Subgroups — C₁₃₇×#8177 #46857563

Order 35072.

618 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2293×C₁₃₇	C₂	1	1	yes	maximal, characteristic
2	#1767×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄²×C₈×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×D₄⋊C₈×C₁₃₇	C₂	1	2	yes	maximal
5	#1279×C₁₃₇	C₂	1	2	yes	maximal, complemented
6	C₂×C₄×D₄×C₁₃₇	C₂²	1	1	yes	characteristic
7	C₂×#117×C₁₃₇	—	2	1	no	—
8	C₂×C₄⋊C₈×C₁₃₇	C₂²	1	2	yes	—
9	C₂×#97×C₁₃₇	—	2	1	no	—
10	C₂×C₄×C₈×C₁₃₇	—	2	2	no	—
11	C₂×C₄×C₈×C₁₃₇	C₂²	1	2	yes	—
12	#390×C₁₃₇	C₄	1	2	yes	—
13	#389×C₁₃₇	C₄	1	2	yes	—
14	#388×C₁₃₇	—	2	2	no	—
15	C₄×C₄⋊C₄×C₁₃₇	C₂²	1	1	yes	characteristic
16	C₄×C₂²⋊C₄×C₁₃₇	—	2	1	no	—
17	C₄³×C₁₃₇	C₂²	1	1	yes	characteristic
18	#336×C₁₃₇	—	2	1	no	—
19	D₄⋊C₈×C₁₃₇	C₄	1	8	yes	complemented
20	C₂²×D₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
21	C₂³×C₄×C₁₃₇	—	2	1	no	—
22	C₂²×C₈×C₁₃₇	—	2	2	no	—
23	C₂²×C₈×C₁₃₇	—	2	2	no	—
24	C₄×D₄×C₁₃₇	—	2	2	no	—
25	C₄×D₄×C₁₃₇	C₂×C₄	1	4	yes	—
26	#117×C₁₃₇	—	4	2	no	—
27	#117×C₁₃₇	—	4	2	no	—
28	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
29	C₂×C₄⋊C₄×C₁₃₇	C₂×C₄	1	2	yes	—
30	C₂×C₄⋊C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
31	C₂×C₂²⋊C₄×C₁₃₇	—	2	1	no	—
32	C₂×C₂²⋊C₄×C₁₃₇	—	2	2	no	—
33	C₂×C₂²⋊C₄×C₁₃₇	—	2	2	no	—
34	C₂×C₄²×C₁₃₇	—	2	1	no	—
35	C₂×C₄²×C₁₃₇	C₂×C₄	1	2	yes	—
36	C₂×C₄²×C₁₃₇	D₄	1	2	yes	—
37	C₂×C₄²×C₁₃₇	D₄	1	1	yes	characteristic
38	C₂×C₄²×C₁₃₇	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
40	C₄⋊C₈×C₁₃₇	—	2	2	no	—
41	C₄⋊C₈×C₁₃₇	C₂×C₄	1	4	yes	—
42	#97×C₁₃₇	—	4	2	no	—
43	C₄×C₈×C₁₃₇	—	2	8	no	—
44	C₄×C₈×C₁₃₇	—	2	2	no	—
45	C₄×C₈×C₁₃₇	C₂×C₄	1	4	yes	—
46	#95×C₁₃₇	—	2	2	no	—
47	#95×C₁₃₇	—	2	1	no	—
48	C₂⁴×C₁₃₇	—	2	1	no	—
49	C₂×D₄×C₁₃₇	—	2	1	no	—
50	C₂×D₄×C₁₃₇	—	2	4	no	—
51	C₂×D₄×C₁₃₇	C₄²	1	2	yes	—
52	C₂²×C₄×C₁₃₇	—	2	1	no	—
