Subgroups — C₁₃₇×#28744 #46878130

Order 35072.

794 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#494×C₁₃₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#446×C₁₃₇	C₂	1	1	yes	maximal, characteristic
3	#2568×C₁₃₇	C₂	1	2	yes	maximal
4	#2056×C₁₃₇	C₂	1	4	yes	maximal
5	#1981×C₁₃₇	C₂	1	2	yes	maximal
6	#1971×C₁₃₇	C₂	1	4	yes	maximal
7	C₄×#106×C₁₃₇	C₂	1	1	yes	maximal, characteristic
8	C₂²×Q₁₆×C₁₃₇	C₂²	1	2	yes	—
9	C₂²×Q₁₆×C₁₃₇	C₂²	1	2	yes	—
10	C₂×C₄⋊Q₈×C₁₃₇	C₂²	1	2	yes	—
11	C₂×C₄×Q₈×C₁₃₇	C₂²	1	1	yes	characteristic
12	#494×C₁₃₇	—	2	4	no	—
13	#446×C₁₃₇	—	2	4	no	—
14	C₂×#106×C₁₃₇	C₂²	1	2	yes	—
15	C₂×C₄⋊C₈×C₁₃₇	C₂²	1	2	yes	—
16	C₂×C₄⋊C₈×C₁₃₇	C₂²	1	1	yes	characteristic
17	C₂×#103×C₁₃₇	C₂²	1	8	yes	—
18	C₂×#97×C₁₃₇	C₂²	1	2	yes	—
19	C₂×C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
20	#400×C₁₃₇	—	2	4	no	—
21	#391×C₁₃₇	—	2	2	no	—
22	#391×C₁₃₇	C₂²	1	8	yes	—
23	#382×C₁₃₇	—	2	2	no	—
24	C₄×C₄⋊C₄×C₁₃₇	C₂²	1	2	yes	—
25	#340×C₁₃₇	C₂²	1	2	yes	—
26	C₂²×Q₈×C₁₃₇	D₄	1	1	yes	characteristic
27	C₂²×Q₈×C₁₃₇	C₂³	1	4	yes	—
28	C₂×Q₁₆×C₁₃₇	—	2	4	no	—
29	C₂×Q₁₆×C₁₃₇	—	2	4	no	—
30	C₂×Q₁₆×C₁₃₇	—	2	8	no	—
31	C₂×Q₁₆×C₁₃₇	—	4	4	no	—
32	C₂²×C₈×C₁₃₇	C₂³	1	2	yes	—
33	C₂²×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
34	C₂²×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
35	C₄⋊Q₈×C₁₃₇	—	2	4	no	—
36	C₄⋊Q₈×C₁₃₇	—	2	4	no	—
37	C₄×Q₈×C₁₃₇	—	2	2	no	—
38	C₄×Q₈×C₁₃₇	—	2	2	no	—
39	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
40	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
41	C₂×C₄⋊C₄×C₁₃₇	D₄	1	1	yes	characteristic
42	C₂×C₄⋊C₄×C₁₃₇	D₄	1	2	yes	—
43	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	4	yes	—
44	C₂×C₄²×C₁₃₇	—	2	2	no	—
45	C₂×C₄²×C₁₃₇	C₂³	1	2	yes	—
46	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
47	#106×C₁₃₇	—	2	4	no	—
48	C₄⋊C₈×C₁₃₇	—	2	4	no	—
49	C₄⋊C₈×C₁₃₇	—	2	2	no	—
50	#103×C₁₃₇	—	4	8	no	—
51	#97×C₁₃₇	—	2	4	no	—
52	C₄×C₈×C₁₃₇	D₄	1	4	yes	—
53	#95×C₁₃₇	—	2	2	no	—
54	#95×C₁₃₇	—	2	8	no	—
55	#95×C₁₃₇	—	2	8	no	—
56	C₂×Q₈×C₁₃₇	—	2	2	no	—
57	C₂×Q₈×C₁₃₇	—	2	4	no	—
58	C₂×Q₈×C₁₃₇	—	2	4	no	—
59	C₂×Q₈×C₁₃₇	—	2	4	no	—
60	C₂×Q₈×C₁₃₇	—	2	8	no	—
