Subgroups — #31789690

Order 17536.

456 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22578201	C₂	1	1	yes	maximal, characteristic
2	C₂×#17984094	C₂	1	1	yes	maximal, characteristic, complemented
3	#22577523	C₂	1	4	yes	maximal, complemented
4	#22577518	C₂	1	4	yes	maximal, complemented
5	#22577451	C₂	1	1	yes	maximal, characteristic, complemented
6	#22577420	C₂	1	1	yes	maximal, characteristic, complemented
7	#22577419	C₂	1	1	yes	maximal, characteristic, complemented
8	#22577358	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#131×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#14555510	C₂²	1	1	yes	characteristic
11	#17984119	C₄	1	2	yes	complemented
12	#17984118	C₄	1	2	yes	complemented
13	#17984114	C₄	1	2	yes	complemented
14	#17984113	C₄	1	2	yes	complemented
15	#17984108	C₂²	1	4	yes	—
16	#17984094	—	2	1	no	—
17	#17984094	—	2	1	no	—
18	#17984094	C₂²	1	2	yes	complemented
19	#17984094	C₂²	1	2	yes	complemented
20	C₂×#14555494	C₂²	1	1	yes	characteristic
21	C₂×#14555494	C₂²	1	1	yes	characteristic
22	C₂×#14555493	—	2	1	no	—
23	C₂×C₈×D₁₃₇	—	2	1	no	—
24	#17984047	C₂²	1	1	yes	characteristic
25	#17984023	C₂²	1	4	yes	complemented
26	#17984015	C₂²	1	4	yes	complemented
27	#17984014	C₂²	1	4	yes	complemented
28	#17983998	C₂²	1	2	yes	complemented
29	C₈×Dic₁₃₇	C₂²	1	2	yes	complemented
30	C₂×#41×C₁₃₇	C₂²	1	1	yes	characteristic
31	#131×C₁₃₇	—	2	1	no	—
32	#131×C₁₃₇	—	2	1	no	—
33	#131×C₁₃₇	C₂²	1	2	yes	complemented
34	#131×C₁₃₇	C₂²	1	2	yes	complemented
35	C₂×M₁₆×C₁₃₇	—	2	1	no	—
36	C₂×M₁₆×C₁₃₇	C₂²	1	1	yes	characteristic
37	C₂²×C₈×C₁₃₇	—	2	1	no	—
38	C₂²×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
39	#14555510	—	2	1	no	—
40	#14555510	—	2	1	no	—
41	#14555510	C₂×C₄	1	2	yes	—
42	#14555510	C₂³	1	2	yes	—
43	C₂×Dic₂₇₄	C₂×C₄	1	1	yes	characteristic
44	C₂×D₅₄₈	C₂×C₄	1	1	yes	characteristic
45	C₂×C₂²⋊D₁₃₇	—	2	1	no	—
46	C₂×C₄×D₁₃₇	—	2	1	no	—
47	#14555500	C₂×C₄	1	4	yes	—
48	#14555498	C₂×C₄	1	4	yes	—
49	#14555497	C₂×C₄	1	4	yes	—
50	#14555495	C₂×C₄	1	4	yes	—
51	#14555494	—	2	1	no	—
52	#14555494	C₂×C₄	1	2	yes	—
53	#14555494	C₂×C₄	1	2	yes	—
54	#14555494	C₂³	1	2	yes	—
55	#14555493	—	2	2	no	—
56	#14555493	—	2	2	no	—
57	C₂×#11764040	—	2	1	no	—
58	C₂×#11764040	C₂³	1	2	yes	—
59	C₈×D₁₃₇	—	2	2	no	—
60	C₈×D₁₃₇	—	2	2	no	—
61	C₄⋊Dic₁₃₇	—	2	1	no	—
62	C₂²⋊Dic₁₃₇	—	2	1	no	—
63	C₄×Dic₁₃₇	C₂³	1	2	yes	—
64	#41×C₁₃₇	—	2	1	no	—
65	#41×C₁₃₇	—	2	1	no	—
66	#41×C₁₃₇	C₂×C₄	1	2	yes	—
67	#41×C₁₃₇	C₂³	1	2	yes	—
68	C₂×Q₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
