Subgroups — C₁₃₇×#8924 #46858310

Order 35072.

644 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2366×C₁₃₇	C₂	1	1	yes	maximal, characteristic
2	#1957×C₁₃₇	C₂	1	1	yes	maximal, characteristic
3	#1948×C₁₃₇	C₂	1	2	yes	maximal, complemented
4	C₂×#331×C₁₃₇	C₂	1	1	yes	maximal, characteristic
5	#1279×C₁₃₇	C₂	1	2	yes	maximal, complemented
6	C₂×#127×C₁₃₇	C₂²	1	1	yes	characteristic
7	C₂×C₄×D₄×C₁₃₇	—	2	2	no	—
8	C₂×#106×C₁₃₇	—	2	2	no	—
9	C₂×C₄⋊C₈×C₁₃₇	C₂²	1	2	yes	—
10	C₂×#102×C₁₃₇	—	2	2	no	—
11	C₂×C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
12	#390×C₁₃₇	—	2	2	no	—
13	#386×C₁₃₇	—	2	2	no	—
14	#384×C₁₃₇	C₂²	1	1	yes	characteristic
15	C₄×C₄⋊C₄×C₁₃₇	C₂²	1	2	yes	—
16	#340×C₁₃₇	—	2	2	no	—
17	#331×C₁₃₇	C₄	1	4	yes	complemented
18	C₂²×D₄×C₁₃₇	—	2	2	no	—
19	C₂²×D₄×C₁₃₇	D₄	1	2	yes	—
20	C₂³×C₄×C₁₃₇	—	2	2	no	—
21	C₂²×C₈×C₁₃₇	—	2	1	no	—
22	C₂²×C₈×C₁₃₇	—	2	2	no	—
23	#127×C₁₃₇	—	2	2	no	—
24	#127×C₁₃₇	C₂×C₄	1	2	yes	—
25	#127×C₁₃₇	Q₈	1	2	yes	—
26	C₄×D₄×C₁₃₇	—	4	4	no	—
27	C₄×D₄×C₁₃₇	—	4	4	no	—
28	C₂×C₄⋊C₄×C₁₃₇	—	2	1	no	—
29	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
30	C₂×C₄⋊C₄×C₁₃₇	D₄	1	2	yes	—
31	C₂×C₄⋊C₄×C₁₃₇	D₄	1	2	yes	—
32	C₂×C₂²⋊C₄×C₁₃₇	—	2	4	no	—
33	C₂×C₂²⋊C₄×C₁₃₇	—	2	2	no	—
34	C₂×C₄²×C₁₃₇	—	2	2	no	—
35	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
36	#106×C₁₃₇	—	4	4	no	—
37	C₄⋊C₈×C₁₃₇	—	2	2	no	—
38	C₄⋊C₈×C₁₃₇	C₂×C₄	1	4	yes	complemented
39	#102×C₁₃₇	—	4	4	no	—
40	C₄×C₈×C₁₃₇	—	2	2	no	—
41	#95×C₁₃₇	—	2	2	no	—
42	C₂⁴×C₁₃₇	—	2	2	no	—
43	C₂×D₄×C₁₃₇	—	2	4	no	—
44	C₂×D₄×C₁₃₇	—	2	4	no	—
45	C₂×D₄×C₁₃₇	—	2	2	no	—
46	C₂×D₄×C₁₃₇	—	4	2	no	—
47	C₂×D₄×C₁₃₇	—	4	2	no	—
48	C₂×D₄×C₁₃₇	—	4	2	no	—
49	C₂×D₄×C₁₃₇	—	4	2	no	—
50	C₂×D₄×C₁₃₇	—	4	2	no	—
51	C₂×D₄×C₁₃₇	—	4	2	no	—
52	C₂×D₄×C₁₃₇	C₄⋊C₄	1	4	yes	—
53	C₂²×C₄×C₁₃₇	—	2	2	no	—
54	C₂²×C₄×C₁₃₇	—	2	2	no	—
55	C₂²×C₄×C₁₃₇	—	4	8	no	—
56	C₂²×C₄×C₁₃₇	—	4	1	no	—
57	C₂²×C₄×C₁₃₇	—	8	2	no	—
58	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
60	C₂×C₈×C₁₃₇	—	2	2	no	—
61	C₂×C₈×C₁₃₇	—	2	4	no	—
62	C₂×C₈×C₁₃₇	—	4	1	no	—
63	C₂×C₈×C₁₃₇	—	4	1	no	—
64	C₂×C₈×C₁₃₇	—	4	2	no	—
65	C₂×C₈×C₁₃₇	—	4	2	no	—
66	C₄⋊C₄×C₁₃₇	—	2	2	no	—
67	C₄⋊C₄×C₁₃₇	—	2	2	no	—
68	C₄⋊C₄×C₁₃₇	—	2	2	no	—
69	C₄⋊C₄×C₁₃₇	—	4	2	no	—
