Subgroups — C₂×#1735×C₁₃₇ #46869731

Order 35072.

2398 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×C₈×C₁₃₇	C₂	1	1	yes	maximal, characteristic
2	C₂²×#117×C₁₃₇	C₂	1	2	yes	maximal
3	#1735×C₁₃₇	C₂	1	8	yes	maximal, direct product factor
4	C₂×#340×C₁₃₇	C₂	1	4	yes	maximal, complemented
5	C₂⁴×C₄×C₁₃₇	C₂²	1	1	yes	characteristic
6	C₂³×C₈×C₁₃₇	—	2	8	no	—
7	C₂³×C₈×C₁₃₇	C₄	1	8	yes	complemented
8	C₂³×C₈×C₁₃₇	C₂²	1	2	yes	—
9	C₂³×C₈×C₁₃₇	C₂²	1	4	yes	—
10	C₂×#117×C₁₃₇	—	2	8	no	—
11	C₂×#117×C₁₃₇	—	2	4	no	—
12	C₂×#117×C₁₃₇	C₄	1	16	yes	complemented
13	C₂×#117×C₁₃₇	C₂²	1	8	yes	—
14	C₂²×C₄⋊C₄×C₁₃₇	C₂²	1	4	yes	—
15	C₂²×C₂²⋊C₄×C₁₃₇	—	2	2	no	—
16	C₂²×C₄²×C₁₃₇	—	2	2	no	—
17	#340×C₁₃₇	C₂²	1	16	yes	complemented
18	C₂⁵×C₁₃₇	D₄	1	1	yes	characteristic
19	C₂³×C₄×C₁₃₇	—	2	4	no	—
20	C₂³×C₄×C₁₃₇	C₂×C₄	1	4	yes	—
21	C₂³×C₄×C₁₃₇	C₂×C₄	1	8	yes	—
22	C₂³×C₄×C₁₃₇	D₄	1	2	yes	—
23	C₂³×C₄×C₁₃₇	D₄	1	8	yes	—
24	C₂³×C₄×C₁₃₇	D₄	1	1	yes	characteristic
25	C₂³×C₄×C₁₃₇	Q₈	1	4	yes	—
26	C₂³×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
27	C₂³×C₄×C₁₃₇	C₂³	1	2	yes	—
28	C₂²×C₈×C₁₃₇	—	2	32	no	—
29	C₂²×C₈×C₁₃₇	—	2	8	no	—
30	C₂²×C₈×C₁₃₇	—	2	16	no	—
31	C₂²×C₈×C₁₃₇	C₂×C₄	1	8	yes	—
32	C₂²×C₈×C₁₃₇	C₂×C₄	1	16	yes	complemented
33	C₂²×C₈×C₁₃₇	C₂³	1	4	yes	—
34	#117×C₁₃₇	—	2	32	no	—
35	#117×C₁₃₇	—	2	16	no	—
36	#117×C₁₃₇	C₂×C₄	1	32	yes	complemented
37	C₂×C₄⋊C₄×C₁₃₇	—	2	4	no	—
38	C₂×C₄⋊C₄×C₁₃₇	—	2	8	no	—
39	C₂×C₄⋊C₄×C₁₃₇	C₂×C₄	1	16	yes	—
40	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	8	yes	—
41	C₂×C₂²⋊C₄×C₁₃₇	—	2	16	no	—
42	C₂×C₂²⋊C₄×C₁₃₇	—	2	8	no	—
43	C₂×C₄²×C₁₃₇	—	2	16	no	—
44	C₂×C₄²×C₁₃₇	—	2	8	no	—
45	C₂⁴×C₁₃₇	—	2	8	no	—
46	C₂⁴×C₁₃₇	C₂²⋊C₄	1	8	yes	—
47	C₂⁴×C₁₃₇	C₄⋊C₄	1	4	yes	—
48	C₂⁴×C₁₃₇	C₂×D₄	1	2	yes	—
49	C₂⁴×C₁₃₇	C₂×D₄	1	1	yes	characteristic
50	C₂²×C₄×C₁₃₇	—	2	16	no	—
51	C₂²×C₄×C₁₃₇	—	2	8	no	—
52	C₂²×C₄×C₁₃₇	—	4	16	no	—
53	C₂²×C₄×C₁₃₇	C₄²	1	16	yes	—
54	C₂²×C₄×C₁₃₇	C₂²⋊C₄	1	8	yes	—
55	C₂²×C₄×C₁₃₇	C₂²⋊C₄	1	16	yes	—
56	C₂²×C₄×C₁₃₇	C₂²⋊C₄	1	16	yes	—
