Subgroups — C₁₃₇×#28704 #46878090

Order 35072.

878 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#500×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₈×D₄×C₁₃₇	C₂	1	1	yes	maximal, characteristic
3	#2547×C₁₃₇	C₂	1	2	yes	maximal
4	#2075×C₁₃₇	C₂	1	4	yes	maximal, complemented
5	#1988×C₁₃₇	C₂	1	2	yes	maximal, complemented
6	#1808×C₁₃₇	C₂	1	4	yes	maximal, complemented
7	C₄×#107×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂³×C₈×C₁₃₇	C₂²	1	2	yes	—
9	C₂×C₄⋊Q₈×C₁₃₇	C₂²	1	2	yes	—
10	C₂×#124×C₁₃₇	—	2	2	no	—
11	C₂×C₄×D₄×C₁₃₇	C₂²	1	1	yes	characteristic
12	#500×C₁₃₇	—	2	4	no	—
13	C₈×D₄×C₁₃₇	—	2	4	no	—
14	C₂×#107×C₁₃₇	C₂²	1	2	yes	—
15	C₂×#107×C₁₃₇	C₂²	1	2	yes	—
16	C₂×#107×C₁₃₇	C₂²	1	2	yes	—
17	C₂×C₄⋊C₈×C₁₃₇	C₂²	1	1	yes	characteristic
18	C₂×C₂²⋊C₈×C₁₃₇	C₂²	1	2	yes	—
19	C₂×C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
20	#397×C₁₃₇	—	2	4	no	—
21	#391×C₁₃₇	—	2	2	no	—
22	#380×C₁₃₇	C₂²	1	8	yes	—
23	C₄×C₄⋊C₄×C₁₃₇	C₂²	1	2	yes	—
24	#340×C₁₃₇	C₂²	1	2	yes	—
25	#340×C₁₃₇	C₂²	1	8	yes	—
26	C₂²×Q₈×C₁₃₇	—	2	2	no	—
27	C₂²×D₄×C₁₃₇	D₄	1	1	yes	characteristic
28	C₂³×C₄×C₁₃₇	C₂³	1	2	yes	—
29	C₂²×C₈×C₁₃₇	—	2	4	no	—
30	C₂²×C₈×C₁₃₇	—	2	8	no	—
31	C₂²×C₈×C₁₃₇	C₂³	1	2	yes	—
32	C₂²×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
33	C₂²×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
34	C₄⋊Q₈×C₁₃₇	—	2	4	no	—
35	C₄⋊Q₈×C₁₃₇	—	2	4	no	—
36	#124×C₁₃₇	—	4	8	no	—
37	C₄×D₄×C₁₃₇	—	2	2	no	—
38	C₄×D₄×C₁₃₇	—	2	2	no	—
39	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
40	C₂×C₄⋊C₄×C₁₃₇	D₄	1	1	yes	characteristic
41	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	4	yes	—
42	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	4	yes	—
43	C₂×C₂²⋊C₄×C₁₃₇	—	2	8	no	—
44	C₂×C₂²⋊C₄×C₁₃₇	D₄	1	2	yes	—
45	C₂×C₄²×C₁₃₇	—	2	2	no	—
46	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
47	#107×C₁₃₇	—	2	4	no	—
48	#107×C₁₃₇	—	2	4	no	—
49	#107×C₁₃₇	—	2	4	no	—
50	C₄⋊C₈×C₁₃₇	—	2	2	no	—
51	C₂²⋊C₈×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	C₂⁴×C₁₃₇	C₂×D₄	1	2	yes	—
56	C₂×Q₈×C₁₃₇	—	4	4	no	—
57	C₂×Q₈×C₁₃₇	—	4	4	no	—
58	C₂×Q₈×C₁₃₇	—	4	4	no	—
59	C₂×D₄×C₁₃₇	—	2	2	no	—
60	C₂×D₄×C₁₃₇	—	2	2	no	—
61	C₂×D₄×C₁₃₇	D₈	1	4	yes	—
62	C₂²×C₄×C₁₃₇	—	2	2	no	—
63	C₂²×C₄×C₁₃₇	—	2	2	no	—
64	C₂²×C₄×C₁₃₇	—	2	4	no	—
65	C₂²×C₄×C₁₃₇	—	2	4	no	—
66	C₂²×C₄×C₁₃₇	—	2	4	no	—
67	C₂²×C₄×C₁₃₇	—	2	4	no	—
68	C₂²×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
69	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
72	C₂×C₈×C₁₃₇	—	2	2	no	—
