Subgroups — C₂²×#341×C₁₃₇ #46869735

Order 35072.

2238 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	C₂³×M₁₆×C₁₃₇	C₂	1	2	yes	maximal
2	C₂⁴×C₈×C₁₃₇	C₂	1	1	yes	maximal, characteristic
3	C₂×#341×C₁₃₇	C₂	1	12	yes	maximal, direct product factor
4	C₂⁴×C₄×C₁₃₇	C₂²	1	1	yes	characteristic
5	C₂²×M₁₆×C₁₃₇	—	2	8	no	—
6	C₂²×M₁₆×C₁₃₇	—	2	6	no	—
7	C₂²×M₁₆×C₁₃₇	C₄	1	16	yes	—
8	C₂²×M₁₆×C₁₃₇	C₂²	1	12	yes	—
9	C₂³×C₈×C₁₃₇	—	2	8	no	—
10	C₂³×C₈×C₁₃₇	—	2	2	no	—
11	C₂³×C₈×C₁₃₇	C₄	1	2	yes	—
12	C₂³×C₈×C₁₃₇	C₄	1	6	yes	—
13	C₂³×C₈×C₁₃₇	C₂²	1	6	yes	—
14	#341×C₁₃₇	C₂²	1	16	yes	direct product factor
15	C₂⁵×C₁₃₇	Q₈	1	1	yes	characteristic
16	C₂³×C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
17	C₂³×C₄×C₁₃₇	C₂×C₄	1	3	yes	—
18	C₂³×C₄×C₁₃₇	C₂×C₄	1	8	yes	—
19	C₂³×C₄×C₁₃₇	D₄	1	8	yes	—
20	C₂³×C₄×C₁₃₇	D₄	1	3	yes	—
21	C₂³×C₄×C₁₃₇	D₄	1	1	yes	characteristic
22	C₂³×C₄×C₁₃₇	Q₈	1	3	yes	—
23	C₂³×C₄×C₁₃₇	C₂³	1	3	yes	—
24	C₂×M₁₆×C₁₃₇	—	2	48	no	—
25	C₂×M₁₆×C₁₃₇	—	2	24	no	—
26	C₂×M₁₆×C₁₃₇	—	2	12	no	—
27	C₂×M₁₆×C₁₃₇	C₂×C₄	1	48	yes	—
28	C₂×M₁₆×C₁₃₇	C₂³	1	8	yes	—
29	C₂²×C₈×C₁₃₇	—	2	24	no	—
30	C₂²×C₈×C₁₃₇	—	2	8	no	—
31	C₂²×C₈×C₁₃₇	—	2	16	no	—
32	C₂²×C₈×C₁₃₇	—	2	24	no	—
33	C₂²×C₈×C₁₃₇	—	2	12	no	—
34	C₂²×C₈×C₁₃₇	C₂×C₄	1	12	yes	—
35	C₂²×C₈×C₁₃₇	C₂×C₄	1	12	yes	—
36	C₂²×C₈×C₁₃₇	C₂³	1	4	yes	—
37	C₂⁴×C₁₃₇	—	2	8	no	—
38	C₂⁴×C₁₃₇	C₄⋊C₄	1	8	yes	—
39	C₂⁴×C₁₃₇	C₄⋊C₄	1	3	yes	—
40	C₂⁴×C₁₃₇	C₄⋊C₄	1	1	yes	characteristic
41	C₂⁴×C₁₃₇	C₂×Q₈	1	3	yes	—
42	C₂²×C₄×C₁₃₇	C₄²	1	12	yes	—
43	C₂²×C₄×C₁₃₇	C₄²	1	4	yes	—
44	C₂²×C₄×C₁₃₇	C₂²⋊C₄	1	12	yes	—
45	C₂²×C₄×C₁₃₇	C₂²⋊C₄	1	4	yes	—
46	C₂²×C₄×C₁₃₇	C₂²⋊C₄	1	8	yes	—
47	C₂²×C₄×C₁₃₇	C₂²⋊C₄	1	24	yes	—
48	C₂²×C₄×C₁₃₇	C₄⋊C₄	1	6	yes	—
49	C₂²×C₄×C₁₃₇	C₄⋊C₄	1	24	yes	—