69	C₂×D₄×C₁₃₇	—	2	1	no	—
70	C₂×D₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₁₃₇	—	2	1	no	—
72	C₂²×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
73	M₁₆×C₁₃₇	—	2	1	no	—
74	M₁₆×C₁₃₇	—	2	2	no	—
75	M₁₆×C₁₃₇	—	2	2	no	—
76	M₁₆×C₁₃₇	C₂³	1	2	yes	—
77	C₂×C₈×C₁₃₇	—	2	1	no	—
78	C₂×C₈×C₁₃₇	—	2	2	no	—
79	C₂×C₈×C₁₃₇	—	2	2	no	—
80	C₂×C₈×C₁₃₇	D₄	1	1	yes	characteristic
81	C₂×C₈×C₁₃₇	D₄	1	1	yes	characteristic
82	C₂×C₈×C₁₃₇	D₄	1	1	yes	characteristic
83	C₂×C₈×C₁₃₇	D₄	1	1	yes	characteristic
84	C₂×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
85	C₂×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
86	C₂²×D₁₃₇	—	2	1	no	—
87	Dic₂₇₄	—	2	1	no	—
88	Dic₂₇₄	C₂²×C₄	1	2	yes	—
89	D₅₄₈	—	2	1	no	—
90	D₅₄₈	C₂²×C₄	1	2	yes	—
91	C₂²⋊D₁₃₇	—	2	2	no	—
92	C₂²⋊D₁₃₇	—	2	2	no	—
93	C₂×Dic₁₃₇	—	2	1	no	—
94	C₂×Dic₁₃₇	—	2	2	no	—
95	C₄×D₁₃₇	—	2	2	no	—
96	C₄×D₁₃₇	—	2	2	no	—
97	#11764040	—	2	2	no	—
98	#11764040	C₂²×C₄	1	4	yes	—
99	C₂³×C₁₃₇	—	2	1	no	—
100	C₂³×C₁₃₇	#41	1	1	yes	characteristic
101	Q₈×C₁₃₇	—	2	1	no	—
102	Q₈×C₁₃₇	C₂²×C₄	1	2	yes	—
103	D₄×C₁₃₇	—	2	2	no	—
104	D₄×C₁₃₇	—	2	2	no	—
105	D₄×C₁₃₇	—	2	1	no	—
106	D₄×C₁₃₇	C₂²×C₄	1	2	yes	—
107	C₂×C₄×C₁₃₇	—	2	1	no	—
108	C₂×C₄×C₁₃₇	—	2	2	no	—
109	C₂×C₄×C₁₃₇	—	2	2	no	—
110	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
111	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
112	C₂×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
113	C₂×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
114	C₂×C₄×C₁₃₇	#41	1	1	yes	characteristic
115	C₂×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
116	C₈×C₁₃₇	—	2	2	no	—
117	C₈×C₁₃₇	C₂×D₄	1	1	yes	characteristic
118	C₈×C₁₃₇	C₂×D₄	1	1	yes	characteristic
119	C₈×C₁₃₇	C₂×D₄	1	1	yes	characteristic
120	C₈×C₁₃₇	C₂×D₄	1	1	yes	characteristic
121	C₂²×C₁₃₇	—	2	2	no	—
122	C₂²×C₁₃₇	—	4	1	no	—
123	C₂²×C₁₃₇	—	4	2	no	—
124	C₂²×C₁₃₇	C₄×D₄	1	1	yes	characteristic
125	C₂²×C₁₃₇	C₄×D₄	1	1	yes	characteristic
126	C₂²×C₁₃₇	C₂×#41	1	1	yes	characteristic
127	C₂×D₁₃₇	—	2	2	no	—
128	C₂×D₁₃₇	—	4	2	no	—
129	C₄×C₁₃₇	—	2	2	no	—
130	C₄×C₁₃₇	C₄×D₄	1	1	yes	characteristic
131	C₄×C₁₃₇	C₄×D₄	1	1	yes	characteristic
132	C₄×C₁₃₇	C₂³×C₄	1	1	yes	characteristic
133	C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
134	Dic₁₃₇	—	2	2	no	—
135	Dic₁₃₇	—	4	2	no	—
136	C₂×C₁₃₇	—	2	1	no	—
137	C₂×C₁₃₇	—	2	1	no	—
138	C₂×C₁₃₇	—	2	1	no	—
139	C₂×C₁₃₇	—	4	2	no	—
140	C₂×C₁₃₇	C₂×C₄×D₄	1	1	yes	characteristic
141	D₁₃₇	—	4	2	no	—
142	C₁₃₇	#2955	1	1	yes	characteristic, complemented, Sylow
143	#2955	—	137	1	no	maximal, Sylow
