Subgroups — #31789697

Order 17536.

458 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22578200	C₂	1	1	yes	maximal, characteristic
2	#22578168	C₂	1	2	yes	maximal, complemented
3	#22577526	C₂	1	1	yes	maximal, characteristic, complemented
4	#22577522	C₂	1	2	yes	maximal, complemented
5	#22577520	C₂	1	1	yes	maximal, characteristic, complemented
6	#22577519	C₂	1	1	yes	maximal, characteristic, complemented
7	#22577518	C₂	1	2	yes	maximal, complemented
8	#22577450	C₂	1	1	yes	maximal, characteristic, complemented
9	#104×D₁₃₇	C₂	1	2	yes	maximal, complemented
10	#22577355	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	#17984120	C₄	1	2	yes	complemented
14	#17984117	C₄	1	2	yes	complemented
15	#17984115	C₄	1	2	yes	complemented
16	#17984112	C₄	1	2	yes	complemented
17	#17984110	C₂²	1	1	yes	characteristic
18	#17984108	C₂²	1	2	yes	—
19	#17984096	—	2	2	no	—
20	#17984096	C₂²	1	2	yes	complemented
21	#17984095	C₂²	1	2	yes	complemented
22	#17984094	—	2	2	no	—
23	#17984094	C₂²	1	2	yes	complemented
24	C₂×#14555493	—	2	2	no	—
25	C₂×#14555493	C₂²	1	1	yes	characteristic
26	C₂×#14555493	C₂²	1	1	yes	characteristic
27	M₁₆×D₁₃₇	—	2	2	no	—
28	M₁₆×D₁₃₇	C₂²	1	2	yes	—
29	C₄²×D₁₃₇	C₂²	1	1	yes	characteristic
30	#17984024	C₂²	1	1	yes	characteristic, complemented
31	#17984023	C₂²	1	2	yes	complemented
32	#17984016	C₂²	1	2	yes	complemented
33	#17984015	C₂²	1	2	yes	complemented
34	#17984014	C₂²	1	2	yes	complemented
35	#17983999	C₂²	1	1	yes	characteristic, complemented
36	C₈×Dic₁₃₇	C₂²	1	1	yes	characteristic, complemented
37	C₈×Dic₁₃₇	C₂²	1	1	yes	characteristic, complemented
38	#135×C₁₃₇	C₂²	1	1	yes	characteristic
39	#131×C₁₃₇	C₂²	1	2	yes	complemented
40	#108×C₁₃₇	C₂²	1	1	yes	characteristic, complemented
41	#104×C₁₃₇	C₂²	1	2	yes	complemented
42	#97×C₁₃₇	C₂²	1	1	yes	characteristic, complemented
43	#14555511	C₂×C₄	1	2	yes	—
44	#14555510	—	2	2	no	—
45	#14555510	C₂³	1	2	yes	—
46	#14555509	—	2	2	no	—
47	#14555509	C₂×C₄	1	2	yes	—
48	Q₈×D₁₃₇	C₂×C₄	1	2	yes	—
49	D₄×D₁₃₇	—	2	2	no	—
50	D₄×D₁₃₇	C₂×C₄	1	2	yes	—
51	C₂×C₄×D₁₃₇	—	2	2	no	—
52	C₂×C₄×D₁₃₇	—	2	1	no	—
53	C₂×C₄×D₁₃₇	C₂³	1	1	yes	characteristic
54	Dic₅₄₈	C₂×C₄	1	1	yes	characteristic, complemented
55	#14555500	C₂×C₄	1	2	yes	complemented
56	#14555499	C₂×C₄	1	2	yes	complemented
57	#14555498	C₂×C₄	1	2	yes	complemented
58	#14555497	C₂×C₄	1	2	yes	complemented
59	D₁₀₉₆	C₂×C₄	1	1	yes	characteristic, complemented
60	#14555495	C₂×C₄	1	2	yes	complemented
61	#14555494	—	2	2	no	—
62	#14555494	C₂³	1	2	yes	—
63	#14555493	—	2	2	no	—
64	#14555493	—	2	2	no	—
