Subgroups — #31797563

Order 17536.

458 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22578482	C₂	1	1	yes	maximal, characteristic, complemented
2	#22578481	C₂	1	1	yes	maximal, characteristic, complemented
3	#22578450	C₂	1	2	yes	maximal, complemented
4	#22578448	C₂	1	2	yes	maximal, complemented
5	#22578446	C₂	1	2	yes	maximal, complemented
6	#22578445	C₂	1	2	yes	maximal, complemented
7	#22578418	C₂	1	1	yes	maximal, characteristic, complemented
8	D₁₃₇⋊#104	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#17984152	C₂	1	1	yes	maximal, characteristic, complemented
10	M₁₆×#10587739	C₂	1	1	yes	maximal, characteristic, complemented
11	#22578199	C₂	1	1	yes	maximal, characteristic
12	#17984176	C₂²	1	2	yes	complemented
13	#17984175	—	2	1	no	—
14	#17984175	—	2	1	no	—
15	#17984175	—	2	2	no	—
16	#17984175	C₂²	1	2	yes	complemented
17	#17984175	C₂²	1	2	yes	complemented
18	#17984175	C₂²	1	2	yes	complemented
19	C₂×#14555520	—	2	1	no	—
20	C₂×#14555520	C₂²	1	1	yes	characteristic
21	D₁₃₇⋊M₁₆	—	2	1	no	—
22	D₁₃₇⋊M₁₆	C₂²	1	1	yes	characteristic
23	D₁₃₇⋊M₁₆	C₂²	1	1	yes	characteristic
24	C₂×#14555519	—	2	2	no	—
25	C₂×C₄×#10587739	C₂²	1	1	yes	characteristic
26	#17984159	C₂²	1	2	yes	complemented
27	#17984158	C₂²	1	2	yes	complemented
28	#17984153	C₂²	1	1	yes	characteristic, complemented
29	#17984153	C₂²	1	1	yes	characteristic, complemented
30	#17984152	C₂²	1	1	yes	characteristic, complemented
31	#17984152	C₂²	1	2	yes	complemented
32	#17984152	C₂²	1	2	yes	complemented
33	#17984152	C₂²	1	1	yes	characteristic, complemented
34	#17984142	C₂²	1	2	yes	complemented
35	C₈×#10587739	C₂²	1	2	yes	complemented
36	C₂×#14555509	C₂²	1	1	yes	characteristic
37	#41×D₁₃₇	C₂²	1	1	yes	characteristic
38	#17984121	C₄	1	1	yes	characteristic, complemented
39	#17984118	C₄	1	1	yes	characteristic, complemented
40	#17984117	C₄	1	2	yes	complemented
41	#17984114	C₄	1	1	yes	characteristic, complemented
42	#17984112	C₄	1	2	yes	complemented
43	#17984109	C₂²	1	2	yes	—
44	#17984107	C₂²	1	2	yes	—
45	M₁₆×D₁₃₇	C₂²	1	1	yes	characteristic
46	AGL(1,ℤ/8)×C₁₃₇	C₄	1	1	yes	characteristic, complemented
47	C₂²×#10587739	—	2	1	no	—
48	#14555520	—	2	2	no	—
49	#14555520	—	2	1	no	—
50	#14555520	D₄	1	1	yes	characteristic
51	#14555520	D₄	1	1	yes	characteristic
52	#14555520	C₂³	1	2	yes	—
53	#14555520	C₂³	1	1	yes	characteristic
54	#14555520	C₂³	1	1	yes	characteristic
55	#14555519	—	2	2	no	—
