Subgroups — C₄×#493×C₁₃₇ #46871073

Order 35072.

1422 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#493×C₁₃₇	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×D₈×C₁₃₇	C₂	1	4	yes	maximal, complemented
3	C₄×#127×C₁₃₇	C₂	1	2	yes	maximal, complemented
4	#1970×C₁₃₇	C₂	1	4	yes	maximal, complemented
5	#1957×C₁₃₇	C₂	1	2	yes	maximal, complemented
6	#1845×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄²×C₈×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₈×C₁₃₇	C₂²	1	4	yes	—
9	C₂×#127×C₁₃₇	C₂²	1	2	yes	—
10	C₂×C₄×D₄×C₁₃₇	—	2	4	no	—
11	C₂×C₄×D₄×C₁₃₇	C₂²	1	4	yes	—
12	#493×C₁₃₇	C₄	1	8	yes	direct product factor
13	C₄×D₈×C₁₃₇	—	2	16	no	—
14	C₂×#107×C₁₃₇	C₂²	1	4	yes	—
15	C₂×#102×C₁₃₇	C₂²	1	8	yes	—
16	C₂×C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₈×C₁₃₇	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₈×C₁₃₇	C₂²	1	2	yes	—
20	#390×C₁₃₇	—	2	4	no	—
21	#390×C₁₃₇	C₂²	1	4	yes	—
22	#384×C₁₃₇	C₂²	1	2	yes	—
23	C₄³×C₁₃₇	C₂²	1	1	yes	characteristic
24	C₂²×D₄×C₁₃₇	—	2	4	no	—
25	C₂²×D₄×C₁₃₇	C₂³	1	4	yes	—
26	C₂³×C₄×C₁₃₇	—	2	4	no	—
27	C₂×D₈×C₁₃₇	—	2	8	no	—
28	C₂×D₈×C₁₃₇	—	2	8	no	—
29	C₂×D₈×C₁₃₇	C₂×C₄	1	16	yes	complemented
30	C₂²×C₈×C₁₃₇	C₂³	1	2	yes	—
31	C₂²×C₈×C₁₃₇	C₂³	1	2	yes	—
32	#127×C₁₃₇	—	2	4	no	—
33	#127×C₁₃₇	C₂×C₄	1	8	yes	complemented
34	C₄×D₄×C₁₃₇	—	2	16	no	—
35	C₄×D₄×C₁₃₇	—	4	16	no	—
36	C₂×C₄⋊C₄×C₁₃₇	—	2	4	no	—
37	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	4	yes	—
38	C₂×C₂²⋊C₄×C₁₃₇	—	2	4	no	—
39	C₂×C₂²⋊C₄×C₁₃₇	—	2	8	no	—
40	C₂×C₄²×C₁₃₇	D₄	1	2	yes	—
41	C₂×C₄²×C₁₃₇	D₄	1	2	yes	—
42	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
43	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
44	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
45	#107×C₁₃₇	—	2	8	no	—
46	#102×C₁₃₇	—	2	16	no	—
47	C₄×C₈×C₁₃₇	—	2	4	no	—
48	C₄×C₈×C₁₃₇	—	2	2	no	—
49	C₄×C₈×C₁₃₇	C₂×C₄	1	4	yes	complemented
50	C₄×C₈×C₁₃₇	D₄	1	8	yes	complemented
51	C₂⁴×C₁₃₇	—	2	4	no	—
52	C₂×D₄×C₁₃₇	—	2	16	no	—
53	C₂×D₄×C₁₃₇	—	2	8	no	—
54	C₂×D₄×C₁₃₇	—	2	8	no	—
55	C₂×D₄×C₁₃₇	—	2	4	no	—
