Subgroups — C₁₃₇×#22617 #46872003

Order 35072.

866 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#107×C₁₃₇	C₂	1	1	yes	maximal, characteristic
2	#2344×C₁₃₇	C₂	1	2	yes	maximal
3	#1943×C₁₃₇	C₂	1	2	yes	maximal, complemented
4	#1921×C₁₃₇	C₂	1	4	yes	maximal, complemented
5	#1823×C₁₃₇	C₂	1	2	yes	maximal
6	#1807×C₁₃₇	C₂	1	1	yes	maximal, characteristic
7	#1776×C₁₃₇	C₂	1	2	yes	maximal, complemented
8	#1752×C₁₃₇	C₂	1	1	yes	maximal, characteristic
9	C₂³×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
10	C₂×#117×C₁₃₇	—	2	2	no	—
11	C₂²×C₄⋊C₄×C₁₃₇	C₂²	1	2	yes	—
12	C₂×#107×C₁₃₇	C₄	1	8	yes	complemented
13	C₂×#107×C₁₃₇	C₂²	1	1	yes	characteristic
14	C₂×#107×C₁₃₇	C₂²	1	2	yes	—
15	C₂×#107×C₁₃₇	C₂²	1	1	yes	characteristic
16	C₂×#106×C₁₃₇	C₂²	1	2	yes	—
17	C₂×C₂²⋊C₈×C₁₃₇	C₂²	1	1	yes	characteristic
18	C₂×#97×C₁₃₇	C₂²	1	2	yes	—
19	#389×C₁₃₇	C₂²	1	4	yes	—
20	#387×C₁₃₇	—	2	4	no	—
21	#381×C₁₃₇	—	2	2	no	—
22	#380×C₁₃₇	C₂²	1	2	yes	—
23	C₄×C₄⋊C₄×C₁₃₇	C₂²	1	4	yes	—
24	C₄×C₂²⋊C₄×C₁₃₇	C₂²	1	1	yes	characteristic
25	#340×C₁₃₇	C₂²	1	4	yes	—
26	#340×C₁₃₇	C₂²	1	2	yes	—
27	#340×C₁₃₇	C₂²	1	4	yes	—
28	#336×C₁₃₇	C₂²	1	2	yes	—
29	C₂³×C₄×C₁₃₇	—	2	2	no	—
30	C₂³×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
31	C₂²×C₈×C₁₃₇	—	2	4	no	—
32	C₂²×C₈×C₁₃₇	C₂×C₄	1	4	yes	—
33	C₂²×C₈×C₁₃₇	C₂³	1	2	yes	—
34	C₂²×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
35	C₂²×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
36	#117×C₁₃₇	—	4	8	no	—
37	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
38	C₂×C₄⋊C₄×C₁₃₇	—	2	4	no	—
39	C₂×C₄⋊C₄×C₁₃₇	—	2	4	no	—
40	C₂×C₄⋊C₄×C₁₃₇	C₂×C₄	1	8	yes	—
41	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	4	yes	—
42	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	2	yes	—
43	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	2	yes	—
44	C₂×C₂²⋊C₄×C₁₃₇	—	2	2	no	—
45	C₂×C₂²⋊C₄×C₁₃₇	D₄	1	2	yes	—
46	C₂×C₄²×C₁₃₇	—	2	4	no	—
47	C₂×C₄²×C₁₃₇	C₂³	1	2	yes	—
48	#107×C₁₃₇	C₂×C₄	1	8	yes	—
