Subgroups — C₁₃₇×#28830 #46878216

Order 35072.

606 conjugacy classes of subgroups.

The subgroup lattice has 264 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	#1947×C₁₃₇	C₂	1	2	yes	maximal
4	#1941×C₁₃₇	C₂	1	2	yes	maximal
5	#1846×C₁₃₇	C₂	1	2	yes	maximal
6	#1844×C₁₃₇	C₂	1	1	yes	maximal, characteristic
7	C₄×#106×C₁₃₇	C₂	1	1	yes	maximal, characteristic
8	C₈×C₄⋊C₄×C₁₃₇	C₂	1	1	yes	maximal, characteristic
9	C₂×#107×C₁₃₇	C₂²	1	4	yes	—
10	C₂×#106×C₁₃₇	C₂²	1	1	yes	characteristic
11	C₂×#106×C₁₃₇	C₂²	1	1	yes	characteristic
12	C₂×#106×C₁₃₇	C₂²	1	2	yes	—
13	C₂×#106×C₁₃₇	C₂²	1	2	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	2	yes	—
35	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	2	yes	—
36	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	2	yes	—
37	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	4	yes	—
38	C₂×C₄²×C₁₃₇	—	2	2	no	—
39	C₂×C₄²×C₁₃₇	C₂³	1	2	yes	—
40	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
41	#107×C₁₃₇	—	4	4	no	—
42	#106×C₁₃₇	—	2	2	no	—
43	#106×C₁₃₇	—	2	2	no	—
44	#106×C₁₃₇	—	2	4	no	—
45	#106×C₁₃₇	—	4	2	no	—
46	C₄⋊C₈×C₁₃₇	Q₈	1	4	yes	—
47	C₄×C₈×C₁₃₇	—	2	4	no	—
48	C₄×C₈×C₁₃₇	Q₈	1	4	yes	—
49	#95×C₁₃₇	—	2	2	no	—
50	#95×C₁₃₇	—	2	4	no	—
51	#95×C₁₃₇	—	2	4	no	—
52	#95×C₁₃₇	—	2	4	no	—
53	#95×C₁₃₇	D₄	1	2	yes	—
54	C₂²×C₄×C₁₃₇	—	2	2	no	—
55	C₂²×C₄×C₁₃₇	—	2	2	no	—
56	C₂²×C₄×C₁₃₇	—	2	4	no	—
57	C₂²×C₄×C₁₃₇	C₂×D₄	1	4	yes	—
58	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
61	C₂×C₈×C₁₃₇	—	2	2	no	—
62	C₂×C₈×C₁₃₇	—	2	1	no	—
63	C₂×C₈×C₁₃₇	—	2	2	no	—
64	C₂×C₈×C₁₃₇	—	2	1	no	—
65	C₂×C₈×C₁₃₇	C₂×Q₈	1	2	yes	—
66	C₂×C₈×C₁₃₇	C₂×Q₈	1	4	yes	—
67	C₂×C₈×C₁₃₇	C₂×Q₈	1	2	yes	—
68	C₂×C₈×C₁₃₇	#41	1	2	yes	—
69	C₂×C₈×C₁₃₇	#41	1	2	yes	—
70	C₄⋊C₄×C₁₃₇	—	2	4	no	—
71	C₄⋊C₄×C₁₃₇	—	2	2	no	—
72	C₄⋊C₄×C₁₃₇	—	2	2	no	—
73	C₄⋊C₄×C₁₃₇	—	2	4	no	—
74	C₄⋊C₄×C₁₃₇	—	2	4	no	—
75	C₄⋊C₄×C₁₃₇	—	2	2	no	—
76	C₄⋊C₄×C₁₃₇	—	2	2	no	—
77	C₄⋊C₄×C₁₃₇	—	4	4	no	—
78	C₄⋊C₄×C₁₃₇	—	8	2	no	—
79	C₄⋊C₄×C₁₃₇	—	8	4	no	—
80	C₄²×C₁₃₇	—	2	2	no	—
81	C₄²×C₁₃₇	—	2	2	no	—
82	C₄²×C₁₃₇	—	4	4	no	—