56	C₂×D₄×C₁₃₇	—	4	8	no	—
57	C₂×D₄×C₁₃₇	—	4	8	no	—
58	C₂×D₄×C₁₃₇	C₂²×C₄	1	8	yes	—
59	C₂²×C₄×C₁₃₇	—	2	4	no	—
60	C₂²×C₄×C₁₃₇	—	4	8	no	—
61	C₂²×C₄×C₁₃₇	—	4	16	no	—
62	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
64	C₂²×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
65	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
67	D₈×C₁₃₇	—	2	32	no	—
68	C₂×C₈×C₁₃₇	—	2	2	no	—
69	C₂×C₈×C₁₃₇	—	2	2	no	—
70	C₂×C₈×C₁₃₇	C₂²×C₄	1	4	yes	—
71	C₂×C₈×C₁₃₇	C₂×D₄	1	4	yes	—
72	C₂×C₈×C₁₃₇	C₂×D₄	1	4	yes	—
73	C₂×C₈×C₁₃₇	#41	1	4	yes	—
74	C₄⋊C₄×C₁₃₇	—	2	8	no	—
75	C₄⋊C₄×C₁₃₇	—	4	8	no	—
76	C₂²⋊C₄×C₁₃₇	—	4	8	no	—
77	C₂²⋊C₄×C₁₃₇	—	4	16	no	—
78	C₄²×C₁₃₇	—	2	4	no	—
79	C₄²×C₁₃₇	—	2	2	no	—
80	C₄²×C₁₃₇	D₈	1	8	yes	complemented
81	C₄²×C₁₃₇	C₂²×C₄	1	2	yes	—
82	C₄²×C₁₃₇	C₂×D₄	1	4	yes	—
83	C₄²×C₁₃₇	#41	1	2	yes	—
84	C₂³×C₁₃₇	—	2	8	no	—
85	C₂³×C₁₃₇	—	4	4	no	—
86	C₂³×C₁₃₇	—	4	4	no	—
87	C₂³×C₁₃₇	—	4	8	no	—
88	C₂³×C₁₃₇	—	4	8	no	—
89	C₂³×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
90	D₄×C₁₃₇	—	2	16	no	—
91	D₄×C₁₃₇	—	4	8	no	—
92	D₄×C₁₃₇	—	4	32	no	—
93	C₂×C₄×C₁₃₇	—	2	2	no	—
94	C₂×C₄×C₁₃₇	—	2	2	no	—
95	C₂×C₄×C₁₃₇	—	2	2	no	—
96	C₂×C₄×C₁₃₇	—	4	4	no	—
97	C₂×C₄×C₁₃₇	—	4	8	no	—
98	C₂×C₄×C₁₃₇	—	8	16	no	—
99	C₂×C₄×C₁₃₇	C₄×D₄	1	4	yes	—
100	C₂×C₄×C₁₃₇	#127	1	2	yes	—
101	C₂×C₄×C₁₃₇	#127	1	2	yes	—
102	C₂×C₄×C₁₃₇	C₂×D₈	1	4	yes	—
103	C₂×C₄×C₁₃₇	C₂×D₈	1	4	yes	—
104	C₂×C₄×C₁₃₇	C₂×D₈	1	4	yes	—
105	C₂×C₄×C₁₃₇	#135	1	4	yes	—
106	C₂×C₄×C₁₃₇	C₂³×C₄	1	1	yes	characteristic
107	C₂×C₄×C₁₃₇	C₂²×D₄	1	2	yes	—
108	C₂×C₄×C₁₃₇	C₂×#41	1	2	yes	—
109	C₂×C₄×C₁₃₇	C₂×#41	1	1	yes	characteristic
110	C₈×C₁₃₇	—	2	4	no	—
111	C₈×C₁₃₇	C₄×D₄	1	8	yes	complemented
112	C₂²×C₁₃₇	—	4	8	no	—
113	C₂²×C₁₃₇	—	4	16	no	—
114	C₂²×C₁₃₇	—	8	4	no	—
115	C₂²×C₁₃₇	—	8	4	no	—
116	C₂²×C₁₃₇	—	8	8	no	—
117	C₂²×C₁₃₇	C₂×C₄×D₄	1	1	yes	characteristic
118	C₂²×C₁₃₇	C₂×#127	1	1	yes	characteristic
119	C₂²×C₁₃₇	#532	1	1	yes	characteristic
120	C₂²×C₁₃₇	C₂²×D₈	1	2	yes	—
121	C₂²×C₁₃₇	C₂×#135	1	2	yes	—
122	C₄×C₁₃₇	—	2	2	no	—
123	C₄×C₁₃₇	—	2	4	no	—
124	C₄×C₁₃₇	—	8	4	no	—
