Subgroups — C₁₃₇×#9823 #46859209

Order 35072.

526 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×#500×C₁₃₇	C₂	1	1	yes	maximal, characteristic
2	#1990×C₁₃₇	C₂	1	2	yes	maximal
3	#1956×C₁₃₇	C₂	1	2	yes	maximal
4	#1845×C₁₃₇	C₂	1	1	yes	maximal, characteristic
5	C₂×C₈²×C₁₃₇	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊Q₈×C₁₃₇	C₂²	1	2	yes	—
7	#500×C₁₃₇	—	2	2	no	—
8	#500×C₁₃₇	C₄	1	4	yes	complemented
9	C₂×#107×C₁₃₇	—	2	2	no	—
10	C₂×#107×C₁₃₇	—	2	2	no	—
11	C₂×#103×C₁₃₇	—	2	4	no	—
12	C₂×C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
15	#384×C₁₃₇	C₂²	1	2	yes	—
16	#340×C₁₃₇	—	2	4	no	—
17	C₈²×C₁₃₇	—	2	2	no	—
18	C₂²×Q₈×C₁₃₇	—	2	2	no	—
19	C₂²×C₈×C₁₃₇	D₄	1	2	yes	—
20	C₂²×C₈×C₁₃₇	D₄	1	2	yes	—
21	C₂²×C₈×C₁₃₇	D₄	1	2	yes	—
22	C₄⋊Q₈×C₁₃₇	—	2	2	no	—
23	C₄⋊Q₈×C₁₃₇	—	2	2	no	—
24	C₄⋊Q₈×C₁₃₇	—	2	2	no	—
25	C₄⋊Q₈×C₁₃₇	C₂×C₄	1	4	yes	—
26	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
27	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
28	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
29	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
30	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
31	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
32	#107×C₁₃₇	—	2	8	no	—
33	#107×C₁₃₇	—	4	4	no	—
34	#103×C₁₃₇	—	4	8	no	—
35	C₄×C₈×C₁₃₇	—	2	1	no	—
36	C₄×C₈×C₁₃₇	—	2	1	no	—
37	C₄×C₈×C₁₃₇	—	2	1	no	—
38	C₄×C₈×C₁₃₇	—	2	1	no	—
39	C₄×C₈×C₁₃₇	C₂×C₄	1	2	yes	—
40	C₄×C₈×C₁₃₇	D₄	1	2	yes	—
41	C₂×Q₈×C₁₃₇	—	2	4	no	—
42	C₂×Q₈×C₁₃₇	—	4	2	no	—
43	C₂×Q₈×C₁₃₇	—	4	4	no	—
44	C₂×Q₈×C₁₃₇	—	4	4	no	—
45	C₂²×C₄×C₁₃₇	—	4	2	no	—
46	C₂²×C₄×C₁₃₇	—	4	2	no	—
47	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
48	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
49	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
50	C₂×C₈×C₁₃₇	—	2	2	no	—
51	C₂×C₈×C₁₃₇	—	2	2	no	—
52	C₂×C₈×C₁₃₇	—	2	2	no	—
53	C₂×C₈×C₁₃₇	—	2	2	no	—
54	C₂×C₈×C₁₃₇	C₂²⋊C₄	1	4	yes	—
55	C₂×C₈×C₁₃₇	D₈	1	4	yes	—
56	C₂×C₈×C₁₃₇	SD₁₆	1	4	yes	—
57	C₂×C₈×C₁₃₇	Q₁₆	1	4	yes	—
58	C₂×C₈×C₁₃₇	Q₁₆	1	4	yes	—
59	C₄⋊C₄×C₁₃₇	—	2	4	no	—
60	C₄⋊C₄×C₁₃₇	—	2	4	no	—
61	C₄⋊C₄×C₁₃₇	—	4	2	no	—
