Subgroups — C₁₃₇×#8665 #46858051

Order 35072.

358 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#446×C₁₃₇	C₂	1	1	yes	maximal, characteristic
2	#1837×C₁₃₇	C₂	1	1	yes	maximal, characteristic
3	#1564×C₁₃₇	C₂	1	1	yes	maximal, characteristic
4	#1487×C₁₃₇	C₂	1	2	yes	maximal
5	#1301×C₁₃₇	C₂	1	2	yes	maximal
6	C₂×C₄×Q₈×C₁₃₇	C₂²	1	1	yes	characteristic
7	#446×C₁₃₇	—	2	1	no	—
8	#446×C₁₃₇	—	2	1	no	—
9	#446×C₁₃₇	C₄	1	2	yes	—
10	#446×C₁₃₇	C₄	1	2	yes	—
11	#431×C₁₃₇	C₂²	1	2	yes	—
12	#423×C₁₃₇	C₂²	1	2	yes	—
13	C₂×C₄⋊C₈×C₁₃₇	—	2	1	no	—
14	C₂×C₄⋊C₈×C₁₃₇	C₂²	1	1	yes	characteristic
15	C₂×#97×C₁₃₇	—	2	1	no	—
16	C₂×C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
17	#341×C₁₃₇	—	2	1	no	—
18	#341×C₁₃₇	—	2	1	no	—
19	#335×C₁₃₇	—	2	1	no	—
20	#334×C₁₃₇	—	2	1	no	—
21	Q₈⋊C₈×C₁₃₇	—	2	4	no	—
22	C₂²×Q₈×C₁₃₇	Q₈	1	1	yes	characteristic
23	C₂×M₁₆×C₁₃₇	—	2	2	no	—
24	C₂²×C₈×C₁₃₇	—	2	1	no	—
25	C₂²×C₈×C₁₃₇	D₄	1	1	yes	characteristic
26	C₂²×C₈×C₁₃₇	D₄	1	1	yes	characteristic
27	C₄×Q₈×C₁₃₇	—	2	1	no	—
28	C₄×Q₈×C₁₃₇	—	2	1	no	—
29	C₄×Q₈×C₁₃₇	C₂×C₄	1	2	yes	—
30	C₄×Q₈×C₁₃₇	D₄	1	2	yes	—
31	C₂×C₄⋊C₄×C₁₃₇	—	2	1	no	—
32	C₂×C₄⋊C₄×C₁₃₇	Q₈	1	1	yes	characteristic
33	C₂×C₄²×C₁₃₇	—	2	1	no	—
34	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
35	C₄⋊C₈×C₁₃₇	—	2	2	no	—
36	C₄⋊C₈×C₁₃₇	—	2	2	no	—
37	C₄⋊C₈×C₁₃₇	—	2	1	no	—
38	C₄⋊C₈×C₁₃₇	—	2	2	no	—
39	C₄⋊C₈×C₁₃₇	—	2	2	no	—
40	C₄⋊C₈×C₁₃₇	—	2	2	no	—
41	C₄⋊C₈×C₁₃₇	C₂×C₄	1	2	yes	—
42	C₂²⋊C₈×C₁₃₇	—	2	2	no	—
43	#97×C₁₃₇	—	2	2	no	—
44	#97×C₁₃₇	—	2	2	no	—
45	C₄×C₈×C₁₃₇	—	2	2	no	—
46	C₄×C₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
47	C₄×C₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
48	C₄×C₈×C₁₃₇	D₄	1	1	yes	characteristic
49	C₄×C₈×C₁₃₇	D₄	1	1	yes	characteristic
50	C₂×Q₈×C₁₃₇	—	2	1	no	—
51	C₂×Q₈×C₁₃₇	—	4	2	no	—
52	C₂×Q₈×C₁₃₇	C₄⋊C₄	1	2	yes	—
53	C₂²×C₄×C₁₃₇	—	2	1	no	—
54	C₂²×C₄×C₁₃₇	—	2	1	no	—
55	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₁₃₇	C₂×Q₈	1	1	yes	characteristic
57	C₂²×C₄×C₁₃₇	#41	1	1	yes	characteristic
58	M₁₆×C₁₃₇	—	4	2	no	—
59	M₁₆×C₁₃₇	—	4	2	no	—
60	C₂×C₈×C₁₃₇	—	2	2	no	—
