Subgroups — C₁₃₇×#23666 #46873052

Order 35072.

910 conjugacy classes of subgroups.

The subgroup lattice has 314 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#427×C₁₃₇	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄⋊C₈×C₁₃₇	C₂	1	1	yes	maximal, characteristic
3	#2296×C₁₃₇	C₂	1	2	yes	maximal
4	#2088×C₁₃₇	C₂	1	4	yes	maximal, complemented
5	#2087×C₁₃₇	C₂	1	2	yes	maximal, complemented
6	#2086×C₁₃₇	C₂	1	2	yes	maximal, complemented
7	#1735×C₁₃₇	C₂	1	2	yes	maximal
8	#1735×C₁₃₇	C₂	1	1	yes	maximal, characteristic
9	C₂³×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
10	C₂³×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
11	C₂×#117×C₁₃₇	C₂²	1	2	yes	—
12	C₂×#117×C₁₃₇	C₂²	1	2	yes	—
13	C₂²×C₄²×C₁₃₇	C₂²	1	1	yes	characteristic
14	#427×C₁₃₇	—	2	4	no	—
15	C₂×#107×C₁₃₇	C₂²	1	1	yes	characteristic
16	C₂×#107×C₁₃₇	C₂²	1	1	yes	characteristic
17	C₂×#106×C₁₃₇	C₂²	1	2	yes	—
18	C₂×C₄⋊C₈×C₁₃₇	—	2	4	no	—
19	C₂×C₄⋊C₈×C₁₃₇	C₂²	1	2	yes	—
20	C₂×C₄⋊C₈×C₁₃₇	C₂²	1	2	yes	—
21	#389×C₁₃₇	C₂²	1	4	yes	—
22	#389×C₁₃₇	C₂²	1	4	yes	—
23	#385×C₁₃₇	—	2	2	no	—
24	#385×C₁₃₇	—	2	2	no	—
25	C₄×C₄⋊C₄×C₁₃₇	—	2	4	no	—
26	#340×C₁₃₇	C₂²	1	2	yes	—
27	#340×C₁₃₇	C₂²	1	4	yes	—
28	#340×C₁₃₇	C₂²	1	4	yes	—
29	#340×C₁₃₇	C₂²	1	2	yes	—
30	C₂³×C₄×C₁₃₇	D₄	1	1	yes	characteristic
31	C₂³×C₄×C₁₃₇	D₄	1	1	yes	characteristic
32	C₂³×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
33	C₂²×C₈×C₁₃₇	—	2	4	no	—
34	C₂²×C₈×C₁₃₇	—	2	2	no	—
35	C₂²×C₈×C₁₃₇	—	2	4	no	—
36	C₂²×C₈×C₁₃₇	D₄	1	4	yes	—
37	C₂²×C₈×C₁₃₇	C₂³	1	2	yes	—
38	C₂²×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
39	C₂²×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
40	#117×C₁₃₇	—	2	4	no	—
41	#117×C₁₃₇	—	2	4	no	—
42	#117×C₁₃₇	—	2	4	no	—
43	#117×C₁₃₇	Q₈	1	8	yes	—
44	C₂×C₄⋊C₄×C₁₃₇	—	2	4	no	—
45	C₂×C₄⋊C₄×C₁₃₇	—	2	4	no	—
46	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	2	yes	—
47	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	2	yes	—
48	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	4	yes	—
49	C₂×C₂²⋊C₄×C₁₃₇	—	2	2	no	—
50	C₂×C₂²⋊C₄×C₁₃₇	—	2	2	no	—
51	C₂×C₄²×C₁₃₇	—	2	2	no	—
