Subgroups — C₁₃₇×#19268 #46868654

Order 35072.

702 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#494×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
2	#3079×C₁₃₇	C₂	1	2	yes	maximal
3	C₈×#41×C₁₃₇	C₂	1	1	yes	maximal, characteristic
4	#1670×C₁₃₇	C₂	1	4	yes	maximal, complemented
5	#1669×C₁₃₇	C₂	1	4	yes	maximal, complemented
6	#1567×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
7	#1549×C₁₃₇	C₂	1	2	yes	maximal, complemented
8	C₂²×Q₁₆×C₁₃₇	—	2	2	no	—
9	C₂×C₄⋊Q₈×C₁₃₇	C₂²	1	2	yes	—
10	C₄×#41×C₁₃₇	C₂²	1	1	yes	characteristic
11	#494×C₁₃₇	—	2	1	no	—
12	#494×C₁₃₇	—	2	1	no	—
13	#494×C₁₃₇	C₂²	1	4	yes	—
14	#477×C₁₃₇	—	2	2	no	—
15	#476×C₁₃₇	—	2	2	no	—
16	#475×C₁₃₇	C₂²	1	4	yes	—
17	#474×C₁₃₇	C₂²	1	4	yes	—
18	C₈×Q₈×C₁₃₇	C₂²	1	2	yes	—
19	C₈×D₄×C₁₃₇	—	2	2	no	—
20	C₈×D₄×C₁₃₇	C₂²	1	2	yes	—
21	#431×C₁₃₇	—	2	1	no	—
22	#431×C₁₃₇	C₂²	1	1	yes	characteristic
23	#428×C₁₃₇	—	2	2	no	—
24	#423×C₁₃₇	C₂²	1	2	yes	—
25	#417×C₁₃₇	—	2	2	no	—
26	C₂×C₄×C₈×C₁₃₇	—	2	1	no	—
27	C₂×C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
28	#335×C₁₃₇	C₂²	1	4	yes	—
29	#332×C₁₃₇	C₂²	1	8	yes	—
30	C₂×#41×C₁₃₇	D₄	1	1	yes	characteristic
31	C₂²×Q₈×C₁₃₇	—	2	2	no	—
32	C₂×Q₁₆×C₁₃₇	—	2	4	no	—
33	C₂×Q₁₆×C₁₃₇	—	2	4	no	—
34	C₂×Q₁₆×C₁₃₇	—	4	8	no	—
35	C₂×M₁₆×C₁₃₇	—	2	2	no	—
36	C₂²×C₈×C₁₃₇	—	2	2	no	—
37	C₂²×C₈×C₁₃₇	D₄	1	2	yes	—
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	C₂³	1	4	yes	—
42	C₄×Q₈×C₁₃₇	D₄	1	1	yes	characteristic
43	C₄×Q₈×C₁₃₇	C₂³	1	1	yes	characteristic
44	C₄×D₄×C₁₃₇	—	2	1	no	—
45	C₄×D₄×C₁₃₇	—	2	1	no	—
46	C₄×D₄×C₁₃₇	D₄	1	1	yes	characteristic
47	C₄×D₄×C₁₃₇	C₂³	1	1	yes	characteristic
48	#117×C₁₃₇	—	2	1	no	—
49	#117×C₁₃₇	D₄	1	1	yes	characteristic
50	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
51	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
52	C₂×C₄²×C₁₃₇	—	2	1	no	—
53	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
54	#107×C₁₃₇	—	2	4	no	—
55	#106×C₁₃₇	—	2	4	no	—
56	C₄⋊C₈×C₁₃₇	—	2	2	no	—
57	C₄⋊C₈×C₁₃₇	—	2	2	no	—
58	C₄⋊C₈×C₁₃₇	C₂³	1	4	yes	—
59	C₄⋊C₈×C₁₃₇	C₂³	1	2	yes	—
60	#103×C₁₃₇	—	2	4	no	—
61	#102×C₁₃₇	—	2	4	no	—
62	C₂²⋊C₈×C₁₃₇	—	2	1	no	—
63	C₂²⋊C₈×C₁₃₇	—	2	2	no	—
64	C₄×C₈×C₁₃₇	—	2	2	no	—
65	C₄×C₈×C₁₃₇	—	2	1	no	—
