Subgroups — C₁₃₇×#28813 #46878199

Order 35072.

822 conjugacy classes of subgroups.

The subgroup lattice has 308 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	2	yes	maximal
2	#2551×C₁₃₇	C₂	1	2	yes	maximal
3	#2057×C₁₃₇	C₂	1	4	yes	maximal
4	#1947×C₁₃₇	C₂	1	2	yes	maximal
5	#1942×C₁₃₇	C₂	1	2	yes	maximal
6	#1845×C₁₃₇	C₂	1	1	yes	maximal, characteristic
7	C₄×#107×C₁₃₇	C₂	1	1	yes	maximal, characteristic
8	C₈×C₄⋊C₄×C₁₃₇	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄⋊Q₈×C₁₃₇	—	2	2	no	—
10	C₂×C₄⋊Q₈×C₁₃₇	C₂²	1	4	yes	—
11	#500×C₁₃₇	—	2	8	no	—
12	C₂×#107×C₁₃₇	C₂²	1	4	yes	—
13	C₂×#107×C₁₃₇	C₂²	1	2	yes	—
14	C₂×#107×C₁₃₇	C₂²	1	2	yes	—
15	C₂×#107×C₁₃₇	C₂²	1	1	yes	characteristic
16	C₂×#107×C₁₃₇	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊C₈×C₁₃₇	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₈×C₁₃₇	C₂²	1	2	yes	—
19	C₂×C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
20	#395×C₁₃₇	—	2	4	no	—
21	#389×C₁₃₇	—	2	2	no	—
22	#389×C₁₃₇	C₂²	1	4	yes	—
23	#384×C₁₃₇	C₂²	1	2	yes	—
24	C₄×C₄⋊C₄×C₁₃₇	C₂²	1	2	yes	—
25	C₄×C₄⋊C₄×C₁₃₇	C₂²	1	1	yes	characteristic
26	#340×C₁₃₇	C₂²	1	2	yes	—
27	#340×C₁₃₇	C₂²	1	4	yes	—
28	#336×C₁₃₇	C₂²	1	2	yes	—
29	C₂²×Q₈×C₁₃₇	—	2	4	no	—
30	C₂²×C₈×C₁₃₇	C₂³	1	2	yes	—
31	C₂²×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
32	C₂²×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
33	C₄⋊Q₈×C₁₃₇	—	2	8	no	—
34	C₄⋊Q₈×C₁₃₇	—	4	4	no	—
35	C₄⋊Q₈×C₁₃₇	—	4	8	no	—
36	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
37	C₂×C₄⋊C₄×C₁₃₇	—	2	4	no	—
38	C₂×C₄⋊C₄×C₁₃₇	—	2	4	no	—
39	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
40	C₂×C₄⋊C₄×C₁₃₇	D₄	1	1	yes	characteristic
41	C₂×C₄⋊C₄×C₁₃₇	D₄	1	1	yes	characteristic
42	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	4	yes	—
43	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	2	yes	—
44	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	2	yes	—
45	C₂×C₄²×C₁₃₇	—	2	2	no	—
46	C₂×C₄²×C₁₃₇	C₂³	1	2	yes	—
47	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
48	#107×C₁₃₇	—	2	4	no	—
49	#107×C₁₃₇	—	2	8	no	—
50	#107×C₁₃₇	—	4	2	no	—
51	#107×C₁₃₇	D₄	1	4	yes	—
52	#107×C₁₃₇	Q₈	1	4	yes	complemented
53	C₄⋊C₈×C₁₃₇	Q₈	1	4	yes	complemented
54	C₄×C₈×C₁₃₇	—	2	4	no	—
55	C₄×C₈×C₁₃₇	Q₈	1	4	yes	complemented
