Subgroups — C₁₃₇×#13194 #46862580

Order 35072.

466 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#436×C₁₃₇	C₂	1	1	yes	maximal, characteristic
2	#2150×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#349×C₁₃₇	C₂	1	4	yes	maximal, complemented
4	C₄×M₃₂×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×M₁₆×C₁₃₇	C₂²	1	2	yes	—
6	C₂×C₄×D₄×C₁₃₇	C₄	1	1	yes	characteristic
7	C₂×M₃₂×C₁₃₇	C₂²	1	2	yes	—
8	C₂×M₃₂×C₁₃₇	C₂²	1	2	yes	—
9	#436×C₁₃₇	—	2	1	no	—
10	#436×C₁₃₇	—	2	1	no	—
11	#436×C₁₃₇	—	2	1	no	—
12	#436×C₁₃₇	—	2	1	no	—
13	C₂×C₄⋊C₈×C₁₃₇	C₄	1	1	yes	characteristic
14	C₂×C₂²⋊C₈×C₁₃₇	C₄	1	2	yes	—
15	C₂×C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
16	C₄⋊C₁₆×C₁₃₇	—	2	1	no	—
17	C₄⋊C₁₆×C₁₃₇	—	2	1	no	—
18	#349×C₁₃₇	—	2	8	no	—
19	#346×C₁₃₇	—	2	1	no	—
20	C₄×C₁₆×C₁₃₇	—	2	1	no	—
21	C₂²×D₄×C₁₃₇	C₈	1	1	yes	characteristic
22	C₂³×C₄×C₁₃₇	C₂×C₄	1	2	yes	—
23	C₂×M₁₆×C₁₃₇	—	2	2	no	—
24	C₂×M₁₆×C₁₃₇	—	2	2	no	—
25	C₂×M₁₆×C₁₃₇	—	2	4	no	—
26	C₂×M₁₆×C₁₃₇	—	2	4	no	—
27	C₂²×C₈×C₁₃₇	C₂×C₄	1	2	yes	—
28	C₂²×C₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
30	C₄×D₄×C₁₃₇	—	2	1	no	—
31	C₄×D₄×C₁₃₇	—	2	1	no	—
32	C₄×D₄×C₁₃₇	—	2	1	no	—
33	C₄×D₄×C₁₃₇	—	2	1	no	—
34	C₂×C₄⋊C₄×C₁₃₇	C₈	1	1	yes	characteristic
35	C₂×C₂²⋊C₄×C₁₃₇	C₈	1	2	yes	—
36	C₂×C₄²×C₁₃₇	C₂×C₄	1	1	yes	characteristic
37	M₃₂×C₁₃₇	—	2	2	no	—
38	M₃₂×C₁₃₇	—	2	4	no	—
39	M₃₂×C₁₃₇	D₄	1	4	yes	complemented
40	C₂×C₁₆×C₁₃₇	—	2	2	no	—
41	C₂×C₁₆×C₁₃₇	—	2	2	no	—
42	C₄⋊C₈×C₁₃₇	—	2	1	no	—
43	C₄⋊C₈×C₁₃₇	—	2	1	no	—
44	C₂²⋊C₈×C₁₃₇	—	2	2	no	—
45	C₂²⋊C₈×C₁₃₇	—	2	2	no	—
46	C₄×C₈×C₁₃₇	D₄	1	1	yes	characteristic
47	C₄×C₈×C₁₃₇	D₄	1	1	yes	characteristic
48	C₄×C₈×C₁₃₇	D₄	1	1	yes	characteristic
49	C₄×C₈×C₁₃₇	D₄	1	1	yes	characteristic
50	C₂⁴×C₁₃₇	C₂×C₈	1	2	yes	—
51	C₂×D₄×C₁₃₇	—	2	1	no	—
52	C₂×D₄×C₁₃₇	—	2	1	no	—
53	C₂×D₄×C₁₃₇	—	4	2	no	—
54	C₂²×C₄×C₁₃₇	—	2	2	no	—
55	C₂²×C₄×C₁₃₇	—	2	4	no	—
56	C₂²×C₄×C₁₃₇	—	2	2	no	—