52	C₂×C₄²×C₁₃₇	—	2	2	no	—
53	C₂×C₄²×C₁₃₇	—	2	2	no	—
54	C₂×C₄²×C₁₃₇	—	2	2	no	—
55	C₂×C₄²×C₁₃₇	D₄	1	2	yes	—
56	C₂×C₄²×C₁₃₇	C₂³	1	2	yes	—
57	#107×C₁₃₇	—	2	2	no	—
58	#107×C₁₃₇	—	2	2	no	—
59	#106×C₁₃₇	—	2	4	no	—
60	C₄⋊C₈×C₁₃₇	—	2	4	no	—
61	C₄⋊C₈×C₁₃₇	—	2	4	no	—
62	#95×C₁₃₇	—	2	4	no	—
63	#95×C₁₃₇	—	2	4	no	—
64	C₂⁴×C₁₃₇	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₁₃₇	—	2	2	no	—
66	C₂²×C₄×C₁₃₇	—	2	4	no	—
67	C₂²×C₄×C₁₃₇	—	2	2	no	—
68	C₂²×C₄×C₁₃₇	—	2	4	no	—
69	C₂²×C₄×C₁₃₇	—	2	2	no	—
70	C₂²×C₄×C₁₃₇	—	2	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	2	yes	—
75	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₁₃₇	C₂×Q₈	1	4	yes	—
77	C₂²×C₄×C₁₃₇	#41	1	2	yes	—
78	C₂²×C₄×C₁₃₇	#41	1	4	yes	—
79	C₂²×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
80	C₂×C₈×C₁₃₇	—	2	8	no	—
81	C₂×C₈×C₁₃₇	—	2	2	no	—
82	C₂×C₈×C₁₃₇	—	2	2	no	—
83	C₂×C₈×C₁₃₇	—	2	2	no	—
84	C₂×C₈×C₁₃₇	—	2	4	no	—
85	C₂×C₈×C₁₃₇	—	4	4	no	—
86	C₂×C₈×C₁₃₇	C₂×D₄	1	2	yes	—
87	C₂×C₈×C₁₃₇	C₂×D₄	1	2	yes	—
88	C₄⋊C₄×C₁₃₇	—	2	4	no	—
89	C₄⋊C₄×C₁₃₇	—	2	2	no	—
90	C₄⋊C₄×C₁₃₇	—	2	2	no	—
91	C₄⋊C₄×C₁₃₇	—	2	2	no	—
92	C₄⋊C₄×C₁₃₇	—	2	2	no	—
93	C₄⋊C₄×C₁₃₇	—	4	8	no	—
94	C₄⋊C₄×C₁₃₇	—	4	8	no	—
95	C₄⋊C₄×C₁₃₇	C₂×Q₈	1	8	yes	—
96	C₂²⋊C₄×C₁₃₇	—	2	8	no	—
97	C₂²⋊C₄×C₁₃₇	—	4	4	no	—
98	C₄²×C₁₃₇	—	2	4	no	—
99	C₄²×C₁₃₇	—	2	8	no	—
100	C₄²×C₁₃₇	—	2	2	no	—
101	C₄²×C₁₃₇	—	2	2	no	—
102	C₄²×C₁₃₇	—	4	4	no	—
103	C₂³×C₁₃₇	—	2	2	no	—
104	C₂³×C₁₃₇	—	2	2	no	—
105	C₂³×C₁₃₇	#121	1	2	yes	—
106	C₂³×C₁₃₇	C₂²⋊Q₈	1	4	yes	—
107	C₂³×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
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	—	2	2	no	—
112	C₂×C₄×C₁₃₇	—	2	8	no	—
113	C₂×C₄×C₁₃₇	—	2	4	no	—
114	C₂×C₄×C₁₃₇	—	4	4	no	—
115	C₂×C₄×C₁₃₇	—	4	4	no	—
116	C₂×C₄×C₁₃₇	—	4	4	no	—
117	C₂×C₄×C₁₃₇	—	4	2	no	—
118	C₂×C₄×C₁₃₇	—	4	2	no	—
119	C₂×C₄×C₁₃₇	—	4	4	no	—
120	C₂×C₄×C₁₃₇	—	4	4	no	—
121	C₂×C₄×C₁₃₇	#121	1	2	yes	—
122	C₂×C₄×C₁₃₇	C₂²⋊Q₈	1	4	yes	—
123	C₂×C₄×C₁₃₇	#123	1	4	yes	—
124	C₂×C₄×C₁₃₇	#125	1	8	yes	—
125	C₂×C₄×C₁₃₇	C₄⋊Q₈	1	4	yes	—
126	C₂×C₄×C₁₃₇	#135	1	4	yes	—
127	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
