Subgroups — C₁₃₇×#25817 #46875203

Order 35072.

1022 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#533×C₁₃₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#501×C₁₃₇	C₂	1	2	yes	maximal, complemented
3	#2851×C₁₃₇	C₂	1	1	yes	maximal, characteristic
4	#2059×C₁₃₇	C₂	1	4	yes	maximal, complemented
5	#1993×C₁₃₇	C₂	1	2	yes	maximal, complemented
6	#1854×C₁₃₇	C₂	1	4	yes	maximal
7	#1749×C₁₃₇	C₂	1	1	yes	maximal, characteristic
8	#533×C₁₃₇	—	2	8	no	—
9	C₂×C₄⋊Q₈×C₁₃₇	C₂²	1	2	yes	—
10	C₂×#125×C₁₃₇	C₂²	1	2	yes	—
11	C₂×#125×C₁₃₇	C₂²	1	1	yes	characteristic
12	C₂×C₂²⋊Q₈×C₁₃₇	—	2	2	no	—
13	C₂×C₄×Q₈×C₁₃₇	—	2	2	no	—
14	C₂×#117×C₁₃₇	C₂²	1	2	yes	—
15	C₂²×C₄²×C₁₃₇	C₂²	1	1	yes	characteristic
16	#501×C₁₃₇	—	2	8	no	—
17	C₂×#107×C₁₃₇	C₂²	1	4	yes	—
18	C₂×#106×C₁₃₇	C₂²	1	4	yes	—
19	C₂×C₂²⋊C₈×C₁₃₇	C₂²	1	2	yes	—
20	C₂×#97×C₁₃₇	C₂²	1	2	yes	—
21	#389×C₁₃₇	C₂²	1	2	yes	—
22	#387×C₁₃₇	—	2	2	no	—
23	#385×C₁₃₇	—	2	2	no	—
24	#382×C₁₃₇	C₂²	1	1	yes	characteristic
25	#380×C₁₃₇	C₂²	1	2	yes	—
26	#340×C₁₃₇	C₂²	1	8	yes	—
27	#336×C₁₃₇	C₂²	1	2	yes	—
28	C₂²×Q₈×C₁₃₇	—	2	2	no	—
29	C₂³×C₄×C₁₃₇	D₄	1	2	yes	—
30	C₂³×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
31	C₂²×C₈×C₁₃₇	C₂³	1	4	yes	—
32	C₄⋊Q₈×C₁₃₇	—	2	4	no	—
33	C₄⋊Q₈×C₁₃₇	D₄	1	8	yes	—
34	#125×C₁₃₇	—	2	8	no	—
35	#125×C₁₃₇	—	2	4	no	—
36	C₂²⋊Q₈×C₁₃₇	—	2	16	no	—
37	C₄×Q₈×C₁₃₇	—	2	16	no	—
38	#117×C₁₃₇	—	2	8	no	—
39	C₂×C₄⋊C₄×C₁₃₇	—	2	4	no	—
40	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
41	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
42	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
43	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	2	yes	—
44	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	2	yes	—
45	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	4	yes	—
46	C₂×C₂²⋊C₄×C₁₃₇	—	2	2	no	—
47	C₂×C₂²⋊C₄×C₁₃₇	—	2	2	no	—
48	C₂×C₄²×C₁₃₇	—	2	2	no	—
49	C₂×C₄²×C₁₃₇	—	2	4	no	—
50	C₂×C₄²×C₁₃₇	D₄	1	2	yes	—
51	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