49	#107×C₁₃₇	C₂×C₄	1	4	yes	—
50	#107×C₁₃₇	C₂×C₄	1	4	yes	—
51	#106×C₁₃₇	—	2	4	no	—
52	C₂²⋊C₈×C₁₃₇	—	2	2	no	—
53	#97×C₁₃₇	—	2	4	no	—
54	#95×C₁₃₇	—	2	4	no	—
55	#95×C₁₃₇	—	2	2	no	—
56	#95×C₁₃₇	—	2	4	no	—
57	#95×C₁₃₇	D₄	1	2	yes	—
58	C₂⁴×C₁₃₇	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₁₃₇	—	2	2	no	—
60	C₂²×C₄×C₁₃₇	—	2	2	no	—
61	C₂²×C₄×C₁₃₇	—	2	8	no	—
62	C₂²×C₄×C₁₃₇	—	2	2	no	—
63	C₂²×C₄×C₁₃₇	—	2	2	no	—
64	C₂²×C₄×C₁₃₇	—	2	4	no	—
65	C₂²×C₄×C₁₃₇	—	4	4	no	—
66	C₂²×C₄×C₁₃₇	—	4	4	no	—
67	C₂²×C₄×C₁₃₇	C₂²×C₄	1	2	yes	—
68	C₂²×C₄×C₁₃₇	C₂×D₄	1	4	yes	—
69	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₁₃₇	#41	1	2	yes	—
71	C₂²×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
72	C₂×C₈×C₁₃₇	—	2	2	no	—
73	C₂×C₈×C₁₃₇	—	2	8	no	—
74	C₂×C₈×C₁₃₇	—	2	2	no	—
75	C₂×C₈×C₁₃₇	—	2	2	no	—
76	C₂×C₈×C₁₃₇	—	2	2	no	—
77	C₂×C₈×C₁₃₇	C₂²×C₄	1	2	yes	—
78	C₂×C₈×C₁₃₇	C₂²×C₄	1	2	yes	—
79	C₂×C₈×C₁₃₇	#41	1	4	yes	—
80	C₄⋊C₄×C₁₃₇	—	2	8	no	—
81	C₄⋊C₄×C₁₃₇	—	2	2	no	—
82	C₄⋊C₄×C₁₃₇	—	2	2	no	—
83	C₄⋊C₄×C₁₃₇	—	2	4	no	—
84	C₄⋊C₄×C₁₃₇	—	2	4	no	—
85	C₄⋊C₄×C₁₃₇	—	4	8	no	—
86	C₄⋊C₄×C₁₃₇	C₂²×C₄	1	4	yes	—
87	C₄⋊C₄×C₁₃₇	C₂²×C₄	1	4	yes	—
88	C₂²⋊C₄×C₁₃₇	—	2	4	no	—
89	C₂²⋊C₄×C₁₃₇	—	4	4	no	—
90	C₄²×C₁₃₇	—	2	2	no	—
91	C₄²×C₁₃₇	—	2	2	no	—
92	C₄²×C₁₃₇	—	4	8	no	—
93	C₂³×C₁₃₇	—	2	2	no	—
94	C₂³×C₁₃₇	—	2	2	no	—
95	C₂³×C₁₃₇	C₄×D₄	1	2	yes	—
96	C₂³×C₁₃₇	AGL(1,ℤ/8)	1	4	yes	—
97	C₂³×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
98	C₂×C₄×C₁₃₇	—	2	2	no	—
99	C₂×C₄×C₁₃₇	—	2	2	no	—
100	C₂×C₄×C₁₃₇	—	2	2	no	—
101	C₂×C₄×C₁₃₇	—	2	4	no	—
102	C₂×C₄×C₁₃₇	—	2	4	no	—
103	C₂×C₄×C₁₃₇	—	2	4	no	—
104	C₂×C₄×C₁₃₇	—	2	4	no	—
105	C₂×C₄×C₁₃₇	—	2	4	no	—
106	C₂×C₄×C₁₃₇	—	2	4	no	—
107	C₂×C₄×C₁₃₇	—	4	2	no	—
108	C₂×C₄×C₁₃₇	—	4	2	no	—
109	C₂×C₄×C₁₃₇	—	4	2	no	—
110	C₂×C₄×C₁₃₇	—	4	2	no	—
111	C₂×C₄×C₁₃₇	—	4	16	no	—
112	C₂×C₄×C₁₃₇	—	4	4	no	—
113	C₂×C₄×C₁₃₇	—	4	4	no	—
114	C₂×C₄×C₁₃₇	—	4	4	no	—
115	C₂×C₄×C₁₃₇	C₄×D₄	1	2	yes	—
116	C₂×C₄×C₁₃₇	C₂³×C₄	1	1	yes	characteristic
117	C₂×C₄×C₁₃₇	C₂×#41	1	2	yes	—
