Subgroups — C₁₃₇×#21154 #46870540

Order 35072.

998 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×#427×C₁₃₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#427×C₁₃₇	C₂	1	1	yes	maximal, characteristic
3	C₂²×#97×C₁₃₇	C₂	1	1	yes	maximal, characteristic
4	#2296×C₁₃₇	C₂	1	2	yes	maximal
5	#1944×C₁₃₇	C₂	1	4	yes	maximal, complemented
6	#1943×C₁₃₇	C₂	1	4	yes	maximal, complemented
7	#1735×C₁₃₇	C₂	1	2	yes	maximal
8	C₂³×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
9	C₂³×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
10	C₂×#117×C₁₃₇	C₂²	1	2	yes	—
11	C₂×#117×C₁₃₇	C₂²	1	2	yes	—
12	C₂²×C₄²×C₁₃₇	C₂²	1	1	yes	characteristic
13	#427×C₁₃₇	—	2	4	no	—
14	#427×C₁₃₇	C₄	1	8	yes	complemented
15	C₂×#107×C₁₃₇	C₂²	1	2	yes	—
16	C₂×#107×C₁₃₇	C₂²	1	2	yes	—
17	C₂×#106×C₁₃₇	C₂²	1	2	yes	—
18	C₂×#106×C₁₃₇	C₂²	1	2	yes	—
19	C₂×#97×C₁₃₇	—	2	4	no	—
20	C₂×#97×C₁₃₇	C₂²	1	2	yes	—
21	C₂×#97×C₁₃₇	C₂²	1	2	yes	—
22	#389×C₁₃₇	C₂²	1	8	yes	—
23	#385×C₁₃₇	—	2	4	no	—
24	C₄×C₄⋊C₄×C₁₃₇	—	2	4	no	—
25	#340×C₁₃₇	C₂²	1	8	yes	—
26	C₂³×C₄×C₁₃₇	D₄	1	2	yes	—
27	C₂³×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
28	C₂²×C₈×C₁₃₇	—	2	2	no	—
29	C₂²×C₈×C₁₃₇	—	2	2	no	—
30	C₂²×C₈×C₁₃₇	C₂×C₄	1	4	yes	—
31	C₂²×C₈×C₁₃₇	D₄	1	4	yes	—
32	C₂²×C₈×C₁₃₇	Q₈	1	4	yes	—
33	C₂²×C₈×C₁₃₇	Q₈	1	4	yes	—
34	C₂²×C₈×C₁₃₇	C₂³	1	2	yes	—
35	C₂²×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
36	C₂²×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
37	#117×C₁₃₇	—	2	4	no	—
38	#117×C₁₃₇	—	2	4	no	—
39	#117×C₁₃₇	—	2	4	no	—
40	#117×C₁₃₇	C₂×C₄	1	8	yes	—
41	C₂×C₄⋊C₄×C₁₃₇	—	2	8	no	—
42	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	4	yes	—
43	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	4	yes	—
44	C₂×C₂²⋊C₄×C₁₃₇	—	2	2	no	—
45	C₂×C₂²⋊C₄×C₁₃₇	—	2	2	no	—
46	C₂×C₄²×C₁₃₇	—	2	2	no	—
47	C₂×C₄²×C₁₃₇	—	2	2	no	—
48	C₂×C₄²×C₁₃₇	—	2	2	no	—
49	C₂×C₄²×C₁₃₇	—	2	2	no	—
50	C₂×C₄²×C₁₃₇	D₄	1	2	yes	—
51	C₂×C₄²×C₁₃₇	C₂³	1	2	yes	—
52	#107×C₁₃₇	—	2	4	no	—
53	#107×C₁₃₇	C₂×C₄	1	8	yes	complemented
54	#106×C₁₃₇	—	2	4	no	—
55	#106×C₁₃₇	C₂×C₄	1	8	yes	complemented
