Subgroups — #31797561

Order 17536.

456 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₁₃₇⋊#131	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#17984175	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	#22578401	C₂	1	1	yes	maximal, characteristic, complemented
9	#22578400	C₂	1	1	yes	maximal, characteristic, complemented
10	#22578369	C₂	1	1	yes	maximal, characteristic, complemented
11	#22578199	C₂	1	1	yes	maximal, characteristic
12	#17984176	C₂²	1	1	yes	characteristic, complemented
13	#17984176	C₂²	1	1	yes	characteristic, complemented
14	#17984175	—	2	1	no	—
15	#17984175	—	2	1	no	—
16	#17984175	—	2	1	no	—
17	#17984175	—	2	1	no	—
18	#17984175	C₂²	1	1	yes	characteristic, complemented
19	#17984175	C₂²	1	2	yes	complemented
20	#17984175	C₂²	1	2	yes	complemented
21	#17984175	C₂²	1	1	yes	characteristic, complemented
22	C₂×#14555520	C₂²	1	1	yes	characteristic
23	D₁₃₇⋊M₁₆	—	2	1	no	—
24	D₁₃₇⋊M₁₆	C₂²	1	1	yes	characteristic
25	C₂×#14555519	—	2	1	no	—
26	C₂×#14555518	—	2	1	no	—
27	C₂×#14555518	C₂²	1	1	yes	characteristic
28	C₂²×#11764047	—	2	1	no	—
29	#17984167	C₂²	1	1	yes	characteristic
30	#17984159	C₂²	1	2	yes	complemented
31	#17984158	C₂²	1	2	yes	complemented
32	#17984153	C₂²	1	2	yes	complemented
33	#17984152	C₂²	1	2	yes	complemented
34	#17984152	C₂²	1	2	yes	complemented
35	#17984152	C₂²	1	2	yes	complemented
36	#17984142	C₂²	1	2	yes	complemented
37	C₈×#10587739	C₂²	1	2	yes	complemented
38	C₂×#14555509	C₂²	1	1	yes	characteristic
39	#41×D₁₃₇	C₂²	1	1	yes	characteristic
40	#17984121	C₄	1	1	yes	characteristic, complemented
41	#17984118	C₄	1	1	yes	characteristic, complemented
42	#17984117	C₄	1	2	yes	complemented
43	#17984114	C₄	1	1	yes	characteristic, complemented
44	#17984112	C₄	1	2	yes	complemented
45	#17984109	C₂²	1	2	yes	—
46	#17984107	C₂²	1	2	yes	—
47	M₁₆×D₁₃₇	C₂²	1	1	yes	characteristic
48	AGL(1,ℤ/8)×C₁₃₇	C₄	1	1	yes	characteristic, complemented
49	#14555520	—	2	1	no	—
50	#14555520	—	2	1	no	—
51	#14555520	—	2	1	no	—
52	#14555520	C₂³	1	1	yes	characteristic
53	#14555520	C₂³	1	1	yes	characteristic
54	#14555520	C₂³	1	2	yes	—
55	#14555519	—	2	2	no	—
56	#14555519	—	2	2	no	—
57	#14555519	—	2	1	no	—
58	#14555519	—	2	1	no	—
59	#14555519	—	2	1	no	—
60	#14555518	—	2	1	no	—
61	#14555518	—	2	1	no	—
