Subgroups — C₁₃₇×#20294 #46869680

Order 35072.

608 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#3373×C₁₃₇	C₂	1	2	yes	maximal, complemented
2	#3329×C₁₃₇	C₂	1	2	yes	maximal, complemented
3	#1708×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#1704×C₁₃₇	C₂	1	2	yes	maximal, complemented
5	#1701×C₁₃₇	C₂	1	2	yes	maximal, complemented
6	#1651×C₁₃₇	C₂	1	2	yes	maximal, complemented
7	#1630×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
8	#1624×C₁₃₇	C₂	1	1	yes	maximal, characteristic
9	#1590×C₁₃₇	C₂	1	2	yes	maximal, complemented
10	#559×C₁₃₇	C₂²	1	1	yes	characteristic
11	#555×C₁₃₇	C₂²	1	1	yes	characteristic
12	D₄×Q₈×C₁₃₇	C₂²	1	2	yes	—
13	#498×C₁₃₇	C₂²	1	1	yes	characteristic
14	#492×C₁₃₇	C₂²	1	1	yes	characteristic
15	#489×C₁₃₇	—	2	2	no	—
16	#487×C₁₃₇	C₂²	1	2	yes	—
17	#486×C₁₃₇	C₂²	1	2	yes	—
18	#485×C₁₃₇	—	2	2	no	—
19	#477×C₁₃₇	C₂²	1	2	yes	—
20	#476×C₁₃₇	—	2	2	no	—
21	#475×C₁₃₇	C₂²	1	2	yes	—
22	#468×C₁₃₇	—	2	2	no	—
23	#463×C₁₃₇	—	2	2	no	—
24	#461×C₁₃₇	C₂²	1	2	yes	—
25	#459×C₁₃₇	C₂²	1	2	yes	—
26	#448×C₁₃₇	—	2	2	no	—
27	#446×C₁₃₇	C₂²	1	2	yes	—
28	#442×C₁₃₇	—	2	2	no	—
29	C₄×SD₁₆×C₁₃₇	C₂²	1	2	yes	—
30	#436×C₁₃₇	C₂²	1	2	yes	—
31	#419×C₁₃₇	—	2	2	no	—
32	#334×C₁₃₇	C₂²	1	2	yes	—
33	#333×C₁₃₇	C₂²	1	1	yes	characteristic
34	#332×C₁₃₇	C₂²	1	2	yes	—
35	#331×C₁₃₇	C₂²	1	1	yes	characteristic
36	Q₈⋊C₈×C₁₃₇	C₂²	1	2	yes	—
37	D₄⋊C₈×C₁₃₇	C₂²	1	2	yes	—
38	C₈²×C₁₃₇	C₂²	1	1	yes	characteristic
39	C₂²×Q₈×C₁₃₇	—	2	2	no	—
40	C₂×SD₁₆×C₁₃₇	—	2	2	no	—
41	C₂×SD₁₆×C₁₃₇	—	4	1	no	—
42	C₂×SD₁₆×C₁₃₇	D₄	1	2	yes	—
43	C₂×D₈×C₁₃₇	—	2	2	no	—
44	C₂×M₁₆×C₁₃₇	—	2	2	no	—
45	C₄⋊Q₈×C₁₃₇	—	2	2	no	—
46	C₄⋊Q₈×C₁₃₇	C₂³	1	2	yes	—
47	C₄⋊Q₈×C₁₃₇	C₂³	1	1	yes	characteristic
48	#127×C₁₃₇	C₂³	1	1	yes	characteristic
49	#124×C₁₃₇	—	2	2	no	—
50	#124×C₁₃₇	—	4	1	no	—
51	#123×C₁₃₇	—	4	1	no	—
52	C₂²⋊Q₈×C₁₃₇	—	2	2	no	—
53	C₂²⋊Q₈×C₁₃₇	—	4	2	no	—
54	#121×C₁₃₇	—	2	2	no	—
55	C₄×Q₈×C₁₃₇	C₂³	1	2	yes	—
56	C₄×D₄×C₁₃₇	—	2	2	no	—
57	C₄×D₄×C₁₃₇	—	2	2	no	—
58	C₄×D₄×C₁₃₇	C₂³	1	2	yes	—
59	#117×C₁₃₇	—	2	2	no	—
60	#107×C₁₃₇	—	2	2	no	—
61	#106×C₁₃₇	—	2	2	no	—
