Subgroups — C₁₃₇×#28699 #46878085

Order 35072.

630 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#3165×C₁₃₇	C₂	1	2	yes	maximal
2	C₂×#500×C₁₃₇	C₂	1	1	yes	maximal, characteristic
3	#2960×C₁₃₇	C₂	1	1	yes	maximal, characteristic
4	#2089×C₁₃₇	C₂	1	4	yes	maximal
5	#1997×C₁₃₇	C₂	1	2	yes	maximal
6	#1952×C₁₃₇	C₂	1	4	yes	maximal
7	C₄×#108×C₁₃₇	C₂	1	1	yes	maximal, characteristic
8	#533×C₁₃₇	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊Q₈×C₁₃₇	C₂²	1	2	yes	—
10	#501×C₁₃₇	—	2	4	no	—
11	#500×C₁₃₇	—	2	2	no	—
12	#500×C₁₃₇	—	2	1	no	—
13	#500×C₁₃₇	—	2	1	no	—
14	#500×C₁₃₇	—	2	1	no	—
15	#500×C₁₃₇	—	2	1	no	—
16	#498×C₁₃₇	—	2	2	no	—
17	#446×C₁₃₇	—	2	1	no	—
18	#446×C₁₃₇	—	2	1	no	—
19	C₈×Q₈×C₁₃₇	—	2	1	no	—
20	C₈×Q₈×C₁₃₇	—	2	1	no	—
21	C₂×#108×C₁₃₇	C₂²	1	2	yes	—
22	#428×C₁₃₇	C₂²	1	8	yes	—
23	C₂×#107×C₁₃₇	C₂²	1	2	yes	—
24	C₂×#107×C₁₃₇	C₂²	1	2	yes	—
25	#424×C₁₃₇	C₂²	1	2	yes	—
26	#423×C₁₃₇	C₂²	1	1	yes	characteristic
27	#405×C₁₃₇	C₂²	1	2	yes	—
28	C₄×M₁₆×C₁₃₇	C₂²	1	2	yes	—
29	C₂×C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
30	#343×C₁₃₇	C₂²	1	8	yes	—
31	#341×C₁₃₇	C₂²	1	2	yes	—
32	C₂²×Q₈×C₁₃₇	—	2	2	no	—
33	C₂×M₁₆×C₁₃₇	C₂³	1	2	yes	—
34	C₂×M₁₆×C₁₃₇	C₂³	1	4	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	2	no	—
40	C₄⋊Q₈×C₁₃₇	—	2	2	no	—
41	C₄⋊Q₈×C₁₃₇	D₄	1	1	yes	characteristic
42	C₄⋊Q₈×C₁₃₇	D₄	1	1	yes	characteristic
43	#125×C₁₃₇	D₄	1	2	yes	—
44	C₂²⋊Q₈×C₁₃₇	—	2	1	no	—
45	C₂²⋊Q₈×C₁₃₇	—	2	1	no	—
46	C₄×Q₈×C₁₃₇	—	2	1	no	—
47	C₄×Q₈×C₁₃₇	—	2	1	no	—
48	#117×C₁₃₇	C₂³	1	2	yes	—
49	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
50	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	4	yes	—
51	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
52	#108×C₁₃₇	Q₈	1	8	yes	complemented
53	#107×C₁₃₇	—	2	2	no	—
54	#107×C₁₃₇	—	2	2	no	—
55	#107×C₁₃₇	—	2	2	no	—
56	#107×C₁₃₇	—	2	2	no	—
57	#107×C₁₃₇	—	2	8	no	—
58	#106×C₁₃₇	—	2	8	no	—
59	C₄⋊C₈×C₁₃₇	—	2	2	no	—
60	C₄⋊C₈×C₁₃₇	—	2	2	no	—