83	C₄²×C₁₃₇	C₂×Q₈	1	2	yes	—
84	C₄²×C₁₃₇	C₂×Q₈	1	2	yes	—
85	C₂³×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
86	C₂×C₄×C₁₃₇	—	2	2	no	—
87	C₂×C₄×C₁₃₇	—	2	4	no	—
88	C₂×C₄×C₁₃₇	—	2	4	no	—
89	C₂×C₄×C₁₃₇	—	2	4	no	—
90	C₂×C₄×C₁₃₇	—	4	2	no	—
91	C₂×C₄×C₁₃₇	—	4	4	no	—
92	C₂×C₄×C₁₃₇	—	4	2	no	—
93	C₂×C₄×C₁₃₇	—	4	2	no	—
94	C₂×C₄×C₁₃₇	—	4	2	no	—
95	C₂×C₄×C₁₃₇	—	4	4	no	—
96	C₂×C₄×C₁₃₇	—	4	2	no	—
97	C₂×C₄×C₁₃₇	—	4	4	no	—
98	C₂×C₄×C₁₃₇	—	4	2	no	—
99	C₂×C₄×C₁₃₇	C₄⋊Q₈	1	2	yes	—
100	C₂×C₄×C₁₃₇	C₄⋊Q₈	1	2	yes	—
101	C₂×C₄×C₁₃₇	#135	1	4	yes	—
102	C₂×C₄×C₁₃₇	C₂²×Q₈	1	1	yes	characteristic
103	C₂×C₄×C₁₃₇	C₂²×Q₈	1	1	yes	characteristic
104	C₂×C₄×C₁₃₇	C₂×#41	1	1	yes	characteristic
105	C₂×C₄×C₁₃₇	C₂×#41	1	1	yes	characteristic
106	C₂×C₄×C₁₃₇	H(5,𝔽₂)	1	1	yes	characteristic
107	C₂×C₄×C₁₃₇	2₋¹⁺⁴	1	1	yes	characteristic
108	C₈×C₁₃₇	—	2	2	no	—
109	C₈×C₁₃₇	—	2	4	no	—
110	C₈×C₁₃₇	#125	1	4	yes	—
111	C₂²×C₁₃₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
112	C₂²×C₁₃₇	#547	1	1	yes	characteristic
113	C₂²×C₁₃₇	#550	1	1	yes	characteristic
114	C₂²×C₁₃₇	C₂×#135	1	2	yes	—
115	C₂²×C₁₃₇	#577	1	2	yes	—
116	C₄×C₁₃₇	—	2	2	no	—
117	C₄×C₁₃₇	—	4	4	no	—
118	C₄×C₁₃₇	—	8	2	no	—
119	C₄×C₁₃₇	—	8	2	no	—
120	C₄×C₁₃₇	—	8	4	no	—
121	C₄×C₁₃₇	#499	1	4	yes	—
122	C₄×C₁₃₇	C₂×#125	1	1	yes	characteristic
123	C₄×C₁₃₇	#544	1	1	yes	characteristic
124	C₄×C₁₃₇	#554	1	1	yes	characteristic
125	C₄×C₁₃₇	#557	1	1	yes	characteristic
126	C₂×C₁₃₇	#2552	1	1	yes	characteristic
127	C₂×C₁₃₇	C₂×#499	1	1	yes	characteristic
128	C₂×C₁₃₇	#3163	1	1	yes	characteristic
129	C₂×C₁₃₇	#3340	1	2	yes	—
130	C₂×C₁₃₇	#3394	1	2	yes	—
131	#28830	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
132	C₁₃₇	#28830	1	1	yes	characteristic, direct product factor, Sylow
133	#2571	C₂×C₁₃₇	1	2	yes	—
134	#2085	C₂×C₁₃₇	1	4	yes	—
135	#1947	C₂×C₁₃₇	1	2	yes	—
136	#1941	C₂×C₁₃₇	1	2	yes	—
137	#1846	C₂×C₁₃₇	1	2	yes	—
138	#1844	C₂×C₁₃₇	1	1	yes	characteristic
139	C₄×#106	C₂×C₁₃₇	1	1	yes	characteristic
140	C₈×C₄⋊C₄	C₂×C₁₃₇	1	1	yes	characteristic
141	C₂×#107	C₂²×C₁₃₇	1	4	yes	—