62	#106×C₁₃₇	—	4	1	no	—
63	#106×C₁₃₇	D₄	1	2	yes	—
64	C₄⋊C₈×C₁₃₇	—	2	2	no	—
65	C₄⋊C₈×C₁₃₇	C₂³	1	2	yes	—
66	C₄⋊C₈×C₁₃₇	C₂³	1	2	yes	—
67	#103×C₁₃₇	—	2	2	no	—
68	#103×C₁₃₇	—	2	2	no	—
69	#103×C₁₃₇	—	2	2	no	—
70	#103×C₁₃₇	—	2	2	no	—
71	#103×C₁₃₇	—	2	2	no	—
72	#103×C₁₃₇	D₄	1	2	yes	—
73	#102×C₁₃₇	—	2	2	no	—
74	#102×C₁₃₇	—	2	2	no	—
75	#102×C₁₃₇	—	2	2	no	—
76	#102×C₁₃₇	D₄	1	2	yes	—
77	C₂²⋊C₈×C₁₃₇	—	2	2	no	—
78	#97×C₁₃₇	—	2	2	no	—
79	C₄×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
80	C₄×C₈×C₁₃₇	C₂³	1	2	yes	—
81	C₂×Q₈×C₁₃₇	—	2	2	no	—
82	C₂×Q₈×C₁₃₇	—	2	1	no	—
83	C₂×Q₈×C₁₃₇	—	4	2	no	—
84	C₂×Q₈×C₁₃₇	—	4	4	no	—
85	C₂×Q₈×C₁₃₇	—	8	2	no	—
86	C₂×Q₈×C₁₃₇	C₂×D₄	1	2	yes	—
87	C₂×D₄×C₁₃₇	—	2	2	no	—
88	C₂×D₄×C₁₃₇	—	2	1	no	—
89	C₂×D₄×C₁₃₇	—	4	1	no	—
90	C₂×D₄×C₁₃₇	C₂×D₄	1	2	yes	—
91	C₂²×C₄×C₁₃₇	—	2	2	no	—
92	C₂²×C₄×C₁₃₇	—	4	1	no	—
93	C₂²×C₄×C₁₃₇	—	4	2	no	—
94	SD₁₆×C₁₃₇	—	4	2	no	—
95	SD₁₆×C₁₃₇	—	8	2	no	—
96	SD₁₆×C₁₃₇	—	8	2	no	—
97	D₈×C₁₃₇	—	4	4	no	—
98	M₁₆×C₁₃₇	—	8	2	no	—
99	C₂×C₈×C₁₃₇	—	2	1	no	—
100	C₂×C₈×C₁₃₇	—	2	2	no	—
101	C₂×C₈×C₁₃₇	—	2	2	no	—
102	C₂×C₈×C₁₃₇	C₂×D₄	1	2	yes	—
103	C₂×C₈×C₁₃₇	C₂×D₄	1	2	yes	—
104	C₄⋊C₄×C₁₃₇	—	2	2	no	—
105	C₄⋊C₄×C₁₃₇	—	2	2	no	—
106	C₄⋊C₄×C₁₃₇	—	2	2	no	—
107	C₄⋊C₄×C₁₃₇	—	2	2	no	—
108	C₄⋊C₄×C₁₃₇	—	4	2	no	—
109	C₄⋊C₄×C₁₃₇	C₂×D₄	1	2	yes	—
110	C₄⋊C₄×C₁₃₇	C₂×D₄	1	2	yes	—
111	C₂²⋊C₄×C₁₃₇	—	2	2	no	—
112	C₂²⋊C₄×C₁₃₇	—	4	2	no	—
113	C₂²⋊C₄×C₁₃₇	—	4	1	no	—
114	C₂²⋊C₄×C₁₃₇	—	4	2	no	—
115	C₂²⋊C₄×C₁₃₇	—	4	2	no	—
116	C₄²×C₁₃₇	—	2	2	no	—
117	C₄²×C₁₃₇	—	4	1	no	—
118	C₄²×C₁₃₇	C₂⁴	1	1	yes	characteristic
119	C₂³×C₁₃₇	—	2	2	no	—
120	C₂³×C₁₃₇	—	4	1	no	—
121	Q₈×C₁₃₇	—	2	2	no	—
122	Q₈×C₁₃₇	—	2	2	no	—
123	Q₈×C₁₃₇	—	4	2	no	—
124	Q₈×C₁₃₇	—	4	2	no	—
125	Q₈×C₁₃₇	—	8	1	no	—
126	Q₈×C₁₃₇	—	8	4	no	—
127	D₄×C₁₃₇	—	2	2	no	—
128	D₄×C₁₃₇	—	2	2	no	—
129	D₄×C₁₃₇	—	4	2	no	—
130	D₄×C₁₃₇	—	4	2	no	—
131	D₄×C₁₃₇	—	8	1	no	—
132	D₄×C₁₃₇	—	16	1	no	—
133	C₂×C₄×C₁₃₇	—	2	2	no	—
134	C₂×C₄×C₁₃₇	—	2	2	no	—
135	C₂×C₄×C₁₃₇	—	2	2	no	—
136	C₂×C₄×C₁₃₇	—	4	2	no	—
137	C₂×C₄×C₁₃₇	—	4	2	no	—
138	C₂×C₄×C₁₃₇	—	4	1	no	—