70	C₄⋊C₄×C₁₃₇	—	4	2	no	—
71	C₄⋊C₄×C₁₃₇	—	4	2	no	—
72	C₄⋊C₄×C₁₃₇	SD₁₆	1	4	yes	—
73	C₂²⋊C₄×C₁₃₇	—	4	8	no	—
74	C₂²⋊C₄×C₁₃₇	—	8	2	no	—
75	C₄²×C₁₃₇	—	4	2	no	—
76	C₄²×C₁₃₇	—	4	2	no	—
77	C₄²×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
78	C₄²×C₁₃₇	C₂×Q₈	1	1	yes	characteristic
79	C₄²×C₁₃₇	#41	1	2	yes	—
80	C₂³×C₁₃₇	—	2	4	no	—
81	C₂³×C₁₃₇	—	4	4	no	—
82	C₂³×C₁₃₇	—	4	4	no	—
83	C₂³×C₁₃₇	—	4	2	no	—
84	C₂³×C₁₃₇	—	4	2	no	—
85	C₂³×C₁₃₇	#120	1	1	yes	characteristic
86	D₄×C₁₃₇	—	4	4	no	—
87	D₄×C₁₃₇	—	4	4	no	—
88	D₄×C₁₃₇	—	4	4	no	—
89	D₄×C₁₃₇	—	4	4	no	—
90	D₄×C₁₃₇	—	4	4	no	—
91	D₄×C₁₃₇	—	8	2	no	—
92	C₂×C₄×C₁₃₇	—	2	1	no	—
93	C₂×C₄×C₁₃₇	—	2	1	no	—
94	C₂×C₄×C₁₃₇	—	2	4	no	—
95	C₂×C₄×C₁₃₇	—	4	2	no	—
96	C₂×C₄×C₁₃₇	—	4	2	no	—
97	C₂×C₄×C₁₃₇	—	4	2	no	—
98	C₂×C₄×C₁₃₇	—	4	2	no	—
99	C₂×C₄×C₁₃₇	—	4	2	no	—
100	C₂×C₄×C₁₃₇	—	8	8	no	—
101	C₂×C₄×C₁₃₇	—	8	2	no	—
102	C₂×C₄×C₁₃₇	—	8	1	no	—
103	C₂×C₄×C₁₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
104	C₂×C₄×C₁₃₇	C₄×D₄	1	2	yes	—
105	C₂×C₄×C₁₃₇	C₂²⋊Q₈	1	2	yes	—
106	C₂×C₄×C₁₃₇	#123	1	1	yes	characteristic
107	C₂×C₄×C₁₃₇	C₂×SD₁₆	1	2	yes	—
108	C₂×C₄×C₁₃₇	#135	1	2	yes	—
109	C₂×C₄×C₁₃₇	AGL(1,ℤ/8)	1	2	yes	—
110	C₂×C₄×C₁₃₇	AGL(1,ℤ/8)	1	2	yes	—
111	C₈×C₁₃₇	—	4	2	no	—
112	C₈×C₁₃₇	—	4	4	no	—
113	C₂²×C₁₃₇	—	4	4	no	—
114	C₂²×C₁₃₇	—	4	8	no	—
115	C₂²×C₁₃₇	—	8	2	no	—
116	C₂²×C₁₃₇	—	8	2	no	—
117	C₂²×C₁₃₇	—	8	4	no	—
118	C₂²×C₁₃₇	#385	1	1	yes	characteristic
119	C₂²×C₁₃₇	#449	1	2	yes	—
120	C₂²×C₁₃₇	#450	1	2	yes	—
121	C₂²×C₁₃₇	#453	1	1	yes	characteristic
122	C₂²×C₁₃₇	#455	1	1	yes	characteristic
123	C₄×C₁₃₇	—	2	1	no	—
124	C₄×C₁₃₇	—	2	1	no	—
125	C₄×C₁₃₇	—	4	4	no	—
126	C₄×C₁₃₇	—	8	2	no	—
127	C₄×C₁₃₇	—	16	1	no	—
128	C₄×C₁₃₇	C₄×SD₁₆	1	2	yes	—
129	C₄×C₁₃₇	#442	1	2	yes	—
130	C₄×C₁₃₇	#476	1	2	yes	—
131	C₄×C₁₃₇	#478	1	2	yes	—
132	C₂×C₁₃₇	—	8	4	no	—
133	C₂×C₁₃₇	#1620	1	1	yes	characteristic
134	C₂×C₁₃₇	#1629	1	1	yes	characteristic
135	C₂×C₁₃₇	#1642	1	2	yes	—
136	C₂×C₁₃₇	#1856	1	1	yes	characteristic
137	C₂×C₁₃₇	#1864	1	2	yes	—
138	#8924	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
139	C₁₃₇	#8924	1	1	yes	characteristic, direct product factor, Sylow
140	#2366	C₂×C₁₃₇	1	1	yes	characteristic
141	#1957	C₂×C₁₃₇	1	1	yes	characteristic