118	C₂×C₄×C₁₃₇	C₂×#41	1	1	yes	characteristic
119	C₂×C₄×C₁₃₇	2₋¹⁺⁴	1	2	yes	—
120	C₈×C₁₃₇	—	2	2	no	—
121	C₈×C₁₃₇	—	4	2	no	—
122	C₈×C₁₃₇	#117	1	4	yes	complemented
123	C₂²×C₁₃₇	—	2	8	no	—
124	C₂²×C₁₃₇	—	2	2	no	—
125	C₂²×C₁₃₇	—	2	2	no	—
126	C₂²×C₁₃₇	#442	1	4	yes	—
127	C₂²×C₁₃₇	C₂×C₄×D₄	1	1	yes	characteristic
128	C₂²×C₁₃₇	#547	1	2	yes	—
129	C₂²×C₁₃₇	C₂×AGL(1,ℤ/8)	1	2	yes	—
130	C₂²×C₁₃₇	#578	1	2	yes	—
131	C₄×C₁₃₇	—	2	2	no	—
132	C₄×C₁₃₇	—	4	4	no	—
133	C₄×C₁₃₇	—	4	4	no	—
134	C₄×C₁₃₇	—	4	4	no	—
135	C₄×C₁₃₇	—	8	4	no	—
136	C₄×C₁₃₇	C₂×#117	1	1	yes	characteristic
137	C₄×C₁₃₇	#519	1	1	yes	characteristic
138	C₄×C₁₃₇	#552	1	2	yes	—
139	C₂×C₁₃₇	—	2	4	no	—
140	C₂×C₁₃₇	#2344	1	1	yes	characteristic
141	C₂×C₁₃₇	C₂×#442	1	1	yes	characteristic
142	C₂×C₁₃₇	#2950	1	1	yes	characteristic
143	C₂×C₁₃₇	#3343	1	4	yes	—
144	#22617	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
145	C₁₃₇	#22617	1	1	yes	characteristic, direct product factor, Sylow
146	C₂²×#107	C₂×C₁₃₇	1	1	yes	characteristic
147	#2344	C₂×C₁₃₇	1	2	yes	—
148	#1943	C₂×C₁₃₇	1	2	yes	complemented
149	#1921	C₂×C₁₃₇	1	4	yes	complemented
150	#1823	C₂×C₁₃₇	1	2	yes	—
151	#1807	C₂×C₁₃₇	1	1	yes	characteristic
152	#1776	C₂×C₁₃₇	1	2	yes	complemented
153	#1752	C₂×C₁₃₇	1	1	yes	characteristic
154	C₂³×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
155	C₂×#117	—	2	2	no	—
156	C₂²×C₄⋊C₄	C₂²×C₁₃₇	1	2	yes	—
157	C₂×#107	C₄×C₁₃₇	1	8	yes	complemented
158	C₂×#107	C₂²×C₁₃₇	1	2	yes	—
159	C₂×#107	C₂²×C₁₃₇	1	1	yes	characteristic
160	C₂×#107	C₂²×C₁₃₇	1	1	yes	characteristic
161	C₂×#106	C₂²×C₁₃₇	1	2	yes	—
162	C₂×C₂²⋊C₈	C₂²×C₁₃₇	1	1	yes	characteristic
163	C₂×#97	C₂²×C₁₃₇	1	2	yes	—
164	#389	C₂²×C₁₃₇	1	4	yes	—
165	#387	—	2	4	no	—
166	#381	—	2	2	no	—
167	#380	C₂²×C₁₃₇	1	2	yes	—
168	C₄×C₄⋊C₄	C₂²×C₁₃₇	1	4	yes	—
169	C₄×C₂²⋊C₄	C₂²×C₁₃₇	1	1	yes	characteristic
170	#340	C₂²×C₁₃₇	1	2	yes	—
171	#340	C₂²×C₁₃₇	1	4	yes	—
172	#340	C₂²×C₁₃₇	1	4	yes	—
173	#336	C₂²×C₁₃₇	1	2	yes	—
174	C₂³×C₄	—	2	2	no	—