139	C₂×C₄×C₁₃₇	—	4	2	no	—
140	C₂×C₄×C₁₃₇	—	8	4	no	—
141	C₂×C₄×C₁₃₇	—	16	1	no	—
142	C₂×C₄×C₁₃₇	C₂²×D₄	1	2	yes	—
143	C₂×C₄×C₁₃₇	H(5,𝔽₂)	1	1	yes	characteristic
144	C₈×C₁₃₇	—	2	2	no	—
145	C₈×C₁₃₇	—	2	2	no	—
146	C₈×C₁₃₇	—	2	2	no	—
147	C₈×C₁₃₇	—	4	2	no	—
148	C₈×C₁₃₇	—	4	2	no	—
149	C₂²×C₁₃₇	—	4	2	no	—
150	C₂²×C₁₃₇	—	4	2	no	—
151	C₂²×C₁₃₇	—	4	2	no	—
152	C₂²×C₁₃₇	—	8	2	no	—
153	C₂²×C₁₃₇	—	8	1	no	—
154	C₂²×C₁₃₇	D₄²	1	1	yes	characteristic
155	C₄×C₁₃₇	—	2	1	no	—
156	C₄×C₁₃₇	—	4	2	no	—
157	C₄×C₁₃₇	—	4	2	no	—
158	C₄×C₁₃₇	—	4	2	no	—
159	C₄×C₁₃₇	—	8	2	no	—
160	C₄×C₁₃₇	—	8	1	no	—
161	C₄×C₁₃₇	#575	1	2	yes	—
162	C₄×C₁₃₇	#577	1	2	yes	—
163	C₂×C₁₃₇	—	8	2	no	—
164	C₂×C₁₃₇	—	16	1	no	—
165	C₂×C₁₃₇	#3275	1	2	yes	—
166	C₂×C₁₃₇	#3291	1	1	yes	characteristic
167	#20294	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
168	C₁₃₇	#20294	1	1	yes	characteristic, direct product factor, Sylow
169	#3373	C₂×C₁₃₇	1	2	yes	complemented
170	#3329	C₂×C₁₃₇	1	2	yes	complemented
171	#1708	C₂×C₁₃₇	1	1	yes	characteristic, complemented
172	#1704	C₂×C₁₃₇	1	2	yes	complemented
173	#1701	C₂×C₁₃₇	1	2	yes	complemented
174	#1651	C₂×C₁₃₇	1	2	yes	complemented
175	#1630	C₂×C₁₃₇	1	1	yes	characteristic, complemented
176	#1624	C₂×C₁₃₇	1	1	yes	characteristic
177	#1590	C₂×C₁₃₇	1	2	yes	complemented
178	#559	C₂²×C₁₃₇	1	1	yes	characteristic
179	#555	C₂²×C₁₃₇	1	1	yes	characteristic
180	D₄×Q₈	C₂²×C₁₃₇	1	2	yes	—
181	#498	C₂²×C₁₃₇	1	1	yes	characteristic
182	#492	C₂²×C₁₃₇	1	1	yes	characteristic
183	#489	—	2	2	no	—
184	#487	C₂²×C₁₃₇	1	2	yes	—
185	#486	C₂²×C₁₃₇	1	2	yes	—
186	#485	—	2	2	no	—
187	#477	C₂²×C₁₃₇	1	2	yes	—
188	#476	—	2	2	no	—
189	#475	C₂²×C₁₃₇	1	2	yes	—
190	#468	—	2	2	no	—
191	#463	—	2	2	no	—
192	#461	C₂²×C₁₃₇	1	2	yes	—
193	#459	C₂²×C₁₃₇	1	2	yes	—
194	#448	—	2	2	no	—
195	#446	C₂²×C₁₃₇	1	2	yes	—
196	#442	—	2	2	no	—
197	C₄×SD₁₆	C₂²×C₁₃₇	1	2	yes	—
198	#436	C₂²×C₁₃₇	1	2	yes	—
199	#419	—	2	2	no	—
200	#334	C₂²×C₁₃₇	1	2	yes	—
201	#333	C₂²×C₁₃₇	1	1	yes	characteristic
202	#332	C₂²×C₁₃₇	1	2	yes	—
203	#331	C₂²×C₁₃₇	1	1	yes	characteristic
