Subgroups — C₁₃₇×#28762 #46878148

Order 35072.

726 conjugacy classes of subgroups.

The subgroup lattice has 246 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	#3139×C₁₃₇	C₂	1	2	yes	maximal
3	#2960×C₁₃₇	C₂	1	1	yes	maximal, characteristic
4	#2047×C₁₃₇	C₂	1	4	yes	maximal
5	#1979×C₁₃₇	C₂	1	4	yes	maximal
6	#1966×C₁₃₇	C₂	1	2	yes	maximal
7	C₄×#108×C₁₃₇	C₂	1	1	yes	maximal, characteristic
8	C₂²×Q₁₆×C₁₃₇	C₂²	1	2	yes	—
9	C₂²×Q₁₆×C₁₃₇	C₂²	1	2	yes	—
10	#533×C₁₃₇	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊Q₈×C₁₃₇	C₂²	1	2	yes	—
12	#494×C₁₃₇	—	2	2	no	—
13	#494×C₁₃₇	—	2	2	no	—
14	#491×C₁₃₇	—	2	4	no	—
15	#487×C₁₃₇	—	2	4	no	—
16	#446×C₁₃₇	—	2	2	no	—
17	C₈×Q₈×C₁₃₇	—	2	2	no	—
18	C₂×#108×C₁₃₇	C₂²	1	2	yes	—
19	#424×C₁₃₇	C₂²	1	2	yes	—
20	#423×C₁₃₇	C₂²	1	1	yes	characteristic
21	#419×C₁₃₇	C₂²	1	8	yes	—
22	C₂×#101×C₁₃₇	C₂²	1	8	yes	—
23	#405×C₁₃₇	C₂²	1	2	yes	—
24	C₄×M₁₆×C₁₃₇	C₂²	1	2	yes	—
25	C₂×C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
26	#341×C₁₃₇	C₂²	1	2	yes	—
27	C₂²×Q₈×C₁₃₇	C₂³	1	4	yes	—
28	C₂×Q₁₆×C₁₃₇	—	2	4	no	—
29	C₂×Q₁₆×C₁₃₇	—	2	2	no	—
30	C₂×Q₁₆×C₁₃₇	—	2	2	no	—
31	C₂×Q₁₆×C₁₃₇	—	2	8	no	—
32	C₂×Q₁₆×C₁₃₇	—	4	4	no	—
33	C₂×M₁₆×C₁₃₇	C₂³	1	4	yes	—
34	C₂×M₁₆×C₁₃₇	C₂³	1	2	yes	—
35	C₂²×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
36	C₂²×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
37	C₄⋊Q₈×C₁₃₇	—	2	2	no	—
38	C₄⋊Q₈×C₁₃₇	—	2	2	no	—
39	C₄⋊Q₈×C₁₃₇	—	2	4	no	—
40	C₄⋊Q₈×C₁₃₇	D₄	1	2	yes	—
41	#125×C₁₃₇	D₄	1	2	yes	—
42	C₂²⋊Q₈×C₁₃₇	—	2	2	no	—
43	C₄×Q₈×C₁₃₇	—	2	2	no	—
44	#117×C₁₃₇	C₂³	1	2	yes	—
45	C₂×C₄⋊C₄×C₁₃₇	—	2	4	no	—
46	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
47	#108×C₁₃₇	—	2	4	no	—
48	C₄⋊C₈×C₁₃₇	—	2	4	no	—
49	C₄⋊C₈×C₁₃₇	—	2	2	no	—
50	#103×C₁₃₇	—	2	16	no	—
51	#101×C₁₃₇	—	4	8	no	—
52	#97×C₁₃₇	—	2	2	no	—
53	#97×C₁₃₇	—	2	2	no	—
54	C₄×C₈×C₁₃₇	—	2	2	no	—
55	C₄×C₈×C₁₃₇	—	2	2	no	—
56	C₄×C₈×C₁₃₇	D₄	1	2	yes	—
57	C₄×C₈×C₁₃₇	D₄	1	2	yes	—
58	C₂×Q₈×C₁₃₇	—	2	2	no	—
59	C₂×Q₈×C₁₃₇	—	2	4	no	—