56	#97×C₁₃₇	—	2	4	no	—
57	#97×C₁₃₇	—	2	4	no	—
58	#95×C₁₃₇	—	2	8	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	4	no	—
64	C₂²×C₄×C₁₃₇	C₂²×C₄	1	2	yes	—
65	C₂²×C₄×C₁₃₇	C₂×D₄	1	4	yes	—
66	C₂²×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
67	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₁₃₇	C₂×Q₈	1	2	yes	—
69	C₂²×C₄×C₁₃₇	C₂×Q₈	1	2	yes	—
70	C₂²×C₄×C₁₃₇	#41	1	2	yes	—
71	C₂²×C₄×C₁₃₇	#41	1	2	yes	—
72	C₂²×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
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	C₄⋊C₄	1	16	yes	complemented
77	C₂×C₈×C₁₃₇	C₂²×C₄	1	2	yes	—
78	C₂×C₈×C₁₃₇	C₂²×C₄	1	2	yes	—
79	C₂×C₈×C₁₃₇	C₂×D₄	1	2	yes	—
80	C₂×C₈×C₁₃₇	C₂×D₄	1	2	yes	—
81	C₂×C₈×C₁₃₇	C₂×Q₈	1	4	yes	—
82	C₂×C₈×C₁₃₇	C₂×Q₈	1	2	yes	—
83	C₂×C₈×C₁₃₇	C₂×Q₈	1	2	yes	—
84	C₄⋊C₄×C₁₃₇	—	2	4	no	—
85	C₄⋊C₄×C₁₃₇	—	2	4	no	—
86	C₄⋊C₄×C₁₃₇	—	2	4	no	—
87	C₄⋊C₄×C₁₃₇	—	4	16	no	—
88	C₄⋊C₄×C₁₃₇	C₂²×C₄	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	—	2	2	no	—
95	C₄²×C₁₃₇	—	4	4	no	—
96	C₂³×C₁₃₇	—	2	4	no	—
97	C₂³×C₁₃₇	C₄×D₄	1	2	yes	—
98	C₂³×C₁₃₇	#121	1	2	yes	—
99	C₂³×C₁₃₇	C₄⋊Q₈	1	2	yes	—
100	C₂³×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
101	C₂×C₄×C₁₃₇	—	2	8	no	—
102	C₂×C₄×C₁₃₇	—	2	4	no	—
103	C₂×C₄×C₁₃₇	—	2	8	no	—
104	C₂×C₄×C₁₃₇	—	4	8	no	—
105	C₂×C₄×C₁₃₇	—	4	8	no	—
106	C₂×C₄×C₁₃₇	—	4	4	no	—
107	C₂×C₄×C₁₃₇	—	4	8	no	—
108	C₂×C₄×C₁₃₇	C₂×C₄⋊C₄	1	4	yes	—
109	C₂×C₄×C₁₃₇	C₄×D₄	1	2	yes	—
110	C₂×C₄×C₁₃₇	C₄×Q₈	1	4	yes	—
111	C₂×C₄×C₁₃₇	#121	1	2	yes	—
112	C₂×C₄×C₁₃₇	C₂²⋊Q₈	1	4	yes	—
113	C₂×C₄×C₁₃₇	#125	1	4	yes	—
114	C₂×C₄×C₁₃₇	C₄⋊Q₈	1	2	yes	—
115	C₂×C₄×C₁₃₇	C₂³×C₄	1	1	yes	characteristic
116	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
117	C₂×C₄×C₁₃₇	C₂²×Q₈	1	1	yes	characteristic
118	C₂×C₄×C₁₃₇	C₂²×Q₈	1	1	yes	characteristic
119	C₂×C₄×C₁₃₇	C₂×#41	1	1	yes	characteristic
120	C₂×C₄×C₁₃₇	C₂×#41	1	1	yes	characteristic
121	C₈×C₁₃₇	—	2	4	no	—
122	C₈×C₁₃₇	C₂×C₄⋊C₄	1	4	yes	—
123	C₈×C₁₃₇	C₂×C₄⋊C₄	1	4	yes	—
124	C₂²×C₁₃₇	—	2	4	no	—
125	C₂²×C₁₃₇	—	2	4	no	—
126	C₂²×C₁₃₇	—	2	4	no	—
127	C₂²×C₁₃₇	#389	1	4	yes	—