204	Q₈⋊C₈	C₂²×C₁₃₇	1	2	yes	—
205	D₄⋊C₈	C₂²×C₁₃₇	1	2	yes	—
206	C₈²	C₂²×C₁₃₇	1	1	yes	characteristic
207	C₂²×Q₈	—	2	2	no	—
208	C₂×SD₁₆	—	2	2	no	—
209	C₂×SD₁₆	—	4	1	no	—
210	C₂×SD₁₆	D₄×C₁₃₇	1	2	yes	—
211	C₂×D₈	—	2	2	no	—
212	C₂×M₁₆	—	2	2	no	—
213	C₄⋊Q₈	—	2	2	no	—
214	C₄⋊Q₈	C₂³×C₁₃₇	1	2	yes	—
215	C₄⋊Q₈	C₂³×C₁₃₇	1	1	yes	characteristic
216	#127	C₂³×C₁₃₇	1	1	yes	characteristic
217	#124	—	2	2	no	—
218	#124	—	4	1	no	—
219	#123	—	4	1	no	—
220	C₂²⋊Q₈	—	2	2	no	—
221	C₂²⋊Q₈	—	4	2	no	—
222	#121	—	2	2	no	—
223	C₄×Q₈	C₂³×C₁₃₇	1	2	yes	—
224	C₄×D₄	—	2	2	no	—
225	C₄×D₄	—	2	2	no	—
226	C₄×D₄	C₂³×C₁₃₇	1	2	yes	—
227	#117	—	2	2	no	—
228	#107	—	2	2	no	—
229	#106	—	2	2	no	—
230	#106	—	4	1	no	—
231	#106	D₄×C₁₃₇	1	2	yes	—
232	C₄⋊C₈	—	2	2	no	—
233	C₄⋊C₈	C₂³×C₁₃₇	1	2	yes	—
234	C₄⋊C₈	C₂³×C₁₃₇	1	2	yes	—
235	#103	—	2	2	no	—
236	#103	—	2	2	no	—
237	#103	—	2	2	no	—
238	#103	—	2	2	no	—
239	#103	—	2	2	no	—
240	#103	D₄×C₁₃₇	1	2	yes	—
241	#102	—	2	2	no	—
242	#102	—	2	2	no	—
243	#102	—	2	2	no	—
244	#102	D₄×C₁₃₇	1	2	yes	—
245	C₂²⋊C₈	—	2	2	no	—
246	#97	—	2	2	no	—
247	C₄×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
248	C₄×C₈	C₂³×C₁₃₇	1	2	yes	—
249	C₂×Q₈	—	2	2	no	—
250	C₂×Q₈	—	2	1	no	—
251	C₂×Q₈	—	4	2	no	—
252	C₂×Q₈	—	4	4	no	—
253	C₂×Q₈	—	8	2	no	—
254	C₂×Q₈	C₂×D₄×C₁₃₇	1	2	yes	—
255	C₂×D₄	—	2	2	no	—
256	C₂×D₄	—	2	1	no	—
257	C₂×D₄	—	4	1	no	—
258	C₂×D₄	C₂×D₄×C₁₃₇	1	2	yes	—
259	C₂²×C₄	—	2	2	no	—
260	C₂²×C₄	—	4	1	no	—
261	C₂²×C₄	—	4	2	no	—
262	SD₁₆	—	4	2	no	—
263	SD₁₆	—	8	2	no	—
264	SD₁₆	—	8	2	no	—
265	D₈	—	4	4	no	—
266	M₁₆	—	8	2	no	—
267	C₂×C₈	—	2	1	no	—
268	C₂×C₈	—	2	2	no	—
269	C₂×C₈	—	2	2	no	—
270	C₂×C₈	C₂×D₄×C₁₃₇	1	2	yes	—
271	C₂×C₈	C₂×D₄×C₁₃₇	1	2	yes	—
272	C₄⋊C₄	—	2	2	no	—
273	C₄⋊C₄	—	2	2	no	—
274	C₄⋊C₄	—	2	2	no	—
275	C₄⋊C₄	—	2	2	no	—
276	C₄⋊C₄	—	4	2	no	—
277	C₄⋊C₄	C₂×D₄×C₁₃₇	1	2	yes	—
278	C₄⋊C₄	C₂×D₄×C₁₃₇	1	2	yes	—
279	C₂²⋊C₄	—	2	2	no	—
280	C₂²⋊C₄	—	4	2	no	—
281	C₂²⋊C₄	—	4	1	no	—
282	C₂²⋊C₄	—	4	2	no	—
283	C₂²⋊C₄	—	4	2	no	—
284	C₄²	—	2	2	no	—
285	C₄²	—	4	1	no	—
286	C₄²	C₂⁴×C₁₃₇	1	1	yes	characteristic
287	C₂³	—	2	2	no	—