57	C₂²×C₄×C₁₃₇	—	2	4	no	—
58	C₂²×C₄×C₁₃₇	C₂×C₈	1	2	yes	—
59	C₂²×C₄×C₁₃₇	C₂×C₈	1	1	yes	characteristic
60	C₂²×C₄×C₁₃₇	C₂×C₈	1	1	yes	characteristic
61	C₂²×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
62	M₁₆×C₁₃₇	—	4	4	no	—
63	M₁₆×C₁₃₇	—	4	2	no	—
64	M₁₆×C₁₃₇	—	4	2	no	—
65	C₂×C₈×C₁₃₇	—	2	2	no	—
66	C₂×C₈×C₁₃₇	—	2	2	no	—
67	C₂×C₈×C₁₃₇	—	2	4	no	—
68	C₂×C₈×C₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
69	C₂×C₈×C₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
70	C₂×C₈×C₁₃₇	C₂×D₄	1	1	yes	characteristic
71	C₂×C₈×C₁₃₇	C₂×D₄	1	2	yes	—
72	C₂×C₈×C₁₃₇	C₂×D₄	1	1	yes	characteristic
73	C₂×C₈×C₁₃₇	#41	1	2	yes	—
74	C₄⋊C₄×C₁₃₇	—	2	1	no	—
75	C₄⋊C₄×C₁₃₇	—	2	1	no	—
76	C₂²⋊C₄×C₁₃₇	—	2	2	no	—
77	C₂²⋊C₄×C₁₃₇	—	2	2	no	—
78	C₄²×C₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
79	C₄²×C₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
80	C₄²×C₁₃₇	M₁₆	1	1	yes	characteristic
81	C₄²×C₁₃₇	M₁₆	1	1	yes	characteristic
82	C₁₆×C₁₃₇	—	2	4	no	—
83	C₁₆×C₁₃₇	—	4	2	no	—
84	C₂³×C₁₃₇	—	2	2	no	—
85	C₂³×C₁₃₇	—	2	4	no	—
86	C₂³×C₁₃₇	—	4	2	no	—
87	C₂³×C₁₃₇	—	4	2	no	—
88	C₂³×C₁₃₇	C₂²×C₈	1	1	yes	characteristic
89	D₄×C₁₃₇	—	4	2	no	—
90	D₄×C₁₃₇	—	4	2	no	—
91	C₂×C₄×C₁₃₇	—	2	2	no	—
92	C₂×C₄×C₁₃₇	—	2	2	no	—
93	C₂×C₄×C₁₃₇	—	2	4	no	—
94	C₂×C₄×C₁₃₇	—	4	1	no	—
95	C₂×C₄×C₁₃₇	—	4	2	no	—
96	C₂×C₄×C₁₃₇	—	4	2	no	—
97	C₂×C₄×C₁₃₇	—	4	4	no	—
98	C₂×C₄×C₁₃₇	C₂²⋊C₈	1	1	yes	characteristic
99	C₂×C₄×C₁₃₇	C₂²⋊C₈	1	1	yes	characteristic
100	C₂×C₄×C₁₃₇	C₂²⋊C₈	1	1	yes	characteristic
101	C₂×C₄×C₁₃₇	C₂²⋊C₈	1	1	yes	characteristic
102	C₂×C₄×C₁₃₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
103	C₂×C₄×C₁₃₇	C₄×D₄	1	2	yes	—
104	C₂×C₄×C₁₃₇	C₂×M₁₆	1	1	yes	characteristic
105	C₂×C₄×C₁₃₇	#131	1	2	yes	—
106	C₈×C₁₃₇	—	2	1	no	—
107	C₈×C₁₃₇	—	2	1	no	—
108	C₈×C₁₃₇	—	4	2	no	—
109	C₈×C₁₃₇	#121	1	2	yes	—
110	C₈×C₁₃₇	#124	1	1	yes	characteristic