52	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
53	#107×C₁₃₇	—	2	8	no	—
54	#106×C₁₃₇	—	2	8	no	—
55	C₂²⋊C₈×C₁₃₇	Q₈	1	8	yes	complemented
56	#97×C₁₃₇	—	2	4	no	—
57	#95×C₁₃₇	—	2	2	no	—
58	#95×C₁₃₇	—	2	2	no	—
59	#95×C₁₃₇	—	2	2	no	—
60	C₂⁴×C₁₃₇	C₂×D₄	1	1	yes	characteristic
61	C₂×Q₈×C₁₃₇	—	2	8	no	—
62	C₂×Q₈×C₁₃₇	—	4	4	no	—
63	C₂×Q₈×C₁₃₇	—	4	4	no	—
64	C₂²×C₄×C₁₃₇	—	2	2	no	—
65	C₂²×C₄×C₁₃₇	—	2	2	no	—
66	C₂²×C₄×C₁₃₇	—	2	4	no	—
67	C₂²×C₄×C₁₃₇	—	2	4	no	—
68	C₂²×C₄×C₁₃₇	—	2	2	no	—
69	C₂²×C₄×C₁₃₇	—	2	2	no	—
70	C₂²×C₄×C₁₃₇	—	4	2	no	—
71	C₂²×C₄×C₁₃₇	—	4	2	no	—
72	C₂²×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
73	C₂²×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
74	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₁₃₇	C₂×Q₈	1	4	yes	—
76	C₂²×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
77	C₂×C₈×C₁₃₇	—	2	4	no	—
78	C₂×C₈×C₁₃₇	—	2	4	no	—
79	C₂×C₈×C₁₃₇	C₂×Q₈	1	8	yes	—
80	C₄⋊C₄×C₁₃₇	—	2	16	no	—
81	C₄⋊C₄×C₁₃₇	—	2	8	no	—
82	C₄⋊C₄×C₁₃₇	—	2	4	no	—
83	C₄⋊C₄×C₁₃₇	—	2	2	no	—
84	C₄⋊C₄×C₁₃₇	—	2	2	no	—
85	C₄⋊C₄×C₁₃₇	—	2	4	no	—
86	C₄⋊C₄×C₁₃₇	—	4	4	no	—
87	C₄⋊C₄×C₁₃₇	—	4	4	no	—
88	C₄⋊C₄×C₁₃₇	C₂×D₄	1	8	yes	—
89	C₂²⋊C₄×C₁₃₇	—	2	8	no	—
90	C₂²⋊C₄×C₁₃₇	—	4	4	no	—
91	C₄²×C₁₃₇	—	2	8	no	—
92	C₄²×C₁₃₇	—	2	4	no	—
93	C₄²×C₁₃₇	—	2	2	no	—
94	C₄²×C₁₃₇	C₂×D₄	1	4	yes	—
95	C₂³×C₁₃₇	—	2	1	no	—
96	C₂³×C₁₃₇	—	2	1	no	—
97	C₂³×C₁₃₇	—	2	2	no	—
98	C₂³×C₁₃₇	—	2	2	no	—
99	C₂³×C₁₃₇	C₄⋊Q₈	1	2	yes	—
100	C₂³×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
101	Q₈×C₁₃₇	—	4	16	no	—
102	C₂×C₄×C₁₃₇	—	2	1	no	—
103	C₂×C₄×C₁₃₇	—	2	1	no	—
104	C₂×C₄×C₁₃₇	—	2	2	no	—
105	C₂×C₄×C₁₃₇	—	2	2	no	—
106	C₂×C₄×C₁₃₇	—	2	8	no	—
107	C₂×C₄×C₁₃₇	—	2	2	no	—
108	C₂×C₄×C₁₃₇	—	2	8	no	—
109	C₂×C₄×C₁₃₇	—	2	2	no	—
110	C₂×C₄×C₁₃₇	—	2	2	no	—
111	C₂×C₄×C₁₃₇	—	4	8	no	—
112	C₂×C₄×C₁₃₇	—	4	4	no	—
113	C₂×C₄×C₁₃₇	—	4	4	no	—
114	C₂×C₄×C₁₃₇	—	4	2	no	—
115	C₂×C₄×C₁₃₇	—	4	2	no	—
116	C₂×C₄×C₁₃₇	—	4	2	no	—
117	C₂×C₄×C₁₃₇	—	4	2	no	—