62	C₄⋊C₄×C₁₃₇	—	4	2	no	—
63	C₄⋊C₄×C₁₃₇	—	4	4	no	—
64	C₄⋊C₄×C₁₃₇	—	4	4	no	—
65	C₄⋊C₄×C₁₃₇	—	4	4	no	—
66	C₄²×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
67	C₄²×C₁₃₇	C₂×D₄	1	1	yes	characteristic
68	C₄²×C₁₃₇	#41	1	1	yes	characteristic
69	C₄²×C₁₃₇	#41	1	1	yes	characteristic
70	C₂³×C₁₃₇	#127	1	1	yes	characteristic
71	Q₈×C₁₃₇	—	4	8	no	—
72	Q₈×C₁₃₇	—	8	4	no	—
73	C₂×C₄×C₁₃₇	—	4	4	no	—
74	C₂×C₄×C₁₃₇	—	8	2	no	—
75	C₂×C₄×C₁₃₇	—	8	2	no	—
76	C₂×C₄×C₁₃₇	—	8	2	no	—
77	C₂×C₄×C₁₃₇	—	8	2	no	—
78	C₂×C₄×C₁₃₇	—	8	2	no	—
79	C₂×C₄×C₁₃₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
80	C₂×C₄×C₁₃₇	C₄×D₄	1	1	yes	characteristic
81	C₂×C₄×C₁₃₇	C₄×D₄	1	1	yes	characteristic
82	C₂×C₄×C₁₃₇	#121	1	1	yes	characteristic
83	C₂×C₄×C₁₃₇	#121	1	1	yes	characteristic
84	C₂×C₄×C₁₃₇	#124	1	1	yes	characteristic
85	C₂×C₄×C₁₃₇	C₂×D₈	1	2	yes	—
86	C₂×C₄×C₁₃₇	C₂×SD₁₆	1	2	yes	—
87	C₂×C₄×C₁₃₇	C₂×Q₁₆	1	2	yes	—
88	C₂×C₄×C₁₃₇	C₂×Q₁₆	1	2	yes	—
89	C₂×C₄×C₁₃₇	#135	1	2	yes	—
90	C₂×C₄×C₁₃₇	#135	1	2	yes	—
91	C₈×C₁₃₇	—	2	4	no	—
92	C₈×C₁₃₇	—	2	4	no	—
93	C₈×C₁₃₇	#103	1	8	yes	complemented
94	C₂²×C₁₃₇	#390	1	1	yes	characteristic
95	C₂²×C₁₃₇	#492	1	1	yes	characteristic
96	C₂²×C₁₃₇	#493	1	1	yes	characteristic
97	C₂²×C₁₃₇	#494	1	1	yes	characteristic
98	C₂²×C₁₃₇	#494	1	1	yes	characteristic
99	C₂²×C₁₃₇	#495	1	1	yes	characteristic
100	C₂²×C₁₃₇	#495	1	1	yes	characteristic
101	C₄×C₁₃₇	—	8	4	no	—
102	C₄×C₁₃₇	—	16	2	no	—
103	C₄×C₁₃₇	C₂×#103	1	2	yes	—
104	C₄×C₁₃₇	C₄×D₈	1	2	yes	—
105	C₄×C₁₃₇	C₄×Q₁₆	1	2	yes	—
106	C₄×C₁₃₇	#466	1	2	yes	—
107	C₄×C₁₃₇	#467	1	2	yes	—
108	C₄×C₁₃₇	#487	1	2	yes	—
109	C₂×C₁₃₇	#1706	1	1	yes	characteristic
110	C₂×C₁₃₇	#1707	1	1	yes	characteristic
111	C₂×C₁₃₇	#1710	1	1	yes	characteristic
112	C₂×C₁₃₇	#1713	1	1	yes	characteristic
113	C₂×C₁₃₇	#1956	1	1	yes	characteristic
114	C₂×C₁₃₇	#1969	1	1	yes	characteristic
115	C₂×C₁₃₇	#1970	1	1	yes	characteristic
116	#9823	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
117	C₁₃₇	#9823	1	1	yes	characteristic, direct product factor, Sylow
118	C₂×#500	C₂×C₁₃₇	1	1	yes	characteristic
119	#1990	C₂×C₁₃₇	1	2	yes	—