62	#14555518	—	2	1	no	—
63	#14555518	D₄	1	1	yes	characteristic
64	#14555518	D₄	1	1	yes	characteristic
65	#14555518	C₂³	1	1	yes	characteristic
66	#14555518	C₂³	1	1	yes	characteristic
67	C₂×#11764047	—	2	2	no	—
68	C₂×#11764047	—	2	2	no	—
69	C₂×#11764047	—	2	1	no	—
70	C₂×#11764047	—	2	1	no	—
71	C₂×#11764047	—	2	1	no	—
72	C₂×#11764047	D₄	1	1	yes	characteristic
73	C₂×#11764047	D₄	1	1	yes	characteristic
74	C₄⋊#10587739	—	2	1	no	—
75	C₂²⋊#10587739	—	2	1	no	—
76	C₄×#10587739	C₂³	1	2	yes	—
77	#14555511	C₂³	1	1	yes	characteristic
78	#14555510	—	2	1	no	—
79	#14555510	C₂×C₄	1	1	yes	characteristic
80	#14555509	—	2	1	no	—
81	#14555509	—	2	1	no	—
82	#14555509	—	2	1	no	—
83	#14555509	C₂×C₄	1	2	yes	—
84	#14555509	C₂³	1	1	yes	characteristic
85	#14555509	C₂³	1	2	yes	—
86	C₂×Dic₂₇₄	C₂×C₄	1	1	yes	characteristic
87	Q₈×D₁₃₇	C₂×C₄	1	1	yes	characteristic
88	C₂×C₂²⋊D₁₃₇	—	2	1	no	—
89	D₄×D₁₃₇	—	2	1	no	—
90	D₄×D₁₃₇	C₂×C₄	1	1	yes	characteristic
91	C₂²×Dic₁₃₇	—	2	1	no	—
92	C₂×C₄×D₁₃₇	—	2	1	no	—
93	C₂×C₄×D₁₃₇	C₂³	1	1	yes	characteristic
94	Dic₅₄₈	C₂×C₄	1	2	yes	—
95	#14555500	C₂×C₄	1	2	yes	—
96	#14555499	C₂×C₄	1	2	yes	—
97	#14555498	C₂×C₄	1	2	yes	—
98	#14555498	C₂×C₄	1	2	yes	—
99	#14555495	C₂×C₄	1	2	yes	—
100	#14555494	C₂×C₄	1	1	yes	characteristic
101	#14555493	C₂×C₄	1	2	yes	—
102	C₈×D₁₃₇	C₂³	1	2	yes	—
103	#41×C₁₃₇	C₂×C₄	1	1	yes	characteristic
104	C₂×D₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
105	SD₁₆×C₁₃₇	C₂×C₄	1	2	yes	—
106	D₈×C₁₃₇	C₂×C₄	1	2	yes	—
107	M₁₆×C₁₃₇	C₂×C₄	1	1	yes	characteristic
108	C₂×#10587739	—	2	2	no	—
109	#11764047	—	2	2	no	—
110	#11764047	—	2	1	no	—
111	#11764047	—	2	1	no	—
112	#11764047	C₂×D₄	1	1	yes	characteristic
113	#11764047	C₂×D₄	1	1	yes	characteristic
114	#11764047	C₂×D₄	1	1	yes	characteristic
115	#11764047	C₂×D₄	1	1	yes	characteristic
116	C₂²×D₁₃₇	—	2	1	no	—
117	C₂²×D₁₃₇	#41	1	1	yes	characteristic
118	Dic₂₇₄	—	2	1	no	—
119	Dic₂₇₄	—	2	1	no	—
120	Dic₂₇₄	C₂²×C₄	1	1	yes	characteristic
121	Dic₂₇₄	C₂²×C₄	1	2	yes	—
122	D₅₄₈	—	2	1	no	—
123	D₅₄₈	C₂²×C₄	1	1	yes	characteristic
124	C₂²⋊D₁₃₇	—	2	1	no	—
125	C₂²⋊D₁₃₇	—	2	2	no	—
126	C₂²⋊D₁₃₇	—	2	1	no	—
127	C₂²⋊D₁₃₇	—	2	2	no	—
128	C₂²⋊D₁₃₇	—	2	1	no	—
129	C₂×Dic₁₃₇	—	2	1	no	—
130	C₂×Dic₁₃₇	—	2	2	no	—
131	C₂×Dic₁₃₇	—	2	1	no	—
132	C₂×Dic₁₃₇	—	2	2	no	—