53	C₂²×C₄×C₁₃₇	—	2	4	no	—
54	C₂²×C₄×C₁₃₇	—	2	2	no	—
55	C₂²×C₄×C₁₃₇	—	4	1	no	—
56	C₂²×C₄×C₁₃₇	—	4	1	no	—
57	C₂²×C₄×C₁₃₇	—	4	2	no	—
58	C₂²×C₄×C₁₃₇	C₂²⋊C₄	1	2	yes	—
59	C₂²×C₄×C₁₃₇	C₂²⋊C₄	1	2	yes	—
60	C₂²×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
63	C₂×C₈×C₁₃₇	—	2	8	no	—
64	C₂×C₈×C₁₃₇	—	2	4	no	—
65	C₂×C₈×C₁₃₇	—	2	4	no	—
66	C₂×C₈×C₁₃₇	—	4	2	no	—
67	C₂×C₈×C₁₃₇	—	4	2	no	—
68	C₄⋊C₄×C₁₃₇	—	2	1	no	—
69	C₄⋊C₄×C₁₃₇	—	2	4	no	—
70	C₄⋊C₄×C₁₃₇	—	4	4	no	—
71	C₄⋊C₄×C₁₃₇	C₄²	1	2	yes	—
72	C₂²⋊C₄×C₁₃₇	—	2	4	no	—
73	C₂²⋊C₄×C₁₃₇	—	4	4	no	—
74	C₂²⋊C₄×C₁₃₇	—	4	4	no	—
75	C₄²×C₁₃₇	—	2	4	no	—
76	C₄²×C₁₃₇	—	2	4	no	—
77	C₄²×C₁₃₇	—	2	2	no	—
78	C₄²×C₁₃₇	—	4	1	no	—
79	C₄²×C₁₃₇	—	4	1	no	—
80	C₄²×C₁₃₇	D₈	1	2	yes	—
81	C₄²×C₁₃₇	SD₁₆	1	2	yes	—
82	C₄²×C₁₃₇	C₂²×C₄	1	2	yes	—
83	C₄²×C₁₃₇	#41	1	2	yes	—
84	C₂³×C₁₃₇	—	2	2	no	—
85	C₂³×C₁₃₇	—	4	2	no	—
86	C₂³×C₁₃₇	—	4	1	no	—
87	C₂³×C₁₃₇	—	4	1	no	—
88	C₂³×C₁₃₇	—	4	2	no	—
89	C₂³×C₁₃₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
90	D₄×C₁₃₇	—	2	4	no	—
91	D₄×C₁₃₇	—	4	2	no	—
92	C₂×C₄×C₁₃₇	—	2	2	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	—	2	2	no	—
97	C₂×C₄×C₁₃₇	—	2	2	no	—
98	C₂×C₄×C₁₃₇	—	4	1	no	—
99	C₂×C₄×C₁₃₇	—	4	1	no	—
100	C₂×C₄×C₁₃₇	—	4	4	no	—
101	C₂×C₄×C₁₃₇	—	4	4	no	—
102	C₂×C₄×C₁₃₇	—	4	2	no	—
103	C₂×C₄×C₁₃₇	—	4	2	no	—
104	C₂×C₄×C₁₃₇	—	4	2	no	—
105	C₂×C₄×C₁₃₇	#102	1	4	yes	—
106	C₂×C₄×C₁₃₇	#104	1	4	yes	—
107	C₂×C₄×C₁₃₇	C₂×C₄²	1	1	yes	characteristic
108	C₂×C₄×C₁₃₇	#117	1	1	yes	characteristic
109	C₂×C₄×C₁₃₇	C₄×D₄	1	2	yes	—
110	C₂×C₄×C₁₃₇	C₄×D₄	1	2	yes	—
111	C₂×C₄×C₁₃₇	#131	1	4	yes	—
112	C₂×C₄×C₁₃₇	C₂×D₈	1	1	yes	characteristic
113	C₂×C₄×C₁₃₇	C₂×SD₁₆	1	1	yes	characteristic
114	C₂×C₄×C₁₃₇	#135	1	2	yes	—
115	C₈×C₁₃₇	—	2	8	no	—
116	C₈×C₁₃₇	—	4	4	no	—
117	C₂²×C₁₃₇	—	4	2	no	—
118	C₂²×C₁₃₇	—	4	2	no	—
119	C₂²×C₁₃₇	—	4	2	no	—