65	#14555493	—	2	2	no	—
66	#14555493	C₂×C₄	1	2	yes	—
67	#14555493	C₂×C₄	1	2	yes	—
68	#14555493	D₄	1	2	yes	—
69	#14555493	D₄	1	2	yes	—
70	C₂×#11764040	—	2	2	no	—
71	C₂×#11764040	C₂³	1	1	yes	characteristic
72	C₂×#11764040	C₂³	1	1	yes	characteristic
73	C₈×D₁₃₇	—	2	2	no	—
74	C₈×D₁₃₇	—	2	2	no	—
75	C₈×D₁₃₇	—	2	2	no	—
76	C₄×Dic₁₃₇	—	2	1	no	—
77	C₄×Dic₁₃₇	C₂³	1	1	yes	characteristic
78	#41×C₁₃₇	C₂³	1	2	yes	—
79	Q₁₆×C₁₃₇	C₂×C₄	1	1	yes	characteristic, complemented
80	SD₁₆×C₁₃₇	C₂×C₄	1	2	yes	complemented
81	D₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic, complemented
82	M₁₆×C₁₃₇	—	2	2	no	—
83	M₁₆×C₁₃₇	C₂³	1	2	yes	—
84	C₂×C₈×C₁₃₇	—	2	2	no	—
85	C₂×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
86	C₂×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
87	C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
88	C₂²×D₁₃₇	—	2	2	no	—
89	C₂²×D₁₃₇	#41	1	1	yes	characteristic
90	Dic₂₇₄	—	2	2	no	—
91	Dic₂₇₄	C₂²×C₄	1	2	yes	—
92	D₅₄₈	—	2	2	no	—
93	D₅₄₈	C₂²×C₄	1	2	yes	—
94	C₂²⋊D₁₃₇	—	2	2	no	—
95	C₂²⋊D₁₃₇	—	2	2	no	—
96	C₂²⋊D₁₃₇	—	2	2	no	—
97	C₂×Dic₁₃₇	—	2	1	no	—
98	C₂×Dic₁₃₇	—	2	2	no	—
99	C₂×Dic₁₃₇	#41	1	1	yes	characteristic
100	C₄×D₁₃₇	—	2	2	no	—
101	C₄×D₁₃₇	—	2	2	no	—
102	C₄×D₁₃₇	—	2	2	no	—
103	C₄×D₁₃₇	—	4	2	no	—
104	C₄×D₁₃₇	C₂²×C₄	1	2	yes	—
105	C₄×D₁₃₇	C₂×D₄	1	2	yes	—
106	#11764040	—	2	2	no	—
107	#11764040	C₂²×C₄	1	2	yes	—
108	#11764040	C₂×D₄	1	1	yes	characteristic
109	#11764040	C₂×D₄	1	1	yes	characteristic
110	Q₈×C₁₃₇	C₂²×C₄	1	2	yes	—
111	D₄×C₁₃₇	—	2	2	no	—
112	D₄×C₁₃₇	C₂²×C₄	1	2	yes	—
113	C₂×C₄×C₁₃₇	—	2	2	no	—
114	C₂×C₄×C₁₃₇	—	2	1	no	—
115	C₂×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
116	C₈×C₁₃₇	—	2	2	no	—
117	C₈×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
118	C₈×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
119	C₈×C₁₃₇	C₂×D₄	1	2	yes	—
120	C₂²×C₁₃₇	—	2	2	no	—
121	C₂²×C₁₃₇	C₂×#41	1	1	yes	characteristic
122	C₂×D₁₃₇	—	2	2	no	—
123	C₂×D₁₃₇	—	4	1	no	—
124	C₂×D₁₃₇	—	8	2	no	—
125	C₂×D₁₃₇	C₄×D₄	1	2	yes	—
126	C₄×C₁₃₇	—	2	2	no	—
127	C₄×C₁₃₇	—	4	1	no	—
128	C₄×C₁₃₇	C₂³×C₄	1	1	yes	characteristic
129	C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
130	Dic₁₃₇	—	2	2	no	—
131	Dic₁₃₇	—	2	2	no	—
132	Dic₁₃₇	C₄×D₄	1	2	yes	—
133	C₂×C₁₃₇	—	2	1	no	—
134	C₂×C₁₃₇	—	4	2	no	—
135	C₂×C₁₃₇	C₂×C₄×D₄	1	1	yes	characteristic
136	D₁₃₇	—	2	2	no	—
137	D₁₃₇	—	4	2	no	—
138	C₁₃₇	#2956	1	1	yes	characteristic, complemented, Sylow
139	#2956	—	137	1	no	maximal, Sylow
140	#575	—	137	1	no	—
141	#568	—	137	2	no	—
142	#444	—	137	2	no	—
143	#444	—	137	1	no	—