66	C₄×C₈×C₁₃₇	D₄	1	1	yes	characteristic
67	C₄×C₈×C₁₃₇	D₄	1	1	yes	characteristic
68	C₄×C₈×C₁₃₇	C₂³	1	2	yes	—
69	#41×C₁₃₇	—	2	2	no	—
70	#41×C₁₃₇	Q₁₆	1	4	yes	complemented
71	C₂×Q₈×C₁₃₇	—	2	4	no	—
72	C₂×Q₈×C₁₃₇	—	4	8	no	—
73	C₂×Q₈×C₁₃₇	—	4	2	no	—
74	C₂×Q₈×C₁₃₇	C₂×D₄	1	1	yes	characteristic
75	C₂×D₄×C₁₃₇	—	2	1	no	—
76	C₂×D₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₁₃₇	—	2	1	no	—
78	C₂²×C₄×C₁₃₇	—	2	1	no	—
79	C₂²×C₄×C₁₃₇	—	2	1	no	—
80	C₂²×C₄×C₁₃₇	—	4	2	no	—
81	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₄×C₁₃₇	#41	1	1	yes	characteristic
84	Q₁₆×C₁₃₇	—	4	8	no	—
85	Q₁₆×C₁₃₇	—	4	8	no	—
86	M₁₆×C₁₃₇	—	4	4	no	—
87	C₂×C₈×C₁₃₇	—	2	4	no	—
88	C₂×C₈×C₁₃₇	—	2	2	no	—
89	C₂×C₈×C₁₃₇	—	2	4	no	—
90	C₂×C₈×C₁₃₇	—	4	4	no	—
91	C₂×C₈×C₁₃₇	C₂×D₄	1	2	yes	—
92	C₂×C₈×C₁₃₇	C₂×D₄	1	2	yes	—
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	1	yes	characteristic
100	C₄⋊C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
101	C₂²⋊C₄×C₁₃₇	—	2	1	no	—
102	C₂²⋊C₄×C₁₃₇	—	2	1	no	—
103	C₂²⋊C₄×C₁₃₇	—	2	1	no	—
104	C₄²×C₁₃₇	—	2	1	no	—
105	C₄²×C₁₃₇	—	2	1	no	—
106	C₄²×C₁₃₇	—	2	1	no	—
107	C₄²×C₁₃₇	C₂×D₄	1	1	yes	characteristic
108	C₄²×C₁₃₇	C₂×D₄	1	1	yes	characteristic
109	C₄²×C₁₃₇	#41	1	1	yes	characteristic
110	C₄²×C₁₃₇	C₂⁴	1	1	yes	characteristic
111	C₂³×C₁₃₇	—	2	1	no	—
112	C₂³×C₁₃₇	#121	1	1	yes	characteristic
113	Q₈×C₁₃₇	—	2	1	no	—
114	Q₈×C₁₃₇	—	4	8	no	—
115	Q₈×C₁₃₇	—	8	4	no	—
116	Q₈×C₁₃₇	C₂×Q₁₆	1	2	yes	complemented
117	D₄×C₁₃₇	—	2	4	no	—
118	D₄×C₁₃₇	—	2	1	no	—
119	D₄×C₁₃₇	C₂×Q₁₆	1	2	yes	complemented
120	C₂×C₄×C₁₃₇	—	2	1	no	—
121	C₂×C₄×C₁₃₇	—	2	2	no	—
122	C₂×C₄×C₁₃₇	—	2	4	no	—
123	C₂×C₄×C₁₃₇	—	2	2	no	—
124	C₂×C₄×C₁₃₇	—	2	1	no	—
125	C₂×C₄×C₁₃₇	—	2	1	no	—
126	C₂×C₄×C₁₃₇	—	4	2	no	—
127	C₂×C₄×C₁₃₇	—	4	2	no	—
128	C₂×C₄×C₁₃₇	—	4	4	no	—
129	C₂×C₄×C₁₃₇	—	8	4	no	—
130	C₂×C₄×C₁₃₇	#121	1	1	yes	characteristic
131	C₂×C₄×C₁₃₇	#121	1	1	yes	characteristic
132	C₂×C₄×C₁₃₇	#121	1	1	yes	characteristic
133	C₂×C₄×C₁₃₇	C₂×Q₁₆	1	2	yes	—
134	C₂×C₄×C₁₃₇	#135	1	2	yes	—
135	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
136	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
137	C₂×C₄×C₁₃₇	C₂×#41	1	1	yes	characteristic
138	C₈×C₁₃₇	—	2	2	no	—
139	C₈×C₁₃₇	—	4	4	no	—
140	C₈×C₁₃₇	—	4	2	no	—