120	C₂²×C₁₃₇	—	8	2	no	—
121	C₂²×C₁₃₇	—	8	2	no	—
122	C₂²×C₁₃₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
123	C₂²×C₁₃₇	#408	1	2	yes	—
124	C₂²×C₁₃₇	C₂×#102	1	1	yes	characteristic
125	C₂²×C₁₃₇	#416	1	1	yes	characteristic
126	C₂²×C₁₃₇	C₂×#104	1	2	yes	—
127	C₄×C₁₃₇	—	2	4	no	—
128	C₄×C₁₃₇	—	2	4	no	—
129	C₄×C₁₃₇	—	2	2	no	—
130	C₄×C₁₃₇	—	4	2	no	—
131	C₄×C₁₃₇	—	8	2	no	—
132	C₄×C₁₃₇	#405	1	2	yes	—
133	C₄×C₁₃₇	#433	1	2	yes	—
134	C₄×C₁₃₇	C₄×D₈	1	2	yes	—
135	C₄×C₁₃₇	C₄×SD₁₆	1	2	yes	—
136	C₂×C₁₃₇	—	8	2	no	—
137	C₂×C₁₃₇	#1492	1	2	yes	—
138	C₂×C₁₃₇	#1494	1	2	yes	—
139	C₂×C₁₃₇	#1750	1	1	yes	characteristic
140	C₂×C₁₃₇	C₄×#104	1	1	yes	characteristic
141	C₂×C₁₃₇	C₄×#102	1	1	yes	characteristic
142	#8177	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
143	C₁₃₇	#8177	1	1	yes	characteristic, direct product factor, Sylow
144	#2293	C₂×C₁₃₇	1	1	yes	characteristic
145	#1767	C₂×C₁₃₇	1	1	yes	characteristic, complemented
146	C₄²×C₈	C₂×C₁₃₇	1	1	yes	characteristic, complemented
147	C₂×D₄⋊C₈	C₂×C₁₃₇	1	2	yes	—
148	#1279	C₂×C₁₃₇	1	2	yes	complemented
149	C₂×C₄×D₄	C₂²×C₁₃₇	1	1	yes	characteristic
150	C₂×#117	—	2	1	no	—
151	C₂×C₄⋊C₈	C₂²×C₁₃₇	1	2	yes	—
152	C₂×#97	—	2	1	no	—
153	C₂×C₄×C₈	—	2	2	no	—
154	C₂×C₄×C₈	C₂²×C₁₃₇	1	2	yes	—
155	#390	C₄×C₁₃₇	1	2	yes	—
156	#389	C₄×C₁₃₇	1	2	yes	—
157	#388	—	2	2	no	—
158	C₄×C₄⋊C₄	C₂²×C₁₃₇	1	1	yes	characteristic
159	C₄×C₂²⋊C₄	—	2	1	no	—
160	C₄³	C₂²×C₁₃₇	1	1	yes	characteristic
161	#336	—	2	1	no	—
162	D₄⋊C₈	C₄×C₁₃₇	1	8	yes	complemented
163	C₂²×D₄	C₂×C₄×C₁₃₇	1	1	yes	characteristic
164	C₂³×C₄	—	2	1	no	—
165	C₂²×C₈	—	2	2	no	—
166	C₂²×C₈	—	2	2	no	—
167	C₄×D₄	—	2	2	no	—
168	C₄×D₄	C₂×C₄×C₁₃₇	1	4	yes	—
169	#117	—	4	2	no	—
170	#117	—	4	2	no	—
171	C₂×C₄⋊C₄	—	2	2	no	—
172	C₂×C₄⋊C₄	C₂×C₄×C₁₃₇	1	2	yes	—
173	C₂×C₄⋊C₄	C₂×C₄×C₁₃₇	1	1	yes	characteristic
174	C₂×C₂²⋊C₄	—	2	1	no	—