61	C₂×C₈×C₁₃₇	—	2	2	no	—
62	C₂×C₈×C₁₃₇	—	2	2	no	—
63	C₂×C₈×C₁₃₇	—	4	2	no	—
64	C₂×C₈×C₁₃₇	—	4	2	no	—
65	C₂×C₈×C₁₃₇	—	4	2	no	—
66	C₂×C₈×C₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
67	C₂×C₈×C₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
68	C₂×C₈×C₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
69	C₂×C₈×C₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
70	C₄⋊C₄×C₁₃₇	—	2	2	no	—
71	C₄⋊C₄×C₁₃₇	—	2	2	no	—
72	C₄⋊C₄×C₁₃₇	—	2	1	no	—
73	C₄⋊C₄×C₁₃₇	C₄⋊C₄	1	2	yes	—
74	C₄²×C₁₃₇	—	2	2	no	—
75	C₄²×C₁₃₇	—	2	2	no	—
76	C₄²×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
77	C₄²×C₁₃₇	C₂×D₄	1	1	yes	characteristic
78	C₄²×C₁₃₇	C₂×D₄	1	1	yes	characteristic
79	C₄²×C₁₃₇	#41	1	1	yes	characteristic
80	C₂³×C₁₃₇	C₂²⋊Q₈	1	1	yes	characteristic
81	Q₈×C₁₃₇	—	2	4	no	—
82	Q₈×C₁₃₇	—	4	2	no	—
83	C₂×C₄×C₁₃₇	—	2	2	no	—
84	C₂×C₄×C₁₃₇	—	2	2	no	—
85	C₂×C₄×C₁₃₇	—	2	2	no	—
86	C₂×C₄×C₁₃₇	—	4	1	no	—
87	C₂×C₄×C₁₃₇	—	4	2	no	—
88	C₂×C₄×C₁₃₇	—	4	2	no	—
89	C₂×C₄×C₁₃₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
90	C₂×C₄×C₁₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
91	C₂×C₄×C₁₃₇	#117	1	1	yes	characteristic
92	C₂×C₄×C₁₃₇	#121	1	1	yes	characteristic
93	C₂×C₄×C₁₃₇	#121	1	1	yes	characteristic
94	C₂×C₄×C₁₃₇	C₂²⋊Q₈	1	1	yes	characteristic
95	C₂×C₄×C₁₃₇	#135	1	2	yes	—
96	C₂×C₄×C₁₃₇	#137	1	2	yes	—
97	C₈×C₁₃₇	—	2	1	no	—
98	C₈×C₁₃₇	—	2	1	no	—
99	C₈×C₁₃₇	—	2	1	no	—
100	C₈×C₁₃₇	—	2	1	no	—
101	C₈×C₁₃₇	—	4	2	no	—
102	C₈×C₁₃₇	—	8	2	no	—
103	C₈×C₁₃₇	—	8	2	no	—
104	C₂²×C₁₃₇	—	2	1	no	—
105	C₂²×C₁₃₇	—	2	1	no	—
106	C₂²×C₁₃₇	#380	1	1	yes	characteristic
107	C₂²×C₁₃₇	#479	1	2	yes	—
108	C₄×C₁₃₇	—	2	1	no	—
109	C₄×C₁₃₇	—	2	1	no	—
110	C₄×C₁₃₇	—	2	1	no	—
111	C₄×C₁₃₇	—	4	2	no	—
112	C₄×C₁₃₇	—	4	2	no	—
113	C₄×C₁₃₇	#416	1	1	yes	characteristic
114	C₄×C₁₃₇	#419	1	1	yes	characteristic
115	C₄×C₁₃₇	#430	1	2	yes	—
116	C₄×C₁₃₇	#464	1	2	yes	—
117	C₂×C₁₃₇	—	2	2	no	—
118	C₂×C₁₃₇	#1805	1	1	yes	characteristic
119	C₂×C₁₃₇	#1811	1	1	yes	characteristic
120	C₂×C₁₃₇	#1814	1	1	yes	characteristic
121	#8665	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