142	C₂×#106	C₂²×C₁₃₇	1	2	yes	—
143	C₂×#106	C₂²×C₁₃₇	1	1	yes	characteristic
144	C₂×#106	C₂²×C₁₃₇	1	1	yes	characteristic
145	C₂×#106	C₂²×C₁₃₇	1	2	yes	—
146	C₂×C₄⋊C₈	C₂²×C₁₃₇	1	1	yes	characteristic
147	C₂×C₄×C₈	C₂²×C₁₃₇	1	2	yes	—
148	C₂×C₄×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
149	#400	—	2	4	no	—
150	#389	—	2	2	no	—
151	#389	C₂²×C₁₃₇	1	4	yes	—
152	#384	C₂²×C₁₃₇	1	2	yes	—
153	#382	—	2	2	no	—
154	#382	C₂²×C₁₃₇	1	4	yes	—
155	C₄×C₄⋊C₄	C₂²×C₁₃₇	1	2	yes	—
156	C₄×C₄⋊C₄	C₂²×C₁₃₇	1	1	yes	characteristic
157	#340	C₂²×C₁₃₇	1	2	yes	—
158	#340	C₂²×C₁₃₇	1	4	yes	—
159	#336	C₂²×C₁₃₇	1	2	yes	—
160	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
161	C₂²×C₈	C₂³×C₁₃₇	1	2	yes	—
162	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
163	C₂×C₄⋊C₄	—	2	2	no	—
164	C₂×C₄⋊C₄	—	2	2	no	—
165	C₂×C₄⋊C₄	—	2	4	no	—
166	C₂×C₄⋊C₄	D₄×C₁₃₇	1	2	yes	—
167	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	2	yes	—
168	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	2	yes	—
169	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	4	yes	—
170	C₂×C₄²	—	2	2	no	—
171	C₂×C₄²	C₂³×C₁₃₇	1	2	yes	—
172	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
173	#107	—	4	4	no	—
174	#106	—	2	2	no	—
175	#106	—	2	4	no	—
176	#106	—	2	2	no	—
177	#106	—	4	2	no	—
178	C₄⋊C₈	Q₈×C₁₃₇	1	4	yes	—
179	C₄×C₈	—	2	4	no	—
180	C₄×C₈	Q₈×C₁₃₇	1	4	yes	—
181	#95	—	2	2	no	—
182	#95	—	2	4	no	—
183	#95	—	2	4	no	—
184	#95	—	2	4	no	—
185	#95	D₄×C₁₃₇	1	2	yes	—
186	C₂²×C₄	—	2	2	no	—
187	C₂²×C₄	—	2	2	no	—
188	C₂²×C₄	—	2	4	no	—
189	C₂²×C₄	C₂×D₄×C₁₃₇	1	4	yes	—
190	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
191	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
192	C₂²×C₄	C₂⁴×C₁₃₇	1	1	yes	characteristic
193	C₂×C₈	—	2	2	no	—
194	C₂×C₈	—	2	1	no	—
195	C₂×C₈	—	2	2	no	—
196	C₂×C₈	—	2	1	no	—
197	C₂×C₈	C₂×Q₈×C₁₃₇	1	2	yes	—
198	C₂×C₈	C₂×Q₈×C₁₃₇	1	4	yes	—
199	C₂×C₈	C₂×Q₈×C₁₃₇	1	2	yes	—
200	C₂×C₈	#41×C₁₃₇	1	2	yes	—
201	C₂×C₈	#41×C₁₃₇	1	2	yes	—
202	C₄⋊C₄	—	2	4	no	—
203	C₄⋊C₄	—	2	2	no	—
204	C₄⋊C₄	—	2	2	no	—
