Subgroups — C₂×#2235×C₁₃₇ #46883026

Order 35072.

630 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#467×C₁₃₇	C₂	1	1	yes	maximal, characteristic
2	C₂²×#107×C₁₃₇	C₂	1	1	yes	maximal, characteristic
3	#2235×C₁₃₇	C₂	1	8	yes	maximal, direct product factor
4	C₂×#367×C₁₃₇	C₂	1	2	yes	maximal, complemented
5	C₂×#358×C₁₃₇	C₂	1	2	yes	maximal, complemented
6	C₂×C₂²⋊C₁₆×C₁₃₇	C₂	1	1	yes	maximal, characteristic
7	C₂²×Q₁₆×C₁₃₇	C₂²	1	1	yes	characteristic
8	C₂³×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
9	C₂×C₂²⋊Q₈×C₁₃₇	—	2	1	no	—
10	C₂²×C₄⋊C₄×C₁₃₇	—	2	1	no	—
11	C₂²×C₁₆×C₁₃₇	C₂²	1	2	yes	—
12	#467×C₁₃₇	—	2	2	no	—
13	#467×C₁₃₇	C₂²	1	4	yes	—
14	C₂×#107×C₁₃₇	—	2	1	no	—
15	C₂×#107×C₁₃₇	—	2	2	no	—
16	C₂×#107×C₁₃₇	C₂²	1	1	yes	characteristic
17	C₂×#107×C₁₃₇	C₂²	1	2	yes	—
18	C₂×#107×C₁₃₇	C₂²	1	4	yes	—
19	C₂×#103×C₁₃₇	—	2	1	no	—
20	#367×C₁₃₇	C₂²	1	8	yes	complemented
21	#358×C₁₃₇	C₂²	1	8	yes	complemented
22	C₂²⋊C₁₆×C₁₃₇	C₂²	1	4	yes	—
23	C₂²×Q₈×C₁₃₇	—	2	1	no	—
24	C₂³×C₄×C₁₃₇	—	4	1	no	—
25	C₂³×C₄×C₁₃₇	D₄	1	1	yes	characteristic
26	C₂×Q₁₆×C₁₃₇	—	2	1	no	—
27	C₂×Q₁₆×C₁₃₇	—	2	4	no	—
28	C₂×Q₁₆×C₁₃₇	C₂³	1	2	yes	—
29	C₂²×C₈×C₁₃₇	—	2	4	no	—
30	C₂²×C₈×C₁₃₇	D₄	1	1	yes	characteristic
31	C₂²×C₈×C₁₃₇	D₄	1	2	yes	—
32	C₂²×C₈×C₁₃₇	C₂³	1	2	yes	—
33	C₂²×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
34	C₂²⋊Q₈×C₁₃₇	—	2	4	no	—
35	C₂²⋊Q₈×C₁₃₇	—	4	2	no	—
36	C₂×C₄⋊C₄×C₁₃₇	—	2	1	no	—
37	C₂×C₄⋊C₄×C₁₃₇	—	2	4	no	—
38	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
39	C₂×C₄⋊C₄×C₁₃₇	—	4	2	no	—
40	C₂×C₄⋊C₄×C₁₃₇	—	4	1	no	—
41	C₂×C₄⋊C₄×C₁₃₇	—	4	1	no	—
42	C₂×C₂²⋊C₄×C₁₃₇	—	4	1	no	—
43	C₂×C₁₆×C₁₃₇	—	2	4	no	—
44	C₂×C₁₆×C₁₃₇	C₂³	1	4	yes	—
45	#107×C₁₃₇	—	2	4	no	—
46	#107×C₁₃₇	—	2	2	no	—
47	#107×C₁₃₇	—	2	1	no	—
48	#107×C₁₃₇	C₂³	1	2	yes	—
49	#107×C₁₃₇	C₂³	1	4	yes	—
50	#103×C₁₃₇	—	2	4	no	—
51	C₂⁴×C₁₃₇	D₈	1	1	yes	characteristic
52	C₂×Q₈×C₁₃₇	—	2	2	no	—
53	C₂×Q₈×C₁₃₇	—	4	1	no	—
54	C₂×Q₈×C₁₃₇	—	4	2	no	—
55	C₂×Q₈×C₁₃₇	—	4	2	no	—
56	C₂²×C₄×C₁₃₇	—	2	2	no	—
57	C₂²×C₄×C₁₃₇	—	2	2	no	—
58	C₂²×C₄×C₁₃₇	—	4	4	no	—
59	C₂²×C₄×C₁₃₇	—	4	1	no	—
60	C₂²×C₄×C₁₃₇	—	4	1	no	—
61	C₂²×C₄×C₁₃₇	—	4	2	no	—
62	C₂²×C₄×C₁₃₇	—	8	4	no	—
63	C₂²×C₄×C₁₃₇	D₈	1	2	yes	—
64	C₂²×C₄×C₁₃₇	D₈	1	1	yes	characteristic
65	C₂²×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