175	C₂×C₂²⋊C₄	—	2	2	no	—
176	C₂×C₂²⋊C₄	—	2	2	no	—
177	C₂×C₄²	—	2	1	no	—
178	C₂×C₄²	C₂×C₄×C₁₃₇	1	2	yes	—
179	C₂×C₄²	D₄×C₁₃₇	1	1	yes	characteristic
180	C₂×C₄²	D₄×C₁₃₇	1	2	yes	—
181	C₂×C₄²	D₄×C₁₃₇	1	1	yes	characteristic
182	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
183	C₄⋊C₈	—	2	2	no	—
184	C₄⋊C₈	C₂×C₄×C₁₃₇	1	4	yes	—
185	#97	—	4	2	no	—
186	C₄×C₈	—	2	8	no	—
187	C₄×C₈	—	2	2	no	—
188	C₄×C₈	C₂×C₄×C₁₃₇	1	4	yes	—
189	#95	—	2	2	no	—
190	#95	—	2	1	no	—
191	C₂⁴	—	2	1	no	—
192	C₂×D₄	—	2	1	no	—
193	C₂×D₄	—	2	4	no	—
194	C₂×D₄	C₄²×C₁₃₇	1	2	yes	—
195	C₂²×C₄	—	2	1	no	—
196	C₂²×C₄	—	2	4	no	—
197	C₂²×C₄	—	2	2	no	—
198	C₂²×C₄	—	4	1	no	—
199	C₂²×C₄	—	4	1	no	—
200	C₂²×C₄	—	4	2	no	—
201	C₂²×C₄	C₂²⋊C₄×C₁₃₇	1	2	yes	—
202	C₂²×C₄	C₂²⋊C₄×C₁₃₇	1	2	yes	—
203	C₂²×C₄	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
204	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
205	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
206	C₂×C₈	—	2	8	no	—
207	C₂×C₈	—	2	4	no	—
208	C₂×C₈	—	2	4	no	—
209	C₂×C₈	—	4	2	no	—
210	C₂×C₈	—	4	2	no	—
211	C₄⋊C₄	—	2	1	no	—
212	C₄⋊C₄	—	2	4	no	—
213	C₄⋊C₄	—	4	4	no	—
214	C₄⋊C₄	C₄²×C₁₃₇	1	2	yes	—
215	C₂²⋊C₄	—	2	4	no	—
216	C₂²⋊C₄	—	4	4	no	—
217	C₂²⋊C₄	—	4	4	no	—
218	C₄²	—	2	4	no	—
219	C₄²	—	2	4	no	—
220	C₄²	—	2	2	no	—
221	C₄²	—	4	1	no	—
222	C₄²	—	4	1	no	—
223	C₄²	D₈×C₁₃₇	1	2	yes	—
224	C₄²	SD₁₆×C₁₃₇	1	2	yes	—
225	C₄²	C₂²×C₄×C₁₃₇	1	2	yes	—
226	C₄²	#41×C₁₃₇	1	2	yes	—
227	C₂³	—	2	2	no	—
228	C₂³	—	4	2	no	—
229	C₂³	—	4	1	no	—
230	C₂³	—	4	1	no	—
231	C₂³	—	4	2	no	—
232	C₂³	C₂×C₂²⋊C₄×C₁₃₇	1	1	yes	characteristic
233	D₄	—	2	4	no	—
234	D₄	—	4	2	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	4	1	no	—
242	C₂×C₄	—	4	1	no	—
243	C₂×C₄	—	4	4	no	—
244	C₂×C₄	—	4	4	no	—
245	C₂×C₄	—	4	2	no	—
246	C₂×C₄	—	4	2	no	—
247	C₂×C₄	—	4	2	no	—
248	C₂×C₄	#102×C₁₃₇	1	4	yes	—