205	C₄⋊C₄	—	2	4	no	—
206	C₄⋊C₄	—	2	4	no	—
207	C₄⋊C₄	—	2	2	no	—
208	C₄⋊C₄	—	2	2	no	—
209	C₄⋊C₄	—	4	4	no	—
210	C₄⋊C₄	—	8	2	no	—
211	C₄⋊C₄	—	8	4	no	—
212	C₄²	—	2	2	no	—
213	C₄²	—	2	2	no	—
214	C₄²	—	4	4	no	—
215	C₄²	C₂×Q₈×C₁₃₇	1	2	yes	—
216	C₄²	C₂×Q₈×C₁₃₇	1	2	yes	—
217	C₂³	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	2	4	no	—
220	C₂×C₄	—	2	4	no	—
221	C₂×C₄	—	2	4	no	—
222	C₂×C₄	—	4	2	no	—
223	C₂×C₄	—	4	4	no	—
224	C₂×C₄	—	4	2	no	—
225	C₂×C₄	—	4	2	no	—
226	C₂×C₄	—	4	2	no	—
227	C₂×C₄	—	4	4	no	—
228	C₂×C₄	—	4	2	no	—
229	C₂×C₄	—	4	4	no	—
230	C₂×C₄	—	4	2	no	—
231	C₂×C₄	C₄⋊Q₈×C₁₃₇	1	2	yes	—
232	C₂×C₄	C₄⋊Q₈×C₁₃₇	1	2	yes	—
233	C₂×C₄	#135×C₁₃₇	1	4	yes	—
234	C₂×C₄	C₂²×Q₈×C₁₃₇	1	1	yes	characteristic
235	C₂×C₄	C₂²×Q₈×C₁₃₇	1	1	yes	characteristic
236	C₂×C₄	C₂×#41×C₁₃₇	1	1	yes	characteristic
237	C₂×C₄	C₂×#41×C₁₃₇	1	1	yes	characteristic
238	C₂×C₄	H(5,𝔽₂)×C₁₃₇	1	1	yes	characteristic
239	C₂×C₄	2₋¹⁺⁴×C₁₃₇	1	1	yes	characteristic
240	C₈	—	2	2	no	—
241	C₈	—	2	4	no	—
242	C₈	#125×C₁₃₇	1	4	yes	—
243	C₂²	C₂×C₄⋊Q₈×C₁₃₇	1	1	yes	characteristic
244	C₂²	#547×C₁₃₇	1	1	yes	characteristic
245	C₂²	#550×C₁₃₇	1	1	yes	characteristic
246	C₂²	C₂×#135×C₁₃₇	1	2	yes	—
247	C₂²	#577×C₁₃₇	1	2	yes	—
248	C₄	—	2	2	no	—
249	C₄	—	4	4	no	—
250	C₄	—	8	2	no	—
251	C₄	—	8	2	no	—
252	C₄	—	8	4	no	—
253	C₄	#499×C₁₃₇	1	4	yes	—
254	C₄	C₂×#125×C₁₃₇	1	1	yes	characteristic
255	C₄	#544×C₁₃₇	1	1	yes	characteristic
256	C₄	#554×C₁₃₇	1	1	yes	characteristic
257	C₄	#557×C₁₃₇	1	1	yes	characteristic
258	C₂	#2552×C₁₃₇	1	1	yes	characteristic
259	C₂	C₂×#499×C₁₃₇	1	1	yes	characteristic
260	C₂	#3163×C₁₃₇	1	1	yes	characteristic
261	C₂	#3340×C₁₃₇	1	2	yes	—
262	C₂	#3394×C₁₃₇	1	2	yes	—

	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
131. #28830	C₁₃₇	direct (N × H)
132. C₁₃₇	#28830	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
#1947×C₁₃₇	2
#1941×C₁₃₇	2
#1846×C₁₃₇	2
#1844×C₁₃₇	2
C₄×#106×C₁₃₇	2
C₈×C₄⋊C₄×C₁₃₇	2
#28830	137

Subgroups — C137×#28830 #46878216

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#28830 #46878216