118	C₂×C₄×C₁₃₇	—	4	2	no	—
119	C₂×C₄×C₁₃₇	—	4	2	no	—
120	C₂×C₄×C₁₃₇	#120	1	4	yes	—
121	C₂×C₄×C₁₃₇	C₄⋊Q₈	1	2	yes	—
122	C₂×C₄×C₁₃₇	C₂²×D₄	1	2	yes	—
123	C₂×C₄×C₁₃₇	C₂²×Q₈	1	2	yes	—
124	C₂×C₄×C₁₃₇	2₋¹⁺⁴	1	2	yes	—
125	C₈×C₁₃₇	—	2	8	no	—
126	C₂²×C₁₃₇	—	2	2	no	—
127	C₂²×C₁₃₇	—	2	2	no	—
128	C₂²×C₁₃₇	—	2	2	no	—
129	C₂²×C₁₃₇	—	4	2	no	—
130	C₂²×C₁₃₇	—	4	2	no	—
131	C₂²×C₁₃₇	C₂×#120	1	1	yes	characteristic
132	C₂²×C₁₃₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
133	C₂²×C₁₃₇	#536	1	1	yes	characteristic
134	C₂²×C₁₃₇	#575	1	2	yes	—
135	C₂²×C₁₃₇	#575	1	2	yes	—
136	C₄×C₁₃₇	—	2	2	no	—
137	C₄×C₁₃₇	—	2	4	no	—
138	C₄×C₁₃₇	—	4	2	no	—
139	C₄×C₁₃₇	—	4	8	no	—
140	C₄×C₁₃₇	—	8	2	no	—
141	C₄×C₁₃₇	—	8	2	no	—
142	C₄×C₁₃₇	D₄×Q₈	1	4	yes	—
143	C₂×C₁₃₇	—	4	2	no	—
144	C₂×C₁₃₇	#2434	1	1	yes	characteristic
145	C₂×C₁₃₇	#3005	1	2	yes	—
146	C₂×C₁₃₇	#3162	1	1	yes	characteristic
147	C₂×C₁₃₇	#3166	1	1	yes	characteristic
148	C₂×C₁₃₇	#3240	1	2	yes	—
149	#25817	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
150	C₁₃₇	#25817	1	1	yes	characteristic, direct product factor, Sylow
151	C₂×#533	C₂×C₁₃₇	1	1	yes	characteristic
152	C₂×#501	C₂×C₁₃₇	1	2	yes	complemented
153	#2851	C₂×C₁₃₇	1	1	yes	characteristic
154	#2059	C₂×C₁₃₇	1	4	yes	complemented
155	#1993	C₂×C₁₃₇	1	2	yes	complemented
156	#1854	C₂×C₁₃₇	1	4	yes	—
157	#1749	C₂×C₁₃₇	1	1	yes	characteristic
158	#533	—	2	8	no	—
159	C₂×C₄⋊Q₈	C₂²×C₁₃₇	1	2	yes	—
160	C₂×#125	C₂²×C₁₃₇	1	2	yes	—
161	C₂×#125	C₂²×C₁₃₇	1	1	yes	characteristic
162	C₂×C₂²⋊Q₈	—	2	2	no	—
163	C₂×C₄×Q₈	—	2	2	no	—
164	C₂×#117	C₂²×C₁₃₇	1	2	yes	—
165	C₂²×C₄²	C₂²×C₁₃₇	1	1	yes	characteristic
166	#501	—	2	8	no	—
167	C₂×#107	C₂²×C₁₃₇	1	4	yes	—
168	C₂×#106	C₂²×C₁₃₇	1	4	yes	—
169	C₂×C₂²⋊C₈	C₂²×C₁₃₇	1	2	yes	—
170	C₂×#97	C₂²×C₁₃₇	1	2	yes	—
171	#389	C₂²×C₁₃₇	1	2	yes	—
172	#387	—	2	2	no	—
173	#385	—	2	2	no	—
174	#382	C₂²×C₁₃₇	1	1	yes	characteristic
175	#380	C₂²×C₁₃₇	1	2	yes	—
176	#340	C₂²×C₁₃₇	1	8	yes	—
177	#336	C₂²×C₁₃₇	1	2	yes	—