122	C₁₃₇	#8665	1	1	yes	characteristic, direct product factor, Sylow
123	C₂×#446	C₂×C₁₃₇	1	1	yes	characteristic
124	#1837	C₂×C₁₃₇	1	1	yes	characteristic
125	#1564	C₂×C₁₃₇	1	1	yes	characteristic
126	#1487	C₂×C₁₃₇	1	2	yes	—
127	#1301	C₂×C₁₃₇	1	2	yes	—
128	C₂×C₄×Q₈	C₂²×C₁₃₇	1	1	yes	characteristic
129	#446	—	2	1	no	—
130	#446	—	2	1	no	—
131	#446	C₄×C₁₃₇	1	2	yes	—
132	#446	C₄×C₁₃₇	1	2	yes	—
133	#431	C₂²×C₁₃₇	1	2	yes	—
134	#423	C₂²×C₁₃₇	1	2	yes	—
135	C₂×C₄⋊C₈	—	2	1	no	—
136	C₂×C₄⋊C₈	C₂²×C₁₃₇	1	1	yes	characteristic
137	C₂×#97	—	2	1	no	—
138	C₂×C₄×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
139	#341	—	2	1	no	—
140	#341	—	2	1	no	—
141	#335	—	2	1	no	—
142	#334	—	2	1	no	—
143	Q₈⋊C₈	—	2	4	no	—
144	C₂²×Q₈	Q₈×C₁₃₇	1	1	yes	characteristic
145	C₂×M₁₆	—	2	2	no	—
146	C₂²×C₈	—	2	1	no	—
147	C₂²×C₈	D₄×C₁₃₇	1	1	yes	characteristic
148	C₂²×C₈	D₄×C₁₃₇	1	1	yes	characteristic
149	C₄×Q₈	—	2	1	no	—
150	C₄×Q₈	—	2	1	no	—
151	C₄×Q₈	C₂×C₄×C₁₃₇	1	2	yes	—
152	C₄×Q₈	D₄×C₁₃₇	1	2	yes	—
153	C₂×C₄⋊C₄	—	2	1	no	—
154	C₂×C₄⋊C₄	Q₈×C₁₃₇	1	1	yes	characteristic
155	C₂×C₄²	—	2	1	no	—
156	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
157	C₄⋊C₈	—	2	2	no	—
158	C₄⋊C₈	—	2	2	no	—
159	C₄⋊C₈	—	2	1	no	—
160	C₄⋊C₈	—	2	2	no	—
161	C₄⋊C₈	—	2	2	no	—
162	C₄⋊C₈	—	2	2	no	—
163	C₄⋊C₈	C₂×C₄×C₁₃₇	1	2	yes	—
164	C₂²⋊C₈	—	2	2	no	—
165	#97	—	2	2	no	—
166	#97	—	2	2	no	—
167	C₄×C₈	—	2	2	no	—
168	C₄×C₈	C₂×C₄×C₁₃₇	1	1	yes	characteristic
169	C₄×C₈	C₂×C₄×C₁₃₇	1	1	yes	characteristic
170	C₄×C₈	D₄×C₁₃₇	1	1	yes	characteristic
171	C₄×C₈	D₄×C₁₃₇	1	1	yes	characteristic
172	C₂×Q₈	—	2	1	no	—
173	C₂×Q₈	—	4	2	no	—
174	C₂×Q₈	C₄⋊C₄×C₁₃₇	1	2	yes	—
175	C₂²×C₄	—	2	1	no	—
176	C₂²×C₄	—	2	1	no	—
177	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
178	C₂²×C₄	C₂×Q₈×C₁₃₇	1	1	yes	characteristic
179	C₂²×C₄	#41×C₁₃₇	1	1	yes	characteristic
180	M₁₆	—	4	2	no	—
181	M₁₆	—	4	2	no	—
182	C₂×C₈	—	2	2	no	—
183	C₂×C₈	—	2	2	no	—
184	C₂×C₈	—	2	2	no	—
185	C₂×C₈	—	4	2	no	—
186	C₂×C₈	—	4	2	no	—
187	C₂×C₈	—	4	2	no	—
188	C₂×C₈	C₂²⋊C₄×C₁₃₇	1	1	yes	characteristic