111	C₈×C₁₃₇	#127	1	1	yes	characteristic
112	C₂²×C₁₃₇	—	2	1	no	—
113	C₂²×C₁₃₇	—	2	1	no	—
114	C₂²×C₁₃₇	—	4	2	no	—
115	C₂²×C₁₃₇	—	4	2	no	—
116	C₂²×C₁₃₇	—	4	2	no	—
117	C₂²×C₁₃₇	—	8	4	no	—
118	C₂²×C₁₃₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
119	C₂²×C₁₃₇	C₈×D₄	1	2	yes	—
120	C₄×C₁₃₇	—	2	1	no	—
121	C₄×C₁₃₇	—	2	1	no	—
122	C₄×C₁₃₇	—	2	1	no	—
123	C₄×C₁₃₇	—	4	2	no	—
124	C₄×C₁₃₇	#349	1	2	yes	complemented
125	C₄×C₁₃₇	#390	1	1	yes	characteristic
126	C₄×C₁₃₇	#408	1	1	yes	characteristic
127	C₄×C₁₃₇	#436	1	2	yes	—
128	C₂×C₁₃₇	—	2	2	no	—
129	C₂×C₁₃₇	—	8	2	no	—
130	C₂×C₁₃₇	#1954	1	1	yes	characteristic
131	C₂×C₁₃₇	C₂×#349	1	1	yes	characteristic
132	C₂×C₁₃₇	#2115	1	1	yes	characteristic
133	#13194	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
134	C₁₃₇	#13194	1	1	yes	characteristic, direct product factor, Sylow
135	C₂×#436	C₂×C₁₃₇	1	1	yes	characteristic
136	#2150	C₂×C₁₃₇	1	1	yes	characteristic, complemented
137	C₂×#349	C₂×C₁₃₇	1	4	yes	complemented
138	C₄×M₃₂	C₂×C₁₃₇	1	1	yes	characteristic, complemented
139	C₂²×M₁₆	C₂²×C₁₃₇	1	2	yes	—
140	C₂×C₄×D₄	C₄×C₁₃₇	1	1	yes	characteristic
141	C₂×M₃₂	C₂²×C₁₃₇	1	2	yes	—
142	C₂×M₃₂	C₂²×C₁₃₇	1	2	yes	—
143	#436	—	2	1	no	—
144	#436	—	2	1	no	—
145	#436	—	2	1	no	—
146	#436	—	2	1	no	—
147	C₂×C₄⋊C₈	C₄×C₁₃₇	1	1	yes	characteristic
148	C₂×C₂²⋊C₈	C₄×C₁₃₇	1	2	yes	—
149	C₂×C₄×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
150	C₄⋊C₁₆	—	2	1	no	—
151	C₄⋊C₁₆	—	2	1	no	—
152	#349	—	2	8	no	—
153	#346	—	2	1	no	—
154	C₄×C₁₆	—	2	1	no	—
155	C₂²×D₄	C₈×C₁₃₇	1	1	yes	characteristic
156	C₂³×C₄	C₂×C₄×C₁₃₇	1	2	yes	—
157	C₂×M₁₆	—	2	4	no	—
158	C₂×M₁₆	—	2	2	no	—
159	C₂×M₁₆	—	2	2	no	—
160	C₂×M₁₆	—	2	4	no	—
161	C₂²×C₈	C₂×C₄×C₁₃₇	1	2	yes	—
162	C₂²×C₈	C₂×C₄×C₁₃₇	1	1	yes	characteristic
163	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
164	C₄×D₄	—	2	1	no	—
165	C₄×D₄	—	2	1	no	—
166	C₄×D₄	—	2	1	no	—
167	C₄×D₄	—	2	1	no	—
