Subgroups — C₁₃₇×#10172 #46859558

Order 35072.

562 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#3227×C₁₃₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#498×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
3	#3152×C₁₃₇	C₂	1	1	yes	maximal, characteristic
4	#1997×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
5	#1546×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
6	#1492×C₁₃₇	C₂	1	1	yes	maximal, characteristic
7	#1278×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#137×C₁₃₇	—	2	2	no	—
9	#532×C₁₃₇	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊Q₈×C₁₃₇	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊Q₈×C₁₃₇	C₂²	1	1	yes	characteristic
12	#498×C₁₃₇	—	2	2	no	—
13	#498×C₁₃₇	—	2	1	no	—
14	#498×C₁₃₇	—	2	2	no	—
15	#497×C₁₃₇	—	2	2	no	—
16	#494×C₁₃₇	—	2	1	no	—
17	#492×C₁₃₇	—	2	1	no	—
18	#486×C₁₃₇	—	2	1	no	—
19	#481×C₁₃₇	—	2	2	no	—
20	#476×C₁₃₇	—	2	2	no	—
21	#431×C₁₃₇	C₂²	1	1	yes	characteristic
22	C₂×#106×C₁₃₇	—	2	2	no	—
23	#423×C₁₃₇	C₂²	1	1	yes	characteristic
24	C₄×M₁₆×C₁₃₇	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
26	#343×C₁₃₇	—	2	2	no	—
27	#332×C₁₃₇	—	2	2	no	—
28	D₄⋊C₈×C₁₃₇	—	2	2	no	—
29	C₂×#41×C₁₃₇	—	2	1	no	—
30	C₂²×Q₈×C₁₃₇	—	2	1	no	—
31	C₂²×Q₈×C₁₃₇	—	2	1	no	—
32	#137×C₁₃₇	—	4	4	no	—
33	#137×C₁₃₇	—	4	4	no	—
34	C₂×Q₁₆×C₁₃₇	—	2	4	no	—
35	C₂×SD₁₆×C₁₃₇	—	2	4	no	—
36	C₂×M₁₆×C₁₃₇	—	2	1	no	—
37	C₂×M₁₆×C₁₃₇	D₄	1	2	yes	—
38	C₂²×C₈×C₁₃₇	—	2	1	no	—
39	C₄⋊Q₈×C₁₃₇	—	2	1	no	—
40	C₄⋊Q₈×C₁₃₇	—	2	1	no	—
41	C₄⋊Q₈×C₁₃₇	—	2	1	no	—
42	C₄⋊Q₈×C₁₃₇	—	2	1	no	—
43	C₄⋊Q₈×C₁₃₇	—	2	1	no	—
44	C₄⋊Q₈×C₁₃₇	—	2	1	no	—
45	C₄⋊Q₈×C₁₃₇	—	2	1	no	—
46	C₄⋊Q₈×C₁₃₇	D₄	1	1	yes	characteristic
47	C₄⋊Q₈×C₁₃₇	D₄	1	2	yes	—
48	#127×C₁₃₇	D₄	1	1	yes	characteristic
49	#124×C₁₃₇	—	2	1	no	—
50	#121×C₁₃₇	—	2	1	no	—
51	#121×C₁₃₇	—	2	1	no	—
52	C₄×D₄×C₁₃₇	—	2	1	no	—
53	C₄×D₄×C₁₃₇	—	2	1	no	—
54	C₂×C₄⋊C₄×C₁₃₇	—	2	1	no	—
55	C₂×C₄⋊C₄×C₁₃₇	—	2	1	no	—
56	C₂×C₄⋊C₄×C₁₃₇	—	2	1	no	—
57	C₂×C₄⋊C₄×C₁₃₇	D₄	1	2	yes	—
58	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
59	#106×C₁₃₇	—	2	8	no	—
60	#106×C₁₃₇	—	2	4	no	—
61	C₄⋊C₈×C₁₃₇	—	2	1	no	—
62	C₄⋊C₈×C₁₃₇	—	2	1	no	—
63	C₄⋊C₈×C₁₃₇	D₄	1	2	yes	—
