Subgroups — C₁₃₇×#28755 #46878141

Order 35072.

818 conjugacy classes of subgroups.

The subgroup lattice has 266 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#491×C₁₃₇	C₂	1	2	yes	maximal
2	#2868×C₁₃₇	C₂	1	1	yes	maximal, characteristic
3	C₄×#125×C₁₃₇	C₂	1	1	yes	maximal, characteristic
4	#2043×C₁₃₇	C₂	1	4	yes	maximal
5	#1981×C₁₃₇	C₂	1	4	yes	maximal
6	#1960×C₁₃₇	C₂	1	2	yes	maximal
7	#1767×C₁₃₇	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄⋊Q₈×C₁₃₇	C₂²	1	2	yes	—
9	C₂×C₄⋊Q₈×C₁₃₇	C₂²	1	2	yes	—
10	C₂×C₄⋊Q₈×C₁₃₇	C₂²	1	2	yes	—
11	C₂×#125×C₁₃₇	C₂²	1	1	yes	characteristic
12	#491×C₁₃₇	—	2	8	no	—
13	C₂×C₄⋊C₈×C₁₃₇	C₂²	1	2	yes	—
14	C₂×#103×C₁₃₇	C₂²	1	8	yes	—
15	C₂×#103×C₁₃₇	C₂²	1	8	yes	—
16	C₂×#97×C₁₃₇	C₂²	1	2	yes	—
17	#389×C₁₃₇	C₂²	1	2	yes	—
18	#387×C₁₃₇	—	2	2	no	—
19	#384×C₁₃₇	—	2	2	no	—
20	#384×C₁₃₇	—	2	2	no	—
21	#382×C₁₃₇	C₂²	1	1	yes	characteristic
22	C₄×C₄⋊C₄×C₁₃₇	—	2	2	no	—
23	C₄×C₄⋊C₄×C₁₃₇	C₂²	1	2	yes	—
24	C₄³×C₁₃₇	C₂²	1	1	yes	characteristic
25	#336×C₁₃₇	C₂²	1	2	yes	—
26	C₂²×Q₈×C₁₃₇	C₂³	1	4	yes	—
27	C₂²×C₈×C₁₃₇	C₂³	1	4	yes	—
28	C₄⋊Q₈×C₁₃₇	—	2	4	no	—
29	C₄⋊Q₈×C₁₃₇	—	2	4	no	—
30	C₄⋊Q₈×C₁₃₇	—	2	8	no	—
31	C₄⋊Q₈×C₁₃₇	—	4	4	no	—
32	#125×C₁₃₇	—	2	2	no	—
33	#125×C₁₃₇	D₄	1	4	yes	—
34	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
35	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
36	C₂×C₄⋊C₄×C₁₃₇	—	2	4	no	—
37	C₂×C₄⋊C₄×C₁₃₇	—	2	4	no	—
38	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
39	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
40	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	2	yes	—
41	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	2	yes	—
42	C₂×C₄²×C₁₃₇	—	2	2	no	—
43	C₂×C₄²×C₁₃₇	D₄	1	2	yes	—
44	C₂×C₄²×C₁₃₇	D₄	1	2	yes	—
45	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
46	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
47	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
48	C₄⋊C₈×C₁₃₇	—	2	4	no	—
49	#103×C₁₃₇	—	2	16	no	—
50	#103×C₁₃₇	—	4	8	no	—
51	#97×C₁₃₇	—	2	4	no	—
52	#95×C₁₃₇	—	2	2	no	—
53	#95×C₁₃₇	—	2	2	no	—
54	#95×C₁₃₇	—	2	2	no	—
55	C₂×Q₈×C₁₃₇	—	2	4	no	—
56	C₂×Q₈×C₁₃₇	—	2	4	no	—
57	C₂×Q₈×C₁₃₇	—	2	8	no	—
58	C₂×Q₈×C₁₃₇	—	2	8	no	—
