Subgroups — #31797547

Order 17536.

470 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22578472	C₂	1	1	yes	maximal, characteristic
2	#22578471	C₂	1	1	yes	maximal, characteristic, complemented
3	#22578443	C₂	1	2	yes	maximal, complemented
4	#22578442	C₂	1	2	yes	maximal, complemented
5	Q₁₆×#10587739	C₂	1	2	yes	maximal, complemented
6	SD₁₆×#10587739	C₂	1	2	yes	maximal, complemented
7	#22578414	C₂	1	1	yes	maximal, characteristic, complemented
8	#22578393	C₂	1	1	yes	maximal, characteristic
9	C₂×#17984151	C₂	1	1	yes	maximal, characteristic, complemented
10	#22578369	C₂	1	1	yes	maximal, characteristic, complemented
11	#137×D₁₃₇	C₂	1	1	yes	maximal, characteristic
12	C₂×#14555518	C₂²	1	1	yes	characteristic
13	Q₈×#10587739	—	2	1	no	—
14	Q₈×#10587739	—	2	1	no	—
15	Q₈×#10587739	C₂²	1	2	yes	—
16	Q₈×#10587739	C₂²	1	2	yes	—
17	Q₈×#10587739	C₂²	1	2	yes	—
18	D₄×#10587739	—	2	2	no	—
19	D₄×#10587739	C₂²	1	2	yes	—
20	#17984167	—	2	1	no	—
21	#17984167	—	2	2	no	—
22	#17984167	C₂²	1	1	yes	characteristic
23	#17984167	C₂²	1	1	yes	characteristic
24	C₂×C₄⋊#10587739	—	2	1	no	—
25	C₂×C₄⋊#10587739	C₂²	1	1	yes	characteristic
26	#17984157	C₂²	1	2	yes	—
27	#17984156	C₂²	1	2	yes	—
28	#17984151	C₂²	1	2	yes	—
29	#17984151	C₂²	1	1	yes	characteristic
30	#17984151	C₂²	1	2	yes	—
31	#17984151	C₂²	1	1	yes	characteristic
32	#17984150	C₂²	1	1	yes	characteristic
33	#17984150	C₂²	1	1	yes	characteristic
34	#17984142	C₂²	1	2	yes	—
35	C₈×#10587739	C₂²	1	2	yes	—
36	#41×D₁₃₇	C₂²	1	1	yes	characteristic
37	C₂×Q₈×D₁₃₇	C₂²	1	1	yes	characteristic
38	#17984121	C₄	1	1	yes	characteristic, complemented
39	#17984120	C₄	1	2	yes	complemented
40	#17984119	C₄	1	1	yes	characteristic, complemented
41	#17984118	C₄	1	1	yes	characteristic, complemented
42	#17984117	C₄	1	2	yes	complemented
43	Q₁₆×D₁₃₇	C₂²	1	2	yes	—
44	SD₁₆×D₁₃₇	C₂²	1	2	yes	—
45	M₁₆×D₁₃₇	C₂²	1	1	yes	characteristic
46	#137×C₁₃₇	C₄	1	1	yes	characteristic, complemented
47	C₂²×#10587739	—	2	2	no	—
48	C₂²×#10587739	—	2	1	no	—
49	#14555518	—	2	1	no	—
50	#14555518	C₂³	1	1	yes	characteristic
51	#14555518	C₂³	1	1	yes	characteristic
52	C₂×#11764047	—	2	1	no	—
53	C₄⋊#10587739	—	2	2	no	—
54	C₄⋊#10587739	—	2	2	no	—
55	C₄⋊#10587739	—	2	2	no	—
56	C₄⋊#10587739	—	2	1	no	—
57	C₄⋊#10587739	D₄	1	1	yes	characteristic
58	C₄⋊#10587739	D₄	1	1	yes	characteristic