57	C₂²×C₄×C₁₃₇	C₄⋊C₄	1	8	yes	—
58	C₂²×C₄×C₁₃₇	C₄⋊C₄	1	4	yes	—
59	C₂²×C₄×C₁₃₇	C₄⋊C₄	1	32	yes	—
60	C₂²×C₄×C₁₃₇	C₂²×C₄	1	2	yes	—
61	C₂²×C₄×C₁₃₇	C₂²×C₄	1	4	yes	—
62	C₂²×C₄×C₁₃₇	C₂²×C₄	1	4	yes	—
63	C₂²×C₄×C₁₃₇	C₂²×C₄	1	8	yes	—
64	C₂²×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
65	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₁₃₇	C₂×D₄	1	4	yes	—
67	C₂²×C₄×C₁₃₇	C₂×D₄	1	8	yes	—
68	C₂²×C₄×C₁₃₇	C₂×Q₈	1	4	yes	—
69	C₂²×C₄×C₁₃₇	C₂×Q₈	1	2	yes	—
70	C₂²×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
71	C₂×C₈×C₁₃₇	—	2	16	no	—
72	C₂×C₈×C₁₃₇	—	2	32	no	—
73	C₂×C₈×C₁₃₇	—	2	8	no	—
74	C₂×C₈×C₁₃₇	C₂²×C₄	1	8	yes	—
75	C₄⋊C₄×C₁₃₇	—	2	16	no	—
76	C₄⋊C₄×C₁₃₇	—	2	8	no	—
77	C₄⋊C₄×C₁₃₇	C₂²×C₄	1	16	yes	—
78	C₂²⋊C₄×C₁₃₇	—	2	32	no	—
79	C₄²×C₁₃₇	—	2	32	no	—
80	C₂³×C₁₃₇	—	2	8	no	—
81	C₂³×C₁₃₇	—	2	32	no	—
82	C₂³×C₁₃₇	—	2	16	no	—
83	C₂³×C₁₃₇	#95	1	16	yes	—
84	C₂³×C₁₃₇	C₂×C₂²⋊C₄	1	8	yes	—
85	C₂³×C₁₃₇	C₂×C₂²⋊C₄	1	4	yes	—
86	C₂³×C₁₃₇	C₂×C₄⋊C₄	1	4	yes	—
87	C₂³×C₁₃₇	C₂×C₄⋊C₄	1	2	yes	—
88	C₂³×C₁₃₇	#135	1	8	yes	—
89	C₂³×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
90	C₂×C₄×C₁₃₇	—	2	16	no	—
91	C₂×C₄×C₁₃₇	—	4	16	no	—
92	C₂×C₄×C₁₃₇	—	4	32	no	—
93	C₂×C₄×C₁₃₇	#95	1	32	yes	—
94	C₂×C₄×C₁₃₇	#95	1	16	yes	—
95	C₂×C₄×C₁₃₇	C₂×C₄²	1	8	yes	—
96	C₂×C₄×C₁₃₇	C₂×C₄²	1	4	yes	—
97	C₂×C₄×C₁₃₇	C₂×C₂²⋊C₄	1	8	yes	—
98	C₂×C₄×C₁₃₇	C₂×C₂²⋊C₄	1	4	yes	—
99	C₂×C₄×C₁₃₇	C₂×C₂²⋊C₄	1	8	yes	—
100	C₂×C₄×C₁₃₇	C₂×C₂²⋊C₄	1	4	yes	—
101	C₂×C₄×C₁₃₇	C₂×C₄⋊C₄	1	16	yes	—
102	C₂×C₄×C₁₃₇	C₂×C₄⋊C₄	1	8	yes	—
103	C₂×C₄×C₁₃₇	C₂×C₄⋊C₄	1	2	yes	—
104	C₂×C₄×C₁₃₇	C₂×C₄⋊C₄	1	4	yes	—
105	C₂×C₄×C₁₃₇	C₂³×C₄	1	1	yes	characteristic
106	C₂×C₄×C₁₃₇	C₂³×C₄	1	2	yes	—
107	C₂×C₄×C₁₃₇	C₂²×D₄	1	2	yes	—
108	C₂×C₄×C₁₃₇	C₂²×Q₈	1	1	yes	characteristic
109	C₈×C₁₃₇	—	2	8	no	—
110	C₂²×C₁₃₇	—	2	16	no	—
111	C₂²×C₁₃₇	—	2	8	no	—
112	C₂²×C₁₃₇	—	2	32	no	—
113	C₂²×C₁₃₇	C₂×#95	1	8	yes	—
114	C₂²×C₁₃₇	C₂×#95	1	4	yes	—
115	C₂²×C₁₃₇	#416	1	16	yes	—
116	C₂²×C₁₃₇	#427	1	8	yes	—
117	C₂²×C₁₃₇	C₂²×C₂²⋊C₄	1	2	yes	—