64	#102×C₁₃₇	—	2	4	no	—
65	C₂²⋊C₈×C₁₃₇	—	2	1	no	—
66	#97×C₁₃₇	—	2	1	no	—
67	C₄×C₈×C₁₃₇	—	2	1	no	—
68	C₄×C₈×C₁₃₇	—	2	1	no	—
69	C₄×C₈×C₁₃₇	—	2	1	no	—
70	C₄×C₈×C₁₃₇	D₄	1	2	yes	—
71	#41×C₁₃₇	—	4	2	no	—
72	#41×C₁₃₇	—	4	2	no	—
73	C₂×Q₈×C₁₃₇	—	2	1	no	—
74	C₂×Q₈×C₁₃₇	—	2	2	no	—
75	C₂×Q₈×C₁₃₇	—	4	1	no	—
76	C₂×Q₈×C₁₃₇	—	4	1	no	—
77	C₂×Q₈×C₁₃₇	—	4	1	no	—
78	C₂×Q₈×C₁₃₇	—	4	2	no	—
79	C₂×Q₈×C₁₃₇	—	4	2	no	—
80	C₂×Q₈×C₁₃₇	—	8	2	no	—
81	C₂×D₄×C₁₃₇	—	2	1	no	—
82	C₂×D₄×C₁₃₇	—	2	1	no	—
83	C₂×D₄×C₁₃₇	—	2	1	no	—
84	C₂×D₄×C₁₃₇	—	4	1	no	—
85	C₂²×C₄×C₁₃₇	—	2	2	no	—
86	C₂²×C₄×C₁₃₇	—	4	1	no	—
87	C₂²×C₄×C₁₃₇	—	4	1	no	—
88	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
89	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
90	C₂²×C₄×C₁₃₇	#41	1	1	yes	characteristic
91	Q₁₆×C₁₃₇	—	4	8	no	—
92	SD₁₆×C₁₃₇	—	4	8	no	—
93	M₁₆×C₁₃₇	—	2	2	no	—
94	M₁₆×C₁₃₇	—	4	2	no	—
95	M₁₆×C₁₃₇	SD₁₆	1	4	yes	—
96	C₂×C₈×C₁₃₇	—	2	4	no	—
97	C₂×C₈×C₁₃₇	—	2	2	no	—
98	C₂×C₈×C₁₃₇	—	2	2	no	—
99	C₂×C₈×C₁₃₇	—	2	2	no	—
100	C₂×C₈×C₁₃₇	—	4	1	no	—
101	C₄⋊C₄×C₁₃₇	—	2	2	no	—
102	C₄⋊C₄×C₁₃₇	—	2	2	no	—
103	C₄⋊C₄×C₁₃₇	—	2	2	no	—
104	C₄⋊C₄×C₁₃₇	—	2	1	no	—
105	C₄⋊C₄×C₁₃₇	—	2	2	no	—
106	C₄⋊C₄×C₁₃₇	—	2	1	no	—
107	C₄⋊C₄×C₁₃₇	—	4	1	no	—
108	C₄⋊C₄×C₁₃₇	—	4	1	no	—
109	C₄⋊C₄×C₁₃₇	—	4	1	no	—
110	C₄⋊C₄×C₁₃₇	—	4	1	no	—
111	C₂²⋊C₄×C₁₃₇	—	4	1	no	—
112	C₂²⋊C₄×C₁₃₇	—	4	1	no	—
113	C₄²×C₁₃₇	C₂×D₄	1	1	yes	characteristic
114	C₄²×C₁₃₇	C₂×D₄	1	1	yes	characteristic
115	C₄²×C₁₃₇	C₂×D₄	1	1	yes	characteristic
116	C₄²×C₁₃₇	C₂×D₄	1	1	yes	characteristic
117	C₂³×C₁₃₇	—	4	1	no	—
118	C₂³×C₁₃₇	#121	1	1	yes	characteristic
119	Q₈×C₁₃₇	—	4	4	no	—
120	Q₈×C₁₃₇	—	4	2	no	—
121	Q₈×C₁₃₇	—	8	2	no	—
122	Q₈×C₁₃₇	—	8	2	no	—
123	Q₈×C₁₃₇	—	8	1	no	—
124	Q₈×C₁₃₇	—	8	1	no	—
125	D₄×C₁₃₇	—	4	2	no	—
126	D₄×C₁₃₇	—	4	2	no	—
127	D₄×C₁₃₇	—	4	2	no	—
128	D₄×C₁₃₇	—	8	2	no	—
129	C₂×C₄×C₁₃₇	—	2	2	no	—
130	C₂×C₄×C₁₃₇	—	2	2	no	—
131	C₂×C₄×C₁₃₇	—	4	1	no	—
132	C₂×C₄×C₁₃₇	—	4	2	no	—
133	C₂×C₄×C₁₃₇	—	4	2	no	—
134	C₂×C₄×C₁₃₇	—	8	2	no	—
135	C₂×C₄×C₁₃₇	—	8	1	no	—
136	C₂×C₄×C₁₃₇	—	8	1	no	—
137	C₂×C₄×C₁₃₇	—	8	2	no	—
138	C₂×C₄×C₁₃₇	#120	1	1	yes	characteristic
139	C₂×C₄×C₁₃₇	#120	1	1	yes	characteristic
140	C₂×C₄×C₁₃₇	#120	1	1	yes	characteristic