56	#95×C₁₃₇	—	2	2	no	—
57	#95×C₁₃₇	—	2	4	no	—
58	#95×C₁₃₇	D₄	1	2	yes	—
59	C₂×Q₈×C₁₃₇	—	4	8	no	—
60	C₂×Q₈×C₁₃₇	—	4	8	no	—
61	C₂×Q₈×C₁₃₇	—	8	4	no	—
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	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
67	C₂²×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
68	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
70	C₂×C₈×C₁₃₇	—	2	2	no	—
71	C₂×C₈×C₁₃₇	—	2	1	no	—
72	C₂×C₈×C₁₃₇	—	2	1	no	—
73	C₂×C₈×C₁₃₇	—	2	2	no	—
74	C₂×C₈×C₁₃₇	C₂×D₄	1	2	yes	—
75	C₂×C₈×C₁₃₇	C₂×Q₈	1	2	yes	—
76	C₂×C₈×C₁₃₇	C₂×Q₈	1	4	yes	—
77	C₂×C₈×C₁₃₇	C₂×Q₈	1	2	yes	—
78	C₂×C₈×C₁₃₇	C₂×Q₈	1	2	yes	—
79	C₄⋊C₄×C₁₃₇	—	2	2	no	—
80	C₄⋊C₄×C₁₃₇	—	2	2	no	—
81	C₄⋊C₄×C₁₃₇	—	2	4	no	—
82	C₄⋊C₄×C₁₃₇	—	2	4	no	—
83	C₄⋊C₄×C₁₃₇	—	4	8	no	—
84	C₄⋊C₄×C₁₃₇	—	4	8	no	—
85	C₄⋊C₄×C₁₃₇	—	4	4	no	—
86	C₄⋊C₄×C₁₃₇	—	8	2	no	—
87	C₄⋊C₄×C₁₃₇	D₈	1	4	yes	—
88	C₄⋊C₄×C₁₃₇	Q₁₆	1	4	yes	—
89	C₄⋊C₄×C₁₃₇	C₂×D₄	1	4	yes	—
90	C₄⋊C₄×C₁₃₇	C₂×Q₈	1	4	yes	—
91	C₄²×C₁₃₇	—	2	2	no	—
92	C₄²×C₁₃₇	—	2	2	no	—
93	C₄²×C₁₃₇	—	4	4	no	—
94	C₄²×C₁₃₇	C₂×Q₈	1	2	yes	—
95	C₄²×C₁₃₇	C₂×Q₈	1	2	yes	—
96	C₂³×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
97	Q₈×C₁₃₇	—	8	16	no	—
98	C₂×C₄×C₁₃₇	—	2	2	no	—
99	C₂×C₄×C₁₃₇	—	2	2	no	—
100	C₂×C₄×C₁₃₇	—	2	1	no	—
101	C₂×C₄×C₁₃₇	—	2	2	no	—
102	C₂×C₄×C₁₃₇	—	2	1	no	—
103	C₂×C₄×C₁₃₇	—	2	2	no	—
104	C₂×C₄×C₁₃₇	—	2	4	no	—
105	C₂×C₄×C₁₃₇	—	2	4	no	—
106	C₂×C₄×C₁₃₇	—	4	2	no	—
107	C₂×C₄×C₁₃₇	—	4	2	no	—
108	C₂×C₄×C₁₃₇	—	4	2	no	—
109	C₂×C₄×C₁₃₇	—	4	2	no	—
110	C₂×C₄×C₁₃₇	—	4	4	no	—
111	C₂×C₄×C₁₃₇	—	4	4	no	—
112	C₂×C₄×C₁₃₇	—	4	4	no	—
113	C₂×C₄×C₁₃₇	C₄⋊Q₈	1	2	yes	—
114	C₂×C₄×C₁₃₇	C₄⋊Q₈	1	2	yes	—
115	C₂×C₄×C₁₃₇	C₂×D₈	1	4	yes	—
116	C₂×C₄×C₁₃₇	C₂×D₈	1	2	yes	—
117	C₂×C₄×C₁₃₇	C₂×Q₁₆	1	4	yes	—
118	C₂×C₄×C₁₃₇	C₂×Q₁₆	1	2	yes	—
119	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
120	C₂×C₄×C₁₃₇	C₂²×Q₈	1	1	yes	characteristic
121	C₂×C₄×C₁₃₇	C₂²×Q₈	1	1	yes	characteristic
122	C₂×C₄×C₁₃₇	C₂²×Q₈	1	1	yes	characteristic
123	C₂×C₄×C₁₃₇	H(5,𝔽₂)	1	1	yes	characteristic
124	C₂×C₄×C₁₃₇	2₋¹⁺⁴	1	1	yes	characteristic
125	C₈×C₁₃₇	—	2	2	no	—
126	C₈×C₁₃₇	—	2	4	no	—
127	C₈×C₁₃₇	C₄⋊Q₈	1	4	yes	complemented
128	C₂²×C₁₃₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
129	C₂²×C₁₃₇	D₄²	1	1	yes	characteristic