178	C₂²×Q₈	—	2	2	no	—
179	C₂³×C₄	D₄×C₁₃₇	1	2	yes	—
180	C₂³×C₄	C₂³×C₁₃₇	1	1	yes	characteristic
181	C₂²×C₈	C₂³×C₁₃₇	1	4	yes	—
182	C₄⋊Q₈	—	2	4	no	—
183	C₄⋊Q₈	D₄×C₁₃₇	1	8	yes	—
184	#125	—	2	8	no	—
185	#125	—	2	4	no	—
186	C₂²⋊Q₈	—	2	16	no	—
187	C₄×Q₈	—	2	16	no	—
188	#117	—	2	8	no	—
189	C₂×C₄⋊C₄	—	2	4	no	—
190	C₂×C₄⋊C₄	—	2	2	no	—
191	C₂×C₄⋊C₄	—	2	2	no	—
192	C₂×C₄⋊C₄	—	2	2	no	—
193	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	2	yes	—
194	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	2	yes	—
195	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	4	yes	—
196	C₂×C₂²⋊C₄	—	2	2	no	—
197	C₂×C₂²⋊C₄	—	2	2	no	—
198	C₂×C₄²	—	2	2	no	—
199	C₂×C₄²	—	2	4	no	—
200	C₂×C₄²	D₄×C₁₃₇	1	2	yes	—
201	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
202	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
203	#107	—	2	8	no	—
204	#106	—	2	8	no	—
205	C₂²⋊C₈	Q₈×C₁₃₇	1	8	yes	complemented
206	#97	—	2	4	no	—
207	#95	—	2	2	no	—
208	#95	—	2	2	no	—
209	#95	—	2	2	no	—
210	C₂⁴	C₂×D₄×C₁₃₇	1	1	yes	characteristic
211	C₂×Q₈	—	2	8	no	—
212	C₂×Q₈	—	4	4	no	—
213	C₂×Q₈	—	4	4	no	—
214	C₂²×C₄	—	2	2	no	—
215	C₂²×C₄	—	2	2	no	—
216	C₂²×C₄	—	2	4	no	—
217	C₂²×C₄	—	2	4	no	—
218	C₂²×C₄	—	2	2	no	—
219	C₂²×C₄	—	2	2	no	—
220	C₂²×C₄	—	4	2	no	—
221	C₂²×C₄	—	4	2	no	—
222	C₂²×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
223	C₂²×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
224	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
225	C₂²×C₄	C₂×Q₈×C₁₃₇	1	4	yes	—
226	C₂²×C₄	C₂⁴×C₁₃₇	1	1	yes	characteristic
227	C₂×C₈	—	2	4	no	—
228	C₂×C₈	—	2	4	no	—
229	C₂×C₈	C₂×Q₈×C₁₃₇	1	8	yes	—
230	C₄⋊C₄	—	2	16	no	—
231	C₄⋊C₄	—	2	8	no	—
232	C₄⋊C₄	—	2	4	no	—
233	C₄⋊C₄	—	2	2	no	—
234	C₄⋊C₄	—	2	2	no	—
235	C₄⋊C₄	—	2	4	no	—
236	C₄⋊C₄	—	4	4	no	—
237	C₄⋊C₄	—	4	4	no	—
238	C₄⋊C₄	C₂×D₄×C₁₃₇	1	8	yes	—
239	C₂²⋊C₄	—	2	8	no	—
240	C₂²⋊C₄	—	4	4	no	—
241	C₄²	—	2	8	no	—
242	C₄²	—	2	4	no	—
243	C₄²	—	2	2	no	—
244	C₄²	C₂×D₄×C₁₃₇	1	4	yes	—
245	C₂³	—	2	1	no	—