141	C₈×C₁₃₇	AGL(1,ℤ/8)	1	4	yes	—
142	C₂²×C₁₃₇	—	2	1	no	—
143	C₂²×C₁₃₇	—	2	1	no	—
144	C₂²×C₁₃₇	—	2	2	no	—
145	C₂²×C₁₃₇	—	4	1	no	—
146	C₂²×C₁₃₇	—	4	1	no	—
147	C₂²×C₁₃₇	#467	1	2	yes	—
148	C₂²×C₁₃₇	C₂×#121	1	1	yes	characteristic
149	C₄×C₁₃₇	—	2	1	no	—
150	C₄×C₁₃₇	—	2	2	no	—
151	C₄×C₁₃₇	—	2	1	no	—
152	C₄×C₁₃₇	—	4	1	no	—
153	C₄×C₁₃₇	—	4	1	no	—
154	C₄×C₁₃₇	—	8	4	no	—
155	C₄×C₁₃₇	#467	1	2	yes	—
156	C₄×C₁₃₇	#473	1	2	yes	—
157	C₄×C₁₃₇	C₂²×Q₁₆	1	1	yes	characteristic
158	C₄×C₁₃₇	C₂×#135	1	1	yes	characteristic
159	C₄×C₁₃₇	C₂×AGL(1,ℤ/8)	1	2	yes	—
160	C₂×C₁₃₇	—	2	2	no	—
161	C₂×C₁₃₇	—	4	2	no	—
162	C₂×C₁₃₇	#3038	1	1	yes	characteristic
163	C₂×C₁₃₇	C₂×#467	1	1	yes	characteristic
164	C₂×C₁₃₇	C₂×#473	1	1	yes	characteristic
165	#19268	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
166	C₁₃₇	#19268	1	1	yes	characteristic, direct product factor, Sylow
167	C₂×#494	C₂×C₁₃₇	1	1	yes	characteristic, complemented
168	#3079	C₂×C₁₃₇	1	2	yes	—
169	C₈×#41	C₂×C₁₃₇	1	1	yes	characteristic
170	#1670	C₂×C₁₃₇	1	4	yes	complemented
171	#1669	C₂×C₁₃₇	1	4	yes	complemented
172	#1567	C₂×C₁₃₇	1	1	yes	characteristic, complemented
173	#1549	C₂×C₁₃₇	1	2	yes	complemented
174	C₂²×Q₁₆	—	2	2	no	—
175	C₂×C₄⋊Q₈	C₂²×C₁₃₇	1	2	yes	—
176	C₄×#41	C₂²×C₁₃₇	1	1	yes	characteristic
177	#494	—	2	1	no	—
178	#494	—	2	1	no	—
179	#494	C₂²×C₁₃₇	1	4	yes	—
180	#477	—	2	2	no	—
181	#476	—	2	2	no	—
182	#475	C₂²×C₁₃₇	1	4	yes	—
183	#474	C₂²×C₁₃₇	1	4	yes	—
184	C₈×Q₈	C₂²×C₁₃₇	1	2	yes	—
185	C₈×D₄	—	2	2	no	—
186	C₈×D₄	C₂²×C₁₃₇	1	2	yes	—
187	#431	—	2	1	no	—
188	#431	C₂²×C₁₃₇	1	1	yes	characteristic
189	#428	—	2	2	no	—
190	#423	C₂²×C₁₃₇	1	2	yes	—
191	#417	—	2	2	no	—
192	C₂×C₄×C₈	—	2	1	no	—
193	C₂×C₄×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
194	#335	C₂²×C₁₃₇	1	4	yes	—
195	#332	C₂²×C₁₃₇	1	8	yes	—
196	C₂×#41	D₄×C₁₃₇	1	1	yes	characteristic
197	C₂²×Q₈	—	2	2	no	—
198	C₂×Q₁₆	—	2	4	no	—
199	C₂×Q₁₆	—	2	4	no	—
200	C₂×Q₁₆	—	4	8	no	—
201	C₂×M₁₆	—	2	2	no	—
202	C₂²×C₈	—	2	2	no	—
203	C₂²×C₈	D₄×C₁₃₇	1	2	yes	—
204	C₄⋊Q₈	—	2	2	no	—
205	C₄⋊Q₈	—	2	2	no	—
206	C₄⋊Q₈	—	2	2	no	—
207	C₄⋊Q₈	C₂³×C₁₃₇	1	4	yes	—
208	C₄×Q₈	D₄×C₁₃₇	1	1	yes	characteristic
209	C₄×Q₈	C₂³×C₁₃₇	1	1	yes	characteristic
210	C₄×D₄	—	2	1	no	—
211	C₄×D₄	—	2	1	no	—
