Subgroups — C₁₃₇×#28763 #46878149

Order 35072.

830 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#493×C₁₃₇	C₂	1	1	yes	maximal, characteristic
2	#3138×C₁₃₇	C₂	1	2	yes	maximal, complemented
3	#2960×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#2045×C₁₃₇	C₂	1	4	yes	maximal, complemented
5	#1978×C₁₃₇	C₂	1	4	yes	maximal, complemented
6	#1965×C₁₃₇	C₂	1	2	yes	maximal, complemented
7	C₄×#108×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₈×C₁₃₇	C₂²	1	2	yes	—
9	C₂²×D₈×C₁₃₇	C₂²	1	2	yes	—
10	#533×C₁₃₇	C₂²	1	1	yes	characteristic
11	C₂×#127×C₁₃₇	C₂²	1	2	yes	—
12	#493×C₁₃₇	—	2	2	no	—
13	#493×C₁₃₇	—	2	2	no	—
14	#490×C₁₃₇	—	2	4	no	—
15	#486×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	#418×C₁₃₇	C₂²	1	8	yes	—
22	C₂×#100×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×D₄×C₁₃₇	—	2	4	no	—
28	C₂²×D₄×C₁₃₇	C₂³	1	4	yes	—
29	C₂×D₈×C₁₃₇	—	2	4	no	—
30	C₂×D₈×C₁₃₇	—	2	2	no	—
31	C₂×D₈×C₁₃₇	—	2	2	no	—
32	C₂×D₈×C₁₃₇	—	2	8	no	—
33	C₂×D₈×C₁₃₇	—	4	4	no	—
34	C₂×M₁₆×C₁₃₇	C₂³	1	4	yes	—
35	C₂×M₁₆×C₁₃₇	C₂³	1	2	yes	—
36	C₂²×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
37	C₂²×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
38	C₄⋊Q₈×C₁₃₇	D₄	1	2	yes	—
39	#127×C₁₃₇	—	2	2	no	—
40	#127×C₁₃₇	—	2	2	no	—
41	#127×C₁₃₇	—	2	4	no	—
42	#125×C₁₃₇	D₄	1	2	yes	—
43	C₂²⋊Q₈×C₁₃₇	—	2	2	no	—
44	C₄×Q₈×C₁₃₇	—	2	2	no	—
45	#117×C₁₃₇	C₂³	1	2	yes	—
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	#102×C₁₃₇	—	2	16	no	—
51	#100×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	C₂⁴×C₁₃₇	—	2	4	no	—
59	C₂×Q₈×C₁₃₇	—	2	2	no	—
60	C₂×D₄×C₁₃₇	—	2	4	no	—
61	C₂×D₄×C₁₃₇	—	2	4	no	—
62	C₂×D₄×C₁₃₇	—	2	8	no	—
63	C₂×D₄×C₁₃₇	—	2	8	no	—
64	C₂×D₄×C₁₃₇	—	4	4	no	—
65	C₂×D₄×C₁₃₇	—	4	4	no	—
66	C₂×D₄×C₁₃₇	—	8	8	no	—
67	C₂²×C₄×C₁₃₇	#41	1	2	yes	—
68	C₂²×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
69	D₈×C₁₃₇	—	4	8	no	—
70	D₈×C₁₃₇	—	4	4	no	—
71	D₈×C₁₃₇	—	4	4	no	—
72	M₁₆×C₁₃₇	—	2	4	no	—
73	M₁₆×C₁₃₇	—	2	4	no	—
74	M₁₆×C₁₃₇	—	2	4	no	—
75	C₂×C₈×C₁₃₇	—	2	2	no	—
76	C₂×C₈×C₁₃₇	—	2	4	no	—
77	C₂×C₈×C₁₃₇	—	2	2	no	—
