Subgroups — C₁₃₇×#21366 #46870752

Order 35072.

850 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#432×C₁₃₇	C₂	1	1	yes	maximal, characteristic
2	#2283×C₁₃₇	C₂	1	1	yes	maximal, characteristic
3	C₄×#117×C₁₃₇	C₂	1	1	yes	maximal, characteristic
4	#1949×C₁₃₇	C₂	1	4	yes	maximal, complemented
5	#1854×C₁₃₇	C₂	1	4	yes	maximal
6	#1776×C₁₃₇	C₂	1	4	yes	maximal, complemented
7	C₂×#117×C₁₃₇	C₂²	1	2	yes	—
8	C₂²×C₄²×C₁₃₇	C₂²	1	1	yes	characteristic
9	#432×C₁₃₇	C₄	1	8	yes	complemented
10	C₂×#107×C₁₃₇	C₂²	1	4	yes	—
11	C₂×#106×C₁₃₇	C₂²	1	4	yes	—
12	C₂×C₄⋊C₈×C₁₃₇	C₂²	1	2	yes	—
13	C₂×C₂²⋊C₈×C₁₃₇	C₂²	1	2	yes	—
14	C₂×#97×C₁₃₇	C₂²	1	2	yes	—
15	#383×C₁₃₇	—	2	1	no	—
16	#382×C₁₃₇	C₂²	1	2	yes	—
17	#381×C₁₃₇	—	2	1	no	—
18	#380×C₁₃₇	C₂²	1	2	yes	—
19	C₄×C₄⋊C₄×C₁₃₇	C₂²	1	2	yes	—
20	C₄×C₄⋊C₄×C₁₃₇	C₂²	1	2	yes	—
21	C₄×C₄⋊C₄×C₁₃₇	C₂²	1	2	yes	—
22	C₄×C₂²⋊C₄×C₁₃₇	—	2	1	no	—
23	C₄×C₂²⋊C₄×C₁₃₇	—	2	1	no	—
24	C₄×C₂²⋊C₄×C₁₃₇	—	2	1	no	—
25	#376×C₁₃₇	—	2	1	no	—
26	#376×C₁₃₇	—	2	1	no	—
27	C₄³×C₁₃₇	—	2	1	no	—
28	#340×C₁₃₇	C₂²	1	8	yes	—
29	C₂³×C₄×C₁₃₇	D₄	1	1	yes	characteristic
30	C₂³×C₄×C₁₃₇	D₄	1	1	yes	characteristic
31	C₂³×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
32	C₂²×C₈×C₁₃₇	C₂³	1	4	yes	—
33	#117×C₁₃₇	—	2	8	no	—
34	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	4	yes	—
35	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	4	yes	—
36	C₂×C₂²⋊C₄×C₁₃₇	—	2	2	no	—
37	C₂×C₂²⋊C₄×C₁₃₇	—	2	2	no	—
38	C₂×C₄²×C₁₃₇	—	2	2	no	—
39	C₂×C₄²×C₁₃₇	—	2	2	no	—
40	C₂×C₄²×C₁₃₇	—	2	2	no	—
41	C₂×C₄²×C₁₃₇	—	2	2	no	—
42	C₂×C₄²×C₁₃₇	C₂×C₄	1	4	yes	—
43	C₂×C₄²×C₁₃₇	D₄	1	1	yes	characteristic
44	C₂×C₄²×C₁₃₇	D₄	1	1	yes	characteristic
45	C₂×C₄²×C₁₃₇	Q₈	1	4	yes	—
46	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
47	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
48	#107×C₁₃₇	—	2	8	no	—
49	#106×C₁₃₇	—	2	8	no	—
50	C₄⋊C₈×C₁₃₇	C₂×C₄	1	8	yes	—
51	C₂²⋊C₈×C₁₃₇	C₂×C₄	1	8	yes	—