125	C₄×C₁₃₇	C₄×D₈	1	8	yes	complemented
126	C₄×C₁₃₇	#493	1	4	yes	direct product factor
127	C₄×C₁₃₇	C₂×C₄×D₄	1	2	yes	—
128	C₄×C₁₃₇	#532	1	2	yes	—
129	C₂×C₁₃₇	—	8	8	no	—
130	C₂×C₁₃₇	C₄×#127	1	1	yes	characteristic
131	C₂×C₁₃₇	C₂×C₄×D₈	1	2	yes	—
132	C₂×C₁₃₇	C₂×#493	1	1	yes	characteristic
133	C₂×C₁₃₇	#3147	1	1	yes	characteristic
134	C₂×C₁₃₇	#3169	1	2	yes	—
135	C₄×#493	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
136	C₁₃₇	C₄×#493	1	1	yes	characteristic, direct product factor, Sylow
137	C₂×#493	C₂×C₁₃₇	1	1	yes	characteristic
138	C₂×C₄×D₈	C₂×C₁₃₇	1	4	yes	complemented
139	C₄×#127	C₂×C₁₃₇	1	2	yes	complemented
140	#1970	C₂×C₁₃₇	1	4	yes	complemented
141	#1957	C₂×C₁₃₇	1	2	yes	complemented
142	#1845	C₂×C₁₃₇	1	1	yes	characteristic, complemented
143	C₄²×C₈	C₂×C₁₃₇	1	1	yes	characteristic, complemented
144	C₂²×D₈	C₂²×C₁₃₇	1	4	yes	—
145	C₂×#127	C₂²×C₁₃₇	1	2	yes	—
146	C₂×C₄×D₄	—	2	4	no	—
147	C₂×C₄×D₄	C₂²×C₁₃₇	1	4	yes	—
148	#493	C₄×C₁₃₇	1	8	yes	direct product factor
149	C₄×D₈	—	2	16	no	—
150	C₂×#107	C₂²×C₁₃₇	1	4	yes	—
151	C₂×#102	C₂²×C₁₃₇	1	8	yes	—
152	C₂×C₄×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
153	C₂×C₄×C₈	C₂²×C₁₃₇	1	2	yes	—
154	C₂×C₄×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
155	C₂×C₄×C₈	C₂²×C₁₃₇	1	2	yes	—
156	#390	—	2	4	no	—
157	#390	C₂²×C₁₃₇	1	4	yes	—
158	#384	C₂²×C₁₃₇	1	2	yes	—
159	C₄³	C₂²×C₁₃₇	1	1	yes	characteristic
160	C₂²×D₄	—	2	4	no	—
161	C₂²×D₄	C₂³×C₁₃₇	1	4	yes	—
162	C₂³×C₄	—	2	4	no	—
163	C₂×D₈	—	2	8	no	—
164	C₂×D₈	—	2	8	no	—
165	C₂×D₈	C₂×C₄×C₁₃₇	1	16	yes	complemented
166	C₂²×C₈	C₂³×C₁₃₇	1	2	yes	—
167	C₂²×C₈	C₂³×C₁₃₇	1	2	yes	—
168	#127	—	2	4	no	—
169	#127	C₂×C₄×C₁₃₇	1	8	yes	complemented
170	C₄×D₄	—	2	16	no	—
171	C₄×D₄	—	4	16	no	—
172	C₂×C₄⋊C₄	—	2	4	no	—
173	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	4	yes	—
174	C₂×C₂²⋊C₄	—	2	8	no	—
175	C₂×C₂²⋊C₄	—	2	4	no	—
176	C₂×C₄²	D₄×C₁₃₇	1	2	yes	—
177	C₂×C₄²	D₄×C₁₃₇	1	2	yes	—
178	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
179	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