73	C₂×C₈×C₁₃₇	—	2	2	no	—
74	C₂×C₈×C₁₃₇	—	2	4	no	—
75	C₂×C₈×C₁₃₇	—	4	8	no	—
76	C₂×C₈×C₁₃₇	C₂×D₄	1	2	yes	—
77	C₂×C₈×C₁₃₇	C₂×D₄	1	2	yes	—
78	C₂×C₈×C₁₃₇	#41	1	4	yes	—
79	C₄⋊C₄×C₁₃₇	—	2	2	no	—
80	C₄⋊C₄×C₁₃₇	—	2	4	no	—
81	C₄⋊C₄×C₁₃₇	—	2	4	no	—
82	C₄⋊C₄×C₁₃₇	—	2	4	no	—
83	C₄⋊C₄×C₁₃₇	—	2	4	no	—
84	C₄⋊C₄×C₁₃₇	—	4	4	no	—
85	C₂²⋊C₄×C₁₃₇	—	2	4	no	—
86	C₂²⋊C₄×C₁₃₇	—	4	16	no	—
87	C₄²×C₁₃₇	—	4	4	no	—
88	C₄²×C₁₃₇	C₂×D₄	1	2	yes	—
89	C₄²×C₁₃₇	#41	1	2	yes	—
90	C₂³×C₁₃₇	—	2	2	no	—
91	C₂³×C₁₃₇	—	2	4	no	—
92	C₂³×C₁₃₇	—	2	2	no	—
93	C₂³×C₁₃₇	—	2	4	no	—
94	C₂³×C₁₃₇	C₂×D₈	1	4	yes	—
95	C₂³×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
96	Q₈×C₁₃₇	—	8	8	no	—
97	D₄×C₁₃₇	—	2	8	no	—
98	C₂×C₄×C₁₃₇	—	2	2	no	—
99	C₂×C₄×C₁₃₇	—	2	2	no	—
100	C₂×C₄×C₁₃₇	—	2	4	no	—
101	C₂×C₄×C₁₃₇	—	4	2	no	—
102	C₂×C₄×C₁₃₇	—	4	2	no	—
103	C₂×C₄×C₁₃₇	—	4	4	no	—
104	C₂×C₄×C₁₃₇	—	4	4	no	—
105	C₂×C₄×C₁₃₇	—	4	4	no	—
106	C₂×C₄×C₁₃₇	—	4	4	no	—
107	C₂×C₄×C₁₃₇	—	4	4	no	—
108	C₂×C₄×C₁₃₇	—	4	4	no	—
109	C₂×C₄×C₁₃₇	—	4	4	no	—
110	C₂×C₄×C₁₃₇	#121	1	2	yes	—
111	C₂×C₄×C₁₃₇	#121	1	2	yes	—
112	C₂×C₄×C₁₃₇	C₂×D₈	1	2	yes	—
113	C₂×C₄×C₁₃₇	#135	1	2	yes	—
114	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
115	C₂×C₄×C₁₃₇	C₂×#41	1	2	yes	—
116	C₂×C₄×C₁₃₇	C₂×#41	1	1	yes	characteristic
117	C₂×C₄×C₁₃₇	2₋¹⁺⁴	1	2	yes	—
118	C₈×C₁₃₇	—	2	2	no	—
119	C₈×C₁₃₇	—	4	2	no	—
120	C₈×C₁₃₇	#124	1	4	yes	complemented
121	C₂²×C₁₃₇	—	2	8	no	—
122	C₂²×C₁₃₇	—	2	4	no	—
123	C₂²×C₁₃₇	—	4	4	no	—
124	C₂²×C₁₃₇	—	4	2	no	—
125	C₂²×C₁₃₇	—	4	2	no	—
126	C₂²×C₁₃₇	C₂×#121	1	1	yes	characteristic
127	C₂²×C₁₃₇	#547	1	2	yes	—
128	C₂²×C₁₃₇	C₂²×D₈	1	1	yes	characteristic
129	C₂²×C₁₃₇	C₂×#135	1	1	yes	characteristic
130	C₂²×C₁₃₇	#578	1	2	yes	—
131	C₄×C₁₃₇	—	2	2	no	—
132	C₄×C₁₃₇	—	4	2	no	—
133	C₄×C₁₃₇	—	8	4	no	—
134	C₄×C₁₃₇	—	8	4	no	—
135	C₄×C₁₃₇	#466	1	4	yes	—
136	C₄×C₁₃₇	C₂×#124	1	1	yes	characteristic
137	C₄×C₁₃₇	#536	1	1	yes	characteristic
138	C₄×C₁₃₇	#556	1	2	yes	—
139	C₂×C₁₃₇	—	4	4	no	—
140	C₂×C₁₃₇	#2547	1	1	yes	characteristic
141	C₂×C₁₃₇	C₂×#466	1	1	yes	characteristic
142	C₂×C₁₃₇	#3059	1	1	yes	characteristic
143	C₂×C₁₃₇	#3334	1	2	yes	—
144	C₂×C₁₃₇	#3342	1	2	yes	—
145	#28704	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
146	C₁₃₇	#28704	1	1	yes	characteristic, direct product factor, Sylow
147	C₂×#500	C₂×C₁₃₇	1	1	yes	characteristic, complemented
148	C₂×C₈×D₄	C₂×C₁₃₇	1	1	yes	characteristic
149	#2547	C₂×C₁₃₇	1	2	yes	—
150	#2075	C₂×C₁₃₇	1	4	yes	complemented
151	#1988	C₂×C₁₃₇	1	2	yes	complemented