144	#444	D₁₃₇	1	1	yes	characteristic, complemented
145	#443	—	137	1	no	—
146	#443	—	137	2	no	—
147	#443	—	137	1	no	—
148	#430	—	137	1	no	—
149	C₂×#104	—	137	2	no	—
150	C₄×M₁₆	—	137	1	no	—
151	C₂×#41	—	137	2	no	—
152	#137	—	137	2	no	—
153	#137	—	137	2	no	—
154	AGL(1,ℤ/8)	—	137	2	no	—
155	AGL(1,ℤ/8)	—	137	2	no	—
156	#135	—	137	2	no	—
157	#135	—	137	1	no	—
158	#135	C₂×D₁₃₇	1	1	yes	characteristic
159	#131	—	274	2	no	—
160	#131	—	274	2	no	—
161	#131	—	137	2	no	—
162	#131	—	137	2	no	—
163	#131	—	137	2	no	—
164	#131	C₂×D₁₃₇	1	2	yes	complemented
165	C₂×M₁₆	—	274	2	no	—
166	C₂×M₁₆	—	274	2	no	—
167	C₂×M₁₆	—	137	2	no	—
168	C₂×M₁₆	—	137	1	no	—
169	C₂×M₁₆	—	137	1	no	—
170	C₂×C₄²	—	137	1	no	—
171	#108	—	137	2	no	—
172	#108	—	137	1	no	—
173	#108	C₂×D₁₃₇	1	1	yes	characteristic, complemented
174	#104	—	137	2	no	—
175	#104	—	137	2	no	—
176	#104	—	137	2	no	—
177	#104	C₂×D₁₃₇	1	2	yes	complemented
178	#97	—	137	1	no	—
179	#97	C₂×D₁₃₇	1	1	yes	characteristic, complemented
180	C₄×C₈	—	137	1	no	—
181	C₄×C₈	—	137	1	no	—
182	#41	—	274	2	no	—
183	#41	—	274	2	no	—
184	#41	—	137	2	no	—
185	#41	—	137	2	no	—
186	#41	—	137	2	no	—
187	#41	C₂²×D₁₃₇	1	2	yes	—
188	C₂×Q₈	—	137	2	no	—
189	C₂×D₄	—	274	2	no	—
190	C₂×D₄	—	137	2	no	—
191	C₂²×C₄	—	274	2	no	—
192	C₂²×C₄	—	274	1	no	—
193	C₂²×C₄	—	137	1	no	—
194	Q₁₆	—	137	2	no	—
195	Q₁₆	—	137	1	no	—
196	Q₁₆	C₄×D₁₃₇	1	1	yes	characteristic, complemented
197	SD₁₆	—	137	2	no	—
198	SD₁₆	—	137	2	no	—
199	SD₁₆	—	137	2	no	—
200	SD₁₆	C₄×D₁₃₇	1	2	yes	complemented
201	D₈	—	137	2	no	—
202	D₈	—	137	1	no	—
203	D₈	C₄×D₁₃₇	1	1	yes	characteristic, complemented
204	M₁₆	—	2	2	no	—
205	M₁₆	—	274	2	no	—
206	M₁₆	—	274	2	no	—
207	M₁₆	—	274	2	no	—
208	M₁₆	—	274	2	no	—
209	M₁₆	—	137	2	no	—
210	M₁₆	—	137	2	no	—
211	M₁₆	—	137	2	no	—
212	M₁₆	—	137	2	no	—
213	M₁₆	—	137	2	no	—
214	M₁₆	C₂²×D₁₃₇	1	2	yes	—
215	C₂×C₈	—	2	2	no	—
216	C₂×C₈	—	274	2	no	—
217	C₂×C₈	—	274	2	no	—
218	C₂×C₈	—	274	2	no	—
219	C₂×C₈	—	274	2	no	—
220	C₂×C₈	—	137	1	no	—
221	C₂×C₈	—	137	1	no	—
222	C₂×C₈	C₂²×D₁₃₇	1	1	yes	characteristic
223	C₂×C₈	C₂²×D₁₃₇	1	1	yes	characteristic
224	C₄²	—	274	1	no	—
225	C₄²	—	137	1	no	—
226	C₄²	C₂²×D₁₃₇	1	1	yes	characteristic
227	C₂³	—	274	2	no	—
228	C₂³	—	137	1	no	—
229	Q₈	—	274	2	no	—
230	Q₈	—	137	2	no	—
231	Q₈	C₂×C₄×D₁₃₇	1	2	yes	—
232	D₄	—	2	2	no	—
233	D₄	—	274	2	no	—
234	D₄	—	274	2	no	—
235	D₄	—	274	2	no	—
236	D₄	—	274	2	no	—
237	D₄	—	137	2	no	—
238	D₄	C₂×C₄×D₁₃₇	1	2	yes	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	274	2	no	—
242	C₂×C₄	—	274	2	no	—