56	#14555519	—	2	2	no	—
57	#14555519	—	2	2	no	—
58	#14555519	—	2	1	no	—
59	#14555519	D₄	1	2	yes	—
60	#14555518	—	2	2	no	—
61	#14555518	—	2	1	no	—
62	#14555518	C₂³	1	2	yes	—
63	C₂×#11764047	—	2	2	no	—
64	C₂×#11764047	—	2	2	no	—
65	C₂×#11764047	—	2	2	no	—
66	C₂×#11764047	—	2	1	no	—
67	C₄×#10587739	—	2	1	no	—
68	C₄×#10587739	C₂³	1	1	yes	characteristic
69	C₄×#10587739	C₂³	1	1	yes	characteristic
70	#14555511	C₂³	1	1	yes	characteristic
71	#14555510	—	2	1	no	—
72	#14555510	C₂×C₄	1	1	yes	characteristic
73	#14555509	—	2	1	no	—
74	#14555509	—	2	1	no	—
75	#14555509	—	2	1	no	—
76	#14555509	C₂×C₄	1	2	yes	—
77	#14555509	C₂³	1	1	yes	characteristic
78	#14555509	C₂³	1	2	yes	—
79	C₂×Dic₂₇₄	C₂×C₄	1	1	yes	characteristic
80	Q₈×D₁₃₇	C₂×C₄	1	1	yes	characteristic
81	C₂×C₂²⋊D₁₃₇	—	2	1	no	—
82	D₄×D₁₃₇	—	2	1	no	—
83	D₄×D₁₃₇	C₂×C₄	1	1	yes	characteristic
84	C₂²×Dic₁₃₇	—	2	1	no	—
85	C₂×C₄×D₁₃₇	—	2	1	no	—
86	C₂×C₄×D₁₃₇	C₂³	1	1	yes	characteristic
87	Dic₅₄₈	C₂×C₄	1	2	yes	complemented
88	#14555500	C₂×C₄	1	2	yes	complemented
89	#14555499	C₂×C₄	1	2	yes	complemented
90	#14555498	C₂×C₄	1	2	yes	complemented
91	#14555498	C₂×C₄	1	2	yes	complemented
92	#14555495	C₂×C₄	1	2	yes	complemented
93	#14555494	C₂×C₄	1	1	yes	characteristic
94	#14555493	C₂×C₄	1	2	yes	—
95	C₈×D₁₃₇	C₂³	1	2	yes	—
96	#41×C₁₃₇	C₂×C₄	1	1	yes	characteristic
97	C₂×D₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
98	SD₁₆×C₁₃₇	C₂×C₄	1	2	yes	complemented
99	D₈×C₁₃₇	C₂×C₄	1	2	yes	complemented
100	M₁₆×C₁₃₇	C₂×C₄	1	1	yes	characteristic
101	C₂×#10587739	—	2	1	no	—
102	C₂×#10587739	—	2	1	no	—
103	C₂×#10587739	—	4	2	no	—
104	#11764047	—	2	1	no	—
105	#11764047	—	2	1	no	—
106	#11764047	—	2	2	no	—
107	#11764047	C₂×D₄	1	2	yes	—
108	#11764047	C₂×D₄	1	2	yes	—
109	C₂²×D₁₃₇	—	2	1	no	—
110	C₂²×D₁₃₇	#41	1	1	yes	characteristic
111	Dic₂₇₄	—	2	1	no	—
112	Dic₂₇₄	—	2	1	no	—
113	Dic₂₇₄	C₂²×C₄	1	1	yes	characteristic
114	Dic₂₇₄	C₂²×C₄	1	2	yes	—
115	D₅₄₈	—	2	1	no	—
116	D₅₄₈	C₂²×C₄	1	1	yes	characteristic
117	C₂²⋊D₁₃₇	—	2	1	no	—
118	C₂²⋊D₁₃₇	—	2	2	no	—
119	C₂²⋊D₁₃₇	—	2	1	no	—
120	C₂²⋊D₁₃₇	—	2	2	no	—
121	C₂²⋊D₁₃₇	—	2	1	no	—
122	C₂×Dic₁₃₇	—	2	1	no	—
123	C₂×Dic₁₃₇	—	2	2	no	—
124	C₂×Dic₁₃₇	—	2	1	no	—
125	C₂×Dic₁₃₇	—	2	2	no	—
126	C₂×Dic₁₃₇	C₂×D₄	1	1	yes	characteristic
127	C₄×D₁₃₇	—	2	1	no	—
128	C₄×D₁₃₇	—	2	1	no	—
129	C₄×D₁₃₇	—	2	1	no	—
130	C₄×D₁₃₇	C₂²×C₄	1	1	yes	characteristic
131	C₄×D₁₃₇	C₂×D₄	1	1	yes	characteristic
132	C₄×D₁₃₇	#41	1	1	yes	characteristic