128	C₂×C₄×C₁₃₇	C₂²×Q₈	1	2	yes	—
129	C₂×C₄×C₁₃₇	C₂×#41	1	2	yes	—
130	C₂×C₄×C₁₃₇	C₂×#41	1	1	yes	characteristic
131	C₈×C₁₃₇	—	2	2	no	—
132	C₈×C₁₃₇	—	2	2	no	—
133	C₈×C₁₃₇	—	4	2	no	—
134	C₂²×C₁₃₇	—	2	8	no	—
135	C₂²×C₁₃₇	—	2	2	no	—
136	C₂²×C₁₃₇	—	2	2	no	—
137	C₂²×C₁₃₇	#398	1	4	yes	—
138	C₂²×C₁₃₇	C₂×#121	1	1	yes	characteristic
139	C₂²×C₁₃₇	C₂×C₂²⋊Q₈	1	2	yes	—
140	C₂²×C₁₃₇	C₂×#135	1	2	yes	—
141	C₂²×C₁₃₇	#575	1	2	yes	—
142	C₄×C₁₃₇	—	2	4	no	—
143	C₄×C₁₃₇	—	4	8	no	—
144	C₄×C₁₃₇	—	4	2	no	—
145	C₄×C₁₃₇	—	8	2	no	—
146	C₄×C₁₃₇	—	8	2	no	—
147	C₄×C₁₃₇	#398	1	4	yes	—
148	C₄×C₁₃₇	C₂×#123	1	1	yes	characteristic
149	C₄×C₁₃₇	C₂×#125	1	2	yes	—
150	C₄×C₁₃₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
151	C₂×C₁₃₇	—	2	4	no	—
152	C₂×C₁₃₇	C₂×#398	1	1	yes	characteristic
153	C₂×C₁₃₇	#3050	1	1	yes	characteristic
154	C₂×C₁₃₇	#3054	1	1	yes	characteristic
155	C₂×C₁₃₇	#3076	1	4	yes	—
156	#23666	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
157	C₁₃₇	#23666	1	1	yes	characteristic, direct product factor, Sylow
158	C₂×#427	C₂×C₁₃₇	1	1	yes	characteristic
159	C₂²×C₄⋊C₈	C₂×C₁₃₇	1	1	yes	characteristic
160	#2296	C₂×C₁₃₇	1	2	yes	—
161	#2088	C₂×C₁₃₇	1	4	yes	complemented
162	#2087	C₂×C₁₃₇	1	2	yes	complemented
163	#2086	C₂×C₁₃₇	1	2	yes	complemented
164	#1735	C₂×C₁₃₇	1	2	yes	—
165	#1735	C₂×C₁₃₇	1	1	yes	characteristic
166	C₂³×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
167	C₂³×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
168	C₂×#117	C₂²×C₁₃₇	1	2	yes	—
169	C₂×#117	C₂²×C₁₃₇	1	2	yes	—
170	C₂²×C₄²	C₂²×C₁₃₇	1	1	yes	characteristic
171	#427	—	2	4	no	—
172	C₂×#107	C₂²×C₁₃₇	1	1	yes	characteristic
173	C₂×#107	C₂²×C₁₃₇	1	1	yes	characteristic
174	C₂×#106	C₂²×C₁₃₇	1	2	yes	—
175	C₂×C₄⋊C₈	—	2	4	no	—
176	C₂×C₄⋊C₈	C₂²×C₁₃₇	1	2	yes	—
177	C₂×C₄⋊C₈	C₂²×C₁₃₇	1	2	yes	—
178	#389	C₂²×C₁₃₇	1	4	yes	—
179	#389	C₂²×C₁₃₇	1	4	yes	—
180	#385	—	2	2	no	—
181	#385	—	2	2	no	—
182	C₄×C₄⋊C₄	—	2	4	no	—
183	#340	C₂²×C₁₃₇	1	2	yes	—
184	#340	C₂²×C₁₃₇	1	4	yes	—
185	#340	C₂²×C₁₃₇	1	4	yes	—
186	#340	C₂²×C₁₃₇	1	2	yes	—
187	C₂³×C₄	D₄×C₁₃₇	1	1	yes	characteristic
188	C₂³×C₄	D₄×C₁₃₇	1	1	yes	characteristic
