Subgroups — C₁₃₇×#9844 #46859230

Order 35072.

554 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#3169×C₁₃₇	C₂	1	1	yes	maximal, characteristic
2	#1962×C₁₃₇	C₂	1	2	yes	maximal, complemented
3	C₄²×C₈×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#1567×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
5	#1492×C₁₃₇	C₂	1	2	yes	maximal, complemented
6	#532×C₁₃₇	C₂²	1	2	yes	—
7	#500×C₁₃₇	C₄	1	1	yes	characteristic, complemented
8	#493×C₁₃₇	C₄	1	1	yes	characteristic, complemented
9	#486×C₁₃₇	C₄	1	2	yes	complemented
10	#466×C₁₃₇	—	2	2	no	—
11	C₄×D₈×C₁₃₇	—	2	2	no	—
12	#423×C₁₃₇	C₂²	1	2	yes	—
13	C₂×#104×C₁₃₇	—	2	4	no	—
14	C₂×C₄×C₈×C₁₃₇	—	2	2	no	—
15	C₂×C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
17	C₄³×C₁₃₇	C₂²	1	1	yes	characteristic
18	#335×C₁₃₇	—	2	2	no	—
19	D₄⋊C₈×C₁₃₇	—	2	4	no	—
20	C₂×#41×C₁₃₇	—	2	2	no	—
21	C₂×D₈×C₁₃₇	—	2	2	no	—
22	C₂×M₁₆×C₁₃₇	—	2	2	no	—
23	C₂²×C₈×C₁₃₇	—	2	1	no	—
24	C₂²×C₈×C₁₃₇	D₄	1	2	yes	—
25	C₄⋊Q₈×C₁₃₇	C₂×C₄	1	2	yes	—
26	#127×C₁₃₇	C₂×C₄	1	2	yes	—
27	#124×C₁₃₇	—	2	2	no	—
28	#121×C₁₃₇	—	2	2	no	—
29	#121×C₁₃₇	—	2	2	no	—
30	C₄×D₄×C₁₃₇	—	2	2	no	—
31	C₄×D₄×C₁₃₇	—	2	2	no	—
32	C₂×C₄²×C₁₃₇	—	2	1	no	—
33	C₂×C₄²×C₁₃₇	—	2	1	no	—
34	C₂×C₄²×C₁₃₇	D₄	1	2	yes	—
35	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
36	#107×C₁₃₇	—	2	2	no	—
37	C₄⋊C₈×C₁₃₇	—	2	2	no	—
38	C₄⋊C₈×C₁₃₇	—	2	2	no	—
39	#104×C₁₃₇	—	4	8	no	—
40	#102×C₁₃₇	—	2	4	no	—
41	C₄×C₈×C₁₃₇	—	2	8	no	—
42	C₄×C₈×C₁₃₇	—	2	1	no	—
43	C₄×C₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
44	C₄×C₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
45	C₄×C₈×C₁₃₇	D₄	1	2	yes	—
46	C₄×C₈×C₁₃₇	D₄	1	2	yes	—
47	#41×C₁₃₇	—	4	4	no	—
48	#41×C₁₃₇	—	4	4	no	—
49	C₂×Q₈×C₁₃₇	—	2	2	no	—
50	C₂×D₄×C₁₃₇	—	2	2	no	—
51	C₂×D₄×C₁₃₇	—	2	2	no	—
52	C₂×D₄×C₁₃₇	—	2	2	no	—
53	C₂×D₄×C₁₃₇	—	4	2	no	—
54	C₂²×C₄×C₁₃₇	—	2	2	no	—
55	C₂²×C₄×C₁₃₇	—	4	2	no	—
56	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₁₃₇	#41	1	1	yes	characteristic
59	D₈×C₁₃₇	—	4	4	no	—
60	M₁₆×C₁₃₇	—	4	4	no	—
61	C₂×C₈×C₁₃₇	—	2	4	no	—
62	C₂×C₈×C₁₃₇	—	2	2	no	—
63	C₂×C₈×C₁₃₇	—	2	4	no	—
64	C₂×C₈×C₁₃₇	—	4	2	no	—
65	C₂×C₈×C₁₃₇	C₂²⋊C₄	1	2	yes	—
66	C₂×C₈×C₁₃₇	C₂²⋊C₄	1	2	yes	—
67	C₄⋊C₄×C₁₃₇	—	2	2	no	—
68	C₄⋊C₄×C₁₃₇	—	2	2	no	—
