Subgroups — C₁₃₇×#28828 #46878214

Order 35072.

606 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2571×C₁₃₇	C₂	1	2	yes	maximal
2	#2085×C₁₃₇	C₂	1	4	yes	maximal
3	#1948×C₁₃₇	C₂	1	2	yes	maximal
4	#1942×C₁₃₇	C₂	1	2	yes	maximal
5	#1846×C₁₃₇	C₂	1	2	yes	maximal
6	#1845×C₁₃₇	C₂	1	1	yes	maximal, characteristic
7	C₄×#107×C₁₃₇	C₂	1	1	yes	maximal, characteristic
8	C₈×C₄⋊C₄×C₁₃₇	C₂	1	1	yes	maximal, characteristic
9	C₂×#107×C₁₃₇	C₂²	1	1	yes	characteristic
10	C₂×#107×C₁₃₇	C₂²	1	1	yes	characteristic
11	C₂×#107×C₁₃₇	C₂²	1	2	yes	—
12	C₂×#107×C₁₃₇	C₂²	1	2	yes	—
13	C₂×#106×C₁₃₇	C₂²	1	4	yes	—
14	C₂×C₄⋊C₈×C₁₃₇	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₁₃₇	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
17	#400×C₁₃₇	—	2	4	no	—
18	#389×C₁₃₇	—	2	2	no	—
19	#389×C₁₃₇	C₂²	1	4	yes	—
20	#384×C₁₃₇	C₂²	1	2	yes	—
21	#382×C₁₃₇	—	2	2	no	—
22	#382×C₁₃₇	C₂²	1	4	yes	—
23	C₄×C₄⋊C₄×C₁₃₇	C₂²	1	2	yes	—
24	C₄×C₄⋊C₄×C₁₃₇	C₂²	1	1	yes	characteristic
25	#340×C₁₃₇	C₂²	1	2	yes	—
26	#340×C₁₃₇	C₂²	1	4	yes	—
27	#336×C₁₃₇	C₂²	1	2	yes	—
28	C₂²×C₈×C₁₃₇	C₂³	1	2	yes	—
29	C₂²×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
30	C₂²×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
31	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
32	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
33	C₂×C₄⋊C₄×C₁₃₇	—	2	4	no	—
34	C₂×C₄⋊C₄×C₁₃₇	D₄	1	1	yes	characteristic
35	C₂×C₄⋊C₄×C₁₃₇	D₄	1	1	yes	characteristic
36	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	2	yes	—
37	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	2	yes	—
38	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	4	yes	—
39	C₂×C₄²×C₁₃₇	—	2	2	no	—
40	C₂×C₄²×C₁₃₇	C₂³	1	2	yes	—
41	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
42	#107×C₁₃₇	—	2	2	no	—
43	#107×C₁₃₇	—	2	2	no	—
44	#107×C₁₃₇	—	2	4	no	—
45	#107×C₁₃₇	—	4	2	no	—
46	#106×C₁₃₇	—	4	4	no	—
47	C₄⋊C₈×C₁₃₇	Q₈	1	4	yes	—
48	C₄×C₈×C₁₃₇	—	2	4	no	—
49	C₄×C₈×C₁₃₇	Q₈	1	4	yes	—
50	#95×C₁₃₇	—	2	2	no	—
51	#95×C₁₃₇	—	2	4	no	—
52	#95×C₁₃₇	—	2	4	no	—
53	#95×C₁₃₇	—	2	4	no	—
54	#95×C₁₃₇	D₄	1	2	yes	—
55	C₂²×C₄×C₁₃₇	—	2	2	no	—
56	C₂²×C₄×C₁₃₇	—	2	2	no	—
57	C₂²×C₄×C₁₃₇	—	2	4	no	—
58	C₂²×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
59	C₂²×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
60	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
63	C₂×C₈×C₁₃₇	—	2	2	no	—
64	C₂×C₈×C₁₃₇	—	2	2	no	—
65	C₂×C₈×C₁₃₇	—	2	1	no	—
66	C₂×C₈×C₁₃₇	—	2	1	no	—
67	C₂×C₈×C₁₃₇	C₂×Q₈	1	2	yes	—
68	C₂×C₈×C₁₃₇	C₂×Q₈	1	4	yes	—