78	C₂×C₈×C₁₃₇	C₂×D₄	1	2	yes	—
79	C₂×C₈×C₁₃₇	C₂×D₄	1	1	yes	characteristic
80	C₂×C₈×C₁₃₇	C₂×D₄	1	1	yes	characteristic
81	C₂×C₈×C₁₃₇	#41	1	4	yes	—
82	C₄⋊C₄×C₁₃₇	—	2	4	no	—
83	C₄⋊C₄×C₁₃₇	—	2	2	no	—
84	C₄⋊C₄×C₁₃₇	C₂×D₄	1	4	yes	—
85	C₂²⋊C₄×C₁₃₇	—	2	2	no	—
86	C₄²×C₁₃₇	—	2	2	no	—
87	C₄²×C₁₃₇	C₂×D₄	1	1	yes	characteristic
88	C₄²×C₁₃₇	C₂×D₄	1	2	yes	—
89	C₄²×C₁₃₇	C₂×D₄	1	1	yes	characteristic
90	C₂³×C₁₃₇	—	4	4	no	—
91	C₂³×C₁₃₇	—	4	8	no	—
92	C₂³×C₁₃₇	—	8	8	no	—
93	C₂³×C₁₃₇	C₂×#41	1	1	yes	characteristic
94	Q₈×C₁₃₇	—	4	4	no	—
95	D₄×C₁₃₇	—	4	4	no	—
96	D₄×C₁₃₇	—	4	4	no	—
97	D₄×C₁₃₇	—	4	8	no	—
98	D₄×C₁₃₇	—	8	8	no	—
99	D₄×C₁₃₇	—	8	8	no	—
100	C₂×C₄×C₁₃₇	—	2	4	no	—
101	C₂×C₄×C₁₃₇	—	2	4	no	—
102	C₂×C₄×C₁₃₇	#121	1	2	yes	—
103	C₂×C₄×C₁₃₇	#121	1	2	yes	—
104	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
105	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
106	C₂×C₄×C₁₃₇	C₂×#41	1	2	yes	—
107	C₂×C₄×C₁₃₇	H(5,𝔽₂)	1	2	yes	—
108	C₈×C₁₃₇	—	2	2	no	—
109	C₈×C₁₃₇	—	4	2	no	—
110	C₈×C₁₃₇	—	4	4	no	—
111	C₈×C₁₃₇	#124	1	2	yes	—
112	C₈×C₁₃₇	#124	1	2	yes	—
113	C₂²×C₁₃₇	—	2	2	no	—
114	C₂²×C₁₃₇	—	8	4	no	—
115	C₂²×C₁₃₇	—	8	8	no	—
116	C₂²×C₁₃₇	—	16	8	no	—
117	C₂²×C₁₃₇	C₂×#121	1	1	yes	characteristic
118	C₂²×C₁₃₇	#559	1	2	yes	—
119	C₄×C₁₃₇	—	2	2	no	—
120	C₄×C₁₃₇	—	4	4	no	—
121	C₄×C₁₃₇	#472	1	4	yes	—
122	C₄×C₁₃₇	C₂×#124	1	1	yes	characteristic
123	C₄×C₁₃₇	#535	1	1	yes	characteristic
124	C₄×C₁₃₇	#550	1	2	yes	—
125	C₂×C₁₃₇	—	2	2	no	—
126	C₂×C₁₃₇	—	16	4	no	—
127	C₂×C₁₃₇	#2548	1	1	yes	characteristic
128	C₂×C₁₃₇	C₂×#472	1	2	yes	—
129	#28763	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
130	C₁₃₇	#28763	1	1	yes	characteristic, direct product factor, Sylow
131	C₂×#493	C₂×C₁₃₇	1	1	yes	characteristic
132	#3138	C₂×C₁₃₇	1	2	yes	complemented
133	#2960	C₂×C₁₃₇	1	1	yes	characteristic, complemented
134	#2045	C₂×C₁₃₇	1	4	yes	complemented
135	#1978	C₂×C₁₃₇	1	4	yes	complemented
136	#1965	C₂×C₁₃₇	1	2	yes	complemented
137	C₄×#108	C₂×C₁₃₇	1	1	yes	characteristic, complemented
138	C₂²×D₈	C₂²×C₁₃₇	1	2	yes	—
139	C₂²×D₈	C₂²×C₁₃₇	1	2	yes	—