144	#575	—	137	1	no	—
145	C₂×#131	—	137	1	no	—
146	C₂×#131	D₁₃₇	1	1	yes	characteristic, complemented
147	#444	—	137	4	no	—
148	#443	—	137	4	no	—
149	#430	—	137	1	no	—
150	#421	—	137	1	no	—
151	#421	—	137	1	no	—
152	#405	—	137	1	no	—
153	C₂×#41	—	137	1	no	—
154	C₂×#41	C₂×D₁₃₇	1	1	yes	characteristic
155	#137	—	137	2	no	—
156	#137	—	137	2	no	—
157	AGL(1,ℤ/8)	—	137	2	no	—
158	AGL(1,ℤ/8)	—	137	2	no	—
159	#135	—	137	4	no	—
160	#131	—	2	1	no	—
161	#131	—	2	1	no	—
162	#131	—	274	1	no	—
163	#131	—	274	1	no	—
164	#131	—	137	2	no	—
165	#131	—	137	2	no	—
166	#131	C₂×D₁₃₇	1	2	yes	complemented
167	#131	C₂×D₁₃₇	1	2	yes	complemented
168	C₂×M₁₆	—	2	1	no	—
169	C₂×M₁₆	—	274	1	no	—
170	C₂×M₁₆	—	137	1	no	—
171	C₂×M₁₆	—	137	1	no	—
172	C₂×M₁₆	C₂×D₁₃₇	1	1	yes	characteristic
173	C₂²×C₈	—	2	1	no	—
174	C₂²×C₈	—	274	1	no	—
175	C₂²×C₈	C₂×D₁₃₇	1	1	yes	characteristic
176	#117	—	137	1	no	—
177	#108	—	137	4	no	—
178	#104	—	137	4	no	—
179	#104	—	137	4	no	—
180	#97	—	137	2	no	—
181	C₄×C₈	—	137	2	no	—
182	#41	—	2	1	no	—
183	#41	—	2	1	no	—
184	#41	—	274	1	no	—
185	#41	—	274	1	no	—
186	#41	—	137	2	no	—
187	#41	—	137	2	no	—
188	#41	C₄×D₁₃₇	1	2	yes	—
189	#41	C₂²×D₁₃₇	1	2	yes	—
190	C₂×Q₈	—	137	1	no	—
191	C₂×Q₈	C₄×D₁₃₇	1	1	yes	characteristic
192	C₂×D₄	—	2	1	no	—
193	C₂×D₄	—	274	1	no	—
194	C₂×D₄	—	137	1	no	—
195	C₂×D₄	C₄×D₁₃₇	1	1	yes	characteristic
196	C₂²×C₄	—	2	1	no	—
197	C₂²×C₄	—	274	1	no	—
198	C₂²×C₄	C₂²×D₁₃₇	1	1	yes	characteristic
199	Q₁₆	—	137	4	no	—
200	SD₁₆	—	137	4	no	—
201	SD₁₆	—	137	4	no	—
202	D₈	—	137	4	no	—
203	M₁₆	—	2	1	no	—
204	M₁₆	—	2	2	no	—
205	M₁₆	—	2	2	no	—
206	M₁₆	—	274	1	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	C₂²×D₁₃₇	1	2	yes	—
213	C₂×C₈	—	2	1	no	—
214	C₂×C₈	—	2	2	no	—
215	C₂×C₈	—	2	2	no	—
216	C₂×C₈	—	274	1	no	—
217	C₂×C₈	—	274	2	no	—
218	C₂×C₈	—	274	2	no	—
219	C₂×C₈	—	137	2	no	—
220	C₂×C₈	C₂²⋊D₁₃₇	1	1	yes	characteristic
221	C₂×C₈	C₂²⋊D₁₃₇	1	1	yes	characteristic
222	C₂×C₈	C₂²⋊D₁₃₇	1	1	yes	characteristic
223	C₂×C₈	C₂²⋊D₁₃₇	1	1	yes	characteristic
224	C₂×C₈	C₂²×D₁₃₇	1	1	yes	characteristic
225	C₂×C₈	C₂²×D₁₃₇	1	1	yes	characteristic
226	C₄⋊C₄	—	274	1	no	—
227	C₂²⋊C₄	—	274	1	no	—
228	C₄²	—	137	2	no	—
229	C₂³	—	2	1	no	—
230	C₂³	—	274	1	no	—
231	C₂³	#14555510	1	1	yes	characteristic
232	Q₈	—	2	1	no	—
233	Q₈	—	274	1	no	—
234	Q₈	—	137	2	no	—
235	Q₈	C₂×C₄×D₁₃₇	1	2	yes	—
236	D₄	—	2	2	no	—
237	D₄	—	2	2	no	—
238	D₄	—	2	1	no	—
239	D₄	—	274	2	no	—
240	D₄	—	274	2	no	—
241	D₄	—	274	1	no	—
242	D₄	—	137	2	no	—
243	D₄	C₂×C₄×D₁₃₇	1	2	yes	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	274	1	no	—