243	C₂×C₄	—	274	1	no	—
244	C₂×C₄	—	274	2	no	—
245	C₂×C₄	—	274	2	no	—
246	C₂×C₄	—	548	2	no	—
247	C₂×C₄	—	137	1	no	—
248	C₂×C₄	—	137	2	no	—
249	C₂×C₄	—	137	2	no	—
250	C₂×C₄	C₂³×D₁₃₇	1	1	yes	characteristic
251	C₈	—	2	2	no	—
252	C₈	—	274	2	no	—
253	C₈	—	137	2	no	—
254	C₈	—	137	1	no	—
255	C₈	—	137	1	no	—
256	C₈	C₂×C₄×D₁₃₇	1	1	yes	characteristic
257	C₈	C₂×C₄×D₁₃₇	1	1	yes	characteristic
258	C₈	D₄×D₁₃₇	1	2	yes	—
259	C₂²	—	2	2	no	—
260	C₂²	—	274	2	no	—
261	C₂²	—	548	1	no	—
262	C₂²	—	1096	2	no	—
263	C₂²	—	137	2	no	—
264	C₂²	#41×D₁₃₇	1	1	yes	characteristic
265	C₄	—	2	2	no	—
266	C₄	—	4	1	no	—
267	C₄	—	274	2	no	—
268	C₄	—	274	2	no	—
269	C₄	—	137	2	no	—
270	C₄	C₂²×C₄×D₁₃₇	1	1	yes	characteristic
271	C₄	C₂×D₄×D₁₃₇	1	1	yes	characteristic
272	C₂	—	2	1	no	—
273	C₂	—	4	2	no	—
274	C₂	—	274	2	no	—
275	C₂	—	548	2	no	—
276	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	#22578200	C₂
cosocle	C₂×C₄×C₁₃₇	C₂⁴
hypercenter	#444	D₁₃₇
hypocenter	C₁₃₇	#2956
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. #22577522	C₂	semidirect
5. #22577520	C₂	semidirect
6. #22577519	C₂	semidirect
7. #22577518	C₂	semidirect
8. #22577450	C₂	semidirect
9. #104×D₁₃₇	C₂	semidirect
10. #22577355	C₂	semidirect
11. #444×C₁₃₇	C₂	semidirect
13. #17984120	C₄	semidirect
14. #17984117	C₄	semidirect
15. #17984115	C₄	semidirect
16. #17984112	C₄	semidirect
20. #17984096	C₂²	semidirect
21. #17984095	C₂²	semidirect
23. #17984094	C₂²	semidirect
30. #17984024	C₂²	semidirect
31. #17984023	C₂²	semidirect
32. #17984016	C₂²	semidirect
33. #17984015	C₂²	semidirect
34. #17984014	C₂²	semidirect
35. #17983999	C₂²	semidirect
36. C₈×Dic₁₃₇	C₂²	semidirect
37. C₈×Dic₁₃₇	C₂²	semidirect
39. #131×C₁₃₇	C₂²	semidirect
40. #108×C₁₃₇	C₂²	semidirect
41. #104×C₁₃₇	C₂²	semidirect
42. #97×C₁₃₇	C₂²	semidirect
54. Dic₅₄₈	C₂×C₄	semidirect
55. #14555500	C₂×C₄	semidirect
56. #14555499	C₂×C₄	semidirect
57. #14555498	C₂×C₄	semidirect
58. #14555497	C₂×C₄	semidirect
59. D₁₀₉₆	C₂×C₄	semidirect
60. #14555495	C₂×C₄	semidirect
79. Q₁₆×C₁₃₇	C₂×C₄	semidirect
80. SD₁₆×C₁₃₇	C₂×C₄	semidirect
81. D₈×C₁₃₇	C₂×C₄	semidirect
138. C₁₃₇	#2956	semidirect
144. #444	D₁₃₇	semidirect
164. #131	C₂×D₁₃₇	semidirect
173. #108	C₂×D₁₃₇	semidirect
177. #104	C₂×D₁₃₇	semidirect
179. #97	C₂×D₁₃₇	semidirect
196. Q₁₆	C₄×D₁₃₇	semidirect
200. SD₁₆	C₄×D₁₃₇	semidirect
203. D₈	C₄×D₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
#22578200	2
#22578168	2
#22577526	2
#22577522	2
#22577520	2
#22577519	2
#22577518	2
#22577450	2
#104×D₁₃₇	2
#22577355	2
#444×C₁₃₇	2
#2956	137