69	C₂×C₈×C₁₃₇	C₂×Q₈	1	2	yes	—
70	C₂×C₈×C₁₃₇	#41	1	2	yes	—
71	C₂×C₈×C₁₃₇	#41	1	2	yes	—
72	C₄⋊C₄×C₁₃₇	—	2	2	no	—
73	C₄⋊C₄×C₁₃₇	—	2	2	no	—
74	C₄⋊C₄×C₁₃₇	—	2	2	no	—
75	C₄⋊C₄×C₁₃₇	—	2	2	no	—
76	C₄⋊C₄×C₁₃₇	—	2	4	no	—
77	C₄⋊C₄×C₁₃₇	—	2	2	no	—
78	C₄⋊C₄×C₁₃₇	—	2	4	no	—
79	C₄⋊C₄×C₁₃₇	—	2	2	no	—
80	C₄⋊C₄×C₁₃₇	—	4	4	no	—
81	C₄⋊C₄×C₁₃₇	—	8	2	no	—
82	C₄⋊C₄×C₁₃₇	—	8	4	no	—
83	C₄²×C₁₃₇	—	2	2	no	—
84	C₄²×C₁₃₇	—	2	2	no	—
85	C₄²×C₁₃₇	—	4	4	no	—
86	C₄²×C₁₃₇	C₂×Q₈	1	2	yes	—
87	C₄²×C₁₃₇	C₂×Q₈	1	2	yes	—
88	C₂³×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
89	C₂×C₄×C₁₃₇	—	2	1	no	—
90	C₂×C₄×C₁₃₇	—	2	1	no	—
91	C₂×C₄×C₁₃₇	—	2	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	—	4	2	no	—
98	C₂×C₄×C₁₃₇	—	4	2	no	—
99	C₂×C₄×C₁₃₇	—	4	4	no	—
100	C₂×C₄×C₁₃₇	—	4	2	no	—
101	C₂×C₄×C₁₃₇	—	4	2	no	—
102	C₂×C₄×C₁₃₇	—	4	4	no	—
103	C₂×C₄×C₁₃₇	—	4	2	no	—
104	C₂×C₄×C₁₃₇	—	4	4	no	—
105	C₂×C₄×C₁₃₇	—	4	2	no	—
106	C₂×C₄×C₁₃₇	C₄⋊Q₈	1	2	yes	—
107	C₂×C₄×C₁₃₇	C₄⋊Q₈	1	2	yes	—
108	C₂×C₄×C₁₃₇	#135	1	2	yes	—
109	C₂×C₄×C₁₃₇	#135	1	2	yes	—
110	C₂×C₄×C₁₃₇	C₂²×Q₈	1	1	yes	characteristic
111	C₂×C₄×C₁₃₇	C₂²×Q₈	1	1	yes	characteristic
112	C₂×C₄×C₁₃₇	C₂×#41	1	1	yes	characteristic
113	C₂×C₄×C₁₃₇	C₂×#41	1	1	yes	characteristic
114	C₂×C₄×C₁₃₇	H(5,𝔽₂)	1	1	yes	characteristic
115	C₂×C₄×C₁₃₇	2₋¹⁺⁴	1	1	yes	characteristic
116	C₈×C₁₃₇	—	2	2	no	—
117	C₈×C₁₃₇	—	2	4	no	—
118	C₈×C₁₃₇	#125	1	4	yes	—
119	C₂²×C₁₃₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
120	C₂²×C₁₃₇	#547	1	1	yes	characteristic
121	C₂²×C₁₃₇	#550	1	1	yes	characteristic
122	C₂²×C₁₃₇	C₂×#135	1	1	yes	characteristic
123	C₂²×C₁₃₇	C₂×#135	1	1	yes	characteristic
124	C₂²×C₁₃₇	#576	1	1	yes	characteristic
125	C₂²×C₁₃₇	#578	1	1	yes	characteristic
126	C₄×C₁₃₇	—	2	2	no	—
127	C₄×C₁₃₇	—	4	2	no	—
128	C₄×C₁₃₇	—	4	2	no	—
129	C₄×C₁₃₇	—	8	2	no	—
130	C₄×C₁₃₇	—	8	2	no	—
131	C₄×C₁₃₇	—	8	4	no	—
132	C₄×C₁₃₇	#499	1	4	yes	—
133	C₄×C₁₃₇	C₂×#125	1	1	yes	characteristic
134	C₄×C₁₃₇	#544	1	1	yes	characteristic
135	C₄×C₁₃₇	#554	1	1	yes	characteristic
136	C₄×C₁₃₇	#557	1	1	yes	characteristic
137	C₂×C₁₃₇	#2552	1	1	yes	characteristic
138	C₂×C₁₃₇	C₂×#499	1	1	yes	characteristic
139	C₂×C₁₃₇	#3164	1	1	yes	characteristic
140	C₂×C₁₃₇	#3341	1	1	yes	characteristic
141	C₂×C₁₃₇	#3342	1	1	yes	characteristic
142	C₂×C₁₃₇	#3393	1	1	yes	characteristic
143	C₂×C₁₃₇	#3396	1	1	yes	characteristic
144	#28828	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
145	C₁₃₇	#28828	1	1	yes	characteristic, direct product factor, Sylow
146	#2571	C₂×C₁₃₇	1	2	yes	—
147	#2085	C₂×C₁₃₇	1	4	yes	—