61	C₂×Q₈×C₁₃₇	—	2	8	no	—
62	C₂²×C₄×C₁₃₇	—	2	4	no	—
63	C₂²×C₄×C₁₃₇	—	2	4	no	—
64	C₂²×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
65	C₂²×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
66	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
69	Q₁₆×C₁₃₇	—	4	8	no	—
70	Q₁₆×C₁₃₇	—	4	8	no	—
71	C₂×C₈×C₁₃₇	—	2	2	no	—
72	C₂×C₈×C₁₃₇	—	2	2	no	—
73	C₂×C₈×C₁₃₇	—	2	4	no	—
74	C₂×C₈×C₁₃₇	C₂×D₄	1	2	yes	—
75	C₂×C₈×C₁₃₇	C₂×D₄	1	2	yes	—
76	C₂×C₈×C₁₃₇	#41	1	4	yes	—
77	C₄⋊C₄×C₁₃₇	—	2	2	no	—
78	C₄⋊C₄×C₁₃₇	—	2	4	no	—
79	C₄⋊C₄×C₁₃₇	—	2	4	no	—
80	C₄⋊C₄×C₁₃₇	—	2	4	no	—
81	C₄⋊C₄×C₁₃₇	—	4	4	no	—
82	C₄⋊C₄×C₁₃₇	—	4	4	no	—
83	C₄²×C₁₃₇	—	2	2	no	—
84	C₄²×C₁₃₇	—	2	2	no	—
85	C₄²×C₁₃₇	—	4	4	no	—
86	C₄²×C₁₃₇	C₂×D₄	1	2	yes	—
87	C₄²×C₁₃₇	#41	1	2	yes	—
88	C₂³×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
89	Q₈×C₁₃₇	—	4	4	no	—
90	Q₈×C₁₃₇	—	4	4	no	—
91	Q₈×C₁₃₇	—	4	4	no	—
92	Q₈×C₁₃₇	—	4	8	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	4	no	—
97	C₂×C₄×C₁₃₇	—	2	4	no	—
98	C₂×C₄×C₁₃₇	—	4	4	no	—
99	C₂×C₄×C₁₃₇	—	4	4	no	—
100	C₂×C₄×C₁₃₇	—	4	8	no	—
101	C₂×C₄×C₁₃₇	—	4	4	no	—
102	C₂×C₄×C₁₃₇	—	4	4	no	—
103	C₂×C₄×C₁₃₇	#121	1	2	yes	—
104	C₂×C₄×C₁₃₇	#121	1	2	yes	—
105	C₂×C₄×C₁₃₇	#137	1	4	yes	—
106	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
107	C₂×C₄×C₁₃₇	C₂×#41	1	2	yes	—
108	C₂×C₄×C₁₃₇	C₂×#41	1	1	yes	characteristic
109	C₂×C₄×C₁₃₇	H(5,𝔽₂)	1	2	yes	—
110	C₈×C₁₃₇	—	2	2	no	—
111	C₈×C₁₃₇	—	4	2	no	—
112	C₈×C₁₃₇	#124	1	4	yes	—
113	C₂²×C₁₃₇	C₂×#121	1	1	yes	characteristic
114	C₂²×C₁₃₇	#550	1	2	yes	—
115	C₂²×C₁₃₇	C₂×#137	1	2	yes	—
116	C₂²×C₁₃₇	#578	1	2	yes	—
117	C₄×C₁₃₇	—	2	2	no	—
118	C₄×C₁₃₇	—	4	2	no	—
119	C₄×C₁₃₇	—	4	2	no	—
120	C₄×C₁₃₇	—	8	4	no	—
121	C₄×C₁₃₇	—	8	4	no	—
122	C₄×C₁₃₇	#470	1	4	yes	—
123	C₄×C₁₃₇	C₂×#124	1	1	yes	characteristic
124	C₄×C₁₃₇	#535	1	1	yes	characteristic
125	C₄×C₁₃₇	#559	1	2	yes	—
126	C₂×C₁₃₇	#2548	1	1	yes	characteristic
127	C₂×C₁₃₇	C₂×#470	1	1	yes	characteristic
128	C₂×C₁₃₇	#3059	1	1	yes	characteristic
129	C₂×C₁₃₇	#3398	1	4	yes	—
130	#28744	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
131	C₁₃₇	#28744	1	1	yes	characteristic, direct product factor, Sylow
