Subgroups — C₁₃₇×#9836 #46859222

Order 35072.

510 conjugacy classes of subgroups.

The subgroup lattice has 218 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	#1844×C₁₃₇	C₂	1	1	yes	maximal, characteristic
5	C₂×#322×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×#106×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	#322×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	#106×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	4	no	—
53	C₂×C₈×C₁₃₇	—	2	4	no	—
54	C₂×C₈×C₁₃₇	C₂²⋊C₄	1	4	yes	—
55	C₂×C₈×C₁₃₇	SD₁₆	1	4	yes	—
56	C₂×C₈×C₁₃₇	Q₁₆	1	4	yes	—
57	C₄⋊C₄×C₁₃₇	—	2	4	no	—
58	C₄⋊C₄×C₁₃₇	—	2	4	no	—
59	C₄⋊C₄×C₁₃₇	—	4	2	no	—
60	C₄⋊C₄×C₁₃₇	—	4	2	no	—
61	C₄⋊C₄×C₁₃₇	—	4	4	no	—
62	C₄⋊C₄×C₁₃₇	—	4	4	no	—
63	C₄⋊C₄×C₁₃₇	—	4	4	no	—
64	C₄²×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
65	C₄²×C₁₃₇	C₂×D₄	1	1	yes	characteristic
66	C₄²×C₁₃₇	#41	1	1	yes	characteristic
67	C₄²×C₁₃₇	#41	1	1	yes	characteristic
68	C₂³×C₁₃₇	#127	1	1	yes	characteristic
69	Q₈×C₁₃₇	—	4	8	no	—
70	Q₈×C₁₃₇	—	8	4	no	—
71	C₂×C₄×C₁₃₇	—	4	4	no	—
72	C₂×C₄×C₁₃₇	—	8	2	no	—
73	C₂×C₄×C₁₃₇	—	8	2	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	C₂×C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₄×C₁₃₇	C₄×D₄	1	1	yes	characteristic
79	C₂×C₄×C₁₃₇	C₄×D₄	1	1	yes	characteristic
80	C₂×C₄×C₁₃₇	#121	1	1	yes	characteristic
81	C₂×C₄×C₁₃₇	#121	1	1	yes	characteristic
82	C₂×C₄×C₁₃₇	#124	1	1	yes	characteristic
83	C₂×C₄×C₁₃₇	C₂×SD₁₆	1	2	yes	—
84	C₂×C₄×C₁₃₇	C₂×Q₁₆	1	2	yes	—
85	C₂×C₄×C₁₃₇	AGL(1,ℤ/8)	1	4	yes	—
86	C₂×C₄×C₁₃₇	#137	1	4	yes	—
87	C₈×C₁₃₇	—	4	2	no	—
88	C₈×C₁₃₇	—	4	2	no	—
89	C₈×C₁₃₇	#103	1	8	yes	complemented
90	C₂²×C₁₃₇	#390	1	1	yes	characteristic
91	C₂²×C₁₃₇	#492	1	1	yes	characteristic
92	C₂²×C₁₃₇	#494	1	1	yes	characteristic
93	C₂²×C₁₃₇	#496	1	2	yes	—
94	C₂²×C₁₃₇	#497	1	2	yes	—
95	C₄×C₁₃₇	—	8	4	no	—
96	C₄×C₁₃₇	—	16	2	no	—
97	C₄×C₁₃₇	C₂×#103	1	2	yes	—
98	C₄×C₁₃₇	#441	1	2	yes	—
99	C₄×C₁₃₇	#442	1	2	yes	—
100	C₄×C₁₃₇	#469	1	2	yes	—
101	C₄×C₁₃₇	#470	1	2	yes	—
102	C₄×C₁₃₇	#487	1	2	yes	—
103	C₂×C₁₃₇	#1714	1	2	yes	—
104	C₂×C₁₃₇	#1723	1	2	yes	—
105	C₂×C₁₃₇	#1956	1	1	yes	characteristic
106	C₂×C₁₃₇	#1971	1	1	yes	characteristic
107	C₂×C₁₃₇	#1972	1	1	yes	characteristic
108	#9836	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
109	C₁₃₇	#9836	1	1	yes	characteristic, direct product factor, Sylow
110	C₂×#500	C₂×C₁₃₇	1	1	yes	characteristic
111	#1990	C₂×C₁₃₇	1	2	yes	—
112	#1956	C₂×C₁₃₇	1	2	yes	—
113	#1844	C₂×C₁₃₇	1	1	yes	characteristic
114	C₂×#322	C₂×C₁₃₇	1	1	yes	characteristic
115	C₂×C₄⋊Q₈	C₂²×C₁₃₇	1	2	yes	—
116	#500	—	2	2	no	—
117	#500	C₄×C₁₃₇	1	4	yes	complemented
118	C₂×#107	—	2	2	no	—
119	C₂×#106	—	2	2	no	—
120	C₂×#103	—	2	4	no	—
121	C₂×C₄×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
122	C₂×C₄×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
123	C₂×C₄×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
124	#384	C₂²×C₁₃₇	1	2	yes	—
125	#340	—	2	4	no	—
126	#322	—	2	2	no	—
127	C₂²×Q₈	—	2	2	no	—