61	C₄⋊C₈×C₁₃₇	—	2	1	no	—
62	C₄⋊C₈×C₁₃₇	—	2	1	no	—
63	#97×C₁₃₇	—	2	2	no	—
64	#97×C₁₃₇	—	2	2	no	—
65	C₄×C₈×C₁₃₇	—	2	2	no	—
66	C₄×C₈×C₁₃₇	—	2	2	no	—
67	C₄×C₈×C₁₃₇	D₄	1	1	yes	characteristic
68	C₄×C₈×C₁₃₇	D₄	1	1	yes	characteristic
69	C₄×C₈×C₁₃₇	D₄	1	1	yes	characteristic
70	C₄×C₈×C₁₃₇	D₄	1	1	yes	characteristic
71	C₂×Q₈×C₁₃₇	—	2	1	no	—
72	C₂×Q₈×C₁₃₇	—	2	1	no	—
73	C₂×Q₈×C₁₃₇	—	4	2	no	—
74	C₂×Q₈×C₁₃₇	—	4	2	no	—
75	C₂×Q₈×C₁₃₇	—	8	4	no	—
76	C₂²×C₄×C₁₃₇	—	2	4	no	—
77	C₂²×C₄×C₁₃₇	#41	1	1	yes	characteristic
78	C₂²×C₄×C₁₃₇	#41	1	1	yes	characteristic
79	C₂²×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
80	M₁₆×C₁₃₇	—	2	4	no	—
81	M₁₆×C₁₃₇	—	2	4	no	—
82	M₁₆×C₁₃₇	C₂×Q₈	1	8	yes	—
83	C₂×C₈×C₁₃₇	—	2	4	no	—
84	C₂×C₈×C₁₃₇	—	2	2	no	—
85	C₂×C₈×C₁₃₇	—	2	2	no	—
86	C₂×C₈×C₁₃₇	C₂×D₄	1	1	yes	characteristic
87	C₂×C₈×C₁₃₇	C₂×D₄	1	1	yes	characteristic
88	C₂×C₈×C₁₃₇	C₂×D₄	1	1	yes	characteristic
89	C₂×C₈×C₁₃₇	C₂×D₄	1	1	yes	characteristic
90	C₂×C₈×C₁₃₇	C₂×Q₈	1	4	yes	—
91	C₄⋊C₄×C₁₃₇	—	2	2	no	—
92	C₄⋊C₄×C₁₃₇	—	2	2	no	—
93	C₄⋊C₄×C₁₃₇	—	2	1	no	—
94	C₄⋊C₄×C₁₃₇	—	2	1	no	—
95	C₄⋊C₄×C₁₃₇	—	2	4	no	—
96	C₄⋊C₄×C₁₃₇	—	2	4	no	—
97	C₄⋊C₄×C₁₃₇	—	4	2	no	—
98	C₄⋊C₄×C₁₃₇	—	4	2	no	—
99	C₄⋊C₄×C₁₃₇	C₂×D₄	1	2	yes	—
100	C₄⋊C₄×C₁₃₇	C₂×D₄	1	2	yes	—
101	C₂²⋊C₄×C₁₃₇	—	2	2	no	—
102	C₄²×C₁₃₇	—	2	2	no	—
103	C₄²×C₁₃₇	C₂×D₄	1	1	yes	characteristic
104	C₄²×C₁₃₇	C₂×D₄	1	1	yes	characteristic
105	C₄²×C₁₃₇	C₂×D₄	1	1	yes	characteristic
106	C₄²×C₁₃₇	C₂×D₄	1	1	yes	characteristic
107	C₂³×C₁₃₇	C₂×#41	1	1	yes	characteristic
108	Q₈×C₁₃₇	—	4	2	no	—
109	Q₈×C₁₃₇	—	4	2	no	—
110	Q₈×C₁₃₇	—	8	4	no	—
111	Q₈×C₁₃₇	—	8	4	no	—
112	C₂×C₄×C₁₃₇	—	2	2	no	—
113	C₂×C₄×C₁₃₇	—	2	2	no	—
114	C₂×C₄×C₁₃₇	—	2	2	no	—
115	C₂×C₄×C₁₃₇	—	2	2	no	—
116	C₂×C₄×C₁₃₇	—	4	4	no	—
117	C₂×C₄×C₁₃₇	—	8	4	no	—
118	C₂×C₄×C₁₃₇	#121	1	1	yes	characteristic
119	C₂×C₄×C₁₃₇	#121	1	1	yes	characteristic
120	C₂×C₄×C₁₃₇	#121	1	1	yes	characteristic
121	C₂×C₄×C₁₃₇	#121	1	1	yes	characteristic