66	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
67	Q₁₆×C₁₃₇	—	2	4	no	—
68	Q₁₆×C₁₃₇	—	4	2	no	—
69	C₂×C₈×C₁₃₇	—	2	2	no	—
70	C₂×C₈×C₁₃₇	—	2	4	no	—
71	C₂×C₈×C₁₃₇	—	2	4	no	—
72	C₂×C₈×C₁₃₇	C₂×D₄	1	2	yes	—
73	C₂×C₈×C₁₃₇	C₂×D₄	1	1	yes	characteristic
74	C₂×C₈×C₁₃₇	#41	1	4	yes	—
75	C₂×C₈×C₁₃₇	C₂⁴	1	1	yes	characteristic
76	C₄⋊C₄×C₁₃₇	—	2	2	no	—
77	C₄⋊C₄×C₁₃₇	—	2	4	no	—
78	C₄⋊C₄×C₁₃₇	—	4	4	no	—
79	C₄⋊C₄×C₁₃₇	—	4	2	no	—
80	C₄⋊C₄×C₁₃₇	—	4	1	no	—
81	C₄⋊C₄×C₁₃₇	—	4	4	no	—
82	C₂²⋊C₄×C₁₃₇	—	4	4	no	—
83	C₁₆×C₁₃₇	—	2	4	no	—
84	C₂³×C₁₃₇	—	2	2	no	—
85	C₂³×C₁₃₇	—	2	2	no	—
86	C₂³×C₁₃₇	SD₃₂	1	4	yes	—
87	C₂³×C₁₃₇	C₂×D₈	1	2	yes	—
88	C₂³×C₁₃₇	C₂×D₈	1	1	yes	characteristic
89	Q₈×C₁₃₇	—	4	2	no	—
90	Q₈×C₁₃₇	—	4	2	no	—
91	Q₈×C₁₃₇	—	8	2	no	—
92	C₂×C₄×C₁₃₇	—	2	2	no	—
93	C₂×C₄×C₁₃₇	—	2	2	no	—
94	C₂×C₄×C₁₃₇	—	2	2	no	—
95	C₂×C₄×C₁₃₇	—	2	4	no	—
96	C₂×C₄×C₁₃₇	—	4	2	no	—
97	C₂×C₄×C₁₃₇	—	4	2	no	—
98	C₂×C₄×C₁₃₇	—	4	4	no	—
99	C₂×C₄×C₁₃₇	—	8	2	no	—
100	C₂×C₄×C₁₃₇	—	8	8	no	—
101	C₂×C₄×C₁₃₇	—	8	4	no	—
102	C₂×C₄×C₁₃₇	—	8	2	no	—
103	C₂×C₄×C₁₃₇	#123	1	2	yes	—
104	C₂×C₄×C₁₃₇	C₂×D₈	1	2	yes	—
105	C₂×C₄×C₁₃₇	C₂×D₈	1	1	yes	characteristic
106	C₂×C₄×C₁₃₇	#137	1	2	yes	—
107	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
108	C₈×C₁₃₇	—	2	2	no	—
109	C₈×C₁₃₇	#123	1	2	yes	—
110	C₈×C₁₃₇	C₂×#41	1	2	yes	—
111	C₂²×C₁₃₇	—	2	2	no	—
112	C₂²×C₁₃₇	—	2	2	no	—
113	C₂²×C₁₃₇	—	2	8	no	—
114	C₂²×C₁₃₇	#480	1	2	yes	—
115	C₂²×C₁₃₇	C₂×SD₃₂	1	4	yes	—
116	C₂²×C₁₃₇	C₂×SD₃₂	1	2	yes	—
117	C₂²×C₁₃₇	#510	1	2	yes	—
118	C₂²×C₁₃₇	C₂²×D₈	1	1	yes	characteristic
119	C₄×C₁₃₇	—	2	2	no	—
120	C₄×C₁₃₇	—	8	2	no	—
121	C₄×C₁₃₇	—	8	4	no	—
122	C₄×C₁₃₇	—	8	2	no	—
123	C₄×C₁₃₇	#480	1	2	yes	—
124	C₄×C₁₃₇	C₂×#123	1	1	yes	characteristic
125	C₄×C₁₃₇	C₂×#137	1	1	yes	characteristic
126	C₂×C₁₃₇	—	2	4	no	—
127	C₂×C₁₃₇	#2235	1	4	yes	direct product factor
128	C₂×C₁₃₇	C₂×#480	1	1	yes	characteristic
129	C₂×C₁₃₇	C₂²×SD₃₂	1	1	yes	characteristic
130	C₂×C₁₃₇	C₂×#510	1	1	yes	characteristic
131	C₂×#2235	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
132	C₁₃₇	C₂×#2235	1	1	yes	characteristic, direct product factor, Sylow
133	C₂×#467	C₂×C₁₃₇	1	1	yes	characteristic
134	C₂²×#107	C₂×C₁₃₇	1	1	yes	characteristic
135	#2235	C₂×C₁₃₇	1	8	yes	direct product factor
136	C₂×#367	C₂×C₁₃₇	1	2	yes	complemented