140	#533	C₂²×C₁₃₇	1	1	yes	characteristic
141	C₂×#127	C₂²×C₁₃₇	1	2	yes	—
142	#493	—	2	2	no	—
143	#493	—	2	2	no	—
144	#490	—	2	4	no	—
145	#486	—	2	4	no	—
146	#446	—	2	2	no	—
147	C₈×Q₈	—	2	2	no	—
148	C₂×#108	C₂²×C₁₃₇	1	2	yes	—
149	#424	C₂²×C₁₃₇	1	2	yes	—
150	#423	C₂²×C₁₃₇	1	1	yes	characteristic
151	#418	C₂²×C₁₃₇	1	8	yes	—
152	C₂×#100	C₂²×C₁₃₇	1	8	yes	—
153	#405	C₂²×C₁₃₇	1	2	yes	—
154	C₄×M₁₆	C₂²×C₁₃₇	1	2	yes	—
155	C₂×C₄×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
156	#341	C₂²×C₁₃₇	1	2	yes	—
157	C₂²×D₄	—	2	4	no	—
158	C₂²×D₄	C₂³×C₁₃₇	1	4	yes	—
159	C₂×D₈	—	2	4	no	—
160	C₂×D₈	—	2	2	no	—
161	C₂×D₈	—	2	2	no	—
162	C₂×D₈	—	2	8	no	—
163	C₂×D₈	—	4	4	no	—
164	C₂×M₁₆	C₂³×C₁₃₇	1	4	yes	—
165	C₂×M₁₆	C₂³×C₁₃₇	1	2	yes	—
166	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
167	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
168	C₄⋊Q₈	D₄×C₁₃₇	1	2	yes	—
169	#127	—	2	2	no	—
170	#127	—	2	2	no	—
171	#127	—	2	4	no	—
172	#125	D₄×C₁₃₇	1	2	yes	—
173	C₂²⋊Q₈	—	2	2	no	—
174	C₄×Q₈	—	2	2	no	—
175	#117	C₂³×C₁₃₇	1	2	yes	—
176	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
177	#108	—	2	4	no	—
178	C₄⋊C₈	—	2	4	no	—
179	C₄⋊C₈	—	2	2	no	—
180	#102	—	2	16	no	—
181	#100	—	4	8	no	—
182	#97	—	2	2	no	—
183	#97	—	2	2	no	—
184	C₄×C₈	—	2	2	no	—
185	C₄×C₈	—	2	2	no	—
186	C₄×C₈	D₄×C₁₃₇	1	2	yes	—
187	C₄×C₈	D₄×C₁₃₇	1	2	yes	—
188	C₂⁴	—	2	4	no	—
189	C₂×Q₈	—	2	2	no	—
190	C₂×D₄	—	2	4	no	—
191	C₂×D₄	—	2	4	no	—
192	C₂×D₄	—	2	8	no	—
193	C₂×D₄	—	2	8	no	—
194	C₂×D₄	—	4	4	no	—
195	C₂×D₄	—	4	4	no	—
196	C₂×D₄	—	8	8	no	—
197	C₂²×C₄	#41×C₁₃₇	1	2	yes	—
198	C₂²×C₄	C₂⁴×C₁₃₇	1	1	yes	characteristic
199	D₈	—	4	8	no	—
200	D₈	—	4	4	no	—
201	D₈	—	4	4	no	—
202	M₁₆	—	2	4	no	—
203	M₁₆	—	2	4	no	—
204	M₁₆	—	2	4	no	—
205	C₂×C₈	—	2	2	no	—
206	C₂×C₈	—	2	4	no	—
207	C₂×C₈	—	2	2	no	—
208	C₂×C₈	C₂×D₄×C₁₃₇	1	2	yes	—
209	C₂×C₈	C₂×D₄×C₁₃₇	1	1	yes	characteristic
210	C₂×C₈	C₂×D₄×C₁₃₇	1	1	yes	characteristic
211	C₂×C₈	#41×C₁₃₇	1	4	yes	—
212	C₄⋊C₄	—	2	4	no	—
213	C₄⋊C₄	—	2	2	no	—
214	C₄⋊C₄	C₂×D₄×C₁₃₇	1	4	yes	—
215	C₂²⋊C₄	—	2	2	no	—