248	C₂×C₄	—	274	2	no	—
249	C₂×C₄	—	274	2	no	—
250	C₂×C₄	—	274	2	no	—
251	C₂×C₄	C₂×C₄×D₁₃₇	1	1	yes	characteristic
252	C₂×C₄	C₂×C₄×D₁₃₇	1	1	yes	characteristic
253	C₂×C₄	C₂×C₂²⋊D₁₃₇	1	1	yes	characteristic
254	C₂×C₄	C₂×C₂²⋊D₁₃₇	1	1	yes	characteristic
255	C₂×C₄	#14555510	1	1	yes	characteristic
256	C₂×C₄	C₂³×D₁₃₇	1	1	yes	characteristic
257	C₈	—	2	2	no	—
258	C₈	—	274	2	no	—
259	C₈	—	137	4	no	—
260	C₈	C₂×C₂²⋊D₁₃₇	1	1	yes	characteristic
261	C₈	C₂×C₂²⋊D₁₃₇	1	1	yes	characteristic
262	C₈	C₂×C₂²⋊D₁₃₇	1	1	yes	characteristic
263	C₈	C₂×C₂²⋊D₁₃₇	1	1	yes	characteristic
264	C₂²	—	2	2	no	—
265	C₂²	—	4	1	no	—
266	C₂²	—	4	2	no	—
267	C₂²	—	274	2	no	—
268	C₂²	—	548	2	no	—
269	C₂²	C₄×C₂²⋊D₁₃₇	1	1	yes	characteristic
270	C₂²	C₄×C₂²⋊D₁₃₇	1	1	yes	characteristic
271	C₂²	C₂×#14555510	1	1	yes	characteristic
272	C₄	—	2	2	no	—
273	C₄	—	274	2	no	—
274	C₄	—	548	2	no	—
275	C₄	C₄×C₂²⋊D₁₃₇	1	1	yes	characteristic
276	C₄	C₄×C₂²⋊D₁₃₇	1	1	yes	characteristic
277	C₄	C₂²×C₄×D₁₃₇	1	1	yes	characteristic
278	C₄	C₂²×C₂²⋊D₁₃₇	1	1	yes	characteristic
279	C₂	—	2	1	no	—
280	C₂	—	2	1	no	—
281	C₂	—	2	1	no	—
282	C₂	—	4	2	no	—
283	C₂	—	548	2	no	—
284	C₂	C₂×C₄×C₂²⋊D₁₃₇	1	1	yes	characteristic

	subgroup	G/H
center	C₈	C₂×C₂²⋊D₁₃₇
commutator	C₄×C₁₃₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₃₇
Fitting	C₂×#131×C₁₃₇	C₂
socle	C₂×C₁₃₇	C₂×C₄×D₄
prime radical	#22578201	C₂
cosocle	C₂×C₄×C₁₃₇	C₂⁴
hypercenter	C₂×#131	D₁₃₇
hypocenter	C₁₃₇	#2955
LCS	G ▷ C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ C₂×#131

Complements

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

N	H ≅ G/N	type
2. C₂×#17984094	C₂	semidirect
3. #22577523	C₂	semidirect
4. #22577518	C₂	semidirect
5. #22577451	C₂	semidirect
6. #22577420	C₂	semidirect
7. #22577419	C₂	semidirect
8. #22577358	C₂	semidirect
9. C₂×#131×C₁₃₇	C₂	semidirect
11. #17984119	C₄	semidirect
12. #17984118	C₄	semidirect
13. #17984114	C₄	semidirect
14. #17984113	C₄	semidirect
18. #17984094	C₂²	semidirect
19. #17984094	C₂²	semidirect
25. #17984023	C₂²	semidirect
26. #17984015	C₂²	semidirect
27. #17984014	C₂²	semidirect
28. #17983998	C₂²	semidirect
29. C₈×Dic₁₃₇	C₂²	semidirect
33. #131×C₁₃₇	C₂²	semidirect
34. #131×C₁₃₇	C₂²	semidirect
142. C₁₃₇	#2955	semidirect
146. C₂×#131	D₁₃₇	semidirect
166. #131	C₂×D₁₃₇	semidirect
167. #131	C₂×D₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
#22578201	2
C₂×#17984094	2
#22577523	2
#22577518	2
#22577451	2
#22577420	2
#22577419	2
#22577358	2
C₂×#131×C₁₃₇	2
#2955	137