133	C₂×Dic₁₃₇	C₂×D₄	1	1	yes	characteristic
134	C₄×D₁₃₇	—	2	1	no	—
135	C₄×D₁₃₇	—	2	1	no	—
136	C₄×D₁₃₇	—	2	1	no	—
137	C₄×D₁₃₇	C₂²×C₄	1	1	yes	characteristic
138	C₄×D₁₃₇	C₂×D₄	1	1	yes	characteristic
139	C₄×D₁₃₇	#41	1	1	yes	characteristic
140	C₄×D₁₃₇	C₂⁴	1	1	yes	characteristic
141	#11764040	C₂²×C₄	1	2	yes	—
142	C₂³×C₁₃₇	—	2	1	no	—
143	Q₈×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
144	D₄×C₁₃₇	—	2	1	no	—
145	D₄×C₁₃₇	—	2	1	no	—
146	D₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
147	D₄×C₁₃₇	C₂²×C₄	1	2	yes	—
148	C₂×C₄×C₁₃₇	—	2	1	no	—
149	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
150	C₈×C₁₃₇	C₂²×C₄	1	2	yes	—
151	C₂²×C₁₃₇	—	2	1	no	—
152	C₂²×C₁₃₇	—	2	2	no	—
153	C₂²×C₁₃₇	—	4	2	no	—
154	C₂²×C₁₃₇	C₄×D₄	1	1	yes	characteristic
155	C₂×D₁₃₇	—	2	1	no	—
156	C₂×D₁₃₇	—	4	1	no	—
157	C₂×D₁₃₇	—	4	2	no	—
158	C₂×D₁₃₇	C₄×D₄	1	1	yes	characteristic
159	C₂×D₁₃₇	C₂×#41	1	1	yes	characteristic
160	#10587739	—	4	2	no	—
161	C₄×C₁₃₇	—	2	1	no	—
162	C₄×C₁₃₇	C₄×D₄	1	1	yes	characteristic
163	C₄×C₁₃₇	C₂³×C₄	1	1	yes	characteristic
164	Dic₁₃₇	—	2	1	no	—
165	Dic₁₃₇	—	2	2	no	—
166	Dic₁₃₇	C₄×D₄	1	1	yes	characteristic
167	Dic₁₃₇	C₂²×D₄	1	1	yes	characteristic
168	C₂×C₁₃₇	—	2	1	no	—
169	C₂×C₁₃₇	—	4	1	no	—
170	C₂×C₁₃₇	—	4	2	no	—
171	C₂×C₁₃₇	C₂×C₄×D₄	1	1	yes	characteristic
172	D₁₃₇	—	2	1	no	—
173	D₁₃₇	—	2	1	no	—
174	D₁₃₇	—	4	1	no	—
175	C₁₃₇	#2955	1	1	yes	characteristic, complemented, Sylow
176	#2955	—	137	1	no	maximal, Sylow
177	#575	—	137	1	no	—
178	C₂×#131	—	137	1	no	—
179	C₂×#131	—	137	1	no	—
180	#444	—	137	2	no	—
181	#444	—	137	2	no	—
182	#443	—	137	2	no	—
183	#443	—	137	2	no	—
184	#430	—	137	1	no	—
185	#421	—	137	1	no	—
186	#421	—	137	1	no	—
187	#405	—	137	1	no	—
188	C₂×#41	—	137	1	no	—
189	C₂×#41	—	137	1	no	—
190	#137	—	137	1	no	—
191	#137	—	137	2	no	—
192	#137	—	137	1	no	—
193	AGL(1,ℤ/8)	—	137	1	no	—
194	AGL(1,ℤ/8)	—	137	2	no	—
195	AGL(1,ℤ/8)	#10587739	1	1	yes	characteristic, complemented
196	#135	—	137	2	no	—
197	#135	—	137	2	no	—
198	#131	—	274	1	no	—
199	#131	—	274	1	no	—
200	#131	—	274	1	no	—
201	#131	—	274	1	no	—
202	#131	—	137	2	no	—
203	#131	—	137	1	no	—
204	#131	—	137	2	no	—
205	#131	—	137	1	no	—
206	#131	—	137	1	no	—
207	#131	—	137	1	no	—
208	C₂×M₁₆	—	274	1	no	—
209	C₂×M₁₆	—	274	1	no	—
210	C₂×M₁₆	—	137	1	no	—
211	C₂×M₁₆	—	137	1	no	—