122	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
123	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
124	C₂×C₄×C₁₃₇	C₂²×Q₈	1	2	yes	—
125	C₂×C₄×C₁₃₇	2₋¹⁺⁴	1	2	yes	—
126	C₈×C₁₃₇	—	2	1	no	—
127	C₈×C₁₃₇	—	2	1	no	—
128	C₈×C₁₃₇	—	2	8	no	—
129	C₈×C₁₃₇	—	4	2	no	—
130	C₈×C₁₃₇	C₄⋊Q₈	1	2	yes	—
131	C₈×C₁₃₇	C₄⋊Q₈	1	2	yes	—
132	C₂²×C₁₃₇	—	2	1	no	—
133	C₂²×C₁₃₇	—	2	1	no	—
134	C₂²×C₁₃₇	C₂×#121	1	1	yes	characteristic
135	C₂²×C₁₃₇	#557	1	2	yes	—
136	C₄×C₁₃₇	—	2	1	no	—
137	C₄×C₁₃₇	—	2	1	no	—
138	C₄×C₁₃₇	—	4	2	no	—
139	C₄×C₁₃₇	—	4	2	no	—
140	C₄×C₁₃₇	—	8	4	no	—
141	C₄×C₁₃₇	#472	1	2	yes	—
142	C₄×C₁₃₇	#473	1	2	yes	—
143	C₄×C₁₃₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
144	C₄×C₁₃₇	#536	1	1	yes	characteristic
145	C₄×C₁₃₇	D₄×Q₈	1	2	yes	—
146	C₂×C₁₃₇	—	2	2	no	—
147	C₂×C₁₃₇	#2546	1	1	yes	characteristic
148	C₂×C₁₃₇	C₂×#472	1	1	yes	characteristic
149	C₂×C₁₃₇	C₂×#473	1	1	yes	characteristic
150	#28699	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
151	C₁₃₇	#28699	1	1	yes	characteristic, direct product factor, Sylow
152	#3165	C₂×C₁₃₇	1	2	yes	—
153	C₂×#500	C₂×C₁₃₇	1	1	yes	characteristic
154	#2960	C₂×C₁₃₇	1	1	yes	characteristic
155	#2089	C₂×C₁₃₇	1	4	yes	—
156	#1997	C₂×C₁₃₇	1	2	yes	—
157	#1952	C₂×C₁₃₇	1	4	yes	—
158	C₄×#108	C₂×C₁₃₇	1	1	yes	characteristic
159	#533	C₂²×C₁₃₇	1	1	yes	characteristic
160	C₂×C₄⋊Q₈	C₂²×C₁₃₇	1	2	yes	—
161	#501	—	2	4	no	—
162	#500	—	2	2	no	—
163	#500	—	2	1	no	—
164	#500	—	2	1	no	—
165	#500	—	2	1	no	—
166	#500	—	2	1	no	—
167	#498	—	2	2	no	—
168	#446	—	2	1	no	—
169	#446	—	2	1	no	—
170	C₈×Q₈	—	2	1	no	—
171	C₈×Q₈	—	2	1	no	—
172	C₂×#108	C₂²×C₁₃₇	1	2	yes	—
173	#428	C₂²×C₁₃₇	1	8	yes	—
174	C₂×#107	C₂²×C₁₃₇	1	2	yes	—
175	C₂×#107	C₂²×C₁₃₇	1	2	yes	—
176	#424	C₂²×C₁₃₇	1	2	yes	—
177	#423	C₂²×C₁₃₇	1	1	yes	characteristic
178	#405	C₂²×C₁₃₇	1	2	yes	—
179	C₄×M₁₆	C₂²×C₁₃₇	1	2	yes	—
180	C₂×C₄×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
181	#343	C₂²×C₁₃₇	1	8	yes	—