128	C₂²×C₁₃₇	C₂×C₄×D₄	1	1	yes	characteristic
129	C₂²×C₁₃₇	C₂×#121	1	1	yes	characteristic
130	C₂²×C₁₃₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
131	C₂²×C₁₃₇	#575	1	4	yes	—
132	C₄×C₁₃₇	—	4	8	no	—
133	C₄×C₁₃₇	—	4	4	no	—
134	C₄×C₁₃₇	—	8	4	no	—
135	C₄×C₁₃₇	#389	1	4	yes	—
136	C₄×C₁₃₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
137	C₄×C₁₃₇	C₂×C₄×Q₈	1	1	yes	characteristic
138	C₄×C₁₃₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
139	C₄×C₁₃₇	C₂×#125	1	1	yes	characteristic
140	C₂×C₁₃₇	—	2	4	no	—
141	C₂×C₁₃₇	C₂×#389	1	1	yes	characteristic
142	C₂×C₁₃₇	#2943	1	2	yes	—
143	C₂×C₁₃₇	#3054	1	2	yes	—
144	C₂×C₁₃₇	#3162	1	2	yes	—
145	#21154	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
146	C₁₃₇	#21154	1	1	yes	characteristic, direct product factor, Sylow
147	C₂×#427	C₂×C₁₃₇	1	1	yes	characteristic
148	C₂×#427	C₂×C₁₃₇	1	1	yes	characteristic
149	C₂²×#97	C₂×C₁₃₇	1	1	yes	characteristic
150	#2296	C₂×C₁₃₇	1	2	yes	—
151	#1944	C₂×C₁₃₇	1	4	yes	complemented
152	#1943	C₂×C₁₃₇	1	4	yes	complemented
153	#1735	C₂×C₁₃₇	1	2	yes	—
154	C₂³×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
155	C₂³×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
156	C₂×#117	C₂²×C₁₃₇	1	2	yes	—
157	C₂×#117	C₂²×C₁₃₇	1	2	yes	—
158	C₂²×C₄²	C₂²×C₁₃₇	1	1	yes	characteristic
159	#427	—	2	4	no	—
160	#427	C₄×C₁₃₇	1	8	yes	complemented
161	C₂×#107	C₂²×C₁₃₇	1	2	yes	—
162	C₂×#107	C₂²×C₁₃₇	1	2	yes	—
163	C₂×#106	C₂²×C₁₃₇	1	2	yes	—
164	C₂×#106	C₂²×C₁₃₇	1	2	yes	—
165	C₂×#97	—	2	4	no	—
166	C₂×#97	C₂²×C₁₃₇	1	2	yes	—
167	C₂×#97	C₂²×C₁₃₇	1	2	yes	—
168	#389	C₂²×C₁₃₇	1	8	yes	—
169	#385	—	2	4	no	—
170	C₄×C₄⋊C₄	—	2	4	no	—
171	#340	C₂²×C₁₃₇	1	8	yes	—
172	C₂³×C₄	D₄×C₁₃₇	1	2	yes	—
173	C₂³×C₄	C₂³×C₁₃₇	1	1	yes	characteristic
174	C₂²×C₈	—	2	2	no	—
175	C₂²×C₈	—	2	2	no	—
176	C₂²×C₈	C₂×C₄×C₁₃₇	1	4	yes	—
177	C₂²×C₈	D₄×C₁₃₇	1	4	yes	—
178	C₂²×C₈	Q₈×C₁₃₇	1	4	yes	—
179	C₂²×C₈	Q₈×C₁₃₇	1	4	yes	—
180	C₂²×C₈	C₂³×C₁₃₇	1	2	yes	—
181	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
182	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
183	#117	—	2	4	no	—