50	C₂²×C₄×C₁₃₇	C₄⋊C₄	1	3	yes	—
51	C₂²×C₄×C₁₃₇	C₄⋊C₄	1	3	yes	—
52	C₂²×C₄×C₁₃₇	C₂²×C₄	1	3	yes	—
53	C₂²×C₄×C₁₃₇	C₂²×C₄	1	3	yes	—
54	C₂²×C₄×C₁₃₇	C₂²×C₄	1	12	yes	—
55	C₂²×C₄×C₁₃₇	C₂×D₄	1	3	yes	—
56	C₂²×C₄×C₁₃₇	C₂×D₄	1	12	yes	—
57	C₂²×C₄×C₁₃₇	C₂×D₄	1	3	yes	—
58	C₂²×C₄×C₁₃₇	C₂×Q₈	1	3	yes	—
59	C₂²×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
60	M₁₆×C₁₃₇	—	2	48	no	—
61	M₁₆×C₁₃₇	—	2	8	no	—
62	M₁₆×C₁₃₇	C₂²×C₄	1	16	yes	—
63	C₂×C₈×C₁₃₇	—	2	24	no	—
64	C₂×C₈×C₁₃₇	—	2	24	no	—
65	C₂×C₈×C₁₃₇	—	2	48	no	—
66	C₂×C₈×C₁₃₇	—	2	8	no	—
67	C₂×C₈×C₁₃₇	—	2	8	no	—
68	C₂×C₈×C₁₃₇	C₂²×C₄	1	8	yes	—
69	C₂³×C₁₃₇	—	2	32	no	—
70	C₂³×C₁₃₇	—	2	12	no	—
71	C₂³×C₁₃₇	—	2	12	no	—
72	C₂³×C₁₃₇	#95	1	12	yes	—
73	C₂³×C₁₃₇	#95	1	4	yes	—
74	C₂³×C₁₃₇	#108	1	8	yes	—
75	C₂³×C₁₃₇	C₂×C₄⋊C₄	1	3	yes	—
76	C₂³×C₁₃₇	C₂×C₄⋊C₄	1	12	yes	—
77	C₂³×C₁₃₇	C₂×C₄⋊C₄	1	3	yes	—
78	C₂³×C₁₃₇	C₂²×Q₈	1	1	yes	characteristic
79	C₂×C₄×C₁₃₇	#95	1	24	yes	—
80	C₂×C₄×C₁₃₇	#95	1	12	yes	—
81	C₂×C₄×C₁₃₇	#95	1	12	yes	—
82	C₂×C₄×C₁₃₇	C₂×C₄²	1	6	yes	—
83	C₂×C₄×C₁₃₇	C₂×C₄²	1	6	yes	—
84	C₂×C₄×C₁₃₇	C₂×C₂²⋊C₄	1	6	yes	—
85	C₂×C₄×C₁₃₇	C₂×C₂²⋊C₄	1	6	yes	—
86	C₂×C₄×C₁₃₇	C₂×C₂²⋊C₄	1	12	yes	—
87	C₂×C₄×C₁₃₇	C₂×C₂²⋊C₄	1	12	yes	—
88	C₂×C₄×C₁₃₇	C₂×C₄⋊C₄	1	12	yes	—
89	C₂×C₄×C₁₃₇	C₂×C₄⋊C₄	1	3	yes	—
90	C₂×C₄×C₁₃₇	C₂×C₄⋊C₄	1	3	yes	—
91	C₂×C₄×C₁₃₇	C₂³×C₄	1	1	yes	characteristic
92	C₂×C₄×C₁₃₇	C₂³×C₄	1	2	yes	—
93	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
94	C₂×C₄×C₁₃₇	C₂²×D₄	1	2	yes	—
95	C₈×C₁₃₇	—	2	8	no	—
96	C₈×C₁₃₇	—	2	16	no	—
97	C₂²×C₁₃₇	—	2	6	no	—
98	C₂²×C₁₃₇	—	2	2	no	—
99	C₂²×C₁₃₇	—	2	24	no	—
100	C₂²×C₁₃₇	—	2	24	no	—
101	C₂²×C₁₃₇	#341	1	16	yes	direct product factor
102	C₂²×C₁₃₇	C₂×#95	1	6	yes	—
103	C₂²×C₁₃₇	C₂×#95	1	6	yes	—
104	C₂²×C₁₃₇	C₂×#108	1	12	yes	—
