Subgroups — C₁₃₇×#10291 #46859677

Order 35072.

652 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#3209×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
2	#3208×C₁₃₇	C₂	1	1	yes	maximal, characteristic
3	#2858×C₁₃₇	C₂	1	1	yes	maximal, characteristic
4	#1994×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
5	#1548×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
6	#1545×C₁₃₇	C₂	1	1	yes	maximal, characteristic
7	#1276×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
8	#533×C₁₃₇	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊Q₈×C₁₃₇	C₂²	1	1	yes	characteristic
10	C₂×#127×C₁₃₇	C₂²	1	1	yes	characteristic
11	C₂×#124×C₁₃₇	—	2	2	no	—
12	#494×C₁₃₇	—	2	1	no	—
13	#492×C₁₃₇	—	2	1	no	—
14	#487×C₁₃₇	—	2	1	no	—
15	#486×C₁₃₇	—	2	1	no	—
16	#462×C₁₃₇	—	2	2	no	—
17	#460×C₁₃₇	—	2	2	no	—
18	#451×C₁₃₇	—	2	2	no	—
19	#450×C₁₃₇	—	2	2	no	—
20	#431×C₁₃₇	C₂²	1	1	yes	characteristic
21	#431×C₁₃₇	C₂²	1	1	yes	characteristic
22	#423×C₁₃₇	C₂²	1	1	yes	characteristic
23	#390×C₁₃₇	—	2	2	no	—
24	#390×C₁₃₇	—	2	2	no	—
25	C₄³×C₁₃₇	C₂²	1	1	yes	characteristic
26	#339×C₁₃₇	—	2	2	no	—
27	#333×C₁₃₇	—	2	2	no	—
28	#331×C₁₃₇	—	2	2	no	—
29	C₂²×Q₈×C₁₃₇	D₄	1	2	yes	—
30	C₂²×D₄×C₁₃₇	—	2	1	no	—
31	C₂²×D₄×C₁₃₇	—	2	1	no	—
32	C₂²×D₄×C₁₃₇	D₄	1	2	yes	—
33	C₂×Q₁₆×C₁₃₇	—	2	4	no	—
34	C₂×SD₁₆×C₁₃₇	—	2	4	no	—
35	C₂×M₁₆×C₁₃₇	—	2	1	no	—
36	C₄⋊Q₈×C₁₃₇	—	2	1	no	—
37	C₄⋊Q₈×C₁₃₇	—	2	1	no	—
38	C₄⋊Q₈×C₁₃₇	—	2	1	no	—
39	C₄⋊Q₈×C₁₃₇	—	2	1	no	—
40	C₄⋊Q₈×C₁₃₇	D₄	1	1	yes	characteristic
41	C₄⋊Q₈×C₁₃₇	D₄	1	1	yes	characteristic
42	#127×C₁₃₇	—	2	1	no	—
43	#127×C₁₃₇	—	2	1	no	—
44	#127×C₁₃₇	—	2	1	no	—
45	#127×C₁₃₇	—	2	1	no	—
46	#125×C₁₃₇	—	2	1	no	—
47	#124×C₁₃₇	—	4	8	no	—
48	C₂²⋊Q₈×C₁₃₇	—	2	1	no	—
49	C₂²⋊Q₈×C₁₃₇	—	2	1	no	—
50	C₄×Q₈×C₁₃₇	—	2	1	no	—
51	C₄×Q₈×C₁₃₇	—	2	1	no	—
52	#117×C₁₃₇	—	2	1	no	—
53	C₂×C₄⋊C₄×C₁₃₇	—	2	1	no	—
54	C₂×C₄⋊C₄×C₁₃₇	—	2	1	no	—
55	C₂×C₂²⋊C₄×C₁₃₇	—	2	8	no	—
56	C₂×C₄²×C₁₃₇	—	2	1	no	—
57	C₂×C₄²×C₁₃₇	—	2	1	no	—
58	C₂×C₄²×C₁₃₇	D₄	1	2	yes	—
59	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
60	C₄⋊C₈×C₁₃₇	—	2	1	no	—
61	C₄⋊C₈×C₁₃₇	—	2	1	no	—
62	C₄⋊C₈×C₁₃₇	D₄	1	2	yes	—
63	C₄⋊C₈×C₁₃₇	D₄	1	2	yes	—