130	C₂²×C₁₃₇	D₄×Q₈	1	1	yes	characteristic
131	C₂²×C₁₃₇	C₂²×D₈	1	1	yes	characteristic
132	C₂²×C₁₃₇	C₂²×Q₁₆	1	1	yes	characteristic
133	C₂²×C₁₃₇	#576	1	1	yes	characteristic
134	C₂²×C₁₃₇	#578	1	1	yes	characteristic
135	C₄×C₁₃₇	—	2	2	no	—
136	C₄×C₁₃₇	—	2	4	no	—
137	C₄×C₁₃₇	—	2	4	no	—
138	C₄×C₁₃₇	—	4	4	no	—
139	C₄×C₁₃₇	—	4	4	no	—
140	C₄×C₁₃₇	—	8	4	no	—
141	C₄×C₁₃₇	#500	1	4	yes	complemented
142	C₄×C₁₃₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
143	C₄×C₁₃₇	#544	1	1	yes	characteristic
144	C₄×C₁₃₇	D₄×Q₈	1	1	yes	characteristic
145	C₄×C₁₃₇	Q₈²	1	1	yes	characteristic
146	C₂×C₁₃₇	#2551	1	1	yes	characteristic
147	C₂×C₁₃₇	C₂×#500	1	1	yes	characteristic
148	C₂×C₁₃₇	#3164	1	1	yes	characteristic
149	C₂×C₁₃₇	D₄×D₈	1	1	yes	characteristic
150	C₂×C₁₃₇	D₄×Q₁₆	1	1	yes	characteristic
151	C₂×C₁₃₇	Q₈×D₈	1	1	yes	characteristic
152	C₂×C₁₃₇	Q₈×Q₁₆	1	1	yes	characteristic
153	#28813	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
154	C₁₃₇	#28813	1	1	yes	characteristic, direct product factor, Sylow
155	C₂×#500	C₂×C₁₃₇	1	2	yes	—
156	#2551	C₂×C₁₃₇	1	2	yes	—
157	#2057	C₂×C₁₃₇	1	4	yes	—
158	#1947	C₂×C₁₃₇	1	2	yes	—
159	#1942	C₂×C₁₃₇	1	2	yes	—
160	#1845	C₂×C₁₃₇	1	1	yes	characteristic
161	C₄×#107	C₂×C₁₃₇	1	1	yes	characteristic
162	C₈×C₄⋊C₄	C₂×C₁₃₇	1	1	yes	characteristic
163	C₂×C₄⋊Q₈	—	2	2	no	—
164	C₂×C₄⋊Q₈	C₂²×C₁₃₇	1	4	yes	—
165	#500	—	2	8	no	—
166	C₂×#107	C₂²×C₁₃₇	1	4	yes	—
167	C₂×#107	C₂²×C₁₃₇	1	2	yes	—
168	C₂×#107	C₂²×C₁₃₇	1	2	yes	—
169	C₂×#107	C₂²×C₁₃₇	1	1	yes	characteristic
170	C₂×#107	C₂²×C₁₃₇	1	1	yes	characteristic
171	C₂×C₄⋊C₈	C₂²×C₁₃₇	1	1	yes	characteristic
172	C₂×C₄×C₈	C₂²×C₁₃₇	1	2	yes	—
173	C₂×C₄×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
174	#395	—	2	4	no	—
175	#389	—	2	2	no	—
176	#389	C₂²×C₁₃₇	1	4	yes	—
177	#384	C₂²×C₁₃₇	1	2	yes	—
178	C₄×C₄⋊C₄	C₂²×C₁₃₇	1	2	yes	—
179	C₄×C₄⋊C₄	C₂²×C₁₃₇	1	1	yes	characteristic
180	#340	C₂²×C₁₃₇	1	2	yes	—
181	#340	C₂²×C₁₃₇	1	4	yes	—
182	#336	C₂²×C₁₃₇	1	2	yes	—
183	C₂²×Q₈	—	2	4	no	—
184	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
185	C₂²×C₈	C₂³×C₁₃₇	1	2	yes	—
186	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
187	C₄⋊Q₈	—	2	8	no	—
188	C₄⋊Q₈	—	4	4	no	—
189	C₄⋊Q₈	—	4	8	no	—
190	C₂×C₄⋊C₄	—	2	2	no	—