132	C₂×#494	C₂×C₁₃₇	1	1	yes	characteristic
133	C₂×#446	C₂×C₁₃₇	1	1	yes	characteristic
134	#2568	C₂×C₁₃₇	1	2	yes	—
135	#2056	C₂×C₁₃₇	1	4	yes	—
136	#1981	C₂×C₁₃₇	1	2	yes	—
137	#1971	C₂×C₁₃₇	1	4	yes	—
138	C₄×#106	C₂×C₁₃₇	1	1	yes	characteristic
139	C₂²×Q₁₆	C₂²×C₁₃₇	1	2	yes	—
140	C₂²×Q₁₆	C₂²×C₁₃₇	1	2	yes	—
141	C₂×C₄⋊Q₈	C₂²×C₁₃₇	1	2	yes	—
142	C₂×C₄×Q₈	C₂²×C₁₃₇	1	1	yes	characteristic
143	#494	—	2	4	no	—
144	#446	—	2	4	no	—
145	C₂×#106	C₂²×C₁₃₇	1	2	yes	—
146	C₂×C₄⋊C₈	C₂²×C₁₃₇	1	2	yes	—
147	C₂×C₄⋊C₈	C₂²×C₁₃₇	1	1	yes	characteristic
148	C₂×#103	C₂²×C₁₃₇	1	8	yes	—
149	C₂×#97	C₂²×C₁₃₇	1	2	yes	—
150	C₂×C₄×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
151	#400	—	2	4	no	—
152	#391	—	2	2	no	—
153	#391	C₂²×C₁₃₇	1	8	yes	—
154	#382	—	2	2	no	—
155	C₄×C₄⋊C₄	C₂²×C₁₃₇	1	2	yes	—
156	#340	C₂²×C₁₃₇	1	2	yes	—
157	C₂²×Q₈	D₄×C₁₃₇	1	1	yes	characteristic
158	C₂²×Q₈	C₂³×C₁₃₇	1	4	yes	—
159	C₂×Q₁₆	—	2	4	no	—
160	C₂×Q₁₆	—	2	4	no	—
161	C₂×Q₁₆	—	2	8	no	—
162	C₂×Q₁₆	—	4	4	no	—
163	C₂²×C₈	C₂³×C₁₃₇	1	2	yes	—
164	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
165	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
166	C₄⋊Q₈	—	2	4	no	—
167	C₄⋊Q₈	—	2	4	no	—
168	C₄×Q₈	—	2	2	no	—
169	C₄×Q₈	—	2	2	no	—
170	C₂×C₄⋊C₄	—	2	2	no	—
171	C₂×C₄⋊C₄	—	2	2	no	—
172	C₂×C₄⋊C₄	D₄×C₁₃₇	1	1	yes	characteristic
173	C₂×C₄⋊C₄	D₄×C₁₃₇	1	2	yes	—
174	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	4	yes	—
175	C₂×C₄²	—	2	2	no	—
176	C₂×C₄²	C₂³×C₁₃₇	1	2	yes	—
177	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
178	#106	—	2	4	no	—
179	C₄⋊C₈	—	2	4	no	—
180	C₄⋊C₈	—	2	2	no	—
181	#103	—	4	8	no	—
182	#97	—	2	4	no	—
183	C₄×C₈	D₄×C₁₃₇	1	4	yes	—
184	#95	—	2	2	no	—
185	#95	—	2	8	no	—
186	#95	—	2	8	no	—
187	C₂×Q₈	—	2	2	no	—
188	C₂×Q₈	—	2	4	no	—
189	C₂×Q₈	—	2	4	no	—
190	C₂×Q₈	—	2	4	no	—
191	C₂×Q₈	—	2	8	no	—
192	C₂×Q₈	—	2	8	no	—
193	C₂²×C₄	—	2	4	no	—
194	C₂²×C₄	—	2	4	no	—
195	C₂²×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
196	C₂²×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
197	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