249	C₂×C₄	#104×C₁₃₇	1	4	yes	—
250	C₂×C₄	C₂×C₄²×C₁₃₇	1	1	yes	characteristic
251	C₂×C₄	#117×C₁₃₇	1	1	yes	characteristic
252	C₂×C₄	C₄×D₄×C₁₃₇	1	2	yes	—
253	C₂×C₄	C₄×D₄×C₁₃₇	1	2	yes	—
254	C₂×C₄	#131×C₁₃₇	1	4	yes	—
255	C₂×C₄	C₂×D₈×C₁₃₇	1	1	yes	characteristic
256	C₂×C₄	C₂×SD₁₆×C₁₃₇	1	1	yes	characteristic
257	C₂×C₄	#135×C₁₃₇	1	2	yes	—
258	C₈	—	2	8	no	—
259	C₈	—	4	4	no	—
260	C₂²	—	4	2	no	—
261	C₂²	—	4	2	no	—
262	C₂²	—	4	2	no	—
263	C₂²	—	8	2	no	—
264	C₂²	—	8	2	no	—
265	C₂²	C₄×C₂²⋊C₄×C₁₃₇	1	1	yes	characteristic
266	C₂²	#408×C₁₃₇	1	2	yes	—
267	C₂²	C₂×#102×C₁₃₇	1	1	yes	characteristic
268	C₂²	#416×C₁₃₇	1	1	yes	characteristic
269	C₂²	C₂×#104×C₁₃₇	1	2	yes	—
270	C₄	—	2	2	no	—
271	C₄	—	2	4	no	—
272	C₄	—	2	4	no	—
273	C₄	—	4	2	no	—
274	C₄	—	8	2	no	—
275	C₄	#405×C₁₃₇	1	2	yes	—
276	C₄	#433×C₁₃₇	1	2	yes	—
277	C₄	C₄×D₈×C₁₃₇	1	2	yes	—
278	C₄	C₄×SD₁₆×C₁₃₇	1	2	yes	—
279	C₂	—	8	2	no	—
280	C₂	#1492×C₁₃₇	1	2	yes	—
281	C₂	#1494×C₁₃₇	1	2	yes	—
282	C₂	#1750×C₁₃₇	1	1	yes	characteristic
283	C₂	C₄×#104×C₁₃₇	1	1	yes	characteristic
284	C₂	C₄×#102×C₁₃₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₁₃₇	C₂×D₄
commutator	C₂×C₄	C₂×C₄²×C₁₃₇
Frattini, cosocle	C₂×C₄²	C₂³×C₁₃₇
socle	C₂³×C₁₃₇	C₂×C₂²⋊C₄
prime radical	C₂²×D₄×C₁₃₇	C₂×C₄
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. #1767×C₁₃₇	C₂	semidirect
3. C₄²×C₈×C₁₃₇	C₂	semidirect
5. #1279×C₁₃₇	C₂	semidirect
19. D₄⋊C₈×C₁₃₇	C₄	semidirect
142. #8177	C₁₃₇	direct (N × H)
143. C₁₃₇	#8177	direct (N × H)
145. #1767	C₂×C₁₃₇	semidirect
146. C₄²×C₈	C₂×C₁₃₇	semidirect
148. #1279	C₂×C₁₃₇	semidirect
162. D₄⋊C₈	C₄×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
#2293×C₁₃₇	2
#1767×C₁₃₇	2
C₄²×C₈×C₁₃₇	2
C₂×D₄⋊C₈×C₁₃₇	2
#1279×C₁₃₇	2
#8177	137

Subgroups — C137×#8177 #46857563

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#8177 #46857563