118	C₂²×C₁₃₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
119	C₂²×C₁₃₇	C₂×#135	1	4	yes	—
120	C₄×C₁₃₇	—	4	16	no	—
121	C₄×C₁₃₇	C₂×#95	1	8	yes	—
122	C₄×C₁₃₇	C₂×#95	1	4	yes	—
123	C₄×C₁₃₇	C₂²×C₄²	1	1	yes	characteristic
124	C₄×C₁₃₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
125	C₄×C₁₃₇	C₂²×C₄⋊C₄	1	2	yes	—
126	C₂×C₁₃₇	—	2	8	no	—
127	C₂×C₁₃₇	#1735	1	8	yes	direct product factor
128	C₂×C₁₃₇	C₂²×#95	1	1	yes	characteristic
129	C₂×C₁₃₇	C₂×#416	1	4	yes	—
130	C₂×C₁₃₇	C₂×#427	1	2	yes	—
131	C₂×#1735	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
132	C₁₃₇	C₂×#1735	1	1	yes	characteristic, direct product factor, Sylow
133	C₂⁴×C₈	C₂×C₁₃₇	1	1	yes	characteristic
134	C₂²×#117	C₂×C₁₃₇	1	2	yes	—
135	#1735	C₂×C₁₃₇	1	8	yes	direct product factor
136	C₂×#340	C₂×C₁₃₇	1	4	yes	complemented
137	C₂⁴×C₄	C₂²×C₁₃₇	1	1	yes	characteristic
138	C₂³×C₈	—	2	8	no	—
139	C₂³×C₈	C₄×C₁₃₇	1	8	yes	complemented
140	C₂³×C₈	C₂²×C₁₃₇	1	2	yes	—
141	C₂³×C₈	C₂²×C₁₃₇	1	4	yes	—
142	C₂×#117	—	2	8	no	—
143	C₂×#117	—	2	4	no	—
144	C₂×#117	C₄×C₁₃₇	1	16	yes	complemented
145	C₂×#117	C₂²×C₁₃₇	1	8	yes	—
146	C₂²×C₄⋊C₄	C₂²×C₁₃₇	1	4	yes	—
147	C₂²×C₂²⋊C₄	—	2	2	no	—
148	C₂²×C₄²	—	2	2	no	—
149	#340	C₂²×C₁₃₇	1	16	yes	complemented
150	C₂⁵	D₄×C₁₃₇	1	1	yes	characteristic
151	C₂³×C₄	—	2	4	no	—
152	C₂³×C₄	C₂×C₄×C₁₃₇	1	4	yes	—
153	C₂³×C₄	C₂×C₄×C₁₃₇	1	8	yes	—
154	C₂³×C₄	D₄×C₁₃₇	1	2	yes	—
155	C₂³×C₄	D₄×C₁₃₇	1	1	yes	characteristic
156	C₂³×C₄	D₄×C₁₃₇	1	8	yes	—
157	C₂³×C₄	Q₈×C₁₃₇	1	4	yes	—
158	C₂³×C₄	C₂³×C₁₃₇	1	1	yes	characteristic
159	C₂³×C₄	C₂³×C₁₃₇	1	2	yes	—
160	C₂²×C₈	—	2	32	no	—
161	C₂²×C₈	—	2	8	no	—
162	C₂²×C₈	—	2	16	no	—
163	C₂²×C₈	C₂×C₄×C₁₃₇	1	8	yes	—
164	C₂²×C₈	C₂×C₄×C₁₃₇	1	16	yes	complemented
165	C₂²×C₈	C₂³×C₁₃₇	1	4	yes	—
166	#117	—	2	32	no	—
167	#117	—	2	16	no	—
168	#117	C₂×C₄×C₁₃₇	1	32	yes	complemented
169	C₂×C₄⋊C₄	—	2	4	no	—
170	C₂×C₄⋊C₄	—	2	8	no	—
171	C₂×C₄⋊C₄	C₂×C₄×C₁₃₇	1	16	yes	—
172	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	8	yes	—