69	C₂²⋊C₄×C₁₃₇	—	4	2	no	—
70	C₂²⋊C₄×C₁₃₇	—	4	2	no	—
71	C₄²×C₁₃₇	—	2	4	no	—
72	C₄²×C₁₃₇	—	2	4	no	—
73	C₄²×C₁₃₇	—	2	2	no	—
74	C₄²×C₁₃₇	D₈	1	4	yes	complemented
75	C₄²×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
76	C₄²×C₁₃₇	C₂×D₄	1	1	yes	characteristic
77	C₄²×C₁₃₇	C₂×D₄	1	1	yes	characteristic
78	C₄²×C₁₃₇	#41	1	1	yes	characteristic
79	C₂³×C₁₃₇	—	4	2	no	—
80	C₂³×C₁₃₇	#121	1	1	yes	characteristic
81	Q₈×C₁₃₇	—	4	4	no	—
82	D₄×C₁₃₇	—	4	4	no	—
83	D₄×C₁₃₇	—	4	4	no	—
84	D₄×C₁₃₇	—	4	4	no	—
85	D₄×C₁₃₇	—	8	4	no	—
86	C₂×C₄×C₁₃₇	—	2	4	no	—
87	C₂×C₄×C₁₃₇	—	2	4	no	—
88	C₂×C₄×C₁₃₇	—	2	4	no	—
89	C₂×C₄×C₁₃₇	—	2	2	no	—
90	C₂×C₄×C₁₃₇	—	2	2	no	—
91	C₂×C₄×C₁₃₇	—	4	2	no	—
92	C₂×C₄×C₁₃₇	—	8	2	no	—
93	C₂×C₄×C₁₃₇	—	8	2	no	—
94	C₂×C₄×C₁₃₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
95	C₂×C₄×C₁₃₇	C₄×D₄	1	1	yes	characteristic
96	C₂×C₄×C₁₃₇	C₄×D₄	1	1	yes	characteristic
97	C₂×C₄×C₁₃₇	#121	1	1	yes	characteristic
98	C₂×C₄×C₁₃₇	#124	1	1	yes	characteristic
99	C₂×C₄×C₁₃₇	#127	1	1	yes	characteristic
100	C₂×C₄×C₁₃₇	C₂×D₈	1	2	yes	—
101	C₂×C₄×C₁₃₇	#135	1	2	yes	—
102	C₈×C₁₃₇	—	2	4	no	—
103	C₈×C₁₃₇	—	2	2	no	—
104	C₈×C₁₃₇	—	8	2	no	—
105	C₈×C₁₃₇	#104	1	4	yes	complemented
106	C₂²×C₁₃₇	—	2	1	no	—
107	C₂²×C₁₃₇	—	2	1	no	—
108	C₂²×C₁₃₇	—	8	2	no	—
109	C₂²×C₁₃₇	—	8	2	no	—
110	C₂²×C₁₃₇	—	8	2	no	—
111	C₂²×C₁₃₇	#390	1	1	yes	characteristic
112	C₂²×C₁₃₇	#466	1	2	yes	—
113	C₄×C₁₃₇	—	2	8	no	—
114	C₄×C₁₃₇	—	2	1	no	—
115	C₄×C₁₃₇	—	2	1	no	—
116	C₄×C₁₃₇	—	8	2	no	—
117	C₄×C₁₃₇	C₂×#104	1	2	yes	—
118	C₄×C₁₃₇	C₄×D₈	1	2	yes	—
119	C₄×C₁₃₇	#443	1	2	yes	—
120	C₄×C₁₃₇	#493	1	1	yes	characteristic
121	C₄×C₁₃₇	#495	1	1	yes	characteristic
122	C₂×C₁₃₇	—	2	2	no	—
123	C₂×C₁₃₇	—	16	2	no	—
124	C₂×C₁₃₇	#1962	1	1	yes	characteristic
125	C₂×C₁₃₇	#1970	1	1	yes	characteristic
126	C₂×C₁₃₇	#1974	1	1	yes	characteristic
127	#9844	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
128	C₁₃₇	#9844	1	1	yes	characteristic, direct product factor, Sylow
129	#3169	C₂×C₁₃₇	1	1	yes	characteristic
130	#1962	C₂×C₁₃₇	1	2	yes	complemented
131	C₄²×C₈	C₂×C₁₃₇	1	1	yes	characteristic, complemented
132	#1567	C₂×C₁₃₇	1	1	yes	characteristic, complemented
133	#1492	C₂×C₁₃₇	1	2	yes	complemented
134	#532	C₂²×C₁₃₇	1	2	yes	—
135	#500	C₄×C₁₃₇	1	1	yes	characteristic, complemented