142	#1948	C₂×C₁₃₇	1	2	yes	complemented
143	C₂×#331	C₂×C₁₃₇	1	1	yes	characteristic
144	#1279	C₂×C₁₃₇	1	2	yes	complemented
145	C₂×#127	C₂²×C₁₃₇	1	1	yes	characteristic
146	C₂×C₄×D₄	—	2	2	no	—
147	C₂×#106	—	2	2	no	—
148	C₂×C₄⋊C₈	C₂²×C₁₃₇	1	2	yes	—
149	C₂×#102	—	2	2	no	—
150	C₂×C₄×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
151	#390	—	2	2	no	—
152	#386	—	2	2	no	—
153	#384	C₂²×C₁₃₇	1	1	yes	characteristic
154	C₄×C₄⋊C₄	C₂²×C₁₃₇	1	2	yes	—
155	#340	—	2	2	no	—
156	#331	C₄×C₁₃₇	1	4	yes	complemented
157	C₂²×D₄	—	2	2	no	—
158	C₂²×D₄	D₄×C₁₃₇	1	2	yes	—
159	C₂³×C₄	—	2	2	no	—
160	C₂²×C₈	—	2	2	no	—
161	C₂²×C₈	—	2	1	no	—
162	#127	—	2	2	no	—
163	#127	C₂×C₄×C₁₃₇	1	2	yes	—
164	#127	Q₈×C₁₃₇	1	2	yes	—
165	C₄×D₄	—	4	4	no	—
166	C₄×D₄	—	4	4	no	—
167	C₂×C₄⋊C₄	—	2	2	no	—
168	C₂×C₄⋊C₄	—	2	1	no	—
169	C₂×C₄⋊C₄	D₄×C₁₃₇	1	2	yes	—
170	C₂×C₄⋊C₄	D₄×C₁₃₇	1	2	yes	—
171	C₂×C₂²⋊C₄	—	2	4	no	—
172	C₂×C₂²⋊C₄	—	2	2	no	—
173	C₂×C₄²	—	2	2	no	—
174	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
175	#106	—	4	4	no	—
176	C₄⋊C₈	—	2	2	no	—
177	C₄⋊C₈	C₂×C₄×C₁₃₇	1	4	yes	complemented
178	#102	—	4	4	no	—
179	C₄×C₈	—	2	2	no	—
180	#95	—	2	2	no	—
181	C₂⁴	—	2	2	no	—
182	C₂×D₄	—	2	4	no	—
183	C₂×D₄	—	2	4	no	—
184	C₂×D₄	—	2	2	no	—
185	C₂×D₄	—	4	2	no	—
186	C₂×D₄	—	4	2	no	—
187	C₂×D₄	—	4	2	no	—
188	C₂×D₄	—	4	2	no	—
189	C₂×D₄	—	4	2	no	—
190	C₂×D₄	—	4	2	no	—
191	C₂×D₄	C₄⋊C₄×C₁₃₇	1	4	yes	—
192	C₂²×C₄	—	2	2	no	—
193	C₂²×C₄	—	2	2	no	—
194	C₂²×C₄	—	4	8	no	—
195	C₂²×C₄	—	4	1	no	—
196	C₂²×C₄	—	8	2	no	—
197	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
198	C₂²×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
199	C₂×C₈	—	2	2	no	—
200	C₂×C₈	—	2	4	no	—
201	C₂×C₈	—	4	1	no	—
202	C₂×C₈	—	4	1	no	—
203	C₂×C₈	—	4	2	no	—
204	C₂×C₈	—	4	2	no	—
205	C₄⋊C₄	—	2	2	no	—
206	C₄⋊C₄	—	2	2	no	—
207	C₄⋊C₄	—	2	2	no	—
208	C₄⋊C₄	—	4	2	no	—
209	C₄⋊C₄	—	4	2	no	—
210	C₄⋊C₄	—	4	2	no	—
211	C₄⋊C₄	SD₁₆×C₁₃₇	1	4	yes	—
212	C₂²⋊C₄	—	4	8	no	—
213	C₂²⋊C₄	—	8	2	no	—
214	C₄²	—	4	2	no	—
215	C₄²	—	4	2	no	—
216	C₄²	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
217	C₄²	C₂×Q₈×C₁₃₇	1	1	yes	characteristic
218	C₄²	#41×C₁₃₇	1	2	yes	—
219	C₂³	—	2	4	no	—
220	C₂³	—	4	4	no	—
221	C₂³	—	4	4	no	—
222	C₂³	—	4	2	no	—