137	C₂×#358	C₂×C₁₃₇	1	2	yes	complemented
138	C₂×C₂²⋊C₁₆	C₂×C₁₃₇	1	1	yes	characteristic
139	C₂²×Q₁₆	C₂²×C₁₃₇	1	1	yes	characteristic
140	C₂³×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
141	C₂×C₂²⋊Q₈	—	2	1	no	—
142	C₂²×C₄⋊C₄	—	2	1	no	—
143	C₂²×C₁₆	C₂²×C₁₃₇	1	2	yes	—
144	#467	—	2	2	no	—
145	#467	C₂²×C₁₃₇	1	4	yes	—
146	C₂×#107	—	2	1	no	—
147	C₂×#107	—	2	2	no	—
148	C₂×#107	C₂²×C₁₃₇	1	2	yes	—
149	C₂×#107	C₂²×C₁₃₇	1	4	yes	—
150	C₂×#107	C₂²×C₁₃₇	1	1	yes	characteristic
151	C₂×#103	—	2	1	no	—
152	#367	C₂²×C₁₃₇	1	8	yes	complemented
153	#358	C₂²×C₁₃₇	1	8	yes	complemented
154	C₂²⋊C₁₆	C₂²×C₁₃₇	1	4	yes	—
155	C₂²×Q₈	—	2	1	no	—
156	C₂³×C₄	—	4	1	no	—
157	C₂³×C₄	D₄×C₁₃₇	1	1	yes	characteristic
158	C₂×Q₁₆	—	2	1	no	—
159	C₂×Q₁₆	—	2	4	no	—
160	C₂×Q₁₆	C₂³×C₁₃₇	1	2	yes	—
161	C₂²×C₈	—	2	4	no	—
162	C₂²×C₈	D₄×C₁₃₇	1	1	yes	characteristic
163	C₂²×C₈	D₄×C₁₃₇	1	2	yes	—
164	C₂²×C₈	C₂³×C₁₃₇	1	2	yes	—
165	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
166	C₂²⋊Q₈	—	2	4	no	—
167	C₂²⋊Q₈	—	4	2	no	—
168	C₂×C₄⋊C₄	—	2	4	no	—
169	C₂×C₄⋊C₄	—	2	2	no	—
170	C₂×C₄⋊C₄	—	2	1	no	—
171	C₂×C₄⋊C₄	—	4	1	no	—
172	C₂×C₄⋊C₄	—	4	2	no	—
173	C₂×C₄⋊C₄	—	4	1	no	—
174	C₂×C₂²⋊C₄	—	4	1	no	—
175	C₂×C₁₆	—	2	4	no	—
176	C₂×C₁₆	C₂³×C₁₃₇	1	4	yes	—
177	#107	—	2	4	no	—
178	#107	—	2	2	no	—
179	#107	—	2	1	no	—
180	#107	C₂³×C₁₃₇	1	4	yes	—
181	#107	C₂³×C₁₃₇	1	2	yes	—
182	#103	—	2	4	no	—
183	C₂⁴	D₈×C₁₃₇	1	1	yes	characteristic
184	C₂×Q₈	—	2	2	no	—
185	C₂×Q₈	—	4	1	no	—
186	C₂×Q₈	—	4	2	no	—
187	C₂×Q₈	—	4	2	no	—
188	C₂²×C₄	—	2	2	no	—
189	C₂²×C₄	—	2	2	no	—
190	C₂²×C₄	—	4	1	no	—
191	C₂²×C₄	—	4	1	no	—
192	C₂²×C₄	—	4	4	no	—
193	C₂²×C₄	—	4	2	no	—
194	C₂²×C₄	—	8	4	no	—
195	C₂²×C₄	D₈×C₁₃₇	1	1	yes	characteristic
196	C₂²×C₄	D₈×C₁₃₇	1	2	yes	—
197	C₂²×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
198	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
199	Q₁₆	—	2	4	no	—
200	Q₁₆	—	4	2	no	—
201	C₂×C₈	—	2	2	no	—
202	C₂×C₈	—	2	4	no	—
203	C₂×C₈	—	2	4	no	—
204	C₂×C₈	C₂×D₄×C₁₃₇	1	2	yes	—
205	C₂×C₈	C₂×D₄×C₁₃₇	1	1	yes	characteristic
206	C₂×C₈	#41×C₁₃₇	1	4	yes	—
207	C₂×C₈	C₂⁴×C₁₃₇	1	1	yes	characteristic
208	C₄⋊C₄	—	2	2	no	—
209	C₄⋊C₄	—	2	4	no	—
210	C₄⋊C₄	—	4	4	no	—
211	C₄⋊C₄	—	4	2	no	—
212	C₄⋊C₄	—	4	1	no	—
213	C₄⋊C₄	—	4	4	no	—
214	C₂²⋊C₄	—	4	4	no	—
215	C₁₆	—	2	4	no	—