180	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
181	#107	—	2	8	no	—
182	#102	—	2	16	no	—
183	C₄×C₈	—	2	4	no	—
184	C₄×C₈	—	2	2	no	—
185	C₄×C₈	C₂×C₄×C₁₃₇	1	4	yes	complemented
186	C₄×C₈	D₄×C₁₃₇	1	8	yes	complemented
187	C₂⁴	—	2	4	no	—
188	C₂×D₄	—	2	16	no	—
189	C₂×D₄	—	2	8	no	—
190	C₂×D₄	—	2	8	no	—
191	C₂×D₄	—	2	4	no	—
192	C₂×D₄	—	4	8	no	—
193	C₂×D₄	—	4	8	no	—
194	C₂×D₄	C₂²×C₄×C₁₃₇	1	8	yes	—
195	C₂²×C₄	—	2	4	no	—
196	C₂²×C₄	—	4	8	no	—
197	C₂²×C₄	—	4	16	no	—
198	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
199	C₂²×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
200	C₂²×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
201	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
202	C₂²×C₄	C₂⁴×C₁₃₇	1	1	yes	characteristic
203	D₈	—	2	32	no	—
204	C₂×C₈	—	2	2	no	—
205	C₂×C₈	—	2	2	no	—
206	C₂×C₈	C₂²×C₄×C₁₃₇	1	4	yes	—
207	C₂×C₈	C₂×D₄×C₁₃₇	1	4	yes	—
208	C₂×C₈	C₂×D₄×C₁₃₇	1	4	yes	—
209	C₂×C₈	#41×C₁₃₇	1	4	yes	—
210	C₄⋊C₄	—	2	8	no	—
211	C₄⋊C₄	—	4	8	no	—
212	C₂²⋊C₄	—	4	8	no	—
213	C₂²⋊C₄	—	4	16	no	—
214	C₄²	—	2	4	no	—
215	C₄²	—	2	2	no	—
216	C₄²	D₈×C₁₃₇	1	8	yes	complemented
217	C₄²	C₂²×C₄×C₁₃₇	1	2	yes	—
218	C₄²	C₂×D₄×C₁₃₇	1	4	yes	—
219	C₄²	#41×C₁₃₇	1	2	yes	—
220	C₂³	—	2	8	no	—
221	C₂³	—	4	4	no	—
222	C₂³	—	4	4	no	—
223	C₂³	—	4	8	no	—
224	C₂³	—	4	8	no	—
225	C₂³	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
226	D₄	—	2	16	no	—
227	D₄	—	4	32	no	—
228	D₄	—	4	8	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	4	4	no	—
233	C₂×C₄	—	4	8	no	—
234	C₂×C₄	—	8	16	no	—
235	C₂×C₄	C₄×D₄×C₁₃₇	1	4	yes	—
236	C₂×C₄	#127×C₁₃₇	1	2	yes	—
237	C₂×C₄	#127×C₁₃₇	1	2	yes	—
238	C₂×C₄	C₂×D₈×C₁₃₇	1	4	yes	—
239	C₂×C₄	C₂×D₈×C₁₃₇	1	4	yes	—
240	C₂×C₄	C₂×D₈×C₁₃₇	1	4	yes	—
241	C₂×C₄	#135×C₁₃₇	1	4	yes	—
242	C₂×C₄	C₂³×C₄×C₁₃₇	1	1	yes	characteristic
243	C₂×C₄	C₂²×D₄×C₁₃₇	1	2	yes	—
244	C₂×C₄	C₂×#41×C₁₃₇	1	2	yes	—
245	C₂×C₄	C₂×#41×C₁₃₇	1	1	yes	characteristic
246	C₈	—	2	4	no	—
247	C₈	C₄×D₄×C₁₃₇	1	8	yes	complemented
248	C₂²	—	4	8	no	—
249	C₂²	—	4	16	no	—
250	C₂²	—	8	4	no	—
251	C₂²	—	8	4	no	—