184	#117	—	2	4	no	—
185	#117	—	2	4	no	—
186	#117	C₂×C₄×C₁₃₇	1	8	yes	—
187	C₂×C₄⋊C₄	—	2	8	no	—
188	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	4	yes	—
189	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	4	yes	—
190	C₂×C₂²⋊C₄	—	2	2	no	—
191	C₂×C₂²⋊C₄	—	2	2	no	—
192	C₂×C₄²	—	2	2	no	—
193	C₂×C₄²	—	2	2	no	—
194	C₂×C₄²	—	2	2	no	—
195	C₂×C₄²	—	2	2	no	—
196	C₂×C₄²	D₄×C₁₃₇	1	2	yes	—
197	C₂×C₄²	C₂³×C₁₃₇	1	2	yes	—
198	#107	—	2	4	no	—
199	#107	C₂×C₄×C₁₃₇	1	8	yes	complemented
200	#106	—	2	4	no	—
201	#106	C₂×C₄×C₁₃₇	1	8	yes	complemented
202	#97	—	2	4	no	—
203	#97	—	2	4	no	—
204	#95	—	2	8	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	4	no	—
210	C₂²×C₄	C₂²×C₄×C₁₃₇	1	2	yes	—
211	C₂²×C₄	C₂×D₄×C₁₃₇	1	4	yes	—
212	C₂²×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
213	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
214	C₂²×C₄	C₂×Q₈×C₁₃₇	1	2	yes	—
215	C₂²×C₄	C₂×Q₈×C₁₃₇	1	2	yes	—
216	C₂²×C₄	#41×C₁₃₇	1	2	yes	—
217	C₂²×C₄	#41×C₁₃₇	1	2	yes	—
218	C₂²×C₄	C₂⁴×C₁₃₇	1	1	yes	characteristic
219	C₂×C₈	—	2	8	no	—
220	C₂×C₈	—	2	2	no	—
221	C₂×C₈	—	2	2	no	—
222	C₂×C₈	C₄⋊C₄×C₁₃₇	1	16	yes	complemented
223	C₂×C₈	C₂²×C₄×C₁₃₇	1	2	yes	—
224	C₂×C₈	C₂²×C₄×C₁₃₇	1	2	yes	—
225	C₂×C₈	C₂×D₄×C₁₃₇	1	2	yes	—
226	C₂×C₈	C₂×D₄×C₁₃₇	1	2	yes	—
227	C₂×C₈	C₂×Q₈×C₁₃₇	1	4	yes	—
228	C₂×C₈	C₂×Q₈×C₁₃₇	1	2	yes	—
229	C₂×C₈	C₂×Q₈×C₁₃₇	1	2	yes	—
230	C₄⋊C₄	—	2	4	no	—
231	C₄⋊C₄	—	2	4	no	—
232	C₄⋊C₄	—	2	4	no	—
233	C₄⋊C₄	—	4	16	no	—
234	C₄⋊C₄	C₂²×C₄×C₁₃₇	1	8	yes	—
235	C₂²⋊C₄	—	2	8	no	—
236	C₂²⋊C₄	—	4	4	no	—
237	C₄²	—	2	8	no	—
238	C₄²	—	2	4	no	—
239	C₄²	—	2	2	no	—
240	C₄²	—	2	2	no	—
241	C₄²	—	4	4	no	—
242	C₂³	—	2	4	no	—
243	C₂³	C₄×D₄×C₁₃₇	1	2	yes	—
244	C₂³	#121×C₁₃₇	1	2	yes	—
245	C₂³	C₄⋊Q₈×C₁₃₇	1	2	yes	—
246	C₂³	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
247	C₂×C₄	—	2	8	no	—
248	C₂×C₄	—	2	4	no	—
249	C₂×C₄	—	2	8	no	—
250	C₂×C₄	—	4	8	no	—
251	C₂×C₄	—	4	8	no	—
252	C₂×C₄	—	4	4	no	—
253	C₂×C₄	—	4	8	no	—
254	C₂×C₄	C₂×C₄⋊C₄×C₁₃₇	1	4	yes	—
255	C₂×C₄	C₄×D₄×C₁₃₇	1	2	yes	—
256	C₂×C₄	C₄×Q₈×C₁₃₇	1	4	yes	—
257	C₂×C₄	#121×C₁₃₇	1	2	yes	—
