Subgroups — #31789703

Order 17536.

452 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#135×D₁₃₇	C₂	1	1	yes	maximal, characteristic
2	#22578168	C₂	1	2	yes	maximal, complemented
3	#22577526	C₂	1	1	yes	maximal, characteristic, complemented
4	#22577525	C₂	1	1	yes	maximal, characteristic, complemented
5	#22577524	C₂	1	1	yes	maximal, characteristic, complemented
6	#22577518	C₂	1	2	yes	maximal, complemented
7	#22577517	C₂	1	2	yes	maximal, complemented
8	#108×D₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
9	#22577418	C₂	1	2	yes	maximal, complemented
10	#22577360	C₂	1	1	yes	maximal, characteristic, complemented
11	#444×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
12	#41×D₁₃₇	C₂²	1	2	yes	—
13	#17984111	C₄	1	1	yes	characteristic, complemented
14	#17984110	C₂²	1	1	yes	characteristic
15	#17984109	C₄	1	2	yes	complemented
16	#17984108	C₂²	1	2	yes	—
17	#17984107	C₄	1	1	yes	characteristic, complemented
18	Q₁₆×D₁₃₇	C₄	1	1	yes	characteristic, complemented
19	SD₁₆×D₁₃₇	C₄	1	2	yes	complemented
20	D₈×D₁₃₇	C₄	1	1	yes	characteristic, complemented
21	#17984096	—	2	2	no	—
22	#17984096	C₂²	1	2	yes	complemented
23	#17984095	C₂²	1	2	yes	complemented
24	#17984094	—	2	2	no	—
25	#17984094	C₂²	1	2	yes	complemented
26	C₂×#14555493	—	2	2	no	—
27	C₂×#14555493	C₂²	1	1	yes	characteristic
28	M₁₆×D₁₃₇	—	2	2	no	—
29	M₁₆×D₁₃₇	C₂²	1	2	yes	—
30	C₂×C₈×D₁₃₇	C₂²	1	1	yes	characteristic
31	#17984045	C₂²	1	1	yes	characteristic
32	#17984024	C₂²	1	1	yes	characteristic, complemented
33	#17984023	C₂²	1	2	yes	complemented
34	#17984016	C₂²	1	2	yes	complemented
35	#17984015	C₂²	1	2	yes	complemented
36	#17984014	C₂²	1	2	yes	complemented
37	#17983998	C₂²	1	1	yes	characteristic, complemented
38	C₄×#11764040	C₂²	1	1	yes	characteristic, complemented
39	C₈×Dic₁₃₇	C₂²	1	1	yes	characteristic, complemented
40	#135×C₁₃₇	C₂²	1	1	yes	characteristic
41	#131×C₁₃₇	C₂²	1	2	yes	complemented
42	#108×C₁₃₇	C₂²	1	1	yes	characteristic, complemented
43	#104×C₁₃₇	C₂²	1	2	yes	complemented
44	#97×C₁₃₇	C₂²	1	1	yes	characteristic, complemented
45	#14555511	C₂×C₄	1	2	yes	—
46	#14555510	—	2	2	no	—
47	#14555510	C₂³	1	2	yes	—
48	#14555509	—	2	2	no	—
49	#14555509	C₂×C₄	1	2	yes	—
50	Q₈×D₁₃₇	C₂×C₄	1	2	yes	—
51	D₄×D₁₃₇	—	2	2	no	—
52	D₄×D₁₃₇	C₂×C₄	1	2	yes	—
53	C₂×C₄×D₁₃₇	—	2	2	no	—
54	C₂×C₄×D₁₃₇	C₂³	1	1	yes	characteristic
55	Dic₅₄₈	C₂×C₄	1	1	yes	characteristic
56	#14555500	C₂×C₄	1	2	yes	—
57	#14555499	C₂×C₄	1	2	yes	—
58	#14555498	C₂×C₄	1	2	yes	—
59	#14555497	C₂×C₄	1	2	yes	—
60	D₁₀₉₆	C₂×C₄	1	1	yes	characteristic
61	#14555495	C₂×C₄	1	2	yes	—
62	#14555494	—	2	2	no	—
63	#14555494	C₂³	1	2	yes	—
64	#14555493	—	2	2	no	—
65	#14555493	—	2	2	no	—