288	C₂³	—	4	1	no	—
289	Q₈	—	2	2	no	—
290	Q₈	—	2	2	no	—
291	Q₈	—	4	2	no	—
292	Q₈	—	4	2	no	—
293	Q₈	—	8	1	no	—
294	Q₈	—	8	4	no	—
295	D₄	—	2	2	no	—
296	D₄	—	2	2	no	—
297	D₄	—	4	2	no	—
298	D₄	—	4	2	no	—
299	D₄	—	8	1	no	—
300	D₄	—	16	1	no	—
301	C₂×C₄	—	2	2	no	—
302	C₂×C₄	—	2	2	no	—
303	C₂×C₄	—	2	2	no	—
304	C₂×C₄	—	4	2	no	—
305	C₂×C₄	—	4	2	no	—
306	C₂×C₄	—	4	1	no	—
307	C₂×C₄	—	4	2	no	—
308	C₂×C₄	—	8	4	no	—
309	C₂×C₄	—	16	1	no	—
310	C₂×C₄	C₂²×D₄×C₁₃₇	1	2	yes	—
311	C₂×C₄	H(5,𝔽₂)×C₁₃₇	1	1	yes	characteristic
312	C₈	—	2	2	no	—
313	C₈	—	2	2	no	—
314	C₈	—	2	2	no	—
315	C₈	—	4	2	no	—
316	C₈	—	4	2	no	—
317	C₂²	—	4	2	no	—
318	C₂²	—	4	2	no	—
319	C₂²	—	4	2	no	—
320	C₂²	—	8	2	no	—
321	C₂²	—	8	1	no	—
322	C₂²	D₄²×C₁₃₇	1	1	yes	characteristic
323	C₄	—	2	1	no	—
324	C₄	—	4	2	no	—
325	C₄	—	4	2	no	—
326	C₄	—	4	2	no	—
327	C₄	—	8	2	no	—
328	C₄	—	8	1	no	—
329	C₄	#575×C₁₃₇	1	2	yes	—
330	C₄	#577×C₁₃₇	1	2	yes	—
331	C₂	—	8	2	no	—
332	C₂	—	16	1	no	—
333	C₂	#3275×C₁₃₇	1	2	yes	—
334	C₂	#3291×C₁₃₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃₇	D₄²
commutator, Frattini, cosocle	C₄²	C₂⁴×C₁₃₇
prime radical	#1624×C₁₃₇	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃₇ ◁ C₄²×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. #3373×C₁₃₇	C₂	semidirect
2. #3329×C₁₃₇	C₂	semidirect
3. #1708×C₁₃₇	C₂	semidirect
4. #1704×C₁₃₇	C₂	semidirect
5. #1701×C₁₃₇	C₂	semidirect
6. #1651×C₁₃₇	C₂	semidirect
7. #1630×C₁₃₇	C₂	semidirect
9. #1590×C₁₃₇	C₂	semidirect
167. #20294	C₁₃₇	direct (N × H)
168. C₁₃₇	#20294	direct (N × H)
169. #3373	C₂×C₁₃₇	semidirect
170. #3329	C₂×C₁₃₇	semidirect
171. #1708	C₂×C₁₃₇	semidirect
172. #1704	C₂×C₁₃₇	semidirect
173. #1701	C₂×C₁₃₇	semidirect
174. #1651	C₂×C₁₃₇	semidirect
175. #1630	C₂×C₁₃₇	semidirect
177. #1590	C₂×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
#3373×C₁₃₇	2
#3329×C₁₃₇	2
#1708×C₁₃₇	2
#1704×C₁₃₇	2
#1701×C₁₃₇	2
#1651×C₁₃₇	2
#1630×C₁₃₇	2
#1624×C₁₃₇	2
#1590×C₁₃₇	2
#20294	137

Subgroups — C137×#20294 #46869680

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#20294 #46869680