212	C₄×D₄	D₄×C₁₃₇	1	1	yes	characteristic
213	C₄×D₄	C₂³×C₁₃₇	1	1	yes	characteristic
214	#117	—	2	1	no	—
215	#117	D₄×C₁₃₇	1	1	yes	characteristic
216	C₂×C₄⋊C₄	—	2	2	no	—
217	C₂×C₄⋊C₄	—	2	2	no	—
218	C₂×C₄²	—	2	1	no	—
219	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
220	#107	—	2	4	no	—
221	#106	—	2	4	no	—
222	C₄⋊C₈	—	2	2	no	—
223	C₄⋊C₈	—	2	2	no	—
224	C₄⋊C₈	C₂³×C₁₃₇	1	2	yes	—
225	C₄⋊C₈	C₂³×C₁₃₇	1	4	yes	—
226	#103	—	2	4	no	—
227	#102	—	2	4	no	—
228	C₂²⋊C₈	—	2	1	no	—
229	C₂²⋊C₈	—	2	2	no	—
230	C₄×C₈	—	2	2	no	—
231	C₄×C₈	—	2	1	no	—
232	C₄×C₈	D₄×C₁₃₇	1	1	yes	characteristic
233	C₄×C₈	D₄×C₁₃₇	1	1	yes	characteristic
234	C₄×C₈	C₂³×C₁₃₇	1	2	yes	—
235	#41	—	2	2	no	—
236	#41	Q₁₆×C₁₃₇	1	4	yes	complemented
237	C₂×Q₈	—	2	4	no	—
238	C₂×Q₈	—	4	8	no	—
239	C₂×Q₈	—	4	2	no	—
240	C₂×Q₈	C₂×D₄×C₁₃₇	1	1	yes	characteristic
241	C₂×D₄	—	2	1	no	—
242	C₂×D₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
243	C₂²×C₄	—	2	1	no	—
244	C₂²×C₄	—	2	1	no	—
245	C₂²×C₄	—	2	1	no	—
246	C₂²×C₄	—	4	2	no	—
247	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
248	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
249	C₂²×C₄	#41×C₁₃₇	1	1	yes	characteristic
250	Q₁₆	—	4	8	no	—
251	Q₁₆	—	4	8	no	—
252	M₁₆	—	4	4	no	—
253	C₂×C₈	—	2	4	no	—
254	C₂×C₈	—	2	2	no	—
255	C₂×C₈	—	2	4	no	—
256	C₂×C₈	—	4	4	no	—
257	C₂×C₈	C₂×D₄×C₁₃₇	1	2	yes	—
258	C₂×C₈	C₂×D₄×C₁₃₇	1	2	yes	—
259	C₄⋊C₄	—	2	1	no	—
260	C₄⋊C₄	—	2	1	no	—
261	C₄⋊C₄	—	2	4	no	—
262	C₄⋊C₄	—	2	4	no	—
263	C₄⋊C₄	—	4	2	no	—
264	C₄⋊C₄	—	4	2	no	—
265	C₄⋊C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
266	C₄⋊C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
267	C₂²⋊C₄	—	2	1	no	—
268	C₂²⋊C₄	—	2	1	no	—
269	C₂²⋊C₄	—	2	1	no	—
270	C₄²	—	2	1	no	—
271	C₄²	—	2	1	no	—
272	C₄²	—	2	1	no	—
273	C₄²	C₂×D₄×C₁₃₇	1	1	yes	characteristic
274	C₄²	C₂×D₄×C₁₃₇	1	1	yes	characteristic
275	C₄²	#41×C₁₃₇	1	1	yes	characteristic
276	C₄²	C₂⁴×C₁₃₇	1	1	yes	characteristic
277	C₂³	—	2	1	no	—
278	C₂³	#121×C₁₃₇	1	1	yes	characteristic
279	Q₈	—	2	1	no	—
280	Q₈	—	4	8	no	—
281	Q₈	—	8	4	no	—
282	Q₈	C₂×Q₁₆×C₁₃₇	1	2	yes	complemented
283	D₄	—	2	4	no	—
284	D₄	—	2	1	no	—
285	D₄	C₂×Q₁₆×C₁₃₇	1	2	yes	complemented
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	2	4	no	—