175	C₂³×C₄	C₂³×C₁₃₇	1	1	yes	characteristic
176	C₂²×C₈	—	2	4	no	—
177	C₂²×C₈	C₂×C₄×C₁₃₇	1	4	yes	—
178	C₂²×C₈	C₂³×C₁₃₇	1	2	yes	—
179	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
180	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
181	#117	—	4	8	no	—
182	C₂×C₄⋊C₄	—	2	2	no	—
183	C₂×C₄⋊C₄	—	2	4	no	—
184	C₂×C₄⋊C₄	—	2	4	no	—
185	C₂×C₄⋊C₄	C₂×C₄×C₁₃₇	1	8	yes	—
186	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	4	yes	—
187	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	2	yes	—
188	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	2	yes	—
189	C₂×C₂²⋊C₄	—	2	2	no	—
190	C₂×C₂²⋊C₄	D₄×C₁₃₇	1	2	yes	—
191	C₂×C₄²	—	2	4	no	—
192	C₂×C₄²	C₂³×C₁₃₇	1	2	yes	—
193	#107	C₂×C₄×C₁₃₇	1	8	yes	—
194	#107	C₂×C₄×C₁₃₇	1	4	yes	—
195	#107	C₂×C₄×C₁₃₇	1	4	yes	—
196	#106	—	2	4	no	—
197	C₂²⋊C₈	—	2	2	no	—
198	#97	—	2	4	no	—
199	#95	—	2	4	no	—
200	#95	—	2	2	no	—
201	#95	—	2	4	no	—
202	#95	D₄×C₁₃₇	1	2	yes	—
203	C₂⁴	C₂×D₄×C₁₃₇	1	1	yes	characteristic
204	C₂²×C₄	—	2	2	no	—
205	C₂²×C₄	—	2	2	no	—
206	C₂²×C₄	—	2	8	no	—
207	C₂²×C₄	—	2	2	no	—
208	C₂²×C₄	—	2	2	no	—
209	C₂²×C₄	—	2	4	no	—
210	C₂²×C₄	—	4	4	no	—
211	C₂²×C₄	—	4	4	no	—
212	C₂²×C₄	C₂²×C₄×C₁₃₇	1	2	yes	—
213	C₂²×C₄	C₂×D₄×C₁₃₇	1	4	yes	—
214	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
215	C₂²×C₄	#41×C₁₃₇	1	2	yes	—
216	C₂²×C₄	C₂⁴×C₁₃₇	1	1	yes	characteristic
217	C₂×C₈	—	2	2	no	—
218	C₂×C₈	—	2	8	no	—
219	C₂×C₈	—	2	2	no	—
220	C₂×C₈	—	2	2	no	—
221	C₂×C₈	—	2	2	no	—
222	C₂×C₈	C₂²×C₄×C₁₃₇	1	2	yes	—
223	C₂×C₈	C₂²×C₄×C₁₃₇	1	2	yes	—
224	C₂×C₈	#41×C₁₃₇	1	4	yes	—
225	C₄⋊C₄	—	2	8	no	—
226	C₄⋊C₄	—	2	2	no	—
227	C₄⋊C₄	—	2	2	no	—
228	C₄⋊C₄	—	2	4	no	—
229	C₄⋊C₄	—	2	4	no	—
230	C₄⋊C₄	—	4	8	no	—
231	C₄⋊C₄	C₂²×C₄×C₁₃₇	1	4	yes	—
232	C₄⋊C₄	C₂²×C₄×C₁₃₇	1	4	yes	—
233	C₂²⋊C₄	—	2	4	no	—
234	C₂²⋊C₄	—	4	4	no	—
235	C₄²	—	2	2	no	—
236	C₄²	—	2	2	no	—
237	C₄²	—	4	8	no	—
238	C₂³	—	2	2	no	—
239	C₂³	—	2	2	no	—
240	C₂³	C₄×D₄×C₁₃₇	1	2	yes	—
241	C₂³	AGL(1,ℤ/8)×C₁₃₇	1	4	yes	—