246	C₂³	—	2	1	no	—
247	C₂³	—	2	2	no	—
248	C₂³	—	2	2	no	—
249	C₂³	C₄⋊Q₈×C₁₃₇	1	2	yes	—
250	C₂³	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
251	Q₈	—	4	16	no	—
252	C₂×C₄	—	2	1	no	—
253	C₂×C₄	—	2	1	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	2	8	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	2	8	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	4	8	no	—
262	C₂×C₄	—	4	4	no	—
263	C₂×C₄	—	4	4	no	—
264	C₂×C₄	—	4	2	no	—
265	C₂×C₄	—	4	2	no	—
266	C₂×C₄	—	4	2	no	—
267	C₂×C₄	—	4	2	no	—
268	C₂×C₄	—	4	2	no	—
269	C₂×C₄	—	4	2	no	—
270	C₂×C₄	#120×C₁₃₇	1	4	yes	—
271	C₂×C₄	C₄⋊Q₈×C₁₃₇	1	2	yes	—
272	C₂×C₄	C₂²×D₄×C₁₃₇	1	2	yes	—
273	C₂×C₄	C₂²×Q₈×C₁₃₇	1	2	yes	—
274	C₂×C₄	2₋¹⁺⁴×C₁₃₇	1	2	yes	—
275	C₈	—	2	8	no	—
276	C₂²	—	2	2	no	—
277	C₂²	—	2	2	no	—
278	C₂²	—	2	2	no	—
279	C₂²	—	4	2	no	—
280	C₂²	—	4	2	no	—
281	C₂²	C₂×#120×C₁₃₇	1	1	yes	characteristic
282	C₂²	C₂×C₄⋊Q₈×C₁₃₇	1	1	yes	characteristic
283	C₂²	#536×C₁₃₇	1	1	yes	characteristic
284	C₂²	#575×C₁₃₇	1	2	yes	—
285	C₂²	#575×C₁₃₇	1	2	yes	—
286	C₄	—	2	2	no	—
287	C₄	—	2	4	no	—
288	C₄	—	4	2	no	—
289	C₄	—	4	8	no	—
290	C₄	—	8	2	no	—
291	C₄	—	8	2	no	—
292	C₄	D₄×Q₈×C₁₃₇	1	4	yes	—
293	C₂	—	4	2	no	—
294	C₂	#2434×C₁₃₇	1	1	yes	characteristic
295	C₂	#3005×C₁₃₇	1	2	yes	—
296	C₂	#3162×C₁₃₇	1	1	yes	characteristic
297	C₂	#3166×C₁₃₇	1	1	yes	characteristic
298	C₂	#3240×C₁₃₇	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
2. C₂×#501×C₁₃₇	C₂	semidirect
4. #2059×C₁₃₇	C₂	semidirect
5. #1993×C₁₃₇	C₂	semidirect
55. C₂²⋊C₈×C₁₃₇	Q₈	semidirect
149. #25817	C₁₃₇	direct (N × H)
150. C₁₃₇	#25817	direct (N × H)
152. C₂×#501	C₂×C₁₃₇	semidirect
154. #2059	C₂×C₁₃₇	semidirect
155. #1993	C₂×C₁₃₇	semidirect
205. C₂²⋊C₈	Q₈×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#533×C₁₃₇	2
C₂×#501×C₁₃₇	2
#2851×C₁₃₇	2
#2059×C₁₃₇	2
#1993×C₁₃₇	2
#1854×C₁₃₇	2
#1749×C₁₃₇	2
#25817	137

Subgroups — C137×#25817 #46875203

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#25817 #46875203