189	C₂×C₈	C₂²⋊C₄×C₁₃₇	1	1	yes	characteristic
190	C₂×C₈	C₂²⋊C₄×C₁₃₇	1	1	yes	characteristic
191	C₂×C₈	C₂²⋊C₄×C₁₃₇	1	1	yes	characteristic
192	C₄⋊C₄	—	2	2	no	—
193	C₄⋊C₄	—	2	2	no	—
194	C₄⋊C₄	—	2	1	no	—
195	C₄⋊C₄	C₄⋊C₄×C₁₃₇	1	2	yes	—
196	C₄²	—	2	2	no	—
197	C₄²	—	2	2	no	—
198	C₄²	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
199	C₄²	C₂×D₄×C₁₃₇	1	1	yes	characteristic
200	C₄²	C₂×D₄×C₁₃₇	1	1	yes	characteristic
201	C₄²	#41×C₁₃₇	1	1	yes	characteristic
202	C₂³	C₂²⋊Q₈×C₁₃₇	1	1	yes	characteristic
203	Q₈	—	2	4	no	—
204	Q₈	—	4	2	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	4	1	no	—
209	C₂×C₄	—	4	2	no	—
210	C₂×C₄	—	4	2	no	—
211	C₂×C₄	C₂×C₂²⋊C₄×C₁₃₇	1	1	yes	characteristic
212	C₂×C₄	C₂×C₄⋊C₄×C₁₃₇	1	1	yes	characteristic
213	C₂×C₄	#117×C₁₃₇	1	1	yes	characteristic
214	C₂×C₄	#121×C₁₃₇	1	1	yes	characteristic
215	C₂×C₄	#121×C₁₃₇	1	1	yes	characteristic
216	C₂×C₄	C₂²⋊Q₈×C₁₃₇	1	1	yes	characteristic
217	C₂×C₄	#135×C₁₃₇	1	2	yes	—
218	C₂×C₄	#137×C₁₃₇	1	2	yes	—
219	C₈	—	2	1	no	—
220	C₈	—	2	1	no	—
221	C₈	—	2	1	no	—
222	C₈	—	2	1	no	—
223	C₈	—	4	2	no	—
224	C₈	—	8	2	no	—
225	C₈	—	8	2	no	—
226	C₂²	—	2	1	no	—
227	C₂²	—	2	1	no	—
228	C₂²	#380×C₁₃₇	1	1	yes	characteristic
229	C₂²	#479×C₁₃₇	1	2	yes	—
230	C₄	—	2	1	no	—
231	C₄	—	2	1	no	—
232	C₄	—	2	1	no	—
233	C₄	—	4	2	no	—
234	C₄	—	4	2	no	—
235	C₄	#416×C₁₃₇	1	1	yes	characteristic
236	C₄	#419×C₁₃₇	1	1	yes	characteristic
237	C₄	#430×C₁₃₇	1	2	yes	—
238	C₄	#464×C₁₃₇	1	2	yes	—
239	C₂	—	2	2	no	—
240	C₂	#1805×C₁₃₇	1	1	yes	characteristic
241	C₂	#1811×C₁₃₇	1	1	yes	characteristic
242	C₂	#1814×C₁₃₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₁₃₇	#121
commutator	C₄²	C₂²×C₄×C₁₃₇
Frattini, cosocle	C₂×C₄²	C₂³×C₁₃₇
socle	C₂²×C₁₃₇	#380
prime radical	C₂³×C₁₃₇	C₂²⋊Q₈
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
121. #8665	C₁₃₇	direct (N × H)
122. C₁₃₇	#8665	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×#446×C₁₃₇	2
#1837×C₁₃₇	2
#1564×C₁₃₇	2
#1487×C₁₃₇	2
#1301×C₁₃₇	2
#8665	137

Subgroups — C137×#8665 #46858051

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#8665 #46858051