258	C₂×C₄	C₂²⋊Q₈×C₁₃₇	1	4	yes	—
259	C₂×C₄	#125×C₁₃₇	1	4	yes	—
260	C₂×C₄	C₄⋊Q₈×C₁₃₇	1	2	yes	—
261	C₂×C₄	C₂³×C₄×C₁₃₇	1	1	yes	characteristic
262	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
263	C₂×C₄	C₂²×Q₈×C₁₃₇	1	1	yes	characteristic
264	C₂×C₄	C₂²×Q₈×C₁₃₇	1	1	yes	characteristic
265	C₂×C₄	C₂×#41×C₁₃₇	1	1	yes	characteristic
266	C₂×C₄	C₂×#41×C₁₃₇	1	1	yes	characteristic
267	C₈	—	2	4	no	—
268	C₈	C₂×C₄⋊C₄×C₁₃₇	1	4	yes	—
269	C₈	C₂×C₄⋊C₄×C₁₃₇	1	4	yes	—
270	C₂²	—	2	4	no	—
271	C₂²	—	2	4	no	—
272	C₂²	—	2	4	no	—
273	C₂²	#389×C₁₃₇	1	4	yes	—
274	C₂²	C₂×C₄×D₄×C₁₃₇	1	1	yes	characteristic
275	C₂²	C₂×#121×C₁₃₇	1	1	yes	characteristic
276	C₂²	C₂×C₄⋊Q₈×C₁₃₇	1	1	yes	characteristic
277	C₂²	#575×C₁₃₇	1	4	yes	—
278	C₄	—	4	4	no	—
279	C₄	—	4	8	no	—
280	C₄	—	8	4	no	—
281	C₄	#389×C₁₃₇	1	4	yes	—
282	C₄	C₂²×C₄⋊C₄×C₁₃₇	1	1	yes	characteristic
283	C₄	C₂×C₄×Q₈×C₁₃₇	1	1	yes	characteristic
284	C₄	C₂×C₂²⋊Q₈×C₁₃₇	1	1	yes	characteristic
285	C₄	C₂×#125×C₁₃₇	1	1	yes	characteristic
286	C₂	—	2	4	no	—
287	C₂	C₂×#389×C₁₃₇	1	1	yes	characteristic
288	C₂	#2943×C₁₃₇	1	2	yes	—
289	C₂	#3054×C₁₃₇	1	2	yes	—
290	C₂	#3162×C₁₃₇	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₁₃₇	C₂×D₄
commutator	C₂×C₄	C₂³×C₄×C₁₃₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃₇
socle	C₂³×C₁₃₇	C₂²×D₄
prime radical	C₂⁴×C₁₃₇	C₂×D₄
LCS	G ▷ C₂×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
5. #1944×C₁₃₇	C₂	semidirect
6. #1943×C₁₃₇	C₂	semidirect
14. #427×C₁₃₇	C₄	semidirect
53. #107×C₁₃₇	C₂×C₄	semidirect
55. #106×C₁₃₇	C₂×C₄	semidirect
76. C₂×C₈×C₁₃₇	C₄⋊C₄	semidirect
145. #21154	C₁₃₇	direct (N × H)
146. C₁₃₇	#21154	direct (N × H)
151. #1944	C₂×C₁₃₇	semidirect
152. #1943	C₂×C₁₃₇	semidirect
160. #427	C₄×C₁₃₇	semidirect
199. #107	C₂×C₄×C₁₃₇	semidirect
201. #106	C₂×C₄×C₁₃₇	semidirect
222. C₂×C₈	C₄⋊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×#427×C₁₃₇	2
C₂²×#97×C₁₃₇	2
#2296×C₁₃₇	2
#1944×C₁₃₇	2
#1943×C₁₃₇	2
#1735×C₁₃₇	2
#21154	137

Subgroups — C137×#21154 #46870540

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#21154 #46870540