288	C₂×C₄	—	2	2	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	4	2	no	—
293	C₂×C₄	—	4	2	no	—
294	C₂×C₄	—	4	4	no	—
295	C₂×C₄	—	8	4	no	—
296	C₂×C₄	#121×C₁₃₇	1	1	yes	characteristic
297	C₂×C₄	#121×C₁₃₇	1	1	yes	characteristic
298	C₂×C₄	#121×C₁₃₇	1	1	yes	characteristic
299	C₂×C₄	C₂×Q₁₆×C₁₃₇	1	2	yes	—
300	C₂×C₄	#135×C₁₃₇	1	2	yes	—
301	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
302	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
303	C₂×C₄	C₂×#41×C₁₃₇	1	1	yes	characteristic
304	C₈	—	2	2	no	—
305	C₈	—	4	4	no	—
306	C₈	—	4	2	no	—
307	C₈	AGL(1,ℤ/8)×C₁₃₇	1	4	yes	—
308	C₂²	—	2	1	no	—
309	C₂²	—	2	1	no	—
310	C₂²	—	2	2	no	—
311	C₂²	—	4	1	no	—
312	C₂²	—	4	1	no	—
313	C₂²	#467×C₁₃₇	1	2	yes	—
314	C₂²	C₂×#121×C₁₃₇	1	1	yes	characteristic
315	C₄	—	2	1	no	—
316	C₄	—	2	2	no	—
317	C₄	—	2	1	no	—
318	C₄	—	4	1	no	—
319	C₄	—	4	1	no	—
320	C₄	—	8	4	no	—
321	C₄	#467×C₁₃₇	1	2	yes	—
322	C₄	#473×C₁₃₇	1	2	yes	—
323	C₄	C₂²×Q₁₆×C₁₃₇	1	1	yes	characteristic
324	C₄	C₂×#135×C₁₃₇	1	1	yes	characteristic
325	C₄	C₂×AGL(1,ℤ/8)×C₁₃₇	1	2	yes	—
326	C₂	—	2	2	no	—
327	C₂	—	4	2	no	—
328	C₂	#3038×C₁₃₇	1	1	yes	characteristic
329	C₂	C₂×#467×C₁₃₇	1	1	yes	characteristic
330	C₂	C₂×#473×C₁₃₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃₇	C₂×#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₁₃₇
prime radical	C₂×#41×C₁₃₇	D₄
LCS	G ▷ 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. C₂×#494×C₁₃₇	C₂	semidirect
4. #1670×C₁₃₇	C₂	semidirect
5. #1669×C₁₃₇	C₂	semidirect
6. #1567×C₁₃₇	C₂	semidirect
7. #1549×C₁₃₇	C₂	semidirect
70. #41×C₁₃₇	Q₁₆	semidirect
116. Q₈×C₁₃₇	C₂×Q₁₆	semidirect
119. D₄×C₁₃₇	C₂×Q₁₆	semidirect
165. #19268	C₁₃₇	direct (N × H)
166. C₁₃₇	#19268	direct (N × H)
167. C₂×#494	C₂×C₁₃₇	semidirect
170. #1670	C₂×C₁₃₇	semidirect
171. #1669	C₂×C₁₃₇	semidirect
172. #1567	C₂×C₁₃₇	semidirect
173. #1549	C₂×C₁₃₇	semidirect
236. #41	Q₁₆×C₁₃₇	semidirect
282. Q₈	C₂×Q₁₆×C₁₃₇	semidirect
285. D₄	C₂×Q₁₆×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#494×C₁₃₇	2
#3079×C₁₃₇	2
C₈×#41×C₁₃₇	2
#1670×C₁₃₇	2
#1669×C₁₃₇	2
#1567×C₁₃₇	2
#1549×C₁₃₇	2
#19268	137

Subgroups — C137×#19268 #46868654

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#19268 #46868654