105	C₂²×C₁₃₇	C₂²×C₄⋊C₄	1	2	yes	—
106	C₂²×C₁₃₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
107	C₄×C₁₃₇	C₂×#95	1	6	yes	—
108	C₄×C₁₃₇	C₂×#95	1	6	yes	—
109	C₄×C₁₃₇	C₂²×C₄²	1	1	yes	characteristic
110	C₄×C₁₃₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
111	C₄×C₁₃₇	C₂²×C₂²⋊C₄	1	2	yes	—
112	C₂×C₁₃₇	—	2	2	no	—
113	C₂×C₁₃₇	—	2	6	no	—
114	C₂×C₁₃₇	C₂×#341	1	12	yes	direct product factor
115	C₂×C₁₃₇	C₂²×#95	1	1	yes	characteristic
116	C₂×C₁₃₇	C₂²×#108	1	2	yes	—
117	C₂²×#341	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
118	C₁₃₇	C₂²×#341	1	1	yes	characteristic, direct product factor, Sylow
119	C₂³×M₁₆	C₂×C₁₃₇	1	2	yes	—
120	C₂⁴×C₈	C₂×C₁₃₇	1	1	yes	characteristic
121	C₂×#341	C₂×C₁₃₇	1	12	yes	direct product factor
122	C₂⁴×C₄	C₂²×C₁₃₇	1	1	yes	characteristic
123	C₂²×M₁₆	—	2	8	no	—
124	C₂²×M₁₆	—	2	6	no	—
125	C₂²×M₁₆	C₄×C₁₃₇	1	16	yes	—
126	C₂²×M₁₆	C₂²×C₁₃₇	1	12	yes	—
127	C₂³×C₈	—	2	8	no	—
128	C₂³×C₈	—	2	2	no	—
129	C₂³×C₈	C₄×C₁₃₇	1	6	yes	—
130	C₂³×C₈	C₄×C₁₃₇	1	2	yes	—
131	C₂³×C₈	C₂²×C₁₃₇	1	6	yes	—
132	#341	C₂²×C₁₃₇	1	16	yes	direct product factor
133	C₂⁵	Q₈×C₁₃₇	1	1	yes	characteristic
134	C₂³×C₄	C₂×C₄×C₁₃₇	1	3	yes	—
135	C₂³×C₄	C₂×C₄×C₁₃₇	1	1	yes	characteristic
136	C₂³×C₄	C₂×C₄×C₁₃₇	1	8	yes	—
137	C₂³×C₄	D₄×C₁₃₇	1	1	yes	characteristic
138	C₂³×C₄	D₄×C₁₃₇	1	8	yes	—
139	C₂³×C₄	D₄×C₁₃₇	1	3	yes	—
140	C₂³×C₄	Q₈×C₁₃₇	1	3	yes	—
141	C₂³×C₄	C₂³×C₁₃₇	1	3	yes	—
142	C₂×M₁₆	—	2	48	no	—
143	C₂×M₁₆	—	2	24	no	—
144	C₂×M₁₆	—	2	12	no	—
145	C₂×M₁₆	C₂×C₄×C₁₃₇	1	48	yes	—
146	C₂×M₁₆	C₂³×C₁₃₇	1	8	yes	—
147	C₂²×C₈	—	2	24	no	—
148	C₂²×C₈	—	2	8	no	—
149	C₂²×C₈	—	2	16	no	—
150	C₂²×C₈	—	2	24	no	—
151	C₂²×C₈	—	2	12	no	—
152	C₂²×C₈	C₂×C₄×C₁₃₇	1	12	yes	—