133	C₄×D₁₃₇	C₂⁴	1	1	yes	characteristic
134	#11764040	C₂²×C₄	1	2	yes	—
135	C₂³×C₁₃₇	—	2	1	no	—
136	Q₈×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
137	D₄×C₁₃₇	—	2	1	no	—
138	D₄×C₁₃₇	—	2	1	no	—
139	D₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
140	D₄×C₁₃₇	C₂²×C₄	1	2	yes	—
141	C₂×C₄×C₁₃₇	—	2	1	no	—
142	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
143	C₈×C₁₃₇	C₂²×C₄	1	2	yes	—
144	C₂²×C₁₃₇	—	2	1	no	—
145	C₂²×C₁₃₇	—	2	2	no	—
146	C₂²×C₁₃₇	—	8	1	no	—
147	C₂²×C₁₃₇	C₄×D₄	1	1	yes	characteristic
148	C₂×D₁₃₇	—	2	1	no	—
149	C₂×D₁₃₇	—	4	1	no	—
150	C₂×D₁₃₇	—	8	1	no	—
151	C₂×D₁₃₇	C₄×D₄	1	1	yes	characteristic
152	C₂×D₁₃₇	C₂×#41	1	1	yes	characteristic
153	#10587739	—	2	2	no	—
154	#10587739	—	4	1	no	—
155	C₄×C₁₃₇	—	2	1	no	—
156	C₄×C₁₃₇	C₄×D₄	1	1	yes	characteristic
157	C₄×C₁₃₇	C₂³×C₄	1	1	yes	characteristic
158	Dic₁₃₇	—	2	1	no	—
159	Dic₁₃₇	—	2	2	no	—
160	Dic₁₃₇	C₄×D₄	1	1	yes	characteristic
161	Dic₁₃₇	C₂²×D₄	1	1	yes	characteristic
162	C₂×C₁₃₇	—	2	1	no	—
163	C₂×C₁₃₇	—	4	1	no	—
164	C₂×C₁₃₇	—	4	2	no	—
165	C₂×C₁₃₇	C₂×C₄×D₄	1	1	yes	characteristic
166	D₁₃₇	—	2	1	no	—
167	D₁₃₇	—	2	1	no	—
168	D₁₃₇	—	4	1	no	—
169	C₁₃₇	#2956	1	1	yes	characteristic, complemented, Sylow
170	#2956	—	137	1	no	maximal, Sylow
171	#575	—	137	1	no	—
172	#568	—	137	1	no	—
173	#568	—	137	1	no	—
174	#444	—	137	2	no	—
175	#444	—	137	2	no	—
176	#443	—	137	2	no	—
177	#443	—	137	2	no	—
178	#430	—	137	1	no	—
179	C₂×#104	—	137	1	no	—
180	C₂×#104	—	137	1	no	—
181	C₄×M₁₆	—	137	1	no	—
182	C₂×#41	—	137	1	no	—
183	C₂×#41	—	137	1	no	—
184	#137	—	137	1	no	—
185	#137	—	137	2	no	—
186	#137	—	137	1	no	—
187	AGL(1,ℤ/8)	—	137	1	no	—
188	AGL(1,ℤ/8)	—	137	2	no	—
189	AGL(1,ℤ/8)	#10587739	1	1	yes	characteristic, complemented
190	#135	—	137	2	no	—
191	#135	—	137	2	no	—
192	#131	—	274	2	no	—
193	#131	—	274	1	no	—
194	#131	—	274	1	no	—
195	#131	—	137	2	no	—
196	#131	—	137	2	no	—
197	#131	—	137	2	no	—
198	#131	—	137	2	no	—
199	C₂×M₁₆	—	274	2	no	—
200	C₂×M₁₆	—	274	1	no	—
201	C₂×M₁₆	—	274	1	no	—
202	C₂×M₁₆	—	137	1	no	—
203	C₂×M₁₆	—	137	1	no	—
204	C₂×M₁₆	—	137	1	no	—
205	C₂×M₁₆	—	137	1	no	—
206	C₂×C₄²	—	137	1	no	—
207	#108	—	137	2	no	—
208	#108	—	137	2	no	—
209	#104	—	137	1	no	—
210	#104	—	137	2	no	—
211	#104	—	137	2	no	—
212	#104	—	137	1	no	—
213	#104	—	137	1	no	—
214	#104	—	137	1	no	—
215	#97	—	137	2	no	—
216	C₄×C₈	—	137	2	no	—
217	#41	—	274	1	no	—
218	#41	—	274	1	no	—