252	C₂²	—	8	8	no	—
253	C₂²	C₂×C₄×D₄×C₁₃₇	1	1	yes	characteristic
254	C₂²	C₂×#127×C₁₃₇	1	1	yes	characteristic
255	C₂²	#532×C₁₃₇	1	1	yes	characteristic
256	C₂²	C₂²×D₈×C₁₃₇	1	2	yes	—
257	C₂²	C₂×#135×C₁₃₇	1	2	yes	—
258	C₄	—	2	2	no	—
259	C₄	—	2	4	no	—
260	C₄	—	8	4	no	—
261	C₄	C₄×D₈×C₁₃₇	1	8	yes	complemented
262	C₄	#493×C₁₃₇	1	4	yes	direct product factor
263	C₄	C₂×C₄×D₄×C₁₃₇	1	2	yes	—
264	C₄	#532×C₁₃₇	1	2	yes	—
265	C₂	—	8	8	no	—
266	C₂	C₄×#127×C₁₃₇	1	1	yes	characteristic
267	C₂	C₂×C₄×D₈×C₁₃₇	1	2	yes	—
268	C₂	C₂×#493×C₁₃₇	1	1	yes	characteristic
269	C₂	#3147×C₁₃₇	1	1	yes	characteristic
270	C₂	#3169×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	C₂×#493×C₁₃₇	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₁₃₇ ◁ C₄³×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×D₈×C₁₃₇	C₂	semidirect
3. C₄×#127×C₁₃₇	C₂	semidirect
4. #1970×C₁₃₇	C₂	semidirect
5. #1957×C₁₃₇	C₂	semidirect
6. #1845×C₁₃₇	C₂	semidirect
7. C₄²×C₈×C₁₃₇	C₂	semidirect
12. #493×C₁₃₇	C₄	direct (N × H)
29. C₂×D₈×C₁₃₇	C₂×C₄	semidirect
33. #127×C₁₃₇	C₂×C₄	semidirect
49. C₄×C₈×C₁₃₇	C₂×C₄	semidirect
50. C₄×C₈×C₁₃₇	D₄	semidirect
80. C₄²×C₁₃₇	D₈	semidirect
111. C₈×C₁₃₇	C₄×D₄	semidirect
125. C₄×C₁₃₇	C₄×D₈	semidirect
126. C₄×C₁₃₇	#493	direct (N × H)
135. C₄×#493	C₁₃₇	direct (N × H)
136. C₁₃₇	C₄×#493	direct (N × H)
138. C₂×C₄×D₈	C₂×C₁₃₇	semidirect
139. C₄×#127	C₂×C₁₃₇	semidirect
140. #1970	C₂×C₁₃₇	semidirect
141. #1957	C₂×C₁₃₇	semidirect
142. #1845	C₂×C₁₃₇	semidirect
143. C₄²×C₈	C₂×C₁₃₇	semidirect
148. #493	C₄×C₁₃₇	direct (N × H)
165. C₂×D₈	C₂×C₄×C₁₃₇	semidirect
169. #127	C₂×C₄×C₁₃₇	semidirect
185. C₄×C₈	C₂×C₄×C₁₃₇	semidirect
186. C₄×C₈	D₄×C₁₃₇	semidirect
216. C₄²	D₈×C₁₃₇	semidirect
247. C₈	C₄×D₄×C₁₃₇	semidirect
261. C₄	C₄×D₈×C₁₃₇	semidirect
262. C₄	#493×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
C₂×#493×C₁₃₇	2
C₂×C₄×D₈×C₁₃₇	2
C₄×#127×C₁₃₇	2
#1970×C₁₃₇	2
#1957×C₁₃₇	2
#1845×C₁₃₇	2
C₄²×C₈×C₁₃₇	2
C₄×#493	137

Subgroups — C4×#493×C137 #46871073

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#493×C₁₃₇ #46871073