148	#1948	C₂×C₁₃₇	1	2	yes	—
149	#1942	C₂×C₁₃₇	1	2	yes	—
150	#1846	C₂×C₁₃₇	1	2	yes	—
151	#1845	C₂×C₁₃₇	1	1	yes	characteristic
152	C₄×#107	C₂×C₁₃₇	1	1	yes	characteristic
153	C₈×C₄⋊C₄	C₂×C₁₃₇	1	1	yes	characteristic
154	C₂×#107	C₂²×C₁₃₇	1	2	yes	—
155	C₂×#107	C₂²×C₁₃₇	1	1	yes	characteristic
156	C₂×#107	C₂²×C₁₃₇	1	1	yes	characteristic
157	C₂×#107	C₂²×C₁₃₇	1	2	yes	—
158	C₂×#106	C₂²×C₁₃₇	1	4	yes	—
159	C₂×C₄⋊C₈	C₂²×C₁₃₇	1	1	yes	characteristic
160	C₂×C₄×C₈	C₂²×C₁₃₇	1	2	yes	—
161	C₂×C₄×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
162	#400	—	2	4	no	—
163	#389	—	2	2	no	—
164	#389	C₂²×C₁₃₇	1	4	yes	—
165	#384	C₂²×C₁₃₇	1	2	yes	—
166	#382	—	2	2	no	—
167	#382	C₂²×C₁₃₇	1	4	yes	—
168	C₄×C₄⋊C₄	C₂²×C₁₃₇	1	2	yes	—
169	C₄×C₄⋊C₄	C₂²×C₁₃₇	1	1	yes	characteristic
170	#340	C₂²×C₁₃₇	1	2	yes	—
171	#340	C₂²×C₁₃₇	1	4	yes	—
172	#336	C₂²×C₁₃₇	1	2	yes	—
173	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
174	C₂²×C₈	C₂³×C₁₃₇	1	2	yes	—
175	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
176	C₂×C₄⋊C₄	—	2	2	no	—
177	C₂×C₄⋊C₄	—	2	2	no	—
178	C₂×C₄⋊C₄	—	2	4	no	—
179	C₂×C₄⋊C₄	D₄×C₁₃₇	1	1	yes	characteristic
180	C₂×C₄⋊C₄	D₄×C₁₃₇	1	1	yes	characteristic
181	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	2	yes	—
182	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	2	yes	—
183	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	4	yes	—
184	C₂×C₄²	—	2	2	no	—
185	C₂×C₄²	C₂³×C₁₃₇	1	2	yes	—
186	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
187	#107	—	2	2	no	—
188	#107	—	2	4	no	—
189	#107	—	2	2	no	—
190	#107	—	4	2	no	—
191	#106	—	4	4	no	—
192	C₄⋊C₈	Q₈×C₁₃₇	1	4	yes	—
193	C₄×C₈	—	2	4	no	—
194	C₄×C₈	Q₈×C₁₃₇	1	4	yes	—
195	#95	—	2	2	no	—
196	#95	—	2	4	no	—
197	#95	—	2	4	no	—
198	#95	—	2	4	no	—
199	#95	D₄×C₁₃₇	1	2	yes	—
200	C₂²×C₄	—	2	2	no	—
201	C₂²×C₄	—	2	2	no	—
202	C₂²×C₄	—	2	4	no	—
203	C₂²×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
204	C₂²×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
205	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
206	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
207	C₂²×C₄	C₂⁴×C₁₃₇	1	1	yes	characteristic
208	C₂×C₈	—	2	2	no	—
209	C₂×C₈	—	2	2	no	—
210	C₂×C₈	—	2	1	no	—
211	C₂×C₈	—	2	1	no	—
212	C₂×C₈	C₂×Q₈×C₁₃₇	1	2	yes	—
213	C₂×C₈	C₂×Q₈×C₁₃₇	1	4	yes	—
214	C₂×C₈	C₂×Q₈×C₁₃₇	1	2	yes	—
215	C₂×C₈	#41×C₁₃₇	1	2	yes	—
216	C₂×C₈	#41×C₁₃₇	1	2	yes	—
217	C₄⋊C₄	—	2	2	no	—
218	C₄⋊C₄	—	2	2	no	—
219	C₄⋊C₄	—	2	2	no	—
220	C₄⋊C₄	—	2	2	no	—
221	C₄⋊C₄	—	2	4	no	—
222	C₄⋊C₄	—	2	2	no	—