182	#341	C₂²×C₁₃₇	1	2	yes	—
183	C₂²×Q₈	—	2	2	no	—
184	C₂×M₁₆	C₂³×C₁₃₇	1	2	yes	—
185	C₂×M₁₆	C₂³×C₁₃₇	1	4	yes	—
186	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
187	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
188	C₄⋊Q₈	—	2	2	no	—
189	C₄⋊Q₈	—	2	2	no	—
190	C₄⋊Q₈	—	2	2	no	—
191	C₄⋊Q₈	—	2	2	no	—
192	C₄⋊Q₈	D₄×C₁₃₇	1	1	yes	characteristic
193	C₄⋊Q₈	D₄×C₁₃₇	1	1	yes	characteristic
194	#125	D₄×C₁₃₇	1	2	yes	—
195	C₂²⋊Q₈	—	2	1	no	—
196	C₂²⋊Q₈	—	2	1	no	—
197	C₄×Q₈	—	2	1	no	—
198	C₄×Q₈	—	2	1	no	—
199	#117	C₂³×C₁₃₇	1	2	yes	—
200	C₂×C₄⋊C₄	—	2	2	no	—
201	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	4	yes	—
202	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
203	#108	Q₈×C₁₃₇	1	8	yes	complemented
204	#107	—	2	2	no	—
205	#107	—	2	2	no	—
206	#107	—	2	2	no	—
207	#107	—	2	2	no	—
208	#107	—	2	8	no	—
209	#106	—	2	8	no	—
210	C₄⋊C₈	—	2	2	no	—
211	C₄⋊C₈	—	2	2	no	—
212	C₄⋊C₈	—	2	1	no	—
213	C₄⋊C₈	—	2	1	no	—
214	#97	—	2	2	no	—
215	#97	—	2	2	no	—
216	C₄×C₈	—	2	2	no	—
217	C₄×C₈	—	2	2	no	—
218	C₄×C₈	D₄×C₁₃₇	1	1	yes	characteristic
219	C₄×C₈	D₄×C₁₃₇	1	1	yes	characteristic
220	C₄×C₈	D₄×C₁₃₇	1	1	yes	characteristic
221	C₄×C₈	D₄×C₁₃₇	1	1	yes	characteristic
222	C₂×Q₈	—	2	1	no	—
223	C₂×Q₈	—	2	1	no	—
224	C₂×Q₈	—	4	2	no	—
225	C₂×Q₈	—	4	2	no	—
226	C₂×Q₈	—	8	4	no	—
227	C₂²×C₄	—	2	4	no	—
228	C₂²×C₄	#41×C₁₃₇	1	1	yes	characteristic
229	C₂²×C₄	#41×C₁₃₇	1	1	yes	characteristic
230	C₂²×C₄	C₂⁴×C₁₃₇	1	1	yes	characteristic
231	M₁₆	—	2	4	no	—
232	M₁₆	—	2	4	no	—
233	M₁₆	C₂×Q₈×C₁₃₇	1	8	yes	—
234	C₂×C₈	—	2	4	no	—
235	C₂×C₈	—	2	2	no	—
236	C₂×C₈	—	2	2	no	—
237	C₂×C₈	C₂×D₄×C₁₃₇	1	1	yes	characteristic
238	C₂×C₈	C₂×D₄×C₁₃₇	1	1	yes	characteristic
239	C₂×C₈	C₂×D₄×C₁₃₇	1	1	yes	characteristic
240	C₂×C₈	C₂×D₄×C₁₃₇	1	1	yes	characteristic
241	C₂×C₈	C₂×Q₈×C₁₃₇	1	4	yes	—
242	C₄⋊C₄	—	2	2	no	—
243	C₄⋊C₄	—	2	2	no	—
244	C₄⋊C₄	—	2	1	no	—
245	C₄⋊C₄	—	2	1	no	—
246	C₄⋊C₄	—	2	4	no	—
247	C₄⋊C₄	—	2	4	no	—
248	C₄⋊C₄	—	4	2	no	—