153	C₂²×C₈	C₂×C₄×C₁₃₇	1	12	yes	—
154	C₂²×C₈	C₂³×C₁₃₇	1	4	yes	—
155	C₂⁴	—	2	8	no	—
156	C₂⁴	C₄⋊C₄×C₁₃₇	1	3	yes	—
157	C₂⁴	C₄⋊C₄×C₁₃₇	1	8	yes	—
158	C₂⁴	C₄⋊C₄×C₁₃₇	1	1	yes	characteristic
159	C₂⁴	C₂×Q₈×C₁₃₇	1	3	yes	—
160	C₂²×C₄	C₄²×C₁₃₇	1	12	yes	—
161	C₂²×C₄	C₄²×C₁₃₇	1	4	yes	—
162	C₂²×C₄	C₂²⋊C₄×C₁₃₇	1	12	yes	—
163	C₂²×C₄	C₂²⋊C₄×C₁₃₇	1	4	yes	—
164	C₂²×C₄	C₂²⋊C₄×C₁₃₇	1	8	yes	—
165	C₂²×C₄	C₂²⋊C₄×C₁₃₇	1	24	yes	—
166	C₂²×C₄	C₄⋊C₄×C₁₃₇	1	6	yes	—
167	C₂²×C₄	C₄⋊C₄×C₁₃₇	1	24	yes	—
168	C₂²×C₄	C₄⋊C₄×C₁₃₇	1	3	yes	—
169	C₂²×C₄	C₄⋊C₄×C₁₃₇	1	3	yes	—
170	C₂²×C₄	C₂²×C₄×C₁₃₇	1	3	yes	—
171	C₂²×C₄	C₂²×C₄×C₁₃₇	1	3	yes	—
172	C₂²×C₄	C₂²×C₄×C₁₃₇	1	12	yes	—
173	C₂²×C₄	C₂×D₄×C₁₃₇	1	3	yes	—
174	C₂²×C₄	C₂×D₄×C₁₃₇	1	12	yes	—
175	C₂²×C₄	C₂×D₄×C₁₃₇	1	3	yes	—
176	C₂²×C₄	C₂×Q₈×C₁₃₇	1	3	yes	—
177	C₂²×C₄	C₂⁴×C₁₃₇	1	1	yes	characteristic
178	M₁₆	—	2	48	no	—
179	M₁₆	—	2	8	no	—
180	M₁₆	C₂²×C₄×C₁₃₇	1	16	yes	—
181	C₂×C₈	—	2	24	no	—
182	C₂×C₈	—	2	24	no	—
183	C₂×C₈	—	2	48	no	—
184	C₂×C₈	—	2	8	no	—
185	C₂×C₈	—	2	8	no	—
186	C₂×C₈	C₂²×C₄×C₁₃₇	1	8	yes	—
187	C₂³	—	2	12	no	—
188	C₂³	—	2	12	no	—
189	C₂³	—	2	32	no	—
190	C₂³	#95×C₁₃₇	1	12	yes	—
191	C₂³	#95×C₁₃₇	1	4	yes	—
192	C₂³	#108×C₁₃₇	1	8	yes	—
193	C₂³	C₂×C₄⋊C₄×C₁₃₇	1	3	yes	—
194	C₂³	C₂×C₄⋊C₄×C₁₃₇	1	12	yes	—
195	C₂³	C₂×C₄⋊C₄×C₁₃₇	1	3	yes	—
196	C₂³	C₂²×Q₈×C₁₃₇	1	1	yes	characteristic
197	C₂×C₄	#95×C₁₃₇	1	24	yes	—
198	C₂×C₄	#95×C₁₃₇	1	12	yes	—
199	C₂×C₄	#95×C₁₃₇	1	12	yes	—
200	C₂×C₄	C₂×C₄²×C₁₃₇	1	6	yes	—
201	C₂×C₄	C₂×C₄²×C₁₃₇	1	6	yes	—
202	C₂×C₄	C₂×C₂²⋊C₄×C₁₃₇	1	6	yes	—
203	C₂×C₄	C₂×C₂²⋊C₄×C₁₃₇	1	6	yes	—
204	C₂×C₄	C₂×C₂²⋊C₄×C₁₃₇	1	12	yes	—
205	C₂×C₄	C₂×C₂²⋊C₄×C₁₃₇	1	12	yes	—
206	C₂×C₄	C₂×C₄⋊C₄×C₁₃₇	1	12	yes	—
207	C₂×C₄	C₂×C₄⋊C₄×C₁₃₇	1	3	yes	—
208	C₂×C₄	C₂×C₄⋊C₄×C₁₃₇	1	3	yes	—