191	C₂×C₄⋊C₄	—	2	4	no	—
192	C₂×C₄⋊C₄	—	2	4	no	—
193	C₂×C₄⋊C₄	—	2	2	no	—
194	C₂×C₄⋊C₄	D₄×C₁₃₇	1	1	yes	characteristic
195	C₂×C₄⋊C₄	D₄×C₁₃₇	1	1	yes	characteristic
196	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	4	yes	—
197	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	2	yes	—
198	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	2	yes	—
199	C₂×C₄²	—	2	2	no	—
200	C₂×C₄²	C₂³×C₁₃₇	1	2	yes	—
201	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
202	#107	—	2	4	no	—
203	#107	—	2	8	no	—
204	#107	—	4	2	no	—
205	#107	D₄×C₁₃₇	1	4	yes	—
206	#107	Q₈×C₁₃₇	1	4	yes	complemented
207	C₄⋊C₈	Q₈×C₁₃₇	1	4	yes	complemented
208	C₄×C₈	—	2	4	no	—
209	C₄×C₈	Q₈×C₁₃₇	1	4	yes	complemented
210	#95	—	2	2	no	—
211	#95	—	2	4	no	—
212	#95	D₄×C₁₃₇	1	2	yes	—
213	C₂×Q₈	—	4	8	no	—
214	C₂×Q₈	—	4	8	no	—
215	C₂×Q₈	—	8	4	no	—
216	C₂²×C₄	—	2	2	no	—
217	C₂²×C₄	—	2	2	no	—
218	C₂²×C₄	—	2	4	no	—
219	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
220	C₂²×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
221	C₂²×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
222	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
223	C₂²×C₄	C₂⁴×C₁₃₇	1	1	yes	characteristic
224	C₂×C₈	—	2	2	no	—
225	C₂×C₈	—	2	1	no	—
226	C₂×C₈	—	2	1	no	—
227	C₂×C₈	—	2	2	no	—
228	C₂×C₈	C₂×D₄×C₁₃₇	1	2	yes	—
229	C₂×C₈	C₂×Q₈×C₁₃₇	1	2	yes	—
230	C₂×C₈	C₂×Q₈×C₁₃₇	1	4	yes	—
231	C₂×C₈	C₂×Q₈×C₁₃₇	1	2	yes	—
232	C₂×C₈	C₂×Q₈×C₁₃₇	1	2	yes	—
233	C₄⋊C₄	—	2	2	no	—
234	C₄⋊C₄	—	2	2	no	—
235	C₄⋊C₄	—	2	4	no	—
236	C₄⋊C₄	—	2	4	no	—
237	C₄⋊C₄	—	4	8	no	—
238	C₄⋊C₄	—	4	8	no	—
239	C₄⋊C₄	—	4	4	no	—
240	C₄⋊C₄	—	8	2	no	—
241	C₄⋊C₄	D₈×C₁₃₇	1	4	yes	—
242	C₄⋊C₄	Q₁₆×C₁₃₇	1	4	yes	—
243	C₄⋊C₄	C₂×D₄×C₁₃₇	1	4	yes	—
244	C₄⋊C₄	C₂×Q₈×C₁₃₇	1	4	yes	—
245	C₄²	—	2	2	no	—
246	C₄²	—	2	2	no	—
247	C₄²	—	4	4	no	—
248	C₄²	C₂×Q₈×C₁₃₇	1	2	yes	—
249	C₄²	C₂×Q₈×C₁₃₇	1	2	yes	—
250	C₂³	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
251	Q₈	—	8	16	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	1	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	2	1	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	2	4	no	—
259	C₂×C₄	—	2	4	no	—
260	C₂×C₄	—	4	2	no	—
261	C₂×C₄	—	4	2	no	—
262	C₂×C₄	—	4	2	no	—
263	C₂×C₄	—	4	2	no	—
264	C₂×C₄	—	4	4	no	—
265	C₂×C₄	—	4	4	no	—