152	#1808	C₂×C₁₃₇	1	4	yes	complemented
153	C₄×#107	C₂×C₁₃₇	1	1	yes	characteristic, complemented
154	C₂³×C₈	C₂²×C₁₃₇	1	2	yes	—
155	C₂×C₄⋊Q₈	C₂²×C₁₃₇	1	2	yes	—
156	C₂×#124	—	2	2	no	—
157	C₂×C₄×D₄	C₂²×C₁₃₇	1	1	yes	characteristic
158	#500	—	2	4	no	—
159	C₈×D₄	—	2	4	no	—
160	C₂×#107	C₂²×C₁₃₇	1	2	yes	—
161	C₂×#107	C₂²×C₁₃₇	1	2	yes	—
162	C₂×#107	C₂²×C₁₃₇	1	2	yes	—
163	C₂×C₄⋊C₈	C₂²×C₁₃₇	1	1	yes	characteristic
164	C₂×C₂²⋊C₈	C₂²×C₁₃₇	1	2	yes	—
165	C₂×C₄×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
166	#397	—	2	4	no	—
167	#391	—	2	2	no	—
168	#380	C₂²×C₁₃₇	1	8	yes	—
169	C₄×C₄⋊C₄	C₂²×C₁₃₇	1	2	yes	—
170	#340	C₂²×C₁₃₇	1	8	yes	—
171	#340	C₂²×C₁₃₇	1	2	yes	—
172	C₂²×Q₈	—	2	2	no	—
173	C₂²×D₄	D₄×C₁₃₇	1	1	yes	characteristic
174	C₂³×C₄	C₂³×C₁₃₇	1	2	yes	—
175	C₂²×C₈	—	2	4	no	—
176	C₂²×C₈	—	2	8	no	—
177	C₂²×C₈	C₂³×C₁₃₇	1	2	yes	—
178	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
179	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
180	C₄⋊Q₈	—	2	4	no	—
181	C₄⋊Q₈	—	2	4	no	—
182	#124	—	4	8	no	—
183	C₄×D₄	—	2	2	no	—
184	C₄×D₄	—	2	2	no	—
185	C₂×C₄⋊C₄	—	2	2	no	—
186	C₂×C₄⋊C₄	D₄×C₁₃₇	1	1	yes	characteristic
187	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	4	yes	—
188	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	4	yes	—
189	C₂×C₂²⋊C₄	—	2	8	no	—
190	C₂×C₂²⋊C₄	D₄×C₁₃₇	1	2	yes	—
191	C₂×C₄²	—	2	2	no	—
192	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
193	#107	—	2	4	no	—
194	#107	—	2	4	no	—
195	#107	—	2	4	no	—
196	C₄⋊C₈	—	2	2	no	—
197	C₂²⋊C₈	—	2	4	no	—
198	C₄×C₈	D₄×C₁₃₇	1	4	yes	—
199	#95	—	2	2	no	—
200	#95	—	2	8	no	—
201	C₂⁴	C₂×D₄×C₁₃₇	1	2	yes	—
202	C₂×Q₈	—	4	4	no	—
203	C₂×Q₈	—	4	4	no	—
204	C₂×Q₈	—	4	4	no	—
205	C₂×D₄	—	2	2	no	—
206	C₂×D₄	—	2	2	no	—
207	C₂×D₄	D₈×C₁₃₇	1	4	yes	—
208	C₂²×C₄	—	2	2	no	—
209	C₂²×C₄	—	2	2	no	—
210	C₂²×C₄	—	2	4	no	—
211	C₂²×C₄	—	2	4	no	—
212	C₂²×C₄	—	2	4	no	—
213	C₂²×C₄	—	2	4	no	—
214	C₂²×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
215	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
216	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
217	C₂²×C₄	C₂⁴×C₁₃₇	1	1	yes	characteristic
218	C₂×C₈	—	2	2	no	—
219	C₂×C₈	—	2	2	no	—
220	C₂×C₈	—	2	4	no	—
221	C₂×C₈	—	4	8	no	—
222	C₂×C₈	C₂×D₄×C₁₃₇	1	2	yes	—
223	C₂×C₈	C₂×D₄×C₁₃₇	1	2	yes	—
224	C₂×C₈	#41×C₁₃₇	1	4	yes	—
225	C₄⋊C₄	—	2	2	no	—
226	C₄⋊C₄	—	2	4	no	—
227	C₄⋊C₄	—	2	4	no	—
228	C₄⋊C₄	—	2	4	no	—
229	C₄⋊C₄	—	2	4	no	—
230	C₄⋊C₄	—	4	4	no	—
231	C₂²⋊C₄	—	2	4	no	—
232	C₂²⋊C₄	—	4	16	no	—
233	C₄²	—	4	4	no	—
234	C₄²	C₂×D₄×C₁₃₇	1	2	yes	—