66	#14555493	—	2	2	no	—
67	#14555493	D₄	1	2	yes	—
68	#14555493	D₄	1	2	yes	—
69	C₂×#11764040	—	2	2	no	—
70	C₂×#11764040	C₂³	1	1	yes	characteristic
71	C₂×#11764040	C₂³	1	1	yes	characteristic
72	C₈×D₁₃₇	—	2	2	no	—
73	C₈×D₁₃₇	—	2	2	no	—
74	C₈×D₁₃₇	—	2	2	no	—
75	C₈×D₁₃₇	C₂×C₄	1	1	yes	characteristic
76	C₈×D₁₃₇	C₂×C₄	1	1	yes	characteristic
77	C₈×D₁₃₇	C₂×C₄	1	1	yes	characteristic
78	C₈×D₁₃₇	C₂×C₄	1	1	yes	characteristic
79	#14555489	—	2	1	no	—
80	#14555487	—	2	1	no	—
81	C₄×Dic₁₃₇	C₂³	1	1	yes	characteristic
82	#41×C₁₃₇	C₂³	1	2	yes	—
83	Q₁₆×C₁₃₇	C₂×C₄	1	1	yes	characteristic
84	SD₁₆×C₁₃₇	C₂×C₄	1	2	yes	—
85	D₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
86	M₁₆×C₁₃₇	—	2	2	no	—
87	M₁₆×C₁₃₇	C₂³	1	2	yes	—
88	C₂×C₈×C₁₃₇	—	2	2	no	—
89	C₂×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
90	C₂×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
91	C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
92	C₂²×D₁₃₇	—	2	2	no	—
93	C₂²×D₁₃₇	#41	1	1	yes	characteristic
94	Dic₂₇₄	—	2	2	no	—
95	Dic₂₇₄	C₂²×C₄	1	2	yes	—
96	D₅₄₈	—	2	2	no	—
97	D₅₄₈	C₂²×C₄	1	2	yes	—
98	C₂²⋊D₁₃₇	—	2	2	no	—
99	C₂²⋊D₁₃₇	—	2	2	no	—
100	C₂²⋊D₁₃₇	—	2	2	no	—
101	C₂×Dic₁₃₇	—	2	2	no	—
102	C₂×Dic₁₃₇	—	2	1	no	—
103	C₂×Dic₁₃₇	#41	1	1	yes	characteristic
104	C₄×D₁₃₇	—	2	2	no	—
105	C₄×D₁₃₇	—	2	2	no	—
106	C₄×D₁₃₇	—	2	2	no	—
107	C₄×D₁₃₇	C₂²×C₄	1	1	yes	characteristic
108	C₄×D₁₃₇	C₂²×C₄	1	1	yes	characteristic
109	C₄×D₁₃₇	C₂×D₄	1	1	yes	characteristic
110	C₄×D₁₃₇	C₂×D₄	1	1	yes	characteristic
111	#11764040	—	2	2	no	—
112	#11764040	C₂²×C₄	1	1	yes	characteristic
113	#11764040	C₂²×C₄	1	1	yes	characteristic
114	#11764040	C₂×D₄	1	2	yes	—
115	Q₈×C₁₃₇	C₂²×C₄	1	2	yes	—
116	D₄×C₁₃₇	—	2	2	no	—
117	D₄×C₁₃₇	C₂²×C₄	1	2	yes	—
118	C₂×C₄×C₁₃₇	—	2	2	no	—
119	C₂×C₄×C₁₃₇	—	2	1	no	—
120	C₂×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
121	C₈×C₁₃₇	—	2	2	no	—
122	C₈×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
123	C₈×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
124	C₈×C₁₃₇	C₂×D₄	1	2	yes	—
125	C₂²×C₁₃₇	—	2	2	no	—
126	C₂²×C₁₃₇	C₂×#41	1	1	yes	characteristic
127	C₂×D₁₃₇	—	2	2	no	—
128	C₂×D₁₃₇	—	4	1	no	—
129	C₂×D₁₃₇	—	8	2	no	—
130	C₂×D₁₃₇	C₄×D₄	1	1	yes	characteristic
131	C₂×D₁₃₇	C₄×D₄	1	1	yes	characteristic
132	C₄×C₁₃₇	—	2	2	no	—
133	C₄×C₁₃₇	—	4	1	no	—
134	C₄×C₁₃₇	C₂³×C₄	1	1	yes	characteristic
135	C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
136	Dic₁₃₇	—	2	2	no	—
137	Dic₁₃₇	—	4	1	no	—
138	Dic₁₃₇	C₄×D₄	1	1	yes	characteristic
139	Dic₁₃₇	C₄×D₄	1	1	yes	characteristic
140	C₂×C₁₃₇	—	2	1	no	—
141	C₂×C₁₃₇	—	4	2	no	—
142	C₂×C₁₃₇	C₂×C₄×D₄	1	1	yes	characteristic
143	D₁₃₇	—	2	1	no	—
144	D₁₃₇	—	2	1	no	—
145	D₁₃₇	—	4	2	no	—
146	C₁₃₇	#2957	1	1	yes	characteristic, complemented, Sylow