64	#103×C₁₃₇	—	2	4	no	—
65	#102×C₁₃₇	—	2	4	no	—
66	C₂²⋊C₈×C₁₃₇	—	2	1	no	—
67	C₂²⋊C₈×C₁₃₇	—	2	1	no	—
68	C₄×C₈×C₁₃₇	—	2	1	no	—
69	C₄×C₈×C₁₃₇	—	2	1	no	—
70	C₂⁴×C₁₃₇	—	2	2	no	—
71	C₂×Q₈×C₁₃₇	—	2	2	no	—
72	C₂×Q₈×C₁₃₇	—	2	1	no	—
73	C₂×Q₈×C₁₃₇	—	2	1	no	—
74	C₂×Q₈×C₁₃₇	—	2	2	no	—
75	C₂×Q₈×C₁₃₇	—	4	4	no	—
76	C₂×D₄×C₁₃₇	—	2	2	no	—
77	C₂×D₄×C₁₃₇	—	2	2	no	—
78	C₂×D₄×C₁₃₇	—	4	4	no	—
79	C₂×D₄×C₁₃₇	—	4	4	no	—
80	C₂×D₄×C₁₃₇	—	4	1	no	—
81	C₂×D₄×C₁₃₇	—	4	1	no	—
82	C₂×D₄×C₁₃₇	—	4	1	no	—
83	C₂×D₄×C₁₃₇	—	4	1	no	—
84	C₂×D₄×C₁₃₇	—	4	2	no	—
85	C₂×D₄×C₁₃₇	—	4	2	no	—
86	C₂²×C₄×C₁₃₇	—	2	2	no	—
87	C₂²×C₄×C₁₃₇	—	2	2	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	C₂×D₄	1	1	yes	characteristic
91	Q₁₆×C₁₃₇	—	8	4	no	—
92	SD₁₆×C₁₃₇	—	8	4	no	—
93	M₁₆×C₁₃₇	—	4	2	no	—
94	C₂×C₈×C₁₃₇	—	2	2	no	—
95	C₂×C₈×C₁₃₇	—	2	2	no	—
96	C₂×C₈×C₁₃₇	—	4	1	no	—
97	C₄⋊C₄×C₁₃₇	—	2	1	no	—
98	C₄⋊C₄×C₁₃₇	—	2	1	no	—
99	C₄⋊C₄×C₁₃₇	—	2	1	no	—
100	C₄⋊C₄×C₁₃₇	—	2	1	no	—
101	C₄⋊C₄×C₁₃₇	—	4	1	no	—
102	C₄⋊C₄×C₁₃₇	—	4	1	no	—
103	C₄⋊C₄×C₁₃₇	—	4	1	no	—
104	C₄⋊C₄×C₁₃₇	—	4	1	no	—
105	C₄⋊C₄×C₁₃₇	—	4	1	no	—
106	C₄⋊C₄×C₁₃₇	—	4	1	no	—
107	C₂²⋊C₄×C₁₃₇	—	4	16	no	—
108	C₂²⋊C₄×C₁₃₇	—	4	1	no	—
109	C₄²×C₁₃₇	—	2	4	no	—
110	C₄²×C₁₃₇	—	2	2	no	—
111	C₄²×C₁₃₇	—	2	2	no	—
112	C₄²×C₁₃₇	—	4	2	no	—
113	C₄²×C₁₃₇	—	4	1	no	—
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	C₄²×C₁₃₇	#41	1	1	yes	characteristic
118	C₂³×C₁₃₇	—	4	4	no	—
119	C₂³×C₁₃₇	—	4	4	no	—
120	C₂³×C₁₃₇	—	4	2	no	—
121	C₂³×C₁₃₇	—	4	4	no	—
122	C₂³×C₁₃₇	#120	1	1	yes	characteristic
123	Q₈×C₁₃₇	—	4	2	no	—
124	Q₈×C₁₃₇	—	4	4	no	—
125	Q₈×C₁₃₇	—	4	2	no	—
126	Q₈×C₁₃₇	—	8	2	no	—
127	D₄×C₁₃₇	—	4	4	no	—
128	D₄×C₁₃₇	—	8	2	no	—
129	D₄×C₁₃₇	—	8	2	no	—
130	D₄×C₁₃₇	—	8	2	no	—
131	D₄×C₁₃₇	—	8	2	no	—
132	D₄×C₁₃₇	—	8	1	no	—
133	D₄×C₁₃₇	—	8	1	no	—
134	C₂×C₄×C₁₃₇	—	2	2	no	—
135	C₂×C₄×C₁₃₇	—	2	2	no	—
136	C₂×C₄×C₁₃₇	—	2	4	no	—
137	C₂×C₄×C₁₃₇	—	4	2	no	—
138	C₂×C₄×C₁₃₇	—	4	2	no	—
139	C₂×C₄×C₁₃₇	—	4	4	no	—
140	C₂×C₄×C₁₃₇	—	4	1	no	—
141	C₂×C₄×C₁₃₇	—	4	1	no	—
142	C₂×C₄×C₁₃₇	—	4	2	no	—
143	C₂×C₄×C₁₃₇	#120	1	1	yes	characteristic
144	C₂×C₄×C₁₃₇	#120	1	1	yes	characteristic
145	C₂×C₄×C₁₃₇	#121	1	1	yes	characteristic