136	#493	C₄×C₁₃₇	1	1	yes	characteristic, complemented
137	#486	C₄×C₁₃₇	1	2	yes	complemented
138	#466	—	2	2	no	—
139	C₄×D₈	—	2	2	no	—
140	#423	C₂²×C₁₃₇	1	2	yes	—
141	C₂×#104	—	2	4	no	—
142	C₂×C₄×C₈	—	2	2	no	—
143	C₂×C₄×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
144	C₂×C₄×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
145	C₄³	C₂²×C₁₃₇	1	1	yes	characteristic
146	#335	—	2	2	no	—
147	D₄⋊C₈	—	2	4	no	—
148	C₂×#41	—	2	2	no	—
149	C₂×D₈	—	2	2	no	—
150	C₂×M₁₆	—	2	2	no	—
151	C₂²×C₈	—	2	1	no	—
152	C₂²×C₈	D₄×C₁₃₇	1	2	yes	—
153	C₄⋊Q₈	C₂×C₄×C₁₃₇	1	2	yes	—
154	#127	C₂×C₄×C₁₃₇	1	2	yes	—
155	#124	—	2	2	no	—
156	#121	—	2	2	no	—
157	#121	—	2	2	no	—
158	C₄×D₄	—	2	2	no	—
159	C₄×D₄	—	2	2	no	—
160	C₂×C₄²	—	2	1	no	—
161	C₂×C₄²	—	2	1	no	—
162	C₂×C₄²	D₄×C₁₃₇	1	2	yes	—
163	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
164	#107	—	2	2	no	—
165	C₄⋊C₈	—	2	2	no	—
166	C₄⋊C₈	—	2	2	no	—
167	#104	—	4	8	no	—
168	#102	—	2	4	no	—
169	C₄×C₈	—	2	8	no	—
170	C₄×C₈	—	2	1	no	—
171	C₄×C₈	C₂×C₄×C₁₃₇	1	1	yes	characteristic
172	C₄×C₈	C₂×C₄×C₁₃₇	1	1	yes	characteristic
173	C₄×C₈	D₄×C₁₃₇	1	2	yes	—
174	C₄×C₈	D₄×C₁₃₇	1	2	yes	—
175	#41	—	4	4	no	—
176	#41	—	4	4	no	—
177	C₂×Q₈	—	2	2	no	—
178	C₂×D₄	—	2	2	no	—
179	C₂×D₄	—	2	2	no	—
180	C₂×D₄	—	2	2	no	—
181	C₂×D₄	—	4	2	no	—
182	C₂²×C₄	—	2	2	no	—
183	C₂²×C₄	—	4	2	no	—
184	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
185	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
186	C₂²×C₄	#41×C₁₃₇	1	1	yes	characteristic
187	D₈	—	4	4	no	—
188	M₁₆	—	4	4	no	—
189	C₂×C₈	—	2	4	no	—
190	C₂×C₈	—	2	2	no	—
191	C₂×C₈	—	2	4	no	—
192	C₂×C₈	—	4	2	no	—
193	C₂×C₈	C₂²⋊C₄×C₁₃₇	1	2	yes	—
194	C₂×C₈	C₂²⋊C₄×C₁₃₇	1	2	yes	—
195	C₄⋊C₄	—	2	2	no	—
196	C₄⋊C₄	—	2	2	no	—
197	C₂²⋊C₄	—	4	2	no	—
198	C₂²⋊C₄	—	4	2	no	—
199	C₄²	—	2	4	no	—
200	C₄²	—	2	4	no	—
201	C₄²	—	2	2	no	—
202	C₄²	D₈×C₁₃₇	1	4	yes	complemented
203	C₄²	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
204	C₄²	C₂×D₄×C₁₃₇	1	1	yes	characteristic
205	C₄²	C₂×D₄×C₁₃₇	1	1	yes	characteristic
206	C₄²	#41×C₁₃₇	1	1	yes	characteristic
207	C₂³	—	4	2	no	—
208	C₂³	#121×C₁₃₇	1	1	yes	characteristic
209	Q₈	—	4	4	no	—
210	D₄	—	4	4	no	—
211	D₄	—	4	4	no	—
212	D₄	—	4	4	no	—
213	D₄	—	8	4	no	—