242	C₂³	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	4	no	—
247	C₂×C₄	—	2	4	no	—
248	C₂×C₄	—	2	4	no	—
249	C₂×C₄	—	2	4	no	—
250	C₂×C₄	—	2	4	no	—
251	C₂×C₄	—	2	4	no	—
252	C₂×C₄	—	4	16	no	—
253	C₂×C₄	—	4	2	no	—
254	C₂×C₄	—	4	2	no	—
255	C₂×C₄	—	4	2	no	—
256	C₂×C₄	—	4	2	no	—
257	C₂×C₄	—	4	4	no	—
258	C₂×C₄	—	4	4	no	—
259	C₂×C₄	—	4	4	no	—
260	C₂×C₄	C₄×D₄×C₁₃₇	1	2	yes	—
261	C₂×C₄	C₂³×C₄×C₁₃₇	1	1	yes	characteristic
262	C₂×C₄	C₂×#41×C₁₃₇	1	2	yes	—
263	C₂×C₄	C₂×#41×C₁₃₇	1	1	yes	characteristic
264	C₂×C₄	2₋¹⁺⁴×C₁₃₇	1	2	yes	—
265	C₈	—	2	2	no	—
266	C₈	—	4	2	no	—
267	C₈	#117×C₁₃₇	1	4	yes	complemented
268	C₂²	—	2	8	no	—
269	C₂²	—	2	2	no	—
270	C₂²	—	2	2	no	—
271	C₂²	#442×C₁₃₇	1	4	yes	—
272	C₂²	C₂×C₄×D₄×C₁₃₇	1	1	yes	characteristic
273	C₂²	#547×C₁₃₇	1	2	yes	—
274	C₂²	C₂×AGL(1,ℤ/8)×C₁₃₇	1	2	yes	—
275	C₂²	#578×C₁₃₇	1	2	yes	—
276	C₄	—	2	2	no	—
277	C₄	—	4	4	no	—
278	C₄	—	4	4	no	—
279	C₄	—	4	4	no	—
280	C₄	—	8	4	no	—
281	C₄	C₂×#117×C₁₃₇	1	1	yes	characteristic
282	C₄	#519×C₁₃₇	1	1	yes	characteristic
283	C₄	#552×C₁₃₇	1	2	yes	—
284	C₂	—	2	4	no	—
285	C₂	#2344×C₁₃₇	1	1	yes	characteristic
286	C₂	C₂×#442×C₁₃₇	1	1	yes	characteristic
287	C₂	#2950×C₁₃₇	1	1	yes	characteristic
288	C₂	#3343×C₁₃₇	1	4	yes	—

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

Complements

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

N	H ≅ G/N	type
3. #1943×C₁₃₇	C₂	semidirect
4. #1921×C₁₃₇	C₂	semidirect
7. #1776×C₁₃₇	C₂	semidirect
12. C₂×#107×C₁₃₇	C₄	semidirect
122. C₈×C₁₃₇	#117	semidirect
144. #22617	C₁₃₇	direct (N × H)
145. C₁₃₇	#22617	direct (N × H)
148. #1943	C₂×C₁₃₇	semidirect
149. #1921	C₂×C₁₃₇	semidirect
152. #1776	C₂×C₁₃₇	semidirect
157. C₂×#107	C₄×C₁₃₇	semidirect
267. C₈	#117×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#107×C₁₃₇	2
#2344×C₁₃₇	2
#1943×C₁₃₇	2
#1921×C₁₃₇	2
#1823×C₁₃₇	2
#1807×C₁₃₇	2
#1776×C₁₃₇	2
#1752×C₁₃₇	2
#22617	137

Subgroups — C137×#22617 #46872003

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#22617 #46872003