60	C₂×Q₈×C₁₃₇	—	2	4	no	—
61	C₂×Q₈×C₁₃₇	—	2	8	no	—
62	C₂×Q₈×C₁₃₇	—	2	8	no	—
63	C₂²×C₄×C₁₃₇	—	2	4	no	—
64	C₂²×C₄×C₁₃₇	#41	1	2	yes	—
65	C₂²×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
66	Q₁₆×C₁₃₇	—	4	8	no	—
67	Q₁₆×C₁₃₇	—	4	4	no	—
68	Q₁₆×C₁₃₇	—	4	4	no	—
69	M₁₆×C₁₃₇	—	2	4	no	—
70	M₁₆×C₁₃₇	—	2	4	no	—
71	M₁₆×C₁₃₇	—	2	4	no	—
72	C₂×C₈×C₁₃₇	—	2	2	no	—
73	C₂×C₈×C₁₃₇	—	2	4	no	—
74	C₂×C₈×C₁₃₇	—	2	2	no	—
75	C₂×C₈×C₁₃₇	C₂×D₄	1	2	yes	—
76	C₂×C₈×C₁₃₇	C₂×D₄	1	1	yes	characteristic
77	C₂×C₈×C₁₃₇	C₂×D₄	1	1	yes	characteristic
78	C₂×C₈×C₁₃₇	#41	1	4	yes	—
79	C₄⋊C₄×C₁₃₇	—	2	4	no	—
80	C₄⋊C₄×C₁₃₇	—	2	2	no	—
81	C₄⋊C₄×C₁₃₇	—	4	4	no	—
82	C₄⋊C₄×C₁₃₇	—	4	4	no	—
83	C₄⋊C₄×C₁₃₇	C₂×D₄	1	4	yes	—
84	C₂²⋊C₄×C₁₃₇	—	2	2	no	—
85	C₄²×C₁₃₇	—	2	2	no	—
86	C₄²×C₁₃₇	C₂×D₄	1	1	yes	characteristic
87	C₄²×C₁₃₇	C₂×D₄	1	2	yes	—
88	C₄²×C₁₃₇	C₂×D₄	1	1	yes	characteristic
89	C₂³×C₁₃₇	C₂×#41	1	1	yes	characteristic
90	Q₈×C₁₃₇	—	4	4	no	—
91	Q₈×C₁₃₇	—	4	4	no	—
92	Q₈×C₁₃₇	—	4	4	no	—
93	Q₈×C₁₃₇	—	4	8	no	—
94	C₂×C₄×C₁₃₇	—	2	4	no	—
95	C₂×C₄×C₁₃₇	—	2	4	no	—
96	C₂×C₄×C₁₃₇	—	4	4	no	—
97	C₂×C₄×C₁₃₇	—	4	8	no	—
98	C₂×C₄×C₁₃₇	#121	1	2	yes	—
99	C₂×C₄×C₁₃₇	#121	1	2	yes	—
100	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
101	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
102	C₂×C₄×C₁₃₇	C₂×#41	1	2	yes	—
103	C₂×C₄×C₁₃₇	H(5,𝔽₂)	1	2	yes	—
104	C₈×C₁₃₇	—	2	2	no	—
105	C₈×C₁₃₇	—	4	2	no	—
106	C₈×C₁₃₇	—	4	4	no	—
107	C₈×C₁₃₇	#124	1	2	yes	—
108	C₈×C₁₃₇	#124	1	2	yes	—
109	C₂²×C₁₃₇	—	2	2	no	—
110	C₂²×C₁₃₇	C₂×#121	1	1	yes	characteristic
111	C₂²×C₁₃₇	#559	1	2	yes	—
112	C₄×C₁₃₇	—	2	2	no	—
113	C₄×C₁₃₇	—	4	4	no	—
114	C₄×C₁₃₇	—	8	4	no	—
115	C₄×C₁₃₇	#473	1	4	yes	—
116	C₄×C₁₃₇	C₂×#124	1	1	yes	characteristic
117	C₄×C₁₃₇	#535	1	1	yes	characteristic
118	C₄×C₁₃₇	#550	1	2	yes	—
119	C₂×C₁₃₇	—	2	2	no	—
120	C₂×C₁₃₇	#2548	1	1	yes	characteristic
121	C₂×C₁₃₇	C₂×#473	1	2	yes	—
122	#28762	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
123	C₁₃₇	#28762	1	1	yes	characteristic, direct product factor, Sylow