141	C₂×C₄×C₁₃₇	#121	1	1	yes	characteristic
142	C₂×C₄×C₁₃₇	#121	1	1	yes	characteristic
143	C₂×C₄×C₁₃₇	#127	1	1	yes	characteristic
144	C₂×C₄×C₁₃₇	C₂×SD₁₆	1	2	yes	—
145	C₂×C₄×C₁₃₇	AGL(1,ℤ/8)	1	2	yes	—
146	C₈×C₁₃₇	—	2	4	no	—
147	C₈×C₁₃₇	—	2	4	no	—
148	C₈×C₁₃₇	—	4	2	no	—
149	C₈×C₁₃₇	—	8	1	no	—
150	C₂²×C₁₃₇	—	2	1	no	—
151	C₂²×C₁₃₇	—	2	1	no	—
152	C₂²×C₁₃₇	—	8	1	no	—
153	C₂²×C₁₃₇	—	8	1	no	—
154	C₂²×C₁₃₇	—	8	1	no	—
155	C₂²×C₁₃₇	#392	1	1	yes	characteristic
156	C₂²×C₁₃₇	#460	1	2	yes	—
157	C₄×C₁₃₇	—	2	1	no	—
158	C₄×C₁₃₇	—	2	1	no	—
159	C₄×C₁₃₇	—	8	1	no	—
160	C₄×C₁₃₇	—	8	2	no	—
161	C₄×C₁₃₇	—	8	2	no	—
162	C₄×C₁₃₇	#450	1	2	yes	—
163	C₄×C₁₃₇	#456	1	2	yes	—
164	C₄×C₁₃₇	#471	1	2	yes	—
165	C₄×C₁₃₇	#492	1	1	yes	characteristic
166	C₄×C₁₃₇	#496	1	1	yes	characteristic
167	C₂×C₁₃₇	—	2	2	no	—
168	C₂×C₁₃₇	—	16	1	no	—
169	C₂×C₁₃₇	#2005	1	1	yes	characteristic
170	C₂×C₁₃₇	#2014	1	1	yes	characteristic
171	C₂×C₁₃₇	#2019	1	1	yes	characteristic
172	#10172	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
173	C₁₃₇	#10172	1	1	yes	characteristic, direct product factor, Sylow
174	#3227	C₂×C₁₃₇	1	1	yes	characteristic
175	C₂×#498	C₂×C₁₃₇	1	1	yes	characteristic, complemented
176	#3152	C₂×C₁₃₇	1	1	yes	characteristic
177	#1997	C₂×C₁₃₇	1	1	yes	characteristic, complemented
178	#1546	C₂×C₁₃₇	1	1	yes	characteristic, complemented
179	#1492	C₂×C₁₃₇	1	1	yes	characteristic
180	#1278	C₂×C₁₃₇	1	1	yes	characteristic, complemented
181	C₂×#137	—	2	2	no	—
182	#532	C₂²×C₁₃₇	1	1	yes	characteristic
183	C₂×C₄⋊Q₈	C₂²×C₁₃₇	1	1	yes	characteristic
184	C₂×C₄⋊Q₈	C₂²×C₁₃₇	1	1	yes	characteristic
185	#498	—	2	2	no	—
186	#498	—	2	1	no	—
187	#498	—	2	2	no	—
188	#497	—	2	2	no	—
189	#494	—	2	1	no	—
190	#492	—	2	1	no	—
191	#486	—	2	1	no	—
192	#481	—	2	2	no	—
193	#476	—	2	2	no	—
194	#431	C₂²×C₁₃₇	1	1	yes	characteristic
195	C₂×#106	—	2	2	no	—
196	#423	C₂²×C₁₃₇	1	1	yes	characteristic
197	C₄×M₁₆	C₂²×C₁₃₇	1	1	yes	characteristic
198	C₂×C₄×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
199	#343	—	2	2	no	—
200	#332	—	2	2	no	—
201	D₄⋊C₈	—	2	2	no	—
202	C₂×#41	—	2	1	no	—
203	C₂²×Q₈	—	2	1	no	—
204	C₂²×Q₈	—	2	1	no	—
205	#137	—	4	4	no	—