249	C₄⋊C₄	—	4	2	no	—
250	C₄⋊C₄	C₂×D₄×C₁₃₇	1	2	yes	—
251	C₄⋊C₄	C₂×D₄×C₁₃₇	1	2	yes	—
252	C₂²⋊C₄	—	2	2	no	—
253	C₄²	—	2	2	no	—
254	C₄²	C₂×D₄×C₁₃₇	1	1	yes	characteristic
255	C₄²	C₂×D₄×C₁₃₇	1	1	yes	characteristic
256	C₄²	C₂×D₄×C₁₃₇	1	1	yes	characteristic
257	C₄²	C₂×D₄×C₁₃₇	1	1	yes	characteristic
258	C₂³	C₂×#41×C₁₃₇	1	1	yes	characteristic
259	Q₈	—	4	2	no	—
260	Q₈	—	4	2	no	—
261	Q₈	—	8	4	no	—
262	Q₈	—	8	4	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	4	4	no	—
268	C₂×C₄	—	8	4	no	—
269	C₂×C₄	#121×C₁₃₇	1	1	yes	characteristic
270	C₂×C₄	#121×C₁₃₇	1	1	yes	characteristic
271	C₂×C₄	#121×C₁₃₇	1	1	yes	characteristic
272	C₂×C₄	#121×C₁₃₇	1	1	yes	characteristic
273	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
274	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
275	C₂×C₄	C₂²×Q₈×C₁₃₇	1	2	yes	—
276	C₂×C₄	2₋¹⁺⁴×C₁₃₇	1	2	yes	—
277	C₈	—	2	1	no	—
278	C₈	—	2	1	no	—
279	C₈	—	2	8	no	—
280	C₈	—	4	2	no	—
281	C₈	C₄⋊Q₈×C₁₃₇	1	2	yes	—
282	C₈	C₄⋊Q₈×C₁₃₇	1	2	yes	—
283	C₂²	—	2	1	no	—
284	C₂²	—	2	1	no	—
285	C₂²	C₂×#121×C₁₃₇	1	1	yes	characteristic
286	C₂²	#557×C₁₃₇	1	2	yes	—
287	C₄	—	2	1	no	—
288	C₄	—	2	1	no	—
289	C₄	—	4	2	no	—
290	C₄	—	4	2	no	—
291	C₄	—	8	4	no	—
292	C₄	#472×C₁₃₇	1	2	yes	—
293	C₄	#473×C₁₃₇	1	2	yes	—
294	C₄	C₂×C₄⋊Q₈×C₁₃₇	1	1	yes	characteristic
295	C₄	#536×C₁₃₇	1	1	yes	characteristic
296	C₄	D₄×Q₈×C₁₃₇	1	2	yes	—
297	C₂	—	2	2	no	—
298	C₂	#2546×C₁₃₇	1	1	yes	characteristic
299	C₂	C₂×#472×C₁₃₇	1	1	yes	characteristic
300	C₂	C₂×#473×C₁₃₇	1	1	yes	characteristic

	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
52. #108×C₁₃₇	Q₈	semidirect
150. #28699	C₁₃₇	direct (N × H)
151. C₁₃₇	#28699	direct (N × H)
203. #108	Q₈×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
#3165×C₁₃₇	2
C₂×#500×C₁₃₇	2
#2960×C₁₃₇	2
#2089×C₁₃₇	2
#1997×C₁₃₇	2
#1952×C₁₃₇	2
C₄×#108×C₁₃₇	2
#28699	137

Subgroups — C137×#28699 #46878085

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#28699 #46878085