209	C₂×C₄	C₂³×C₄×C₁₃₇	1	1	yes	characteristic
210	C₂×C₄	C₂³×C₄×C₁₃₇	1	2	yes	—
211	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
212	C₂×C₄	C₂²×D₄×C₁₃₇	1	2	yes	—
213	C₈	—	2	8	no	—
214	C₈	—	2	16	no	—
215	C₂²	—	2	6	no	—
216	C₂²	—	2	2	no	—
217	C₂²	—	2	24	no	—
218	C₂²	—	2	24	no	—
219	C₂²	#341×C₁₃₇	1	16	yes	direct product factor
220	C₂²	C₂×#95×C₁₃₇	1	6	yes	—
221	C₂²	C₂×#95×C₁₃₇	1	6	yes	—
222	C₂²	C₂×#108×C₁₃₇	1	12	yes	—
223	C₂²	C₂²×C₄⋊C₄×C₁₃₇	1	2	yes	—
224	C₂²	C₂²×C₄⋊C₄×C₁₃₇	1	1	yes	characteristic
225	C₄	C₂×#95×C₁₃₇	1	6	yes	—
226	C₄	C₂×#95×C₁₃₇	1	6	yes	—
227	C₄	C₂²×C₄²×C₁₃₇	1	1	yes	characteristic
228	C₄	C₂²×C₂²⋊C₄×C₁₃₇	1	1	yes	characteristic
229	C₄	C₂²×C₂²⋊C₄×C₁₃₇	1	2	yes	—
230	C₂	—	2	6	no	—
231	C₂	—	2	2	no	—
232	C₂	C₂×#341×C₁₃₇	1	12	yes	direct product factor
233	C₂	C₂²×#95×C₁₃₇	1	1	yes	characteristic
234	C₂	C₂²×#108×C₁₃₇	1	2	yes	—

	subgroup	G/H
center	C₂³×C₄×C₁₃₇	D₄
commutator	C₄	C₂²×C₄²×C₁₃₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃₇
socle	C₂⁴×C₁₃₇	C₄⋊C₄
prime radical	C₂⁵×C₁₃₇	Q₈
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₄×C₁₃₇ ◁ C₂⁴×C₄×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂×#341×C₁₃₇	C₂	direct (N × H)
14. #341×C₁₃₇	C₂²	direct (N × H)
101. C₂²×C₁₃₇	#341	direct (N × H)
114. C₂×C₁₃₇	C₂×#341	direct (N × H)
117. C₂²×#341	C₁₃₇	direct (N × H)
118. C₁₃₇	C₂²×#341	direct (N × H)
121. C₂×#341	C₂×C₁₃₇	direct (N × H)
132. #341	C₂²×C₁₃₇	direct (N × H)
219. C₂²	#341×C₁₃₇	direct (N × H)
232. C₂	C₂×#341×C₁₃₇	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×M₁₆×C₁₃₇	2
C₂⁴×C₈×C₁₃₇	2
C₂×#341×C₁₃₇	2
C₂²×#341	137

Subgroups — C22×#341×C137 #46869735

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#341×C₁₃₇ #46869735