147	#2957	—	137	1	no	maximal, Sylow
148	C₂×#135	—	137	1	no	—
149	#568	—	137	2	no	—
150	#444	—	137	1	no	—
151	#444	—	137	1	no	—
152	#444	—	137	1	no	—
153	#444	D₁₃₇	1	1	yes	characteristic, complemented
154	#443	—	137	2	no	—
155	#443	—	137	2	no	—
156	C₂×#108	—	137	1	no	—
157	#421	—	137	2	no	—
158	#405	—	137	1	no	—
159	C₂×#41	—	137	2	no	—
160	#135	—	137	2	no	—
161	#135	—	137	1	no	—
162	#135	—	137	2	no	—
163	#135	—	137	1	no	—
164	#135	—	137	1	no	—
165	#135	C₂×D₁₃₇	1	1	yes	characteristic
166	C₂×Q₁₆	—	137	1	no	—
167	C₂×SD₁₆	—	137	2	no	—
168	C₂×D₈	—	137	1	no	—
169	#131	—	274	2	no	—
170	#131	—	274	2	no	—
171	#131	—	137	2	no	—
172	#131	—	137	2	no	—
173	#131	—	137	2	no	—
174	#131	C₂×D₁₃₇	1	2	yes	complemented
175	C₂×M₁₆	—	274	2	no	—
176	C₂×M₁₆	—	274	2	no	—
177	C₂×M₁₆	—	137	2	no	—
178	C₂×M₁₆	—	137	1	no	—
179	C₂²×C₈	—	137	1	no	—
180	#117	—	137	1	no	—
181	#108	—	137	2	no	—
182	#108	—	137	1	no	—
183	#108	C₂×D₁₃₇	1	1	yes	characteristic, complemented
184	#104	—	137	2	no	—
185	#104	—	137	2	no	—
186	#104	—	137	2	no	—
187	#104	C₂×D₁₃₇	1	2	yes	complemented
188	#97	—	137	1	no	—
189	#97	C₂×D₁₃₇	1	1	yes	characteristic, complemented
190	C₄×C₈	—	137	1	no	—
191	C₄×C₈	—	137	1	no	—
192	#41	—	274	2	no	—
193	#41	—	274	2	no	—
194	#41	—	137	2	no	—
195	#41	—	137	2	no	—
196	#41	—	137	2	no	—
197	#41	C₂²×D₁₃₇	1	2	yes	—
198	C₂×Q₈	—	137	2	no	—
199	C₂×D₄	—	274	2	no	—
200	C₂×D₄	—	137	2	no	—
201	C₂²×C₄	—	274	2	no	—
202	C₂²×C₄	—	137	1	no	—
203	Q₁₆	—	137	2	no	—
204	Q₁₆	—	137	1	no	—
205	Q₁₆	C₄×D₁₃₇	1	1	yes	characteristic
206	SD₁₆	—	137	2	no	—
207	SD₁₆	—	137	2	no	—
208	SD₁₆	—	137	2	no	—
209	SD₁₆	C₄×D₁₃₇	1	2	yes	—
210	D₈	—	137	2	no	—
211	D₈	—	137	1	no	—
212	D₈	C₄×D₁₃₇	1	1	yes	characteristic
213	M₁₆	—	2	2	no	—
214	M₁₆	—	274	2	no	—
215	M₁₆	—	274	2	no	—
216	M₁₆	—	274	2	no	—
217	M₁₆	—	274	2	no	—
218	M₁₆	—	137	2	no	—
219	M₁₆	—	137	2	no	—
220	M₁₆	—	137	2	no	—
221	M₁₆	C₂²×D₁₃₇	1	2	yes	—
222	C₂×C₈	—	2	2	no	—
223	C₂×C₈	—	274	2	no	—
224	C₂×C₈	—	274	2	no	—
225	C₂×C₈	—	274	2	no	—
226	C₂×C₈	—	274	2	no	—
227	C₂×C₈	—	137	1	no	—
228	C₂×C₈	—	137	1	no	—
229	C₂×C₈	—	137	1	no	—
230	C₂×C₈	—	137	1	no	—
231	C₂×C₈	—	137	1	no	—
232	C₂×C₈	—	137	1	no	—
233	C₂×C₈	C₂²×D₁₃₇	1	1	yes	characteristic
234	C₂×C₈	C₂²×D₁₃₇	1	1	yes	characteristic
235	C₄⋊C₄	—	274	1	no	—
236	C₂²⋊C₄	—	274	1	no	—
237	C₄²	—	137	1	no	—
238	C₄²	C₂²×D₁₃₇	1	1	yes	characteristic
239	C₂³	—	274	2	no	—
240	C₂³	—	137	1	no	—
241	Q₈	—	274	2	no	—
242	Q₈	—	137	2	no	—
243	Q₈	C₂×C₄×D₁₃₇	1	2	yes	—
244	D₄	—	2	2	no	—
245	D₄	—	274	2	no	—
246	D₄	—	274	2	no	—
247	D₄	—	274	2	no	—
248	D₄	—	274	2	no	—
249	D₄	—	137	2	no	—
250	D₄	C₂×C₄×D₁₃₇	1	2	yes	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	2	1	no	—