198	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
199	C₂²×C₄	C₂⁴×C₁₃₇	1	1	yes	characteristic
200	Q₁₆	—	4	8	no	—
201	Q₁₆	—	4	8	no	—
202	C₂×C₈	—	2	2	no	—
203	C₂×C₈	—	2	2	no	—
204	C₂×C₈	—	2	4	no	—
205	C₂×C₈	C₂×D₄×C₁₃₇	1	2	yes	—
206	C₂×C₈	C₂×D₄×C₁₃₇	1	2	yes	—
207	C₂×C₈	#41×C₁₃₇	1	4	yes	—
208	C₄⋊C₄	—	2	2	no	—
209	C₄⋊C₄	—	2	4	no	—
210	C₄⋊C₄	—	2	4	no	—
211	C₄⋊C₄	—	2	4	no	—
212	C₄⋊C₄	—	4	4	no	—
213	C₄⋊C₄	—	4	4	no	—
214	C₄²	—	2	2	no	—
215	C₄²	—	2	2	no	—
216	C₄²	—	4	4	no	—
217	C₄²	C₂×D₄×C₁₃₇	1	2	yes	—
218	C₄²	#41×C₁₃₇	1	2	yes	—
219	C₂³	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
220	Q₈	—	4	4	no	—
221	Q₈	—	4	4	no	—
222	Q₈	—	4	4	no	—
223	Q₈	—	4	8	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	—	2	4	no	—
229	C₂×C₄	—	4	4	no	—
230	C₂×C₄	—	4	4	no	—
231	C₂×C₄	—	4	8	no	—
232	C₂×C₄	—	4	4	no	—
233	C₂×C₄	—	4	4	no	—
234	C₂×C₄	#121×C₁₃₇	1	2	yes	—
235	C₂×C₄	#121×C₁₃₇	1	2	yes	—
236	C₂×C₄	#137×C₁₃₇	1	4	yes	—
237	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
238	C₂×C₄	C₂×#41×C₁₃₇	1	2	yes	—
239	C₂×C₄	C₂×#41×C₁₃₇	1	1	yes	characteristic
240	C₂×C₄	H(5,𝔽₂)×C₁₃₇	1	2	yes	—
241	C₈	—	2	2	no	—
242	C₈	—	4	2	no	—
243	C₈	#124×C₁₃₇	1	4	yes	—
244	C₂²	C₂×#121×C₁₃₇	1	1	yes	characteristic
245	C₂²	#550×C₁₃₇	1	2	yes	—
246	C₂²	C₂×#137×C₁₃₇	1	2	yes	—
247	C₂²	#578×C₁₃₇	1	2	yes	—
248	C₄	—	2	2	no	—
249	C₄	—	4	2	no	—
250	C₄	—	4	2	no	—
251	C₄	—	8	4	no	—
252	C₄	—	8	4	no	—
253	C₄	#470×C₁₃₇	1	4	yes	—
254	C₄	C₂×#124×C₁₃₇	1	1	yes	characteristic
255	C₄	#535×C₁₃₇	1	1	yes	characteristic
256	C₄	#559×C₁₃₇	1	2	yes	—
257	C₂	#2548×C₁₃₇	1	1	yes	characteristic
258	C₂	C₂×#470×C₁₃₇	1	1	yes	characteristic
259	C₂	#3059×C₁₃₇	1	1	yes	characteristic
260	C₂	#3398×C₁₃₇	1	4	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 ◁ C₂×C₄×Q₈×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
130. #28744	C₁₃₇	direct (N × H)
131. C₁₃₇	#28744	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×#494×C₁₃₇	2
C₂×#446×C₁₃₇	2
#2568×C₁₃₇	2
#2056×C₁₃₇	2
#1981×C₁₃₇	2
#1971×C₁₃₇	2
C₄×#106×C₁₃₇	2
#28744	137

Subgroups — C137×#28744 #46878130

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#28744 #46878130