59	C₂²×C₄×C₁₃₇	—	2	2	no	—
60	C₂²×C₄×C₁₃₇	—	2	2	no	—
61	C₂²×C₄×C₁₃₇	—	2	4	no	—
62	C₂²×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
63	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
65	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
67	C₂×C₈×C₁₃₇	—	2	4	no	—
68	C₂×C₈×C₁₃₇	—	2	4	no	—
69	C₂×C₈×C₁₃₇	—	2	4	no	—
70	C₄⋊C₄×C₁₃₇	—	2	8	no	—
71	C₄⋊C₄×C₁₃₇	—	2	2	no	—
72	C₄⋊C₄×C₁₃₇	—	2	4	no	—
73	C₄⋊C₄×C₁₃₇	—	4	8	no	—
74	C₄⋊C₄×C₁₃₇	—	4	8	no	—
75	C₄⋊C₄×C₁₃₇	—	4	4	no	—
76	C₄⋊C₄×C₁₃₇	—	4	4	no	—
77	C₄⋊C₄×C₁₃₇	—	4	4	no	—
78	C₄⋊C₄×C₁₃₇	C₂×D₄	1	4	yes	—
79	C₄²×C₁₃₇	—	2	4	no	—
80	C₄²×C₁₃₇	—	2	4	no	—
81	C₄²×C₁₃₇	—	2	2	no	—
82	C₄²×C₁₃₇	—	4	4	no	—
83	C₄²×C₁₃₇	C₂×D₄	1	2	yes	—
84	C₄²×C₁₃₇	#41	1	2	yes	—
85	C₄²×C₁₃₇	#41	1	4	yes	—
86	C₂³×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
87	Q₈×C₁₃₇	—	4	8	no	—
88	Q₈×C₁₃₇	—	4	8	no	—
89	C₂×C₄×C₁₃₇	—	2	2	no	—
90	C₂×C₄×C₁₃₇	—	2	2	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	4	no	—
95	C₂×C₄×C₁₃₇	—	4	2	no	—
96	C₂×C₄×C₁₃₇	—	4	2	no	—
97	C₂×C₄×C₁₃₇	—	4	4	no	—
98	C₂×C₄×C₁₃₇	—	4	4	no	—
99	C₂×C₄×C₁₃₇	—	4	2	no	—
100	C₂×C₄×C₁₃₇	—	4	4	no	—
101	C₂×C₄×C₁₃₇	—	4	2	no	—
102	C₂×C₄×C₁₃₇	—	4	2	no	—
103	C₂×C₄×C₁₃₇	#121	1	4	yes	—
104	C₂×C₄×C₁₃₇	#124	1	2	yes	—
105	C₂×C₄×C₁₃₇	#124	1	2	yes	—
106	C₂×C₄×C₁₃₇	#137	1	4	yes	—
107	C₂×C₄×C₁₃₇	#137	1	4	yes	—
108	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
109	C₂×C₄×C₁₃₇	C₂×#41	1	2	yes	—
110	C₂×C₄×C₁₃₇	C₂×#41	1	1	yes	characteristic
111	C₂×C₄×C₁₃₇	H(5,𝔽₂)	1	2	yes	—
112	C₈×C₁₃₇	—	4	4	no	—
113	C₂²×C₁₃₇	C₂×#121	1	1	yes	characteristic
114	C₂²×C₁₃₇	C₂×#124	1	1	yes	characteristic
115	C₂²×C₁₃₇	#535	1	1	yes	characteristic
116	C₂²×C₁₃₇	C₂×#137	1	2	yes	—
117	C₂²×C₁₃₇	C₂×#137	1	2	yes	—
118	C₄×C₁₃₇	—	2	2	no	—
119	C₄×C₁₃₇	—	2	4	no	—
120	C₄×C₁₃₇	—	2	4	no	—
121	C₄×C₁₃₇	—	4	4	no	—
122	C₄×C₁₃₇	—	8	2	no	—
123	C₄×C₁₃₇	—	8	4	no	—
124	C₄×C₁₃₇	#491	1	4	yes	complemented
125	C₄×C₁₃₇	#550	1	2	yes	—
126	C₄×C₁₃₇	#559	1	2	yes	—
127	C₂×C₁₃₇	#2548	1	1	yes	characteristic
128	C₂×C₁₃₇	#3039	1	2	yes	—
129	C₂×C₁₃₇	C₂×#491	1	1	yes	characteristic
130	C₂×C₁₃₇	#3139	1	1	yes	characteristic
131	C₂×C₁₃₇	#3218	1	2	yes	—
132	#28755	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