253	C₂×C₄	—	274	2	no	—
254	C₂×C₄	—	274	2	no	—
255	C₂×C₄	—	274	2	no	—
256	C₂×C₄	—	274	2	no	—
257	C₂×C₄	—	274	1	no	—
258	C₂×C₄	—	137	1	no	—
259	C₂×C₄	—	137	1	no	—
260	C₂×C₄	—	137	1	no	—
261	C₂×C₄	—	137	1	no	—
262	C₂×C₄	—	137	1	no	—
263	C₂×C₄	C₂³×D₁₃₇	1	1	yes	characteristic
264	C₈	—	2	2	no	—
265	C₈	—	274	2	no	—
266	C₈	—	137	1	no	—
267	C₈	—	137	1	no	—
268	C₈	—	137	2	no	—
269	C₈	C₂×C₄×D₁₃₇	1	1	yes	characteristic
270	C₈	C₂×C₄×D₁₃₇	1	1	yes	characteristic
271	C₈	D₄×D₁₃₇	1	2	yes	—
272	C₂²	—	2	2	no	—
273	C₂²	—	274	2	no	—
274	C₂²	—	548	1	no	—
275	C₂²	—	1096	2	no	—
276	C₂²	—	137	1	no	—
277	C₂²	—	137	1	no	—
278	C₂²	#41×D₁₃₇	1	1	yes	characteristic
279	C₄	—	2	2	no	—
280	C₄	—	4	1	no	—
281	C₄	—	274	2	no	—
282	C₄	—	548	1	no	—
283	C₄	—	137	1	no	—
284	C₄	—	137	1	no	—
285	C₄	C₂²×C₄×D₁₃₇	1	1	yes	characteristic
286	C₄	C₂×D₄×D₁₃₇	1	1	yes	characteristic
287	C₂	—	2	1	no	—
288	C₂	—	4	2	no	—
289	C₂	—	274	1	no	—
290	C₂	—	274	1	no	—
291	C₂	—	548	2	no	—
292	C₂	C₄×D₄×D₁₃₇	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×D₄×D₁₃₇
commutator	C₄×C₁₃₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₃₇
Fitting	#444×C₁₃₇	C₂
socle	C₂×C₁₃₇	C₂×C₄×D₄
prime radical	#135×D₁₃₇	C₂
cosocle	C₂×C₄×C₁₃₇	C₂⁴
hypercenter	#444	D₁₃₇
hypocenter	C₁₃₇	#2957
LCS	G ▷ C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ #444

Complements

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

N	H ≅ G/N	type
2. #22578168	C₂	semidirect
3. #22577526	C₂	semidirect
4. #22577525	C₂	semidirect
5. #22577524	C₂	semidirect
6. #22577518	C₂	semidirect
7. #22577517	C₂	semidirect
8. #108×D₁₃₇	C₂	semidirect
9. #22577418	C₂	semidirect
10. #22577360	C₂	semidirect
11. #444×C₁₃₇	C₂	semidirect
13. #17984111	C₄	semidirect
15. #17984109	C₄	semidirect
17. #17984107	C₄	semidirect
18. Q₁₆×D₁₃₇	C₄	semidirect
19. SD₁₆×D₁₃₇	C₄	semidirect
20. D₈×D₁₃₇	C₄	semidirect
22. #17984096	C₂²	semidirect
23. #17984095	C₂²	semidirect
25. #17984094	C₂²	semidirect
32. #17984024	C₂²	semidirect
33. #17984023	C₂²	semidirect
34. #17984016	C₂²	semidirect
35. #17984015	C₂²	semidirect
36. #17984014	C₂²	semidirect
37. #17983998	C₂²	semidirect
38. C₄×#11764040	C₂²	semidirect
39. C₈×Dic₁₃₇	C₂²	semidirect
41. #131×C₁₃₇	C₂²	semidirect
42. #108×C₁₃₇	C₂²	semidirect
43. #104×C₁₃₇	C₂²	semidirect
44. #97×C₁₃₇	C₂²	semidirect
146. C₁₃₇	#2957	semidirect
153. #444	D₁₃₇	semidirect
174. #131	C₂×D₁₃₇	semidirect
183. #108	C₂×D₁₃₇	semidirect
187. #104	C₂×D₁₃₇	semidirect
189. #97	C₂×D₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
#135×D₁₃₇	2
#22578168	2
#22577526	2
#22577525	2
#22577524	2
#22577518	2
#22577517	2
#108×D₁₃₇	2
#22577418	2
#22577360	2
#444×C₁₃₇	2
#2957	137