212	C₂×M₁₆	—	137	1	no	—
213	C₂²×C₈	—	274	1	no	—
214	C₂²×C₈	—	274	1	no	—
215	C₂²×C₈	—	137	1	no	—
216	#117	—	137	1	no	—
217	#108	—	137	2	no	—
218	#108	—	137	2	no	—
219	#104	—	137	2	no	—
220	#104	—	137	2	no	—
221	#104	—	137	2	no	—
222	#104	—	137	2	no	—
223	#97	—	137	2	no	—
224	C₄×C₈	—	137	2	no	—
225	#41	—	274	1	no	—
226	#41	—	274	1	no	—
227	#41	—	274	1	no	—
228	#41	—	274	1	no	—
229	#41	—	137	1	no	—
230	#41	—	137	2	no	—
231	#41	—	137	1	no	—
232	#41	—	137	2	no	—
233	#41	—	137	1	no	—
234	#41	C₂×#10587739	1	1	yes	characteristic
235	C₂×Q₈	—	137	1	no	—
236	C₂×Q₈	—	137	1	no	—
237	C₂×D₄	—	274	1	no	—
238	C₂×D₄	—	274	1	no	—
239	C₂×D₄	—	137	1	no	—
240	C₂×D₄	C₂×#10587739	1	1	yes	characteristic
241	C₂²×C₄	—	274	1	no	—
242	C₂²×C₄	—	274	1	no	—
243	C₂²×C₄	—	137	1	no	—
244	Q₁₆	—	137	2	no	—
245	Q₁₆	—	137	2	no	—
246	SD₁₆	—	137	2	no	—
247	SD₁₆	—	137	2	no	—
248	SD₁₆	—	137	2	no	—
249	SD₁₆	C₂×#10587739	1	2	yes	—
250	D₈	—	137	2	no	—
251	D₈	C₂×#10587739	1	2	yes	—
252	M₁₆	—	274	1	no	—
253	M₁₆	—	274	1	no	—
254	M₁₆	—	274	2	no	—
255	M₁₆	—	274	1	no	—
256	M₁₆	—	274	2	no	—
257	M₁₆	—	274	1	no	—
258	M₁₆	—	274	1	no	—
259	M₁₆	—	274	1	no	—
260	M₁₆	—	137	1	no	—
261	M₁₆	—	137	1	no	—
262	M₁₆	—	137	2	no	—
263	M₁₆	—	137	1	no	—
264	M₁₆	—	137	2	no	—
265	M₁₆	C₂×#10587739	1	1	yes	characteristic
266	C₂×C₈	—	274	1	no	—
267	C₂×C₈	—	274	1	no	—
268	C₂×C₈	—	274	2	no	—
269	C₂×C₈	—	274	1	no	—
270	C₂×C₈	—	274	2	no	—
271	C₂×C₈	—	274	1	no	—
272	C₂×C₈	—	274	1	no	—
273	C₂×C₈	—	274	1	no	—
274	C₂×C₈	—	137	1	no	—
275	C₂×C₈	—	137	1	no	—
276	C₂×C₈	—	137	1	no	—
277	C₂×C₈	—	137	1	no	—
278	C₂×C₈	—	137	1	no	—
279	C₂×C₈	—	137	1	no	—
280	C₂×C₈	—	137	2	no	—
281	C₄⋊C₄	—	274	1	no	—
282	C₂²⋊C₄	—	274	1	no	—
283	C₄²	—	137	2	no	—
284	C₂³	—	2	1	no	—
285	C₂³	—	274	1	no	—
286	C₂³	—	137	1	no	—
287	Q₈	—	274	1	no	—
288	Q₈	—	274	1	no	—
289	Q₈	—	137	1	no	—
290	Q₈	—	137	2	no	—
291	Q₈	C₂²×#10587739	1	1	yes	characteristic
292	D₄	—	2	1	no	—
293	D₄	—	2	1	no	—
294	D₄	—	274	1	no	—
295	D₄	—	274	1	no	—
296	D₄	—	274	2	no	—
297	D₄	—	274	2	no	—
298	D₄	—	274	1	no	—
299	D₄	—	274	1	no	—
300	D₄	—	137	1	no	—
301	D₄	C₂²×#10587739	1	2	yes	—
302	D₄	C₂²×#10587739	1	1	yes	characteristic
303	C₂×C₄	—	2	1	no	—
304	C₂×C₄	—	274	1	no	—
305	C₂×C₄	—	274	1	no	—
306	C₂×C₄	—	274	1	no	—
307	C₂×C₄	—	274	1	no	—