128	C₂²×C₈	D₄×C₁₃₇	1	2	yes	—
129	C₂²×C₈	D₄×C₁₃₇	1	2	yes	—
130	C₂²×C₈	D₄×C₁₃₇	1	2	yes	—
131	C₄⋊Q₈	—	2	2	no	—
132	C₄⋊Q₈	—	2	2	no	—
133	C₄⋊Q₈	—	2	2	no	—
134	C₄⋊Q₈	C₂×C₄×C₁₃₇	1	4	yes	—
135	C₂×C₄⋊C₄	—	2	2	no	—
136	C₂×C₄⋊C₄	—	2	2	no	—
137	C₂×C₄⋊C₄	—	2	2	no	—
138	C₂×C₄⋊C₄	—	2	2	no	—
139	C₂×C₄⋊C₄	—	2	2	no	—
140	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
141	#107	—	2	8	no	—
142	#106	—	4	4	no	—
143	#103	—	4	8	no	—
144	C₄×C₈	—	2	1	no	—
145	C₄×C₈	—	2	1	no	—
146	C₄×C₈	—	2	1	no	—
147	C₄×C₈	—	2	1	no	—
148	C₄×C₈	C₂×C₄×C₁₃₇	1	2	yes	—
149	C₄×C₈	D₄×C₁₃₇	1	2	yes	—
150	C₂×Q₈	—	2	4	no	—
151	C₂×Q₈	—	4	2	no	—
152	C₂×Q₈	—	4	4	no	—
153	C₂×Q₈	—	4	4	no	—
154	C₂²×C₄	—	4	2	no	—
155	C₂²×C₄	—	4	2	no	—
156	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
157	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
158	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
159	C₂×C₈	—	2	2	no	—
160	C₂×C₈	—	2	2	no	—
161	C₂×C₈	—	2	4	no	—
162	C₂×C₈	—	2	4	no	—
163	C₂×C₈	C₂²⋊C₄×C₁₃₇	1	4	yes	—
164	C₂×C₈	SD₁₆×C₁₃₇	1	4	yes	—
165	C₂×C₈	Q₁₆×C₁₃₇	1	4	yes	—
166	C₄⋊C₄	—	2	4	no	—
167	C₄⋊C₄	—	2	4	no	—
168	C₄⋊C₄	—	4	2	no	—
169	C₄⋊C₄	—	4	2	no	—
170	C₄⋊C₄	—	4	4	no	—
171	C₄⋊C₄	—	4	4	no	—
172	C₄⋊C₄	—	4	4	no	—
173	C₄²	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
174	C₄²	C₂×D₄×C₁₃₇	1	1	yes	characteristic
175	C₄²	#41×C₁₃₇	1	1	yes	characteristic
176	C₄²	#41×C₁₃₇	1	1	yes	characteristic
177	C₂³	#127×C₁₃₇	1	1	yes	characteristic
178	Q₈	—	4	8	no	—
179	Q₈	—	8	4	no	—
180	C₂×C₄	—	4	4	no	—
181	C₂×C₄	—	8	2	no	—
182	C₂×C₄	—	8	2	no	—
183	C₂×C₄	—	8	2	no	—
184	C₂×C₄	—	8	2	no	—
185	C₂×C₄	—	8	2	no	—
186	C₂×C₄	C₂×C₂²⋊C₄×C₁₃₇	1	1	yes	characteristic
187	C₂×C₄	C₄×D₄×C₁₃₇	1	1	yes	characteristic
188	C₂×C₄	C₄×D₄×C₁₃₇	1	1	yes	characteristic
189	C₂×C₄	#121×C₁₃₇	1	1	yes	characteristic
190	C₂×C₄	#121×C₁₃₇	1	1	yes	characteristic
191	C₂×C₄	#124×C₁₃₇	1	1	yes	characteristic
192	C₂×C₄	C₂×SD₁₆×C₁₃₇	1	2	yes	—
193	C₂×C₄	C₂×Q₁₆×C₁₃₇	1	2	yes	—
194	C₂×C₄	AGL(1,ℤ/8)×C₁₃₇	1	4	yes	—
195	C₂×C₄	#137×C₁₃₇	1	4	yes	—
196	C₈	—	4	2	no	—
197	C₈	—	4	2	no	—
198	C₈	#103×C₁₃₇	1	8	yes	complemented
199	C₂²	#390×C₁₃₇	1	1	yes	characteristic
200	C₂²	#492×C₁₃₇	1	1	yes	characteristic
201	C₂²	#494×C₁₃₇	1	1	yes	characteristic
202	C₂²	#496×C₁₃₇	1	2	yes	—
203	C₂²	#497×C₁₃₇	1	2	yes	—
204	C₄	—	8	4	no	—
205	C₄	—	16	2	no	—
206	C₄	C₂×#103×C₁₃₇	1	2	yes	—
207	C₄	#441×C₁₃₇	1	2	yes	—
208	C₄	#442×C₁₃₇	1	2	yes	—
209	C₄	#469×C₁₃₇	1	2	yes	—
210	C₄	#470×C₁₃₇	1	2	yes	—
211	C₄	#487×C₁₃₇	1	2	yes	—
212	C₂	#1714×C₁₃₇	1	2	yes	—
213	C₂	#1723×C₁₃₇	1	2	yes	—
214	C₂	#1956×C₁₃₇	1	1	yes	characteristic
215	C₂	#1971×C₁₃₇	1	1	yes	characteristic
216	C₂	#1972×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
89. C₈×C₁₃₇	#103	semidirect
108. #9836	C₁₃₇	direct (N × H)
109. C₁₃₇	#9836	direct (N × H)
117. #500	C₄×C₁₃₇	semidirect
198. 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
#1844×C₁₃₇	2
C₂×#322×C₁₃₇	2
#9836	137

Subgroups — C137×#9836 #46859222

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#9836 #46859222