146	C₂×C₄×C₁₃₇	#121	1	1	yes	characteristic
147	C₂×C₄×C₁₃₇	#121	1	1	yes	characteristic
148	C₂×C₄×C₁₃₇	#127	1	1	yes	characteristic
149	C₂×C₄×C₁₃₇	#135	1	2	yes	—
150	C₂×C₄×C₁₃₇	#137	1	2	yes	—
151	C₈×C₁₃₇	—	4	2	no	—
152	C₈×C₁₃₇	—	4	2	no	—
153	C₈×C₁₃₇	—	8	1	no	—
154	C₂²×C₁₃₇	—	2	1	no	—
155	C₂²×C₁₃₇	—	2	1	no	—
156	C₂²×C₁₃₇	—	8	8	no	—
157	C₂²×C₁₃₇	—	8	2	no	—
158	C₂²×C₁₃₇	—	8	2	no	—
159	C₂²×C₁₃₇	—	8	2	no	—
160	C₂²×C₁₃₇	#392	1	1	yes	characteristic
161	C₂²×C₁₃₇	#452	1	2	yes	—
162	C₄×C₁₃₇	—	2	1	no	—
163	C₄×C₁₃₇	—	2	1	no	—
164	C₄×C₁₃₇	—	4	4	no	—
165	C₄×C₁₃₇	—	8	2	no	—
166	C₄×C₁₃₇	—	8	1	no	—
167	C₄×C₁₃₇	—	8	1	no	—
168	C₄×C₁₃₇	#453	1	2	yes	—
169	C₄×C₁₃₇	#455	1	2	yes	—
170	C₄×C₁₃₇	#464	1	2	yes	—
171	C₄×C₁₃₇	#495	1	1	yes	characteristic
172	C₄×C₁₃₇	#497	1	1	yes	characteristic
173	C₂×C₁₃₇	—	2	2	no	—
174	C₂×C₁₃₇	—	16	2	no	—
175	C₂×C₁₃₇	#2002	1	1	yes	characteristic
176	C₂×C₁₃₇	#2004	1	1	yes	characteristic
177	C₂×C₁₃₇	#2015	1	1	yes	characteristic
178	#10291	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
179	C₁₃₇	#10291	1	1	yes	characteristic, direct product factor, Sylow
180	#3209	C₂×C₁₃₇	1	1	yes	characteristic, complemented
181	#3208	C₂×C₁₃₇	1	1	yes	characteristic
182	#2858	C₂×C₁₃₇	1	1	yes	characteristic
183	#1994	C₂×C₁₃₇	1	1	yes	characteristic, complemented
184	#1548	C₂×C₁₃₇	1	1	yes	characteristic, complemented
185	#1545	C₂×C₁₃₇	1	1	yes	characteristic
186	#1276	C₂×C₁₃₇	1	1	yes	characteristic, complemented
187	#533	C₂²×C₁₃₇	1	1	yes	characteristic
188	C₂×C₄⋊Q₈	C₂²×C₁₃₇	1	1	yes	characteristic
189	C₂×#127	C₂²×C₁₃₇	1	1	yes	characteristic
190	C₂×#124	—	2	2	no	—
191	#494	—	2	1	no	—
192	#492	—	2	1	no	—
193	#487	—	2	1	no	—
194	#486	—	2	1	no	—
195	#462	—	2	2	no	—
196	#460	—	2	2	no	—
197	#451	—	2	2	no	—
198	#450	—	2	2	no	—
199	#431	C₂²×C₁₃₇	1	1	yes	characteristic
200	#431	C₂²×C₁₃₇	1	1	yes	characteristic
201	#423	C₂²×C₁₃₇	1	1	yes	characteristic
202	#390	—	2	2	no	—
203	#390	—	2	2	no	—
204	C₄³	C₂²×C₁₃₇	1	1	yes	characteristic
205	#339	—	2	2	no	—
206	#333	—	2	2	no	—
207	#331	—	2	2	no	—
208	C₂²×Q₈	D₄×C₁₃₇	1	2	yes	—
209	C₂²×D₄	—	2	1	no	—
210	C₂²×D₄	—	2	1	no	—