168	C₂×C₄⋊C₄	C₈×C₁₃₇	1	1	yes	characteristic
169	C₂×C₂²⋊C₄	C₈×C₁₃₇	1	2	yes	—
170	C₂×C₄²	C₂×C₄×C₁₃₇	1	1	yes	characteristic
171	M₃₂	—	2	2	no	—
172	M₃₂	—	2	4	no	—
173	M₃₂	D₄×C₁₃₇	1	4	yes	complemented
174	C₂×C₁₆	—	2	2	no	—
175	C₂×C₁₆	—	2	2	no	—
176	C₄⋊C₈	—	2	1	no	—
177	C₄⋊C₈	—	2	1	no	—
178	C₂²⋊C₈	—	2	2	no	—
179	C₂²⋊C₈	—	2	2	no	—
180	C₄×C₈	D₄×C₁₃₇	1	1	yes	characteristic
181	C₄×C₈	D₄×C₁₃₇	1	1	yes	characteristic
182	C₄×C₈	D₄×C₁₃₇	1	1	yes	characteristic
183	C₄×C₈	D₄×C₁₃₇	1	1	yes	characteristic
184	C₂⁴	C₂×C₈×C₁₃₇	1	2	yes	—
185	C₂×D₄	—	2	1	no	—
186	C₂×D₄	—	2	1	no	—
187	C₂×D₄	—	4	2	no	—
188	C₂²×C₄	—	2	2	no	—
189	C₂²×C₄	—	2	4	no	—
190	C₂²×C₄	—	2	2	no	—
191	C₂²×C₄	—	2	4	no	—
192	C₂²×C₄	C₂×C₈×C₁₃₇	1	2	yes	—
193	C₂²×C₄	C₂×C₈×C₁₃₇	1	1	yes	characteristic
194	C₂²×C₄	C₂×C₈×C₁₃₇	1	1	yes	characteristic
195	C₂²×C₄	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
196	M₁₆	—	4	4	no	—
197	M₁₆	—	4	2	no	—
198	M₁₆	—	4	2	no	—
199	C₂×C₈	—	2	2	no	—
200	C₂×C₈	—	2	2	no	—
201	C₂×C₈	—	2	4	no	—
202	C₂×C₈	C₂²⋊C₄×C₁₃₇	1	1	yes	characteristic
203	C₂×C₈	C₂²⋊C₄×C₁₃₇	1	1	yes	characteristic
204	C₂×C₈	C₂×D₄×C₁₃₇	1	1	yes	characteristic
205	C₂×C₈	C₂×D₄×C₁₃₇	1	2	yes	—
206	C₂×C₈	C₂×D₄×C₁₃₇	1	1	yes	characteristic
207	C₂×C₈	#41×C₁₃₇	1	2	yes	—
208	C₄⋊C₄	—	2	1	no	—
209	C₄⋊C₄	—	2	1	no	—
210	C₂²⋊C₄	—	2	2	no	—
211	C₂²⋊C₄	—	2	2	no	—
212	C₄²	C₂²⋊C₄×C₁₃₇	1	1	yes	characteristic
213	C₄²	C₂²⋊C₄×C₁₃₇	1	1	yes	characteristic
214	C₄²	M₁₆×C₁₃₇	1	1	yes	characteristic
215	C₄²	M₁₆×C₁₃₇	1	1	yes	characteristic
216	C₁₆	—	2	4	no	—
217	C₁₆	—	4	2	no	—
218	C₂³	—	2	2	no	—
219	C₂³	—	2	4	no	—
220	C₂³	—	4	2	no	—
221	C₂³	—	4	2	no	—
222	C₂³	C₂²×C₈×C₁₃₇	1	1	yes	characteristic
223	D₄	—	4	2	no	—
224	D₄	—	4	2	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	4	no	—
228	C₂×C₄	—	4	1	no	—
229	C₂×C₄	—	4	2	no	—
230	C₂×C₄	—	4	2	no	—
231	C₂×C₄	—	4	4	no	—
232	C₂×C₄	C₂²⋊C₈×C₁₃₇	1	1	yes	characteristic