223	C₄⋊C₄	—	2	4	no	—
224	C₄⋊C₄	—	2	2	no	—
225	C₄⋊C₄	—	4	4	no	—
226	C₄⋊C₄	—	8	2	no	—
227	C₄⋊C₄	—	8	4	no	—
228	C₄²	—	2	2	no	—
229	C₄²	—	2	2	no	—
230	C₄²	—	4	4	no	—
231	C₄²	C₂×Q₈×C₁₃₇	1	2	yes	—
232	C₄²	C₂×Q₈×C₁₃₇	1	2	yes	—
233	C₂³	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	1	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	—	2	2	no	—
242	C₂×C₄	—	4	2	no	—
243	C₂×C₄	—	4	2	no	—
244	C₂×C₄	—	4	4	no	—
245	C₂×C₄	—	4	2	no	—
246	C₂×C₄	—	4	2	no	—
247	C₂×C₄	—	4	4	no	—
248	C₂×C₄	—	4	2	no	—
249	C₂×C₄	—	4	4	no	—
250	C₂×C₄	—	4	2	no	—
251	C₂×C₄	C₄⋊Q₈×C₁₃₇	1	2	yes	—
252	C₂×C₄	C₄⋊Q₈×C₁₃₇	1	2	yes	—
253	C₂×C₄	#135×C₁₃₇	1	2	yes	—
254	C₂×C₄	#135×C₁₃₇	1	2	yes	—
255	C₂×C₄	C₂²×Q₈×C₁₃₇	1	1	yes	characteristic
256	C₂×C₄	C₂²×Q₈×C₁₃₇	1	1	yes	characteristic
257	C₂×C₄	C₂×#41×C₁₃₇	1	1	yes	characteristic
258	C₂×C₄	C₂×#41×C₁₃₇	1	1	yes	characteristic
259	C₂×C₄	H(5,𝔽₂)×C₁₃₇	1	1	yes	characteristic
260	C₂×C₄	2₋¹⁺⁴×C₁₃₇	1	1	yes	characteristic
261	C₈	—	2	2	no	—
262	C₈	—	2	4	no	—
263	C₈	#125×C₁₃₇	1	4	yes	—
264	C₂²	C₂×C₄⋊Q₈×C₁₃₇	1	1	yes	characteristic
265	C₂²	#547×C₁₃₇	1	1	yes	characteristic
266	C₂²	#550×C₁₃₇	1	1	yes	characteristic
267	C₂²	C₂×#135×C₁₃₇	1	1	yes	characteristic
268	C₂²	C₂×#135×C₁₃₇	1	1	yes	characteristic
269	C₂²	#576×C₁₃₇	1	1	yes	characteristic
270	C₂²	#578×C₁₃₇	1	1	yes	characteristic
271	C₄	—	2	2	no	—
272	C₄	—	4	2	no	—
273	C₄	—	4	2	no	—
274	C₄	—	8	2	no	—
275	C₄	—	8	2	no	—
276	C₄	—	8	4	no	—
277	C₄	#499×C₁₃₇	1	4	yes	—
278	C₄	C₂×#125×C₁₃₇	1	1	yes	characteristic
279	C₄	#544×C₁₃₇	1	1	yes	characteristic
280	C₄	#554×C₁₃₇	1	1	yes	characteristic
281	C₄	#557×C₁₃₇	1	1	yes	characteristic
282	C₂	#2552×C₁₃₇	1	1	yes	characteristic
283	C₂	C₂×#499×C₁₃₇	1	1	yes	characteristic
284	C₂	#3164×C₁₃₇	1	1	yes	characteristic
285	C₂	#3341×C₁₃₇	1	1	yes	characteristic
286	C₂	#3342×C₁₃₇	1	1	yes	characteristic
287	C₂	#3393×C₁₃₇	1	1	yes	characteristic
288	C₂	#3396×C₁₃₇	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₃₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃₇
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃₇ ◁ C₄×C₄⋊C₄×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
144. #28828	C₁₃₇	direct (N × H)
145. C₁₃₇	#28828	direct (N × H)

Maximal subgroups

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

subgroup	index
#2571×C₁₃₇	2
#2085×C₁₃₇	2
#1948×C₁₃₇	2
#1942×C₁₃₇	2
#1846×C₁₃₇	2
#1845×C₁₃₇	2
C₄×#107×C₁₃₇	2
C₈×C₄⋊C₄×C₁₃₇	2
#28828	137

Subgroups — C137×#28828 #46878214

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#28828 #46878214