189	C₂³×C₄	C₂³×C₁₃₇	1	1	yes	characteristic
190	C₂²×C₈	—	2	4	no	—
191	C₂²×C₈	—	2	2	no	—
192	C₂²×C₈	—	2	4	no	—
193	C₂²×C₈	D₄×C₁₃₇	1	4	yes	—
194	C₂²×C₈	C₂³×C₁₃₇	1	2	yes	—
195	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
196	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
197	#117	—	2	4	no	—
198	#117	—	2	4	no	—
199	#117	—	2	4	no	—
200	#117	Q₈×C₁₃₇	1	8	yes	—
201	C₂×C₄⋊C₄	—	2	4	no	—
202	C₂×C₄⋊C₄	—	2	4	no	—
203	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	2	yes	—
204	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	2	yes	—
205	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	4	yes	—
206	C₂×C₂²⋊C₄	—	2	2	no	—
207	C₂×C₂²⋊C₄	—	2	2	no	—
208	C₂×C₄²	—	2	2	no	—
209	C₂×C₄²	—	2	2	no	—
210	C₂×C₄²	—	2	2	no	—
211	C₂×C₄²	—	2	2	no	—
212	C₂×C₄²	D₄×C₁₃₇	1	2	yes	—
213	C₂×C₄²	C₂³×C₁₃₇	1	2	yes	—
214	#107	—	2	2	no	—
215	#107	—	2	2	no	—
216	#106	—	2	4	no	—
217	C₄⋊C₈	—	2	4	no	—
218	C₄⋊C₈	—	2	4	no	—
219	#95	—	2	4	no	—
220	#95	—	2	4	no	—
221	C₂⁴	C₂×D₄×C₁₃₇	1	1	yes	characteristic
222	C₂²×C₄	—	2	2	no	—
223	C₂²×C₄	—	2	4	no	—
224	C₂²×C₄	—	2	2	no	—
225	C₂²×C₄	—	2	4	no	—
226	C₂²×C₄	—	2	2	no	—
227	C₂²×C₄	—	2	2	no	—
228	C₂²×C₄	—	4	2	no	—
229	C₂²×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
230	C₂²×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
231	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
232	C₂²×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
233	C₂²×C₄	C₂×Q₈×C₁₃₇	1	4	yes	—
234	C₂²×C₄	#41×C₁₃₇	1	2	yes	—
235	C₂²×C₄	#41×C₁₃₇	1	4	yes	—
236	C₂²×C₄	C₂⁴×C₁₃₇	1	1	yes	characteristic
237	C₂×C₈	—	2	8	no	—
238	C₂×C₈	—	2	2	no	—
239	C₂×C₈	—	2	2	no	—
240	C₂×C₈	—	2	2	no	—
241	C₂×C₈	—	2	4	no	—
242	C₂×C₈	—	4	4	no	—
243	C₂×C₈	C₂×D₄×C₁₃₇	1	2	yes	—
244	C₂×C₈	C₂×D₄×C₁₃₇	1	2	yes	—
245	C₄⋊C₄	—	2	4	no	—
246	C₄⋊C₄	—	2	2	no	—
247	C₄⋊C₄	—	2	2	no	—
248	C₄⋊C₄	—	2	2	no	—
249	C₄⋊C₄	—	2	2	no	—
250	C₄⋊C₄	—	4	8	no	—
251	C₄⋊C₄	—	4	8	no	—
252	C₄⋊C₄	C₂×Q₈×C₁₃₇	1	8	yes	—
253	C₂²⋊C₄	—	2	8	no	—
254	C₂²⋊C₄	—	4	4	no	—
255	C₄²	—	2	4	no	—
256	C₄²	—	2	8	no	—
257	C₄²	—	2	2	no	—
258	C₄²	—	2	2	no	—
259	C₄²	—	4	4	no	—