233	C₂×C₄	C₂²⋊C₈×C₁₃₇	1	1	yes	characteristic
234	C₂×C₄	C₂²⋊C₈×C₁₃₇	1	1	yes	characteristic
235	C₂×C₄	C₂²⋊C₈×C₁₃₇	1	1	yes	characteristic
236	C₂×C₄	C₂×C₂²⋊C₄×C₁₃₇	1	1	yes	characteristic
237	C₂×C₄	C₄×D₄×C₁₃₇	1	2	yes	—
238	C₂×C₄	C₂×M₁₆×C₁₃₇	1	1	yes	characteristic
239	C₂×C₄	#131×C₁₃₇	1	2	yes	—
240	C₈	—	2	1	no	—
241	C₈	—	2	1	no	—
242	C₈	—	4	2	no	—
243	C₈	#121×C₁₃₇	1	2	yes	—
244	C₈	#124×C₁₃₇	1	1	yes	characteristic
245	C₈	#127×C₁₃₇	1	1	yes	characteristic
246	C₂²	—	2	1	no	—
247	C₂²	—	2	1	no	—
248	C₂²	—	4	2	no	—
249	C₂²	—	4	2	no	—
250	C₂²	—	4	2	no	—
251	C₂²	—	8	4	no	—
252	C₂²	C₂×C₂²⋊C₈×C₁₃₇	1	1	yes	characteristic
253	C₂²	C₈×D₄×C₁₃₇	1	2	yes	—
254	C₄	—	2	1	no	—
255	C₄	—	2	1	no	—
256	C₄	—	2	1	no	—
257	C₄	—	4	2	no	—
258	C₄	#349×C₁₃₇	1	2	yes	complemented
259	C₄	#390×C₁₃₇	1	1	yes	characteristic
260	C₄	#408×C₁₃₇	1	1	yes	characteristic
261	C₄	#436×C₁₃₇	1	2	yes	—
262	C₂	—	2	2	no	—
263	C₂	—	8	2	no	—
264	C₂	#1954×C₁₃₇	1	1	yes	characteristic
265	C₂	C₂×#349×C₁₃₇	1	1	yes	characteristic
266	C₂	#2115×C₁₃₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₁₃₇	C₂×C₂²⋊C₄
commutator	C₂³	C₂²×C₈×C₁₃₇
Frattini, cosocle	C₂²×C₈	C₂³×C₁₃₇
socle	C₂²×C₁₃₇	C₂×C₂²⋊C₈
prime radical	C₂²×D₄×C₁₃₇	C₈
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₃₇ ◁ C₂×C₄×C₈×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. #2150×C₁₃₇	C₂	semidirect
3. C₂×#349×C₁₃₇	C₂	semidirect
4. C₄×M₃₂×C₁₃₇	C₂	semidirect
39. M₃₂×C₁₃₇	D₄	semidirect
124. C₄×C₁₃₇	#349	semidirect
133. #13194	C₁₃₇	direct (N × H)
134. C₁₃₇	#13194	direct (N × H)
136. #2150	C₂×C₁₃₇	semidirect
137. C₂×#349	C₂×C₁₃₇	semidirect
138. C₄×M₃₂	C₂×C₁₃₇	semidirect
173. M₃₂	D₄×C₁₃₇	semidirect
258. C₄	#349×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#436×C₁₃₇	2
#2150×C₁₃₇	2
C₂×#349×C₁₃₇	2
C₄×M₃₂×C₁₃₇	2
#13194	137

Subgroups — C137×#13194 #46862580

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#13194 #46862580