206	#137	—	4	4	no	—
207	C₂×Q₁₆	—	2	4	no	—
208	C₂×SD₁₆	—	2	4	no	—
209	C₂×M₁₆	—	2	1	no	—
210	C₂×M₁₆	D₄×C₁₃₇	1	2	yes	—
211	C₂²×C₈	—	2	1	no	—
212	C₄⋊Q₈	—	2	1	no	—
213	C₄⋊Q₈	—	2	1	no	—
214	C₄⋊Q₈	—	2	1	no	—
215	C₄⋊Q₈	—	2	1	no	—
216	C₄⋊Q₈	—	2	1	no	—
217	C₄⋊Q₈	—	2	1	no	—
218	C₄⋊Q₈	—	2	1	no	—
219	C₄⋊Q₈	D₄×C₁₃₇	1	2	yes	—
220	C₄⋊Q₈	D₄×C₁₃₇	1	1	yes	characteristic
221	#127	D₄×C₁₃₇	1	1	yes	characteristic
222	#124	—	2	1	no	—
223	#121	—	2	1	no	—
224	#121	—	2	1	no	—
225	C₄×D₄	—	2	1	no	—
226	C₄×D₄	—	2	1	no	—
227	C₂×C₄⋊C₄	—	2	1	no	—
228	C₂×C₄⋊C₄	—	2	1	no	—
229	C₂×C₄⋊C₄	—	2	1	no	—
230	C₂×C₄⋊C₄	D₄×C₁₃₇	1	2	yes	—
231	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
232	#106	—	2	8	no	—
233	#106	—	2	4	no	—
234	C₄⋊C₈	—	2	1	no	—
235	C₄⋊C₈	—	2	1	no	—
236	C₄⋊C₈	D₄×C₁₃₇	1	2	yes	—
237	#102	—	2	4	no	—
238	C₂²⋊C₈	—	2	1	no	—
239	#97	—	2	1	no	—
240	C₄×C₈	—	2	1	no	—
241	C₄×C₈	—	2	1	no	—
242	C₄×C₈	—	2	1	no	—
243	C₄×C₈	D₄×C₁₃₇	1	2	yes	—
244	#41	—	4	2	no	—
245	#41	—	4	2	no	—
246	C₂×Q₈	—	2	1	no	—
247	C₂×Q₈	—	2	2	no	—
248	C₂×Q₈	—	4	1	no	—
249	C₂×Q₈	—	4	1	no	—
250	C₂×Q₈	—	4	1	no	—
251	C₂×Q₈	—	4	2	no	—
252	C₂×Q₈	—	4	2	no	—
253	C₂×Q₈	—	8	2	no	—
254	C₂×D₄	—	2	1	no	—
255	C₂×D₄	—	2	1	no	—
256	C₂×D₄	—	2	1	no	—
257	C₂×D₄	—	4	1	no	—
258	C₂²×C₄	—	2	2	no	—
259	C₂²×C₄	—	4	1	no	—
260	C₂²×C₄	—	4	1	no	—
261	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
262	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
263	C₂²×C₄	#41×C₁₃₇	1	1	yes	characteristic
264	Q₁₆	—	4	8	no	—
265	SD₁₆	—	4	8	no	—
266	M₁₆	—	2	2	no	—
267	M₁₆	—	4	2	no	—
268	M₁₆	SD₁₆×C₁₃₇	1	4	yes	—
269	C₂×C₈	—	2	4	no	—
270	C₂×C₈	—	2	2	no	—
271	C₂×C₈	—	2	2	no	—
272	C₂×C₈	—	2	2	no	—
273	C₂×C₈	—	4	1	no	—
274	C₄⋊C₄	—	2	2	no	—
275	C₄⋊C₄	—	2	2	no	—
276	C₄⋊C₄	—	2	2	no	—
277	C₄⋊C₄	—	2	1	no	—
278	C₄⋊C₄	—	2	2	no	—
279	C₄⋊C₄	—	2	1	no	—
280	C₄⋊C₄	—	4	1	no	—
281	C₄⋊C₄	—	4	1	no	—
282	C₄⋊C₄	—	4	1	no	—
283	C₄⋊C₄	—	4	1	no	—
284	C₂²⋊C₄	—	4	1	no	—
285	C₂²⋊C₄	—	4	1	no	—
286	C₄²	C₂×D₄×C₁₃₇	1	1	yes	characteristic
287	C₄²	C₂×D₄×C₁₃₇	1	1	yes	characteristic
288	C₄²	C₂×D₄×C₁₃₇	1	1	yes	characteristic
289	C₄²	C₂×D₄×C₁₃₇	1	1	yes	characteristic
290	C₂³	—	4	1	no	—
291	C₂³	#121×C₁₃₇	1	1	yes	characteristic