120	#1956	C₂×C₁₃₇	1	2	yes	—
121	#1845	C₂×C₁₃₇	1	1	yes	characteristic
122	C₂×C₈²	C₂×C₁₃₇	1	1	yes	characteristic
123	C₂×C₄⋊Q₈	C₂²×C₁₃₇	1	2	yes	—
124	#500	—	2	2	no	—
125	#500	C₄×C₁₃₇	1	4	yes	complemented
126	C₂×#107	—	2	2	no	—
127	C₂×#107	—	2	2	no	—
128	C₂×#103	—	2	4	no	—
129	C₂×C₄×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
130	C₂×C₄×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
131	C₂×C₄×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
132	#384	C₂²×C₁₃₇	1	2	yes	—
133	#340	—	2	4	no	—
134	C₈²	—	2	2	no	—
135	C₂²×Q₈	—	2	2	no	—
136	C₂²×C₈	D₄×C₁₃₇	1	2	yes	—
137	C₂²×C₈	D₄×C₁₃₇	1	2	yes	—
138	C₂²×C₈	D₄×C₁₃₇	1	2	yes	—
139	C₄⋊Q₈	—	2	2	no	—
140	C₄⋊Q₈	—	2	2	no	—
141	C₄⋊Q₈	—	2	2	no	—
142	C₄⋊Q₈	C₂×C₄×C₁₃₇	1	4	yes	—
143	C₂×C₄⋊C₄	—	2	2	no	—
144	C₂×C₄⋊C₄	—	2	2	no	—
145	C₂×C₄⋊C₄	—	2	2	no	—
146	C₂×C₄⋊C₄	—	2	2	no	—
147	C₂×C₄⋊C₄	—	2	2	no	—
148	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
149	#107	—	2	8	no	—
150	#107	—	4	4	no	—
151	#103	—	4	8	no	—
152	C₄×C₈	—	2	1	no	—
153	C₄×C₈	—	2	1	no	—
154	C₄×C₈	—	2	1	no	—
155	C₄×C₈	—	2	1	no	—
156	C₄×C₈	C₂×C₄×C₁₃₇	1	2	yes	—
157	C₄×C₈	D₄×C₁₃₇	1	2	yes	—
158	C₂×Q₈	—	2	4	no	—
159	C₂×Q₈	—	4	2	no	—
160	C₂×Q₈	—	4	4	no	—
161	C₂×Q₈	—	4	4	no	—
162	C₂²×C₄	—	4	2	no	—
163	C₂²×C₄	—	4	2	no	—
164	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
165	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
166	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
167	C₂×C₈	—	2	2	no	—
168	C₂×C₈	—	2	2	no	—
169	C₂×C₈	—	2	2	no	—
170	C₂×C₈	—	2	2	no	—
171	C₂×C₈	C₂²⋊C₄×C₁₃₇	1	4	yes	—
172	C₂×C₈	D₈×C₁₃₇	1	4	yes	—
173	C₂×C₈	SD₁₆×C₁₃₇	1	4	yes	—
174	C₂×C₈	Q₁₆×C₁₃₇	1	4	yes	—
175	C₂×C₈	Q₁₆×C₁₃₇	1	4	yes	—
176	C₄⋊C₄	—	2	4	no	—
177	C₄⋊C₄	—	2	4	no	—
178	C₄⋊C₄	—	4	2	no	—
179	C₄⋊C₄	—	4	2	no	—
180	C₄⋊C₄	—	4	4	no	—
181	C₄⋊C₄	—	4	4	no	—
182	C₄⋊C₄	—	4	4	no	—
183	C₄²	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
184	C₄²	C₂×D₄×C₁₃₇	1	1	yes	characteristic
185	C₄²	#41×C₁₃₇	1	1	yes	characteristic
186	C₄²	#41×C₁₃₇	1	1	yes	characteristic
187	C₂³	#127×C₁₃₇	1	1	yes	characteristic
188	Q₈	—	4	8	no	—
189	Q₈	—	8	4	no	—