235	C₄²	#41×C₁₃₇	1	2	yes	—
236	C₂³	—	2	2	no	—
237	C₂³	—	2	4	no	—
238	C₂³	—	2	2	no	—
239	C₂³	—	2	4	no	—
240	C₂³	C₂×D₈×C₁₃₇	1	4	yes	—
241	C₂³	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
242	Q₈	—	8	8	no	—
243	D₄	—	2	8	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	4	no	—
247	C₂×C₄	—	4	2	no	—
248	C₂×C₄	—	4	2	no	—
249	C₂×C₄	—	4	4	no	—
250	C₂×C₄	—	4	4	no	—
251	C₂×C₄	—	4	4	no	—
252	C₂×C₄	—	4	4	no	—
253	C₂×C₄	—	4	4	no	—
254	C₂×C₄	—	4	4	no	—
255	C₂×C₄	—	4	4	no	—
256	C₂×C₄	#121×C₁₃₇	1	2	yes	—
257	C₂×C₄	#121×C₁₃₇	1	2	yes	—
258	C₂×C₄	C₂×D₈×C₁₃₇	1	2	yes	—
259	C₂×C₄	#135×C₁₃₇	1	2	yes	—
260	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
261	C₂×C₄	C₂×#41×C₁₃₇	1	2	yes	—
262	C₂×C₄	C₂×#41×C₁₃₇	1	1	yes	characteristic
263	C₂×C₄	2₋¹⁺⁴×C₁₃₇	1	2	yes	—
264	C₈	—	2	2	no	—
265	C₈	—	4	2	no	—
266	C₈	#124×C₁₃₇	1	4	yes	complemented
267	C₂²	—	2	4	no	—
268	C₂²	—	2	8	no	—
269	C₂²	—	4	2	no	—
270	C₂²	—	4	2	no	—
271	C₂²	—	4	4	no	—
272	C₂²	C₂×#121×C₁₃₇	1	1	yes	characteristic
273	C₂²	#547×C₁₃₇	1	2	yes	—
274	C₂²	C₂²×D₈×C₁₃₇	1	1	yes	characteristic
275	C₂²	C₂×#135×C₁₃₇	1	1	yes	characteristic
276	C₂²	#578×C₁₃₇	1	2	yes	—
277	C₄	—	2	2	no	—
278	C₄	—	4	2	no	—
279	C₄	—	8	4	no	—
280	C₄	—	8	4	no	—
281	C₄	#466×C₁₃₇	1	4	yes	—
282	C₄	C₂×#124×C₁₃₇	1	1	yes	characteristic
283	C₄	#536×C₁₃₇	1	1	yes	characteristic
284	C₄	#556×C₁₃₇	1	2	yes	—
285	C₂	—	4	4	no	—
286	C₂	#2547×C₁₃₇	1	1	yes	characteristic
287	C₂	C₂×#466×C₁₃₇	1	1	yes	characteristic
288	C₂	#3059×C₁₃₇	1	1	yes	characteristic
289	C₂	#3334×C₁₃₇	1	2	yes	—
290	C₂	#3342×C₁₃₇	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₃₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃₇
prime radical	C₂²×D₄×C₁₃₇	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃₇ ◁ C₂×C₄×D₄×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#500×C₁₃₇	C₂	semidirect
4. #2075×C₁₃₇	C₂	semidirect
5. #1988×C₁₃₇	C₂	semidirect
6. #1808×C₁₃₇	C₂	semidirect
7. C₄×#107×C₁₃₇	C₂	semidirect
120. C₈×C₁₃₇	#124	semidirect
145. #28704	C₁₃₇	direct (N × H)
146. C₁₃₇	#28704	direct (N × H)
147. C₂×#500	C₂×C₁₃₇	semidirect
150. #2075	C₂×C₁₃₇	semidirect
151. #1988	C₂×C₁₃₇	semidirect
152. #1808	C₂×C₁₃₇	semidirect
153. C₄×#107	C₂×C₁₃₇	semidirect
266. C₈	#124×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#500×C₁₃₇	2
C₂×C₈×D₄×C₁₃₇	2
#2547×C₁₃₇	2
#2075×C₁₃₇	2
#1988×C₁₃₇	2
#1808×C₁₃₇	2
C₄×#107×C₁₃₇	2
#28704	137

Subgroups — C137×#28704 #46878090

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#28704 #46878090