214	C₂×C₄	—	2	4	no	—
215	C₂×C₄	—	2	4	no	—
216	C₂×C₄	—	2	4	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	4	2	no	—
220	C₂×C₄	—	8	2	no	—
221	C₂×C₄	—	8	2	no	—
222	C₂×C₄	C₂×C₂²⋊C₄×C₁₃₇	1	1	yes	characteristic
223	C₂×C₄	C₄×D₄×C₁₃₇	1	1	yes	characteristic
224	C₂×C₄	C₄×D₄×C₁₃₇	1	1	yes	characteristic
225	C₂×C₄	#121×C₁₃₇	1	1	yes	characteristic
226	C₂×C₄	#124×C₁₃₇	1	1	yes	characteristic
227	C₂×C₄	#127×C₁₃₇	1	1	yes	characteristic
228	C₂×C₄	C₂×D₈×C₁₃₇	1	2	yes	—
229	C₂×C₄	#135×C₁₃₇	1	2	yes	—
230	C₈	—	2	4	no	—
231	C₈	—	2	2	no	—
232	C₈	—	8	2	no	—
233	C₈	#104×C₁₃₇	1	4	yes	complemented
234	C₂²	—	2	1	no	—
235	C₂²	—	2	1	no	—
236	C₂²	—	8	2	no	—
237	C₂²	—	8	2	no	—
238	C₂²	—	8	2	no	—
239	C₂²	#390×C₁₃₇	1	1	yes	characteristic
240	C₂²	#466×C₁₃₇	1	2	yes	—
241	C₄	—	2	8	no	—
242	C₄	—	2	1	no	—
243	C₄	—	2	1	no	—
244	C₄	—	8	2	no	—
245	C₄	C₂×#104×C₁₃₇	1	2	yes	—
246	C₄	C₄×D₈×C₁₃₇	1	2	yes	—
247	C₄	#443×C₁₃₇	1	2	yes	—
248	C₄	#493×C₁₃₇	1	1	yes	characteristic
249	C₄	#495×C₁₃₇	1	1	yes	characteristic
250	C₂	—	2	2	no	—
251	C₂	—	16	2	no	—
252	C₂	#1962×C₁₃₇	1	1	yes	characteristic
253	C₂	#1970×C₁₃₇	1	1	yes	characteristic
254	C₂	#1974×C₁₃₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₁₃₇	#127
commutator	C₄²	C₂²×C₄×C₁₃₇
Frattini, cosocle	C₂×C₄²	C₂³×C₁₃₇
socle	C₂²×C₁₃₇	#390
prime radical	#3169×C₁₃₇	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₃₇ ◁ C₂×C₄²×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. #1962×C₁₃₇	C₂	semidirect
3. C₄²×C₈×C₁₃₇	C₂	semidirect
4. #1567×C₁₃₇	C₂	semidirect
5. #1492×C₁₃₇	C₂	semidirect
7. #500×C₁₃₇	C₄	semidirect
8. #493×C₁₃₇	C₄	semidirect
9. #486×C₁₃₇	C₄	semidirect
74. C₄²×C₁₃₇	D₈	semidirect
105. C₈×C₁₃₇	#104	semidirect
127. #9844	C₁₃₇	direct (N × H)
128. C₁₃₇	#9844	direct (N × H)
130. #1962	C₂×C₁₃₇	semidirect
131. C₄²×C₈	C₂×C₁₃₇	semidirect
132. #1567	C₂×C₁₃₇	semidirect
133. #1492	C₂×C₁₃₇	semidirect
135. #500	C₄×C₁₃₇	semidirect
136. #493	C₄×C₁₃₇	semidirect
137. #486	C₄×C₁₃₇	semidirect
202. C₄²	D₈×C₁₃₇	semidirect
233. C₈	#104×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
#3169×C₁₃₇	2
#1962×C₁₃₇	2
C₄²×C₈×C₁₃₇	2
#1567×C₁₃₇	2
#1492×C₁₃₇	2
#9844	137

Subgroups — C137×#9844 #46859230

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#9844 #46859230