219	#41	—	274	1	no	—
220	#41	—	274	1	no	—
221	#41	—	137	1	no	—
222	#41	—	137	2	no	—
223	#41	—	137	1	no	—
224	#41	—	137	2	no	—
225	#41	—	137	1	no	—
226	#41	C₂×#10587739	1	1	yes	characteristic
227	C₂×Q₈	—	137	1	no	—
228	C₂×Q₈	—	137	1	no	—
229	C₂×D₄	—	274	1	no	—
230	C₂×D₄	—	274	1	no	—
231	C₂×D₄	—	137	1	no	—
232	C₂×D₄	C₂×#10587739	1	1	yes	characteristic
233	C₂²×C₄	—	274	1	no	—
234	C₂²×C₄	—	274	1	no	—
235	C₂²×C₄	—	274	1	no	—
236	C₂²×C₄	—	137	1	no	—
237	Q₁₆	—	137	2	no	—
238	Q₁₆	—	137	2	no	—
239	SD₁₆	—	137	2	no	—
240	SD₁₆	—	137	2	no	—
241	SD₁₆	—	137	2	no	—
242	SD₁₆	C₂×#10587739	1	2	yes	complemented
243	D₈	—	137	2	no	—
244	D₈	C₂×#10587739	1	2	yes	complemented
245	M₁₆	—	274	1	no	—
246	M₁₆	—	274	1	no	—
247	M₁₆	—	274	2	no	—
248	M₁₆	—	274	2	no	—
249	M₁₆	—	274	2	no	—
250	M₁₆	—	274	2	no	—
251	M₁₆	—	137	2	no	—
252	M₁₆	—	137	1	no	—
253	M₁₆	—	137	1	no	—
254	M₁₆	—	137	2	no	—
255	M₁₆	—	137	1	no	—
256	M₁₆	—	137	1	no	—
257	M₁₆	—	137	1	no	—
258	M₁₆	—	137	2	no	—
259	M₁₆	C₂×#10587739	1	1	yes	characteristic
260	C₂×C₈	—	274	1	no	—
261	C₂×C₈	—	274	1	no	—
262	C₂×C₈	—	274	2	no	—
263	C₂×C₈	—	274	2	no	—
264	C₂×C₈	—	274	2	no	—
265	C₂×C₈	—	274	2	no	—
266	C₂×C₈	—	137	2	no	—
267	C₂×C₈	—	137	2	no	—
268	C₄²	—	274	1	no	—
269	C₄²	—	137	1	no	—
270	C₄²	—	137	1	no	—
271	C₂³	—	2	1	no	—
272	C₂³	—	274	1	no	—
273	C₂³	—	137	1	no	—
274	Q₈	—	274	1	no	—
275	Q₈	—	274	1	no	—
276	Q₈	—	137	1	no	—
277	Q₈	—	137	2	no	—
278	Q₈	C₂²×#10587739	1	1	yes	characteristic
279	D₄	—	2	1	no	—
280	D₄	—	2	1	no	—
281	D₄	—	274	1	no	—
282	D₄	—	274	1	no	—
283	D₄	—	274	2	no	—
284	D₄	—	274	2	no	—
285	D₄	—	274	1	no	—
286	D₄	—	274	1	no	—
287	D₄	—	137	1	no	—
288	D₄	C₂²×#10587739	1	2	yes	—
289	D₄	C₂²×#10587739	1	1	yes	characteristic
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	274	1	no	—
292	C₂×C₄	—	274	1	no	—
293	C₂×C₄	—	274	1	no	—
294	C₂×C₄	—	274	1	no	—
295	C₂×C₄	—	274	1	no	—
296	C₂×C₄	—	274	2	no	—
297	C₂×C₄	—	274	2	no	—
298	C₂×C₄	—	274	1	no	—
299	C₂×C₄	—	274	1	no	—
300	C₂×C₄	—	548	2	no	—
301	C₂×C₄	—	137	1	no	—
302	C₂×C₄	—	137	1	no	—
303	C₂×C₄	—	137	1	no	—
304	C₂×C₄	—	137	1	no	—
305	C₂×C₄	—	137	1	no	—
306	C₂×C₄	C₂²×#10587739	1	1	yes	characteristic
307	C₈	—	274	2	no	—
308	C₈	—	274	1	no	—
309	C₈	—	274	1	no	—
310	C₈	—	137	2	no	—
311	C₈	—	137	2	no	—
312	C₈	—	137	2	no	—
313	C₈	C₂²×#10587739	1	2	yes	—
314	C₂²	—	2	1	no	—
315	C₂²	—	2	2	no	—
316	C₂²	—	8	1	no	—
317	C₂²	—	274	1	no	—
318	C₂²	—	548	1	no	—