124	C₂×#494	C₂×C₁₃₇	1	1	yes	characteristic
125	#3139	C₂×C₁₃₇	1	2	yes	—
126	#2960	C₂×C₁₃₇	1	1	yes	characteristic
127	#2047	C₂×C₁₃₇	1	4	yes	—
128	#1979	C₂×C₁₃₇	1	4	yes	—
129	#1966	C₂×C₁₃₇	1	2	yes	—
130	C₄×#108	C₂×C₁₃₇	1	1	yes	characteristic
131	C₂²×Q₁₆	C₂²×C₁₃₇	1	2	yes	—
132	C₂²×Q₁₆	C₂²×C₁₃₇	1	2	yes	—
133	#533	C₂²×C₁₃₇	1	1	yes	characteristic
134	C₂×C₄⋊Q₈	C₂²×C₁₃₇	1	2	yes	—
135	#494	—	2	2	no	—
136	#494	—	2	2	no	—
137	#491	—	2	4	no	—
138	#487	—	2	4	no	—
139	#446	—	2	2	no	—
140	C₈×Q₈	—	2	2	no	—
141	C₂×#108	C₂²×C₁₃₇	1	2	yes	—
142	#424	C₂²×C₁₃₇	1	2	yes	—
143	#423	C₂²×C₁₃₇	1	1	yes	characteristic
144	#419	C₂²×C₁₃₇	1	8	yes	—
145	C₂×#101	C₂²×C₁₃₇	1	8	yes	—
146	#405	C₂²×C₁₃₇	1	2	yes	—
147	C₄×M₁₆	C₂²×C₁₃₇	1	2	yes	—
148	C₂×C₄×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
149	#341	C₂²×C₁₃₇	1	2	yes	—
150	C₂²×Q₈	C₂³×C₁₃₇	1	4	yes	—
151	C₂×Q₁₆	—	2	4	no	—
152	C₂×Q₁₆	—	2	2	no	—
153	C₂×Q₁₆	—	2	2	no	—
154	C₂×Q₁₆	—	2	8	no	—
155	C₂×Q₁₆	—	4	4	no	—
156	C₂×M₁₆	C₂³×C₁₃₇	1	4	yes	—
157	C₂×M₁₆	C₂³×C₁₃₇	1	2	yes	—
158	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
159	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
160	C₄⋊Q₈	—	2	2	no	—
161	C₄⋊Q₈	—	2	2	no	—
162	C₄⋊Q₈	—	2	4	no	—
163	C₄⋊Q₈	D₄×C₁₃₇	1	2	yes	—
164	#125	D₄×C₁₃₇	1	2	yes	—
165	C₂²⋊Q₈	—	2	2	no	—
166	C₄×Q₈	—	2	2	no	—
167	#117	C₂³×C₁₃₇	1	2	yes	—
168	C₂×C₄⋊C₄	—	2	4	no	—
169	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
170	#108	—	2	4	no	—
171	C₄⋊C₈	—	2	4	no	—
172	C₄⋊C₈	—	2	2	no	—
173	#103	—	2	16	no	—
174	#101	—	4	8	no	—
175	#97	—	2	2	no	—
176	#97	—	2	2	no	—
177	C₄×C₈	—	2	2	no	—
178	C₄×C₈	—	2	2	no	—
179	C₄×C₈	D₄×C₁₃₇	1	2	yes	—
180	C₄×C₈	D₄×C₁₃₇	1	2	yes	—
181	C₂×Q₈	—	2	2	no	—
182	C₂×Q₈	—	2	4	no	—
183	C₂×Q₈	—	2	4	no	—
184	C₂×Q₈	—	2	8	no	—
185	C₂×Q₈	—	2	8	no	—
186	C₂²×C₄	—	2	4	no	—
187	C₂²×C₄	#41×C₁₃₇	1	2	yes	—
188	C₂²×C₄	C₂⁴×C₁₃₇	1	1	yes	characteristic
189	Q₁₆	—	4	8	no	—
190	Q₁₆	—	4	4	no	—
191	Q₁₆	—	4	4	no	—
192	M₁₆	—	2	4	no	—
193	M₁₆	—	2	4	no	—
194	M₁₆	—	2	4	no	—