173	C₂×C₂²⋊C₄	—	2	16	no	—
174	C₂×C₂²⋊C₄	—	2	8	no	—
175	C₂×C₄²	—	2	16	no	—
176	C₂×C₄²	—	2	8	no	—
177	C₂⁴	—	2	8	no	—
178	C₂⁴	C₂²⋊C₄×C₁₃₇	1	8	yes	—
179	C₂⁴	C₄⋊C₄×C₁₃₇	1	4	yes	—
180	C₂⁴	C₂×D₄×C₁₃₇	1	2	yes	—
181	C₂⁴	C₂×D₄×C₁₃₇	1	1	yes	characteristic
182	C₂²×C₄	—	2	16	no	—
183	C₂²×C₄	—	2	8	no	—
184	C₂²×C₄	—	4	16	no	—
185	C₂²×C₄	C₄²×C₁₃₇	1	16	yes	—
186	C₂²×C₄	C₂²⋊C₄×C₁₃₇	1	8	yes	—
187	C₂²×C₄	C₂²⋊C₄×C₁₃₇	1	16	yes	—
188	C₂²×C₄	C₂²⋊C₄×C₁₃₇	1	16	yes	—
189	C₂²×C₄	C₄⋊C₄×C₁₃₇	1	8	yes	—
190	C₂²×C₄	C₄⋊C₄×C₁₃₇	1	4	yes	—
191	C₂²×C₄	C₄⋊C₄×C₁₃₇	1	32	yes	—
192	C₂²×C₄	C₂²×C₄×C₁₃₇	1	2	yes	—
193	C₂²×C₄	C₂²×C₄×C₁₃₇	1	4	yes	—
194	C₂²×C₄	C₂²×C₄×C₁₃₇	1	4	yes	—
195	C₂²×C₄	C₂²×C₄×C₁₃₇	1	8	yes	—
196	C₂²×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
197	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
198	C₂²×C₄	C₂×D₄×C₁₃₇	1	4	yes	—
199	C₂²×C₄	C₂×D₄×C₁₃₇	1	8	yes	—
200	C₂²×C₄	C₂×Q₈×C₁₃₇	1	4	yes	—
201	C₂²×C₄	C₂×Q₈×C₁₃₇	1	2	yes	—
202	C₂²×C₄	C₂⁴×C₁₃₇	1	1	yes	characteristic
203	C₂×C₈	—	2	16	no	—
204	C₂×C₈	—	2	32	no	—
205	C₂×C₈	—	2	8	no	—
206	C₂×C₈	C₂²×C₄×C₁₃₇	1	8	yes	—
207	C₄⋊C₄	—	2	16	no	—
208	C₄⋊C₄	—	2	8	no	—
209	C₄⋊C₄	C₂²×C₄×C₁₃₇	1	16	yes	—
210	C₂²⋊C₄	—	2	32	no	—
211	C₄²	—	2	32	no	—
212	C₂³	—	2	32	no	—
213	C₂³	—	2	16	no	—
214	C₂³	—	2	8	no	—
215	C₂³	#95×C₁₃₇	1	16	yes	—
216	C₂³	C₂×C₂²⋊C₄×C₁₃₇	1	8	yes	—
217	C₂³	C₂×C₂²⋊C₄×C₁₃₇	1	4	yes	—
218	C₂³	C₂×C₄⋊C₄×C₁₃₇	1	4	yes	—
219	C₂³	C₂×C₄⋊C₄×C₁₃₇	1	2	yes	—
220	C₂³	#135×C₁₃₇	1	8	yes	—
221	C₂³	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
222	C₂×C₄	—	2	16	no	—
223	C₂×C₄	—	4	32	no	—
224	C₂×C₄	—	4	16	no	—
225	C₂×C₄	#95×C₁₃₇	1	32	yes	—
226	C₂×C₄	#95×C₁₃₇	1	16	yes	—
227	C₂×C₄	C₂×C₄²×C₁₃₇	1	8	yes	—
228	C₂×C₄	C₂×C₄²×C₁₃₇	1	4	yes	—
229	C₂×C₄	C₂×C₂²⋊C₄×C₁₃₇	1	8	yes	—
230	C₂×C₄	C₂×C₂²⋊C₄×C₁₃₇	1	4	yes	—
231	C₂×C₄	C₂×C₂²⋊C₄×C₁₃₇	1	8	yes	—
232	C₂×C₄	C₂×C₂²⋊C₄×C₁₃₇	1	4	yes	—
233	C₂×C₄	C₂×C₄⋊C₄×C₁₃₇	1	2	yes	—
234	C₂×C₄	C₂×C₄⋊C₄×C₁₃₇	1	4	yes	—
235	C₂×C₄	C₂×C₄⋊C₄×C₁₃₇	1	16	yes	—