292	Q₈	—	4	4	no	—
293	Q₈	—	4	2	no	—
294	Q₈	—	8	2	no	—
295	Q₈	—	8	2	no	—
296	Q₈	—	8	1	no	—
297	Q₈	—	8	1	no	—
298	D₄	—	4	2	no	—
299	D₄	—	4	2	no	—
300	D₄	—	4	2	no	—
301	D₄	—	8	2	no	—
302	C₂×C₄	—	2	2	no	—
303	C₂×C₄	—	2	2	no	—
304	C₂×C₄	—	4	1	no	—
305	C₂×C₄	—	4	2	no	—
306	C₂×C₄	—	4	2	no	—
307	C₂×C₄	—	8	2	no	—
308	C₂×C₄	—	8	1	no	—
309	C₂×C₄	—	8	1	no	—
310	C₂×C₄	—	8	2	no	—
311	C₂×C₄	#120×C₁₃₇	1	1	yes	characteristic
312	C₂×C₄	#120×C₁₃₇	1	1	yes	characteristic
313	C₂×C₄	#120×C₁₃₇	1	1	yes	characteristic
314	C₂×C₄	#121×C₁₃₇	1	1	yes	characteristic
315	C₂×C₄	#121×C₁₃₇	1	1	yes	characteristic
316	C₂×C₄	#127×C₁₃₇	1	1	yes	characteristic
317	C₂×C₄	C₂×SD₁₆×C₁₃₇	1	2	yes	—
318	C₂×C₄	AGL(1,ℤ/8)×C₁₃₇	1	2	yes	—
319	C₈	—	2	4	no	—
320	C₈	—	2	4	no	—
321	C₈	—	4	2	no	—
322	C₈	—	8	1	no	—
323	C₂²	—	2	1	no	—
324	C₂²	—	2	1	no	—
325	C₂²	—	8	1	no	—
326	C₂²	—	8	1	no	—
327	C₂²	—	8	1	no	—
328	C₂²	#392×C₁₃₇	1	1	yes	characteristic
329	C₂²	#460×C₁₃₇	1	2	yes	—
330	C₄	—	2	1	no	—
331	C₄	—	2	1	no	—
332	C₄	—	8	1	no	—
333	C₄	—	8	2	no	—
334	C₄	—	8	2	no	—
335	C₄	#450×C₁₃₇	1	2	yes	—
336	C₄	#456×C₁₃₇	1	2	yes	—
337	C₄	#471×C₁₃₇	1	2	yes	—
338	C₄	#492×C₁₃₇	1	1	yes	characteristic
339	C₄	#496×C₁₃₇	1	1	yes	characteristic
340	C₂	—	2	2	no	—
341	C₂	—	16	1	no	—
342	C₂	#2005×C₁₃₇	1	1	yes	characteristic
343	C₂	#2014×C₁₃₇	1	1	yes	characteristic
344	C₂	#2019×C₁₃₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃₇	#392
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₁₃₇
prime radical	#532×C₁₃₇	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ 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. C₂×#498×C₁₃₇	C₂	semidirect
4. #1997×C₁₃₇	C₂	semidirect
5. #1546×C₁₃₇	C₂	semidirect
7. #1278×C₁₃₇	C₂	semidirect
172. #10172	C₁₃₇	direct (N × H)
173. C₁₃₇	#10172	direct (N × H)
175. C₂×#498	C₂×C₁₃₇	semidirect
177. #1997	C₂×C₁₃₇	semidirect
178. #1546	C₂×C₁₃₇	semidirect
180. #1278	C₂×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
#3227×C₁₃₇	2
C₂×#498×C₁₃₇	2
#3152×C₁₃₇	2
#1997×C₁₃₇	2
#1546×C₁₃₇	2
#1492×C₁₃₇	2
#1278×C₁₃₇	2
#10172	137

Subgroups — C137×#10172 #46859558

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#10172 #46859558