216	C₄²	—	2	2	no	—
217	C₄²	C₂×D₄×C₁₃₇	1	1	yes	characteristic
218	C₄²	C₂×D₄×C₁₃₇	1	2	yes	—
219	C₄²	C₂×D₄×C₁₃₇	1	1	yes	characteristic
220	C₂³	—	4	4	no	—
221	C₂³	—	4	8	no	—
222	C₂³	—	8	8	no	—
223	C₂³	C₂×#41×C₁₃₇	1	1	yes	characteristic
224	Q₈	—	4	4	no	—
225	D₄	—	4	4	no	—
226	D₄	—	4	4	no	—
227	D₄	—	4	8	no	—
228	D₄	—	8	8	no	—
229	D₄	—	8	8	no	—
230	C₂×C₄	—	2	4	no	—
231	C₂×C₄	—	2	4	no	—
232	C₂×C₄	#121×C₁₃₇	1	2	yes	—
233	C₂×C₄	#121×C₁₃₇	1	2	yes	—
234	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
235	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
236	C₂×C₄	C₂×#41×C₁₃₇	1	2	yes	—
237	C₂×C₄	H(5,𝔽₂)×C₁₃₇	1	2	yes	—
238	C₈	—	2	2	no	—
239	C₈	—	4	2	no	—
240	C₈	—	4	4	no	—
241	C₈	#124×C₁₃₇	1	2	yes	—
242	C₈	#124×C₁₃₇	1	2	yes	—
243	C₂²	—	2	2	no	—
244	C₂²	—	8	4	no	—
245	C₂²	—	8	8	no	—
246	C₂²	—	16	8	no	—
247	C₂²	C₂×#121×C₁₃₇	1	1	yes	characteristic
248	C₂²	#559×C₁₃₇	1	2	yes	—
249	C₄	—	2	2	no	—
250	C₄	—	4	4	no	—
251	C₄	#472×C₁₃₇	1	4	yes	—
252	C₄	C₂×#124×C₁₃₇	1	1	yes	characteristic
253	C₄	#535×C₁₃₇	1	1	yes	characteristic
254	C₄	#550×C₁₃₇	1	2	yes	—
255	C₂	—	2	2	no	—
256	C₂	—	16	4	no	—
257	C₂	#2548×C₁₃₇	1	1	yes	characteristic
258	C₂	C₂×#472×C₁₃₇	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₃₇	C₂×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃₇
prime radical	C₂×#493×C₁₃₇	C₂
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
2. #3138×C₁₃₇	C₂	semidirect
3. #2960×C₁₃₇	C₂	semidirect
4. #2045×C₁₃₇	C₂	semidirect
5. #1978×C₁₃₇	C₂	semidirect
6. #1965×C₁₃₇	C₂	semidirect
7. C₄×#108×C₁₃₇	C₂	semidirect
129. #28763	C₁₃₇	direct (N × H)
130. C₁₃₇	#28763	direct (N × H)
132. #3138	C₂×C₁₃₇	semidirect
133. #2960	C₂×C₁₃₇	semidirect
134. #2045	C₂×C₁₃₇	semidirect
135. #1978	C₂×C₁₃₇	semidirect
136. #1965	C₂×C₁₃₇	semidirect
137. C₄×#108	C₂×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#493×C₁₃₇	2
#3138×C₁₃₇	2
#2960×C₁₃₇	2
#2045×C₁₃₇	2
#1978×C₁₃₇	2
#1965×C₁₃₇	2
C₄×#108×C₁₃₇	2
#28763	137

Subgroups — C137×#28763 #46878149

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#28763 #46878149