211	C₂²×D₄	D₄×C₁₃₇	1	2	yes	—
212	C₂×Q₁₆	—	2	4	no	—
213	C₂×SD₁₆	—	2	4	no	—
214	C₂×M₁₆	—	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	1	yes	characteristic
220	C₄⋊Q₈	D₄×C₁₃₇	1	1	yes	characteristic
221	#127	—	2	1	no	—
222	#127	—	2	1	no	—
223	#127	—	2	1	no	—
224	#127	—	2	1	no	—
225	#125	—	2	1	no	—
226	#124	—	4	8	no	—
227	C₂²⋊Q₈	—	2	1	no	—
228	C₂²⋊Q₈	—	2	1	no	—
229	C₄×Q₈	—	2	1	no	—
230	C₄×Q₈	—	2	1	no	—
231	#117	—	2	1	no	—
232	C₂×C₄⋊C₄	—	2	1	no	—
233	C₂×C₄⋊C₄	—	2	1	no	—
234	C₂×C₂²⋊C₄	—	2	8	no	—
235	C₂×C₄²	—	2	1	no	—
236	C₂×C₄²	—	2	1	no	—
237	C₂×C₄²	D₄×C₁₃₇	1	2	yes	—
238	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
239	C₄⋊C₈	—	2	1	no	—
240	C₄⋊C₈	—	2	1	no	—
241	C₄⋊C₈	D₄×C₁₃₇	1	2	yes	—
242	C₄⋊C₈	D₄×C₁₃₇	1	2	yes	—
243	#103	—	2	4	no	—
244	#102	—	2	4	no	—
245	C₂²⋊C₈	—	2	1	no	—
246	C₂²⋊C₈	—	2	1	no	—
247	C₄×C₈	—	2	1	no	—
248	C₄×C₈	—	2	1	no	—
249	C₂⁴	—	2	2	no	—
250	C₂×Q₈	—	2	2	no	—
251	C₂×Q₈	—	2	1	no	—
252	C₂×Q₈	—	2	1	no	—
253	C₂×Q₈	—	2	2	no	—
254	C₂×Q₈	—	4	4	no	—
255	C₂×D₄	—	2	2	no	—
256	C₂×D₄	—	2	2	no	—
257	C₂×D₄	—	4	4	no	—
258	C₂×D₄	—	4	4	no	—
259	C₂×D₄	—	4	1	no	—
260	C₂×D₄	—	4	1	no	—
261	C₂×D₄	—	4	1	no	—
262	C₂×D₄	—	4	1	no	—
263	C₂×D₄	—	4	2	no	—
264	C₂×D₄	—	4	2	no	—
265	C₂²×C₄	—	2	2	no	—
266	C₂²×C₄	—	2	2	no	—
267	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
268	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
269	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
270	Q₁₆	—	8	4	no	—
271	SD₁₆	—	8	4	no	—
272	M₁₆	—	4	2	no	—
273	C₂×C₈	—	2	2	no	—
274	C₂×C₈	—	2	2	no	—
275	C₂×C₈	—	4	1	no	—
276	C₄⋊C₄	—	2	1	no	—
277	C₄⋊C₄	—	2	1	no	—
278	C₄⋊C₄	—	2	1	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	C₄⋊C₄	—	4	1	no	—
285	C₄⋊C₄	—	4	1	no	—
286	C₂²⋊C₄	—	4	16	no	—
287	C₂²⋊C₄	—	4	1	no	—
288	C₄²	—	2	4	no	—
289	C₄²	—	2	2	no	—
290	C₄²	—	2	2	no	—
291	C₄²	—	4	2	no	—
292	C₄²	—	4	1	no	—
293	C₄²	C₂×D₄×C₁₃₇	1	1	yes	characteristic
294	C₄²	C₂×D₄×C₁₃₇	1	1	yes	characteristic
295	C₄²	C₂×D₄×C₁₃₇	1	1	yes	characteristic
296	C₄²	#41×C₁₃₇	1	1	yes	characteristic
297	C₂³	—	4	4	no	—
298	C₂³	—	4	4	no	—
299	C₂³	—	4	2	no	—
300	C₂³	—	4	4	no	—
301	C₂³	#120×C₁₃₇	1	1	yes	characteristic