190	C₂×C₄	—	4	4	no	—
191	C₂×C₄	—	8	2	no	—
192	C₂×C₄	—	8	2	no	—
193	C₂×C₄	—	8	2	no	—
194	C₂×C₄	—	8	2	no	—
195	C₂×C₄	—	8	2	no	—
196	C₂×C₄	C₂×C₂²⋊C₄×C₁₃₇	1	1	yes	characteristic
197	C₂×C₄	C₄×D₄×C₁₃₇	1	1	yes	characteristic
198	C₂×C₄	C₄×D₄×C₁₃₇	1	1	yes	characteristic
199	C₂×C₄	#121×C₁₃₇	1	1	yes	characteristic
200	C₂×C₄	#121×C₁₃₇	1	1	yes	characteristic
201	C₂×C₄	#124×C₁₃₇	1	1	yes	characteristic
202	C₂×C₄	C₂×D₈×C₁₃₇	1	2	yes	—
203	C₂×C₄	C₂×SD₁₆×C₁₃₇	1	2	yes	—
204	C₂×C₄	C₂×Q₁₆×C₁₃₇	1	2	yes	—
205	C₂×C₄	C₂×Q₁₆×C₁₃₇	1	2	yes	—
206	C₂×C₄	#135×C₁₃₇	1	2	yes	—
207	C₂×C₄	#135×C₁₃₇	1	2	yes	—
208	C₈	—	2	4	no	—
209	C₈	—	2	4	no	—
210	C₈	#103×C₁₃₇	1	8	yes	complemented
211	C₂²	#390×C₁₃₇	1	1	yes	characteristic
212	C₂²	#492×C₁₃₇	1	1	yes	characteristic
213	C₂²	#493×C₁₃₇	1	1	yes	characteristic
214	C₂²	#494×C₁₃₇	1	1	yes	characteristic
215	C₂²	#494×C₁₃₇	1	1	yes	characteristic
216	C₂²	#495×C₁₃₇	1	1	yes	characteristic
217	C₂²	#495×C₁₃₇	1	1	yes	characteristic
218	C₄	—	8	4	no	—
219	C₄	—	16	2	no	—
220	C₄	C₂×#103×C₁₃₇	1	2	yes	—
221	C₄	C₄×D₈×C₁₃₇	1	2	yes	—
222	C₄	C₄×Q₁₆×C₁₃₇	1	2	yes	—
223	C₄	#466×C₁₃₇	1	2	yes	—
224	C₄	#467×C₁₃₇	1	2	yes	—
225	C₄	#487×C₁₃₇	1	2	yes	—
226	C₂	#1706×C₁₃₇	1	1	yes	characteristic
227	C₂	#1707×C₁₃₇	1	1	yes	characteristic
228	C₂	#1710×C₁₃₇	1	1	yes	characteristic
229	C₂	#1713×C₁₃₇	1	1	yes	characteristic
230	C₂	#1956×C₁₃₇	1	1	yes	characteristic
231	C₂	#1969×C₁₃₇	1	1	yes	characteristic
232	C₂	#1970×C₁₃₇	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₃₇	#127
commutator	C₄²	C₂²×C₄×C₁₃₇
Frattini, cosocle	C₂×C₄²	C₂³×C₁₃₇
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃₇ ◁ C₂×C₄²×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
8. #500×C₁₃₇	C₄	semidirect
93. C₈×C₁₃₇	#103	semidirect
116. #9823	C₁₃₇	direct (N × H)
117. C₁₃₇	#9823	direct (N × H)
125. #500	C₄×C₁₃₇	semidirect
210. C₈	#103×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#500×C₁₃₇	2
#1990×C₁₃₇	2
#1956×C₁₃₇	2
#1845×C₁₃₇	2
C₂×C₈²×C₁₃₇	2
#9823	137

Subgroups — C137×#9823 #46859209

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#9823 #46859209