260	C₂³	—	2	2	no	—
261	C₂³	—	2	2	no	—
262	C₂³	#121×C₁₃₇	1	2	yes	—
263	C₂³	C₂²⋊Q₈×C₁₃₇	1	4	yes	—
264	C₂³	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
265	C₂×C₄	—	2	8	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	2	8	no	—
270	C₂×C₄	—	2	4	no	—
271	C₂×C₄	—	4	4	no	—
272	C₂×C₄	—	4	4	no	—
273	C₂×C₄	—	4	4	no	—
274	C₂×C₄	—	4	2	no	—
275	C₂×C₄	—	4	2	no	—
276	C₂×C₄	—	4	4	no	—
277	C₂×C₄	—	4	4	no	—
278	C₂×C₄	#121×C₁₃₇	1	2	yes	—
279	C₂×C₄	C₂²⋊Q₈×C₁₃₇	1	4	yes	—
280	C₂×C₄	#123×C₁₃₇	1	4	yes	—
281	C₂×C₄	#125×C₁₃₇	1	8	yes	—
282	C₂×C₄	C₄⋊Q₈×C₁₃₇	1	4	yes	—
283	C₂×C₄	#135×C₁₃₇	1	4	yes	—
284	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
285	C₂×C₄	C₂²×Q₈×C₁₃₇	1	2	yes	—
286	C₂×C₄	C₂×#41×C₁₃₇	1	2	yes	—
287	C₂×C₄	C₂×#41×C₁₃₇	1	1	yes	characteristic
288	C₈	—	2	2	no	—
289	C₈	—	2	2	no	—
290	C₈	—	4	2	no	—
291	C₂²	—	2	2	no	—
292	C₂²	—	2	2	no	—
293	C₂²	—	2	8	no	—
294	C₂²	#398×C₁₃₇	1	4	yes	—
295	C₂²	C₂×#121×C₁₃₇	1	1	yes	characteristic
296	C₂²	C₂×C₂²⋊Q₈×C₁₃₇	1	2	yes	—
297	C₂²	C₂×#135×C₁₃₇	1	2	yes	—
298	C₂²	#575×C₁₃₇	1	2	yes	—
299	C₄	—	2	4	no	—
300	C₄	—	4	2	no	—
301	C₄	—	4	8	no	—
302	C₄	—	8	2	no	—
303	C₄	—	8	2	no	—
304	C₄	#398×C₁₃₇	1	4	yes	—
305	C₄	C₂×#123×C₁₃₇	1	1	yes	characteristic
306	C₄	C₂×#125×C₁₃₇	1	2	yes	—
307	C₄	C₂×C₄⋊Q₈×C₁₃₇	1	1	yes	characteristic
308	C₂	—	2	4	no	—
309	C₂	C₂×#398×C₁₃₇	1	1	yes	characteristic
310	C₂	#3050×C₁₃₇	1	1	yes	characteristic
311	C₂	#3054×C₁₃₇	1	1	yes	characteristic
312	C₂	#3076×C₁₃₇	1	4	yes	—

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

Complements

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

N	H ≅ G/N	type
4. #2088×C₁₃₇	C₂	semidirect
5. #2087×C₁₃₇	C₂	semidirect
6. #2086×C₁₃₇	C₂	semidirect
156. #23666	C₁₃₇	direct (N × H)
157. C₁₃₇	#23666	direct (N × H)
161. #2088	C₂×C₁₃₇	semidirect
162. #2087	C₂×C₁₃₇	semidirect
163. #2086	C₂×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#427×C₁₃₇	2
C₂²×C₄⋊C₈×C₁₃₇	2
#2296×C₁₃₇	2
#2088×C₁₃₇	2
#2087×C₁₃₇	2
#2086×C₁₃₇	2
#1735×C₁₃₇	2
#23666	137

Subgroups — C137×#23666 #46873052

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#23666 #46873052