319	C₂²	—	1096	1	no	—
320	C₂²	—	137	1	no	—
321	C₂²	—	137	1	no	—
322	C₂²	D₄×#10587739	1	1	yes	characteristic
323	C₄	—	2	1	no	—
324	C₄	—	274	2	no	—
325	C₄	—	274	1	no	—
326	C₄	—	274	2	no	—
327	C₄	—	548	1	no	—
328	C₄	—	137	1	no	—
329	C₄	—	137	1	no	—
330	C₄	D₄×#10587739	1	1	yes	characteristic
331	C₄	C₂³×#10587739	1	1	yes	characteristic
332	C₂	—	2	1	no	—
333	C₂	—	4	2	no	—
334	C₂	—	4	1	no	—
335	C₂	—	274	1	no	—
336	C₂	—	274	1	no	—
337	C₂	—	548	1	no	—
338	C₂	C₂×D₄×#10587739	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×D₄×#10587739
commutator	C₄×C₁₃₇	C₂³×C₄
Frattini	C₄	C₂³×#10587739
Fitting	AGL(1,ℤ/8)×C₁₃₇	C₄
socle	C₂×C₁₃₇	C₂×C₄×D₄
prime radical	#22578199	C₂
cosocle	C₄×D₁₃₇	C₂⁴
hypercenter	AGL(1,ℤ/8)	#10587739
hypocenter	C₁₃₇	#2956
LCS	G ▷ C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ AGL(1,ℤ/8)

Complements

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

N	H ≅ G/N	type
1. #22578482	C₂	semidirect
2. #22578481	C₂	semidirect
3. #22578450	C₂	semidirect
4. #22578448	C₂	semidirect
5. #22578446	C₂	semidirect
6. #22578445	C₂	semidirect
7. #22578418	C₂	semidirect
8. D₁₃₇⋊#104	C₂	semidirect
9. C₂×#17984152	C₂	semidirect
10. M₁₆×#10587739	C₂	semidirect
12. #17984176	C₂²	semidirect
16. #17984175	C₂²	semidirect
17. #17984175	C₂²	semidirect
18. #17984175	C₂²	semidirect
26. #17984159	C₂²	semidirect
27. #17984158	C₂²	semidirect
28. #17984153	C₂²	semidirect
29. #17984153	C₂²	semidirect
30. #17984152	C₂²	semidirect
31. #17984152	C₂²	semidirect
32. #17984152	C₂²	semidirect
33. #17984152	C₂²	semidirect
34. #17984142	C₂²	semidirect
35. C₈×#10587739	C₂²	semidirect
38. #17984121	C₄	semidirect
39. #17984118	C₄	semidirect
40. #17984117	C₄	semidirect
41. #17984114	C₄	semidirect
42. #17984112	C₄	semidirect
46. AGL(1,ℤ/8)×C₁₃₇	C₄	semidirect
87. Dic₅₄₈	C₂×C₄	semidirect
88. #14555500	C₂×C₄	semidirect
89. #14555499	C₂×C₄	semidirect
90. #14555498	C₂×C₄	semidirect
91. #14555498	C₂×C₄	semidirect
92. #14555495	C₂×C₄	semidirect
98. SD₁₆×C₁₃₇	C₂×C₄	semidirect
99. D₈×C₁₃₇	C₂×C₄	semidirect
169. C₁₃₇	#2956	semidirect
189. AGL(1,ℤ/8)	#10587739	semidirect
242. SD₁₆	C₂×#10587739	semidirect
244. D₈	C₂×#10587739	semidirect

Maximal subgroups

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

subgroup	index
#22578482	2
#22578481	2
#22578450	2
#22578448	2
#22578446	2
#22578445	2
#22578418	2
D₁₃₇⋊#104	2
C₂×#17984152	2
M₁₆×#10587739	2
#22578199	2
#2956	137