133	C₁₃₇	#28755	1	1	yes	characteristic, direct product factor, Sylow
134	C₂×#491	C₂×C₁₃₇	1	2	yes	—
135	#2868	C₂×C₁₃₇	1	1	yes	characteristic
136	C₄×#125	C₂×C₁₃₇	1	1	yes	characteristic
137	#2043	C₂×C₁₃₇	1	4	yes	—
138	#1981	C₂×C₁₃₇	1	4	yes	—
139	#1960	C₂×C₁₃₇	1	2	yes	—
140	#1767	C₂×C₁₃₇	1	1	yes	characteristic
141	C₂×C₄⋊Q₈	C₂²×C₁₃₇	1	2	yes	—
142	C₂×C₄⋊Q₈	C₂²×C₁₃₇	1	2	yes	—
143	C₂×C₄⋊Q₈	C₂²×C₁₃₇	1	2	yes	—
144	C₂×#125	C₂²×C₁₃₇	1	1	yes	characteristic
145	#491	—	2	8	no	—
146	C₂×C₄⋊C₈	C₂²×C₁₃₇	1	2	yes	—
147	C₂×#103	C₂²×C₁₃₇	1	8	yes	—
148	C₂×#103	C₂²×C₁₃₇	1	8	yes	—
149	C₂×#97	C₂²×C₁₃₇	1	2	yes	—
150	#389	C₂²×C₁₃₇	1	2	yes	—
151	#387	—	2	2	no	—
152	#384	—	2	2	no	—
153	#384	—	2	2	no	—
154	#382	C₂²×C₁₃₇	1	1	yes	characteristic
155	C₄×C₄⋊C₄	—	2	2	no	—
156	C₄×C₄⋊C₄	C₂²×C₁₃₇	1	2	yes	—
157	C₄³	C₂²×C₁₃₇	1	1	yes	characteristic
158	#336	C₂²×C₁₃₇	1	2	yes	—
159	C₂²×Q₈	C₂³×C₁₃₇	1	4	yes	—
160	C₂²×C₈	C₂³×C₁₃₇	1	4	yes	—
161	C₄⋊Q₈	—	2	4	no	—
162	C₄⋊Q₈	—	2	4	no	—
163	C₄⋊Q₈	—	2	8	no	—
164	C₄⋊Q₈	—	4	4	no	—
165	#125	—	2	2	no	—
166	#125	D₄×C₁₃₇	1	4	yes	—
167	C₂×C₄⋊C₄	—	2	2	no	—
168	C₂×C₄⋊C₄	—	2	4	no	—
169	C₂×C₄⋊C₄	—	2	4	no	—
170	C₂×C₄⋊C₄	—	2	2	no	—
171	C₂×C₄⋊C₄	—	2	2	no	—
172	C₂×C₄⋊C₄	—	2	2	no	—
173	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	2	yes	—
174	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	2	yes	—
175	C₂×C₄²	—	2	2	no	—
176	C₂×C₄²	D₄×C₁₃₇	1	2	yes	—
177	C₂×C₄²	D₄×C₁₃₇	1	2	yes	—
178	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
179	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
180	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
181	C₄⋊C₈	—	2	4	no	—
182	#103	—	2	16	no	—
183	#103	—	4	8	no	—
184	#97	—	2	4	no	—
185	#95	—	2	2	no	—
186	#95	—	2	2	no	—
187	#95	—	2	2	no	—
188	C₂×Q₈	—	2	4	no	—
189	C₂×Q₈	—	2	4	no	—
190	C₂×Q₈	—	2	8	no	—
191	C₂×Q₈	—	2	8	no	—
192	C₂²×C₄	—	2	2	no	—
193	C₂²×C₄	—	2	2	no	—
194	C₂²×C₄	—	2	4	no	—
195	C₂²×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
196	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
197	C₂²×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
198	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
199	C₂²×C₄	C₂⁴×C₁₃₇	1	1	yes	characteristic