236	C₂×C₄	C₂×C₄⋊C₄×C₁₃₇	1	8	yes	—
237	C₂×C₄	C₂³×C₄×C₁₃₇	1	1	yes	characteristic
238	C₂×C₄	C₂³×C₄×C₁₃₇	1	2	yes	—
239	C₂×C₄	C₂²×D₄×C₁₃₇	1	2	yes	—
240	C₂×C₄	C₂²×Q₈×C₁₃₇	1	1	yes	characteristic
241	C₈	—	2	8	no	—
242	C₂²	—	2	16	no	—
243	C₂²	—	2	8	no	—
244	C₂²	—	2	32	no	—
245	C₂²	C₂×#95×C₁₃₇	1	8	yes	—
246	C₂²	C₂×#95×C₁₃₇	1	4	yes	—
247	C₂²	#416×C₁₃₇	1	16	yes	—
248	C₂²	#427×C₁₃₇	1	8	yes	—
249	C₂²	C₂²×C₂²⋊C₄×C₁₃₇	1	2	yes	—
250	C₂²	C₂²×C₄⋊C₄×C₁₃₇	1	1	yes	characteristic
251	C₂²	C₂×#135×C₁₃₇	1	4	yes	—
252	C₄	—	4	16	no	—
253	C₄	C₂×#95×C₁₃₇	1	8	yes	—
254	C₄	C₂×#95×C₁₃₇	1	4	yes	—
255	C₄	C₂²×C₄²×C₁₃₇	1	1	yes	characteristic
256	C₄	C₂²×C₂²⋊C₄×C₁₃₇	1	1	yes	characteristic
257	C₄	C₂²×C₄⋊C₄×C₁₃₇	1	2	yes	—
258	C₂	—	2	8	no	—
259	C₂	#1735×C₁₃₇	1	8	yes	direct product factor
260	C₂	C₂²×#95×C₁₃₇	1	1	yes	characteristic
261	C₂	C₂×#416×C₁₃₇	1	4	yes	—
262	C₂	C₂×#427×C₁₃₇	1	2	yes	—

	subgroup	G/H
center	C₂³×C₄×C₁₃₇	D₄
commutator	C₄	C₂²×C₄²×C₁₃₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃₇
socle	C₂⁴×C₁₃₇	C₂×D₄
prime radical	C₂⁵×C₁₃₇	D₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₄×C₁₃₇ ◁ C₂⁴×C₄×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. #1735×C₁₃₇	C₂	direct (N × H)
4. C₂×#340×C₁₃₇	C₂	semidirect
7. C₂³×C₈×C₁₃₇	C₄	semidirect
12. C₂×#117×C₁₃₇	C₄	semidirect
17. #340×C₁₃₇	C₂²	semidirect
32. C₂²×C₈×C₁₃₇	C₂×C₄	semidirect
36. #117×C₁₃₇	C₂×C₄	semidirect
127. C₂×C₁₃₇	#1735	direct (N × H)
131. C₂×#1735	C₁₃₇	direct (N × H)
132. C₁₃₇	C₂×#1735	direct (N × H)
135. #1735	C₂×C₁₃₇	direct (N × H)
136. C₂×#340	C₂×C₁₃₇	semidirect
139. C₂³×C₈	C₄×C₁₃₇	semidirect
144. C₂×#117	C₄×C₁₃₇	semidirect
149. #340	C₂²×C₁₃₇	semidirect
164. C₂²×C₈	C₂×C₄×C₁₃₇	semidirect
168. #117	C₂×C₄×C₁₃₇	semidirect
259. C₂	#1735×C₁₃₇	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×C₈×C₁₃₇	2
C₂²×#117×C₁₃₇	2
#1735×C₁₃₇	2
C₂×#340×C₁₃₇	2
C₂×#1735	137

Subgroups — C2×#1735×C137 #46869731

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#1735×C₁₃₇ #46869731