302	Q₈	—	4	2	no	—
303	Q₈	—	4	4	no	—
304	Q₈	—	4	2	no	—
305	Q₈	—	8	2	no	—
306	D₄	—	4	4	no	—
307	D₄	—	8	2	no	—
308	D₄	—	8	2	no	—
309	D₄	—	8	2	no	—
310	D₄	—	8	2	no	—
311	D₄	—	8	1	no	—
312	D₄	—	8	1	no	—
313	C₂×C₄	—	2	2	no	—
314	C₂×C₄	—	2	2	no	—
315	C₂×C₄	—	2	4	no	—
316	C₂×C₄	—	4	2	no	—
317	C₂×C₄	—	4	2	no	—
318	C₂×C₄	—	4	4	no	—
319	C₂×C₄	—	4	1	no	—
320	C₂×C₄	—	4	1	no	—
321	C₂×C₄	—	4	2	no	—
322	C₂×C₄	#120×C₁₃₇	1	1	yes	characteristic
323	C₂×C₄	#120×C₁₃₇	1	1	yes	characteristic
324	C₂×C₄	#121×C₁₃₇	1	1	yes	characteristic
325	C₂×C₄	#121×C₁₃₇	1	1	yes	characteristic
326	C₂×C₄	#121×C₁₃₇	1	1	yes	characteristic
327	C₂×C₄	#127×C₁₃₇	1	1	yes	characteristic
328	C₂×C₄	#135×C₁₃₇	1	2	yes	—
329	C₂×C₄	#137×C₁₃₇	1	2	yes	—
330	C₈	—	4	2	no	—
331	C₈	—	4	2	no	—
332	C₈	—	8	1	no	—
333	C₂²	—	2	1	no	—
334	C₂²	—	2	1	no	—
335	C₂²	—	8	8	no	—
336	C₂²	—	8	2	no	—
337	C₂²	—	8	2	no	—
338	C₂²	—	8	2	no	—
339	C₂²	#392×C₁₃₇	1	1	yes	characteristic
340	C₂²	#452×C₁₃₇	1	2	yes	—
341	C₄	—	2	1	no	—
342	C₄	—	2	1	no	—
343	C₄	—	4	4	no	—
344	C₄	—	8	2	no	—
345	C₄	—	8	1	no	—
346	C₄	—	8	1	no	—
347	C₄	#453×C₁₃₇	1	2	yes	—
348	C₄	#455×C₁₃₇	1	2	yes	—
349	C₄	#464×C₁₃₇	1	2	yes	—
350	C₄	#495×C₁₃₇	1	1	yes	characteristic
351	C₄	#497×C₁₃₇	1	1	yes	characteristic
352	C₂	—	2	2	no	—
353	C₂	—	16	2	no	—
354	C₂	#2002×C₁₃₇	1	1	yes	characteristic
355	C₂	#2004×C₁₃₇	1	1	yes	characteristic
356	C₂	#2015×C₁₃₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃₇	#392
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₁₃₇
prime radical	C₂×#127×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
1. #3209×C₁₃₇	C₂	semidirect
4. #1994×C₁₃₇	C₂	semidirect
5. #1548×C₁₃₇	C₂	semidirect
7. #1276×C₁₃₇	C₂	semidirect
178. #10291	C₁₃₇	direct (N × H)
179. C₁₃₇	#10291	direct (N × H)
180. #3209	C₂×C₁₃₇	semidirect
183. #1994	C₂×C₁₃₇	semidirect
184. #1548	C₂×C₁₃₇	semidirect
186. #1276	C₂×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
#3209×C₁₃₇	2
#3208×C₁₃₇	2
#2858×C₁₃₇	2
#1994×C₁₃₇	2
#1548×C₁₃₇	2
#1545×C₁₃₇	2
#1276×C₁₃₇	2
#10291	137

Subgroups — C137×#10291 #46859677

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#10291 #46859677