200	C₂×C₈	—	2	4	no	—
201	C₂×C₈	—	2	4	no	—
202	C₂×C₈	—	2	4	no	—
203	C₄⋊C₄	—	2	8	no	—
204	C₄⋊C₄	—	2	2	no	—
205	C₄⋊C₄	—	2	4	no	—
206	C₄⋊C₄	—	4	8	no	—
207	C₄⋊C₄	—	4	8	no	—
208	C₄⋊C₄	—	4	4	no	—
209	C₄⋊C₄	—	4	4	no	—
210	C₄⋊C₄	—	4	4	no	—
211	C₄⋊C₄	C₂×D₄×C₁₃₇	1	4	yes	—
212	C₄²	—	2	4	no	—
213	C₄²	—	2	4	no	—
214	C₄²	—	2	2	no	—
215	C₄²	—	4	4	no	—
216	C₄²	C₂×D₄×C₁₃₇	1	2	yes	—
217	C₄²	#41×C₁₃₇	1	2	yes	—
218	C₄²	#41×C₁₃₇	1	4	yes	—
219	C₂³	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
220	Q₈	—	4	8	no	—
221	Q₈	—	4	8	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	4	no	—
228	C₂×C₄	—	4	2	no	—
229	C₂×C₄	—	4	2	no	—
230	C₂×C₄	—	4	4	no	—
231	C₂×C₄	—	4	4	no	—
232	C₂×C₄	—	4	2	no	—
233	C₂×C₄	—	4	4	no	—
234	C₂×C₄	—	4	2	no	—
235	C₂×C₄	—	4	2	no	—
236	C₂×C₄	#121×C₁₃₇	1	4	yes	—
237	C₂×C₄	#124×C₁₃₇	1	2	yes	—
238	C₂×C₄	#124×C₁₃₇	1	2	yes	—
239	C₂×C₄	#137×C₁₃₇	1	4	yes	—
240	C₂×C₄	#137×C₁₃₇	1	4	yes	—
241	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
242	C₂×C₄	C₂×#41×C₁₃₇	1	2	yes	—
243	C₂×C₄	C₂×#41×C₁₃₇	1	1	yes	characteristic
244	C₂×C₄	H(5,𝔽₂)×C₁₃₇	1	2	yes	—
245	C₈	—	4	4	no	—
246	C₂²	C₂×#121×C₁₃₇	1	1	yes	characteristic
247	C₂²	C₂×#124×C₁₃₇	1	1	yes	characteristic
248	C₂²	#535×C₁₃₇	1	1	yes	characteristic
249	C₂²	C₂×#137×C₁₃₇	1	2	yes	—
250	C₂²	C₂×#137×C₁₃₇	1	2	yes	—
251	C₄	—	2	4	no	—
252	C₄	—	2	2	no	—
253	C₄	—	2	4	no	—
254	C₄	—	4	4	no	—
255	C₄	—	8	4	no	—
256	C₄	—	8	2	no	—
257	C₄	#491×C₁₃₇	1	4	yes	complemented
258	C₄	#550×C₁₃₇	1	2	yes	—
259	C₄	#559×C₁₃₇	1	2	yes	—
260	C₂	#2548×C₁₃₇	1	1	yes	characteristic
261	C₂	#3039×C₁₃₇	1	2	yes	—
262	C₂	C₂×#491×C₁₃₇	1	1	yes	characteristic
263	C₂	#3139×C₁₃₇	1	1	yes	characteristic
264	C₂	#3218×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 ◁ C₄³×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
124. C₄×C₁₃₇	#491	semidirect
132. #28755	C₁₃₇	direct (N × H)
133. C₁₃₇	#28755	direct (N × H)
257. C₄	#491×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#491×C₁₃₇	2
#2868×C₁₃₇	2
C₄×#125×C₁₃₇	2
#2043×C₁₃₇	2
#1981×C₁₃₇	2
#1960×C₁₃₇	2
#1767×C₁₃₇	2
#28755	137

Subgroups — C137×#28755 #46878141

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#28755 #46878141