52	#97×C₁₃₇	C₂×C₄	1	8	yes	—
53	#95×C₁₃₇	—	2	2	no	—
54	#95×C₁₃₇	—	2	2	no	—
55	#95×C₁₃₇	—	2	2	no	—
56	#95×C₁₃₇	—	2	2	no	—
57	#95×C₁₃₇	—	2	2	no	—
58	#95×C₁₃₇	—	2	2	no	—
59	C₂⁴×C₁₃₇	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₁₃₇	—	2	4	no	—
61	C₂²×C₄×C₁₃₇	—	2	4	no	—
62	C₂²×C₄×C₁₃₇	—	2	4	no	—
63	C₂²×C₄×C₁₃₇	—	4	2	no	—
64	C₂²×C₄×C₁₃₇	—	4	2	no	—
65	C₂²×C₄×C₁₃₇	C₄⋊C₄	1	4	yes	—
66	C₂²×C₄×C₁₃₇	C₄⋊C₄	1	4	yes	—
67	C₂²×C₄×C₁₃₇	C₂²×C₄	1	2	yes	—
68	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₁₃₇	C₂×Q₈	1	2	yes	—
74	C₂²×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
75	C₂×C₈×C₁₃₇	—	2	4	no	—
76	C₂×C₈×C₁₃₇	—	2	4	no	—
77	C₂×C₈×C₁₃₇	C₂²×C₄	1	8	yes	—
78	C₄⋊C₄×C₁₃₇	—	2	4	no	—
79	C₄⋊C₄×C₁₃₇	—	2	4	no	—
80	C₄⋊C₄×C₁₃₇	—	2	4	no	—
81	C₄⋊C₄×C₁₃₇	#41	1	8	yes	—
82	C₂²⋊C₄×C₁₃₇	—	2	8	no	—
83	C₂²⋊C₄×C₁₃₇	—	4	4	no	—
84	C₄²×C₁₃₇	—	2	8	no	—
85	C₄²×C₁₃₇	—	4	4	no	—
86	C₄²×C₁₃₇	—	4	4	no	—
87	C₄²×C₁₃₇	—	4	4	no	—
88	C₄²×C₁₃₇	C₄⋊C₄	1	4	yes	—
89	C₄²×C₁₃₇	C₄⋊C₄	1	4	yes	—
90	C₄²×C₁₃₇	C₂²×C₄	1	2	yes	—
91	C₄²×C₁₃₇	C₂²×C₄	1	2	yes	—
92	C₄²×C₁₃₇	C₂×Q₈	1	2	yes	—
93	C₄²×C₁₃₇	C₂×Q₈	1	2	yes	—
94	C₂³×C₁₃₇	—	2	2	no	—
95	C₂³×C₁₃₇	—	2	2	no	—
96	C₂³×C₁₃₇	—	2	2	no	—
97	C₂³×C₁₃₇	C₂×C₄⋊C₄	1	2	yes	—
98	C₂³×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
99	C₂×C₄×C₁₃₇	—	2	2	no	—
100	C₂×C₄×C₁₃₇	—	2	2	no	—
101	C₂×C₄×C₁₃₇	—	2	2	no	—
102	C₂×C₄×C₁₃₇	—	2	2	no	—
103	C₂×C₄×C₁₃₇	—	2	2	no	—
104	C₂×C₄×C₁₃₇	—	2	4	no	—
105	C₂×C₄×C₁₃₇	—	2	4	no	—
106	C₂×C₄×C₁₃₇	—	2	8	no	—
107	C₂×C₄×C₁₃₇	—	4	4	no	—
108	C₂×C₄×C₁₃₇	—	4	4	no	—
109	C₂×C₄×C₁₃₇	—	4	4	no	—
110	C₂×C₄×C₁₃₇	—	4	4	no	—
111	C₂×C₄×C₁₃₇	—	4	4	no	—
112	C₂×C₄×C₁₃₇	—	4	4	no	—
113	C₂×C₄×C₁₃₇	—	4	4	no	—
114	C₂×C₄×C₁₃₇	C₂×C₄⋊C₄	1	2	yes	—
115	C₂×C₄×C₁₃₇	C₂×C₄⋊C₄	1	2	yes	—
116	C₂×C₄×C₁₃₇	C₂×C₄⋊C₄	1	2	yes	—
117	C₂×C₄×C₁₃₇	C₂×C₄⋊C₄	1	2	yes	—
118	C₂×C₄×C₁₃₇	C₂×C₄⋊C₄	1	2	yes	—
119	C₂×C₄×C₁₃₇	C₂³×C₄	1	1	yes	characteristic