216	C₂³	—	2	2	no	—
217	C₂³	—	2	2	no	—
218	C₂³	SD₃₂×C₁₃₇	1	4	yes	—
219	C₂³	C₂×D₈×C₁₃₇	1	2	yes	—
220	C₂³	C₂×D₈×C₁₃₇	1	1	yes	characteristic
221	Q₈	—	4	2	no	—
222	Q₈	—	4	2	no	—
223	Q₈	—	8	2	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	2	4	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	4	2	no	—
229	C₂×C₄	—	4	2	no	—
230	C₂×C₄	—	4	4	no	—
231	C₂×C₄	—	8	2	no	—
232	C₂×C₄	—	8	8	no	—
233	C₂×C₄	—	8	4	no	—
234	C₂×C₄	—	8	2	no	—
235	C₂×C₄	#123×C₁₃₇	1	2	yes	—
236	C₂×C₄	C₂×D₈×C₁₃₇	1	2	yes	—
237	C₂×C₄	C₂×D₈×C₁₃₇	1	1	yes	characteristic
238	C₂×C₄	#137×C₁₃₇	1	2	yes	—
239	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
240	C₈	—	2	2	no	—
241	C₈	#123×C₁₃₇	1	2	yes	—
242	C₈	C₂×#41×C₁₃₇	1	2	yes	—
243	C₂²	—	2	2	no	—
244	C₂²	—	2	2	no	—
245	C₂²	—	2	8	no	—
246	C₂²	#480×C₁₃₇	1	2	yes	—
247	C₂²	C₂×SD₃₂×C₁₃₇	1	4	yes	—
248	C₂²	C₂×SD₃₂×C₁₃₇	1	2	yes	—
249	C₂²	#510×C₁₃₇	1	2	yes	—
250	C₂²	C₂²×D₈×C₁₃₇	1	1	yes	characteristic
251	C₄	—	2	2	no	—
252	C₄	—	8	2	no	—
253	C₄	—	8	4	no	—
254	C₄	—	8	2	no	—
255	C₄	#480×C₁₃₇	1	2	yes	—
256	C₄	C₂×#123×C₁₃₇	1	1	yes	characteristic
257	C₄	C₂×#137×C₁₃₇	1	1	yes	characteristic
258	C₂	—	2	4	no	—
259	C₂	#2235×C₁₃₇	1	4	yes	direct product factor
260	C₂	C₂×#480×C₁₃₇	1	1	yes	characteristic
261	C₂	C₂²×SD₃₂×C₁₃₇	1	1	yes	characteristic
262	C₂	C₂×#510×C₁₃₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₁₃₇	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₁₃₇
prime radical	C₂⁴×C₁₃₇	D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃₇ ◁ C₂³×C₄×C₁₃₇ ◁ C₂³×C₈×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. #2235×C₁₃₇	C₂	direct (N × H)
4. C₂×#367×C₁₃₇	C₂	semidirect
5. C₂×#358×C₁₃₇	C₂	semidirect
20. #367×C₁₃₇	C₂²	semidirect
21. #358×C₁₃₇	C₂²	semidirect
127. C₂×C₁₃₇	#2235	direct (N × H)
131. C₂×#2235	C₁₃₇	direct (N × H)
132. C₁₃₇	C₂×#2235	direct (N × H)
135. #2235	C₂×C₁₃₇	direct (N × H)
136. C₂×#367	C₂×C₁₃₇	semidirect
137. C₂×#358	C₂×C₁₃₇	semidirect
152. #367	C₂²×C₁₃₇	semidirect
153. #358	C₂²×C₁₃₇	semidirect
259. C₂	#2235×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
C₂×#467×C₁₃₇	2
C₂²×#107×C₁₃₇	2
#2235×C₁₃₇	2
C₂×#367×C₁₃₇	2
C₂×#358×C₁₃₇	2
C₂×C₂²⋊C₁₆×C₁₃₇	2
C₂×#2235	137

Subgroups — C2×#2235×C137 #46883026

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#2235×C₁₃₇ #46883026