308	C₂×C₄	—	274	2	no	—
309	C₂×C₄	—	274	2	no	—
310	C₂×C₄	—	274	1	no	—
311	C₂×C₄	—	274	2	no	—
312	C₂×C₄	—	137	1	no	—
313	C₂×C₄	—	137	1	no	—
314	C₂×C₄	—	137	1	no	—
315	C₂×C₄	—	137	1	no	—
316	C₂×C₄	—	137	1	no	—
317	C₂×C₄	C₂²×#10587739	1	1	yes	characteristic
318	C₈	—	274	2	no	—
319	C₈	—	274	1	no	—
320	C₈	—	274	1	no	—
321	C₈	—	137	1	no	—
322	C₈	—	137	1	no	—
323	C₈	—	137	1	no	—
324	C₈	—	137	1	no	—
325	C₈	—	137	2	no	—
326	C₈	C₂²×#10587739	1	2	yes	—
327	C₂²	—	2	1	no	—
328	C₂²	—	2	2	no	—
329	C₂²	—	4	2	no	—
330	C₂²	—	274	1	no	—
331	C₂²	—	548	1	no	—
332	C₂²	—	548	2	no	—
333	C₂²	—	137	1	no	—
334	C₂²	—	137	1	no	—
335	C₂²	D₄×#10587739	1	1	yes	characteristic
336	C₄	—	2	1	no	—
337	C₄	—	274	1	no	—
338	C₄	—	274	2	no	—
339	C₄	—	548	2	no	—
340	C₄	—	137	1	no	—
341	C₄	—	137	1	no	—
342	C₄	D₄×#10587739	1	1	yes	characteristic
343	C₄	C₂³×#10587739	1	1	yes	characteristic
344	C₂	—	2	1	no	—
345	C₂	—	4	2	no	—
346	C₂	—	4	1	no	—
347	C₂	—	274	1	no	—
348	C₂	—	274	1	no	—
349	C₂	—	548	1	no	—
350	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	#2955
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. D₁₃₇⋊#131	C₂	semidirect
2. C₂×#17984175	C₂	semidirect
3. #22578450	C₂	semidirect
4. #22578448	C₂	semidirect
5. #22578446	C₂	semidirect
6. #22578445	C₂	semidirect
7. #22578418	C₂	semidirect
8. #22578401	C₂	semidirect
9. #22578400	C₂	semidirect
10. #22578369	C₂	semidirect
12. #17984176	C₂²	semidirect
13. #17984176	C₂²	semidirect
18. #17984175	C₂²	semidirect
19. #17984175	C₂²	semidirect
20. #17984175	C₂²	semidirect
21. #17984175	C₂²	semidirect
30. #17984159	C₂²	semidirect
31. #17984158	C₂²	semidirect
32. #17984153	C₂²	semidirect
33. #17984152	C₂²	semidirect
34. #17984152	C₂²	semidirect
35. #17984152	C₂²	semidirect
36. #17984142	C₂²	semidirect
37. C₈×#10587739	C₂²	semidirect
40. #17984121	C₄	semidirect
41. #17984118	C₄	semidirect
42. #17984117	C₄	semidirect
43. #17984114	C₄	semidirect
44. #17984112	C₄	semidirect
48. AGL(1,ℤ/8)×C₁₃₇	C₄	semidirect
175. C₁₃₇	#2955	semidirect
195. AGL(1,ℤ/8)	#10587739	semidirect

Maximal subgroups

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

subgroup	index
D₁₃₇⋊#131	2
C₂×#17984175	2
#22578450	2
#22578448	2
#22578446	2
#22578445	2
#22578418	2
#22578401	2
#22578400	2
#22578369	2
#22578199	2
#2955	137