120	C₂×C₄×C₁₃₇	C₂²×Q₈	1	1	yes	characteristic
121	C₂×C₄×C₁₃₇	C₂×#41	1	4	yes	—
122	C₈×C₁₃₇	—	2	8	no	—
123	C₂²×C₁₃₇	—	2	2	no	—
124	C₂²×C₁₃₇	—	2	2	no	—
125	C₂²×C₁₃₇	—	2	2	no	—
126	C₂²×C₁₃₇	—	4	4	no	—
127	C₂²×C₁₃₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
128	C₂²×C₁₃₇	#525	1	2	yes	—
129	C₂²×C₁₃₇	#575	1	2	yes	—
130	C₂²×C₁₃₇	#575	1	2	yes	—
131	C₄×C₁₃₇	—	2	2	no	—
132	C₄×C₁₃₇	—	2	2	no	—
133	C₄×C₁₃₇	—	2	2	no	—
134	C₄×C₁₃₇	—	2	2	no	—
135	C₄×C₁₃₇	—	2	2	no	—
136	C₄×C₁₃₇	—	4	8	no	—
137	C₄×C₁₃₇	—	8	4	no	—
138	C₄×C₁₃₇	C₄×#41	1	2	yes	—
139	C₄×C₁₃₇	#533	1	2	yes	—
140	C₂×C₁₃₇	—	4	2	no	—
141	C₂×C₁₃₇	#2296	1	1	yes	characteristic
142	C₂×C₁₃₇	#2911	1	1	yes	characteristic
143	C₂×C₁₃₇	#2911	1	1	yes	characteristic
144	C₂×C₁₃₇	#3111	1	4	yes	—
145	#21366	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
146	C₁₃₇	#21366	1	1	yes	characteristic, direct product factor, Sylow
147	C₂×#432	C₂×C₁₃₇	1	1	yes	characteristic
148	#2283	C₂×C₁₃₇	1	1	yes	characteristic
149	C₄×#117	C₂×C₁₃₇	1	1	yes	characteristic
150	#1949	C₂×C₁₃₇	1	4	yes	complemented
151	#1854	C₂×C₁₃₇	1	4	yes	—
152	#1776	C₂×C₁₃₇	1	4	yes	complemented
153	C₂×#117	C₂²×C₁₃₇	1	2	yes	—
154	C₂²×C₄²	C₂²×C₁₃₇	1	1	yes	characteristic
155	#432	C₄×C₁₃₇	1	8	yes	complemented
156	C₂×#107	C₂²×C₁₃₇	1	4	yes	—
157	C₂×#106	C₂²×C₁₃₇	1	4	yes	—
158	C₂×C₄⋊C₈	C₂²×C₁₃₇	1	2	yes	—
159	C₂×C₂²⋊C₈	C₂²×C₁₃₇	1	2	yes	—
160	C₂×#97	C₂²×C₁₃₇	1	2	yes	—
161	#383	—	2	1	no	—
162	#382	C₂²×C₁₃₇	1	2	yes	—
163	#381	—	2	1	no	—
164	#380	C₂²×C₁₃₇	1	2	yes	—
165	C₄×C₄⋊C₄	C₂²×C₁₃₇	1	2	yes	—
166	C₄×C₄⋊C₄	C₂²×C₁₃₇	1	2	yes	—
167	C₄×C₄⋊C₄	C₂²×C₁₃₇	1	2	yes	—
168	C₄×C₂²⋊C₄	—	2	1	no	—
169	C₄×C₂²⋊C₄	—	2	1	no	—
170	C₄×C₂²⋊C₄	—	2	1	no	—
171	#376	—	2	1	no	—
172	#376	—	2	1	no	—
173	C₄³	—	2	1	no	—
174	#340	C₂²×C₁₃₇	1	8	yes	—
175	C₂³×C₄	D₄×C₁₃₇	1	1	yes	characteristic
176	C₂³×C₄	D₄×C₁₃₇	1	1	yes	characteristic
177	C₂³×C₄	C₂³×C₁₃₇	1	1	yes	characteristic