223	C₂³	—	4	2	no	—
224	C₂³	#120×C₁₃₇	1	1	yes	characteristic
225	D₄	—	4	4	no	—
226	D₄	—	4	4	no	—
227	D₄	—	4	4	no	—
228	D₄	—	4	4	no	—
229	D₄	—	4	4	no	—
230	D₄	—	8	2	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	4	no	—
234	C₂×C₄	—	4	2	no	—
235	C₂×C₄	—	4	2	no	—
236	C₂×C₄	—	4	2	no	—
237	C₂×C₄	—	4	2	no	—
238	C₂×C₄	—	4	2	no	—
239	C₂×C₄	—	8	8	no	—
240	C₂×C₄	—	8	2	no	—
241	C₂×C₄	—	8	1	no	—
242	C₂×C₄	C₂×C₄⋊C₄×C₁₃₇	1	1	yes	characteristic
243	C₂×C₄	C₄×D₄×C₁₃₇	1	2	yes	—
244	C₂×C₄	C₂²⋊Q₈×C₁₃₇	1	2	yes	—
245	C₂×C₄	#123×C₁₃₇	1	1	yes	characteristic
246	C₂×C₄	C₂×SD₁₆×C₁₃₇	1	2	yes	—
247	C₂×C₄	#135×C₁₃₇	1	2	yes	—
248	C₂×C₄	AGL(1,ℤ/8)×C₁₃₇	1	2	yes	—
249	C₂×C₄	AGL(1,ℤ/8)×C₁₃₇	1	2	yes	—
250	C₈	—	4	2	no	—
251	C₈	—	4	4	no	—
252	C₂²	—	4	4	no	—
253	C₂²	—	4	8	no	—
254	C₂²	—	8	2	no	—
255	C₂²	—	8	2	no	—
256	C₂²	—	8	4	no	—
257	C₂²	#385×C₁₃₇	1	1	yes	characteristic
258	C₂²	#449×C₁₃₇	1	2	yes	—
259	C₂²	#450×C₁₃₇	1	2	yes	—
260	C₂²	#453×C₁₃₇	1	1	yes	characteristic
261	C₂²	#455×C₁₃₇	1	1	yes	characteristic
262	C₄	—	2	1	no	—
263	C₄	—	2	1	no	—
264	C₄	—	4	4	no	—
265	C₄	—	8	2	no	—
266	C₄	—	16	1	no	—
267	C₄	C₄×SD₁₆×C₁₃₇	1	2	yes	—
268	C₄	#442×C₁₃₇	1	2	yes	—
269	C₄	#476×C₁₃₇	1	2	yes	—
270	C₄	#478×C₁₃₇	1	2	yes	—
271	C₂	—	8	4	no	—
272	C₂	#1620×C₁₃₇	1	1	yes	characteristic
273	C₂	#1629×C₁₃₇	1	1	yes	characteristic
274	C₂	#1642×C₁₃₇	1	2	yes	—
275	C₂	#1856×C₁₃₇	1	1	yes	characteristic
276	C₂	#1864×C₁₃₇	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₃₇	#120
commutator	C₄²	C₂²×C₄×C₁₃₇
Frattini, cosocle	C₂×C₄²	C₂³×C₁₃₇
prime radical	C₂×#127×C₁₃₇	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃₇ ◁ C₂×C₄²×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. #1948×C₁₃₇	C₂	semidirect
5. #1279×C₁₃₇	C₂	semidirect
17. #331×C₁₃₇	C₄	semidirect
38. C₄⋊C₈×C₁₃₇	C₂×C₄	semidirect
138. #8924	C₁₃₇	direct (N × H)
139. C₁₃₇	#8924	direct (N × H)
142. #1948	C₂×C₁₃₇	semidirect
144. #1279	C₂×C₁₃₇	semidirect
156. #331	C₄×C₁₃₇	semidirect
177. C₄⋊C₈	C₂×C₄×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
#2366×C₁₃₇	2
#1957×C₁₃₇	2
#1948×C₁₃₇	2
C₂×#331×C₁₃₇	2
#1279×C₁₃₇	2
#8924	137

Subgroups — C137×#8924 #46858310

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#8924 #46858310