266	C₂×C₄	—	4	4	no	—
267	C₂×C₄	C₄⋊Q₈×C₁₃₇	1	2	yes	—
268	C₂×C₄	C₄⋊Q₈×C₁₃₇	1	2	yes	—
269	C₂×C₄	C₂×D₈×C₁₃₇	1	4	yes	—
270	C₂×C₄	C₂×D₈×C₁₃₇	1	2	yes	—
271	C₂×C₄	C₂×Q₁₆×C₁₃₇	1	4	yes	—
272	C₂×C₄	C₂×Q₁₆×C₁₃₇	1	2	yes	—
273	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
274	C₂×C₄	C₂²×Q₈×C₁₃₇	1	1	yes	characteristic
275	C₂×C₄	C₂²×Q₈×C₁₃₇	1	1	yes	characteristic
276	C₂×C₄	C₂²×Q₈×C₁₃₇	1	1	yes	characteristic
277	C₂×C₄	H(5,𝔽₂)×C₁₃₇	1	1	yes	characteristic
278	C₂×C₄	2₋¹⁺⁴×C₁₃₇	1	1	yes	characteristic
279	C₈	—	2	2	no	—
280	C₈	—	2	4	no	—
281	C₈	C₄⋊Q₈×C₁₃₇	1	4	yes	complemented
282	C₂²	C₂×C₄⋊Q₈×C₁₃₇	1	1	yes	characteristic
283	C₂²	D₄²×C₁₃₇	1	1	yes	characteristic
284	C₂²	D₄×Q₈×C₁₃₇	1	1	yes	characteristic
285	C₂²	C₂²×D₈×C₁₃₇	1	1	yes	characteristic
286	C₂²	C₂²×Q₁₆×C₁₃₇	1	1	yes	characteristic
287	C₂²	#576×C₁₃₇	1	1	yes	characteristic
288	C₂²	#578×C₁₃₇	1	1	yes	characteristic
289	C₄	—	2	2	no	—
290	C₄	—	2	4	no	—
291	C₄	—	2	4	no	—
292	C₄	—	4	4	no	—
293	C₄	—	4	4	no	—
294	C₄	—	8	4	no	—
295	C₄	#500×C₁₃₇	1	4	yes	complemented
296	C₄	C₂×C₄⋊Q₈×C₁₃₇	1	1	yes	characteristic
297	C₄	#544×C₁₃₇	1	1	yes	characteristic
298	C₄	D₄×Q₈×C₁₃₇	1	1	yes	characteristic
299	C₄	Q₈²×C₁₃₇	1	1	yes	characteristic
300	C₂	#2551×C₁₃₇	1	1	yes	characteristic
301	C₂	C₂×#500×C₁₃₇	1	1	yes	characteristic
302	C₂	#3164×C₁₃₇	1	1	yes	characteristic
303	C₂	D₄×D₈×C₁₃₇	1	1	yes	characteristic
304	C₂	D₄×Q₁₆×C₁₃₇	1	1	yes	characteristic
305	C₂	Q₈×D₈×C₁₃₇	1	1	yes	characteristic
306	C₂	Q₈×Q₁₆×C₁₃₇	1	1	yes	characteristic

	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 ◁ C₄×C₄⋊C₄×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
52. #107×C₁₃₇	Q₈	semidirect
53. C₄⋊C₈×C₁₃₇	Q₈	semidirect
55. C₄×C₈×C₁₃₇	Q₈	semidirect
127. C₈×C₁₃₇	C₄⋊Q₈	semidirect
141. C₄×C₁₃₇	#500	semidirect
153. #28813	C₁₃₇	direct (N × H)
154. C₁₃₇	#28813	direct (N × H)
206. #107	Q₈×C₁₃₇	semidirect
207. C₄⋊C₈	Q₈×C₁₃₇	semidirect
209. C₄×C₈	Q₈×C₁₃₇	semidirect
281. C₈	C₄⋊Q₈×C₁₃₇	semidirect
295. C₄	#500×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
#2551×C₁₃₇	2
#2057×C₁₃₇	2
#1947×C₁₃₇	2
#1942×C₁₃₇	2
#1845×C₁₃₇	2
C₄×#107×C₁₃₇	2
C₈×C₄⋊C₄×C₁₃₇	2
#28813	137

Subgroups — C137×#28813 #46878199

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#28813 #46878199