178	C₂²×C₈	C₂³×C₁₃₇	1	4	yes	—
179	#117	—	2	8	no	—
180	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	4	yes	—
181	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	4	yes	—
182	C₂×C₂²⋊C₄	—	2	2	no	—
183	C₂×C₂²⋊C₄	—	2	2	no	—
184	C₂×C₄²	—	2	2	no	—
185	C₂×C₄²	—	2	2	no	—
186	C₂×C₄²	—	2	2	no	—
187	C₂×C₄²	—	2	2	no	—
188	C₂×C₄²	C₂×C₄×C₁₃₇	1	4	yes	—
189	C₂×C₄²	D₄×C₁₃₇	1	1	yes	characteristic
190	C₂×C₄²	D₄×C₁₃₇	1	1	yes	characteristic
191	C₂×C₄²	Q₈×C₁₃₇	1	4	yes	—
192	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
193	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
194	#107	—	2	8	no	—
195	#106	—	2	8	no	—
196	C₄⋊C₈	C₂×C₄×C₁₃₇	1	8	yes	—
197	C₂²⋊C₈	C₂×C₄×C₁₃₇	1	8	yes	—
198	#97	C₂×C₄×C₁₃₇	1	8	yes	—
199	#95	—	2	2	no	—
200	#95	—	2	2	no	—
201	#95	—	2	2	no	—
202	#95	—	2	2	no	—
203	#95	—	2	2	no	—
204	#95	—	2	2	no	—
205	C₂⁴	C₂×D₄×C₁₃₇	1	1	yes	characteristic
206	C₂²×C₄	—	2	4	no	—
207	C₂²×C₄	—	2	4	no	—
208	C₂²×C₄	—	2	4	no	—
209	C₂²×C₄	—	4	2	no	—
210	C₂²×C₄	—	4	2	no	—
211	C₂²×C₄	C₄⋊C₄×C₁₃₇	1	4	yes	—
212	C₂²×C₄	C₄⋊C₄×C₁₃₇	1	4	yes	—
213	C₂²×C₄	C₂²×C₄×C₁₃₇	1	2	yes	—
214	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
215	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
216	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
217	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
218	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
219	C₂²×C₄	C₂×Q₈×C₁₃₇	1	2	yes	—
220	C₂²×C₄	C₂⁴×C₁₃₇	1	1	yes	characteristic
221	C₂×C₈	—	2	4	no	—
222	C₂×C₈	—	2	4	no	—
223	C₂×C₈	C₂²×C₄×C₁₃₇	1	8	yes	—
224	C₄⋊C₄	—	2	4	no	—
225	C₄⋊C₄	—	2	4	no	—
226	C₄⋊C₄	—	2	4	no	—
227	C₄⋊C₄	#41×C₁₃₇	1	8	yes	—
228	C₂²⋊C₄	—	2	8	no	—
229	C₂²⋊C₄	—	4	4	no	—
230	C₄²	—	2	8	no	—
231	C₄²	—	4	4	no	—
232	C₄²	—	4	4	no	—
233	C₄²	—	4	4	no	—
234	C₄²	C₄⋊C₄×C₁₃₇	1	4	yes	—
235	C₄²	C₄⋊C₄×C₁₃₇	1	4	yes	—
236	C₄²	C₂²×C₄×C₁₃₇	1	2	yes	—
237	C₄²	C₂²×C₄×C₁₃₇	1	2	yes	—
