes	characteristic
185	C₂²⋊C₄	D₄×C₁₃₇	1	1	yes	characteristic
186	C₄²	—	2	2	no	—
187	C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
188	C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
189	C₂³	—	2	2	no	—
190	C₂³	C₂×D₄×C₁₃₇	1	1	yes	characteristic
191	Q₈	—	2	2	no	—
192	Q₈	—	2	2	no	—
193	Q₈	—	8	2	no	—
194	D₄	—	2	2	no	—
195	D₄	—	2	2	no	—
196	D₄	—	8	2	no	—
197	C₂×C₄	—	2	1	no	—
198	C₂×C₄	—	2	1	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	8	2	no	—
203	C₂×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
204	C₂×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
205	C₂×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
206	C₂×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
207	C₂×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
208	C₂×C₄	C₂⁴×C₁₃₇	1	1	yes	characteristic
209	C₈	—	2	1	no	—
210	C₈	—	2	1	no	—
211	C₈	#41×C₁₃₇	1	2	yes	—
212	C₈	#41×C₁₃₇	1	2	yes	—
213	C₂²	—	2	1	no	—
214	C₂²	—	2	1	no	—
215	C₂²	—	2	1	no	—
216	C₂²	—	4	2	no	—
217	C₂²	—	4	2	no	—
218	C₂²	—	4	2	no	—
219	C₂²	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
220	C₄	—	2	1	no	—
221	C₄	—	2	1	no	—
222	C₄	—	2	1	no	—
223	C₄	—	2	1	no	—
224	C₄	—	2	1	no	—
225	C₄	—	4	2	no	—
226	C₄	—	4	2	no	—
227	C₄	—	4	2	no	—
228	C₄	C₂×#41×C₁₃₇	1	1	yes	characteristic
229	C₄	C₂×#41×C₁₃₇	1	1	yes	characteristic
230	C₂	—	4	1	no	—
231	C₂	—	8	2	no	—
232	C₂	#532×C₁₃₇	1	1	yes	characteristic
233	C₂	#576×C₁₃₇	1	1	yes	characteristic
234	C₂	#578×C₁₃₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃₇	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₁₃₇
prime radical	#466×C₁₃₇	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃₇ ◁ #117×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. #542×C₁₃₇	C₂	semidirect
2. #500×C₁₃₇	C₂	semidirect
3. #495×C₁₃₇	C₂	semidirect
4. #489×C₁₃₇	C₂	semidirect
5. #468×C₁₃₇	C₂	semidirect
6. #467×C₁₃₇	C₂	semidirect
8. #440×C₁₃₇	C₂	semidirect
9. C₄×Q₁₆×C₁₃₇	C₂	semidirect
10. C₄×D₈×C₁₃₇	C₂	semidirect
11. #405×C₁₃₇	C₂	semidirect
117. C₁₃₇	#3175	direct (N × H)
118. #3175	C₁₃₇	direct (N × H)
119. #542	C₂×C₁₃₇	semidirect
120. #500	C₂×C₁₃₇	semidirect
121. #495	C₂×C₁₃₇	semidirect
122. #489	C₂×C₁₃₇	semidirect
123. #468	C₂×C₁₃₇	semidirect
124. #467	C₂×C₁₃₇	semidirect
126. #440	C₂×C₁₃₇	semidirect
127. C₄×Q₁₆	C₂×C₁₃₇	semidirect
128. C₄×D₈	C₂×C₁₃₇	semidirect
129. #405	C₂×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
#542×C₁₃₇	2
#500×C₁₃₇	2
#495×C₁₃₇	2
#489×C₁₃₇	2
#468×C₁₃₇	2
#467×C₁₃₇	2
#466×C₁₃₇	2
#440×C₁₃₇	2
C₄×Q₁₆×C₁₃₇	2
C₄×D₈×C₁₃₇	2
#405×C₁₃₇	2
#3175	137

Subgroups — #3175×C137 #31778186

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #3175×C₁₃₇ #31778186