195	C₂×C₈	—	2	2	no	—
196	C₂×C₈	—	2	4	no	—
197	C₂×C₈	—	2	2	no	—
198	C₂×C₈	C₂×D₄×C₁₃₇	1	2	yes	—
199	C₂×C₈	C₂×D₄×C₁₃₇	1	1	yes	characteristic
200	C₂×C₈	C₂×D₄×C₁₃₇	1	1	yes	characteristic
201	C₂×C₈	#41×C₁₃₇	1	4	yes	—
202	C₄⋊C₄	—	2	4	no	—
203	C₄⋊C₄	—	2	2	no	—
204	C₄⋊C₄	—	4	4	no	—
205	C₄⋊C₄	—	4	4	no	—
206	C₄⋊C₄	C₂×D₄×C₁₃₇	1	4	yes	—
207	C₂²⋊C₄	—	2	2	no	—
208	C₄²	—	2	2	no	—
209	C₄²	C₂×D₄×C₁₃₇	1	1	yes	characteristic
210	C₄²	C₂×D₄×C₁₃₇	1	2	yes	—
211	C₄²	C₂×D₄×C₁₃₇	1	1	yes	characteristic
212	C₂³	C₂×#41×C₁₃₇	1	1	yes	characteristic
213	Q₈	—	4	4	no	—
214	Q₈	—	4	4	no	—
215	Q₈	—	4	4	no	—
216	Q₈	—	4	8	no	—
217	C₂×C₄	—	2	4	no	—
218	C₂×C₄	—	2	4	no	—
219	C₂×C₄	—	4	4	no	—
220	C₂×C₄	—	4	8	no	—
221	C₂×C₄	#121×C₁₃₇	1	2	yes	—
222	C₂×C₄	#121×C₁₃₇	1	2	yes	—
223	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
224	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
225	C₂×C₄	C₂×#41×C₁₃₇	1	2	yes	—
226	C₂×C₄	H(5,𝔽₂)×C₁₃₇	1	2	yes	—
227	C₈	—	2	2	no	—
228	C₈	—	4	2	no	—
229	C₈	—	4	4	no	—
230	C₈	#124×C₁₃₇	1	2	yes	—
231	C₈	#124×C₁₃₇	1	2	yes	—
232	C₂²	—	2	2	no	—
233	C₂²	C₂×#121×C₁₃₇	1	1	yes	characteristic
234	C₂²	#559×C₁₃₇	1	2	yes	—
235	C₄	—	2	2	no	—
236	C₄	—	4	4	no	—
237	C₄	—	8	4	no	—
238	C₄	#473×C₁₃₇	1	4	yes	—
239	C₄	C₂×#124×C₁₃₇	1	1	yes	characteristic
240	C₄	#535×C₁₃₇	1	1	yes	characteristic
241	C₄	#550×C₁₃₇	1	2	yes	—
242	C₂	—	2	2	no	—
243	C₂	#2548×C₁₃₇	1	1	yes	characteristic
244	C₂	C₂×#473×C₁₃₇	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₃₇	C₂×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃₇
prime radical	C₂³×C₁₃₇	C₂×#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃₇ ◁ C₂×C₄²×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
122. #28762	C₁₃₇	direct (N × H)
123. C₁₃₇	#28762	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
#3139×C₁₃₇	2
#2960×C₁₃₇	2
#2047×C₁₃₇	2
#1979×C₁₃₇	2
#1966×C₁₃₇	2
C₄×#108×C₁₃₇	2
#28762	137

Subgroups — C137×#28762 #46878148

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#28762 #46878148