238	C₄²	C₂×Q₈×C₁₃₇	1	2	yes	—
239	C₄²	C₂×Q₈×C₁₃₇	1	2	yes	—
240	C₂³	—	2	2	no	—
241	C₂³	—	2	2	no	—
242	C₂³	—	2	2	no	—
243	C₂³	C₂×C₄⋊C₄×C₁₃₇	1	2	yes	—
244	C₂³	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	2	4	no	—
251	C₂×C₄	—	2	4	no	—
252	C₂×C₄	—	2	8	no	—
253	C₂×C₄	—	4	4	no	—
254	C₂×C₄	—	4	4	no	—
255	C₂×C₄	—	4	4	no	—
256	C₂×C₄	—	4	4	no	—
257	C₂×C₄	—	4	4	no	—
258	C₂×C₄	—	4	4	no	—
259	C₂×C₄	—	4	4	no	—
260	C₂×C₄	C₂×C₄⋊C₄×C₁₃₇	1	2	yes	—
261	C₂×C₄	C₂×C₄⋊C₄×C₁₃₇	1	2	yes	—
262	C₂×C₄	C₂×C₄⋊C₄×C₁₃₇	1	2	yes	—
263	C₂×C₄	C₂×C₄⋊C₄×C₁₃₇	1	2	yes	—
264	C₂×C₄	C₂×C₄⋊C₄×C₁₃₇	1	2	yes	—
265	C₂×C₄	C₂³×C₄×C₁₃₇	1	1	yes	characteristic
266	C₂×C₄	C₂²×Q₈×C₁₃₇	1	1	yes	characteristic
267	C₂×C₄	C₂×#41×C₁₃₇	1	4	yes	—
268	C₈	—	2	8	no	—
269	C₂²	—	2	2	no	—
270	C₂²	—	2	2	no	—
271	C₂²	—	2	2	no	—
272	C₂²	—	4	4	no	—
273	C₂²	C₂²×C₄⋊C₄×C₁₃₇	1	1	yes	characteristic
274	C₂²	#525×C₁₃₇	1	2	yes	—
275	C₂²	#575×C₁₃₇	1	2	yes	—
276	C₂²	#575×C₁₃₇	1	2	yes	—
277	C₄	—	2	2	no	—
278	C₄	—	2	2	no	—
279	C₄	—	2	2	no	—
280	C₄	—	2	2	no	—
281	C₄	—	2	2	no	—
282	C₄	—	4	8	no	—
283	C₄	—	8	4	no	—
284	C₄	C₄×#41×C₁₃₇	1	2	yes	—
285	C₄	#533×C₁₃₇	1	2	yes	—
286	C₂	—	4	2	no	—
287	C₂	#2296×C₁₃₇	1	1	yes	characteristic
288	C₂	#2911×C₁₃₇	1	1	yes	characteristic
289	C₂	#2911×C₁₃₇	1	1	yes	characteristic
290	C₂	#3111×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 ◁ C₂²×C₄²×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
4. #1949×C₁₃₇	C₂	semidirect
6. #1776×C₁₃₇	C₂	semidirect
9. #432×C₁₃₇	C₄	semidirect
145. #21366	C₁₃₇	direct (N × H)
146. C₁₃₇	#21366	direct (N × H)
150. #1949	C₂×C₁₃₇	semidirect
152. #1776	C₂×C₁₃₇	semidirect
155. #432	C₄×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#432×C₁₃₇	2
#2283×C₁₃₇	2
C₄×#117×C₁₃₇	2
#1949×C₁₃₇	2
#1854×C₁₃₇	2
#1776×C₁₃₇	2
#21366	137

Subgroups — C137×#21366 #46870752

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#21366 #46870752