59	C₄⋊#10587739	C₂³	1	1	yes	characteristic
60	C₄⋊#10587739	C₂³	1	1	yes	characteristic
61	C₄⋊#10587739	C₂³	1	2	yes	—
62	C₂²⋊#10587739	—	2	2	no	—
63	C₂²⋊#10587739	—	2	1	no	—
64	C₂²⋊#10587739	D₄	1	2	yes	—
65	C₄×#10587739	—	2	2	no	—
66	C₄×#10587739	—	2	2	no	—
67	C₄×#10587739	—	2	2	no	—
68	C₄×#10587739	—	2	1	no	—
69	C₄×#10587739	C₂³	1	2	yes	—
70	#14555511	C₂×C₄	1	1	yes	characteristic
71	#14555510	—	2	1	no	—
72	#14555510	C₂×C₄	1	1	yes	characteristic
73	#14555509	—	2	1	no	—
74	#14555509	C₂×C₄	1	1	yes	characteristic
75	C₂×Dic₂₇₄	—	2	1	no	—
76	C₂×Dic₂₇₄	C₂×C₄	1	1	yes	characteristic, complemented
77	Q₈×D₁₃₇	—	2	1	no	—
78	Q₈×D₁₃₇	—	2	1	no	—
79	Q₈×D₁₃₇	C₂×C₄	1	2	yes	complemented
80	Q₈×D₁₃₇	C₂³	1	2	yes	—
81	Q₈×D₁₃₇	C₂³	1	1	yes	characteristic
82	D₄×D₁₃₇	—	2	1	no	—
83	D₄×D₁₃₇	C₂³	1	1	yes	characteristic
84	C₂×C₄×D₁₃₇	—	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	#14555500	C₂×C₄	1	2	yes	complemented
90	#14555499	C₂×C₄	1	2	yes	complemented
91	#14555498	C₂×C₄	1	2	yes	complemented
92	#14555497	C₂×C₄	1	2	yes	complemented
93	#14555494	C₂×C₄	1	1	yes	characteristic, complemented
94	#14555493	C₂×C₄	1	2	yes	complemented
95	C₈×D₁₃₇	C₂³	1	2	yes	—
96	#41×C₁₃₇	C₂×C₄	1	1	yes	characteristic
97	C₂×Q₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic, complemented
98	Q₁₆×C₁₃₇	C₂×C₄	1	2	yes	complemented
99	SD₁₆×C₁₃₇	C₂×C₄	1	2	yes	complemented
100	M₁₆×C₁₃₇	C₂×C₄	1	1	yes	characteristic, complemented
101	C₂×#10587739	—	2	1	no	—
102	C₂×#10587739	—	2	1	no	—
103	C₂×#10587739	—	2	2	no	—
104	C₂×#10587739	—	8	2	no	—
105	C₂×#10587739	—	8	1	no	—
106	C₂×#10587739	C₂×D₄	1	2	yes	—
107	C₂×#10587739	C₂×D₄	1	2	yes	—
108	#11764047	—	2	1	no	—
109	#11764047	—	2	1	no	—
110	C₂²×D₁₃₇	—	2	1	no	—
111	C₂²×D₁₃₇	C₂×D₄	1	1	yes	characteristic
112	Dic₂₇₄	—	2	1	no	—
113	Dic₂₇₄	—	2	2	no	—
114	Dic₂₇₄	—	2	2	no	—
115	Dic₂₇₄	—	2	1	no	—
116	Dic₂₇₄	C₂²×C₄	1	2	yes	—
117	Dic₂₇₄	C₂²×C₄	1	1	yes	characteristic
118	D₅₄₈	—	2	1	no	—
119	D₅₄₈	C₂²×C₄	1	1	yes	characteristic
120	C₂²⋊D₁₃₇	—	2	1	no	—
121	C₂²⋊D₁₃₇	—	2	1	no	—
122	C₂²⋊D₁₃₇	—	2	1	no	—
123	C₂×Dic₁₃₇	—	2	1	no	—
124	C₂×Dic₁₃₇	—	2	1	no	—
125	C₂×Dic₁₃₇	#41	1	1	yes	characteristic
126	C₄×D₁₃₇	—	2	2	no	—
127	C₄×D₁₃₇	—	2	1	no	—
128	C₄×D₁₃₇	—	2	2	no	—
129	C₄×D₁₃₇	—	2	1	no	—
130	C₄×D₁₃₇	—	2	1	no	—
131	C₄×D₁₃₇	C₂²×C₄	1	1	yes	characteristic
132	C₄×D₁₃₇	C₂×D₄	1	1	yes	characteristic
133	C₄×D₁₃₇	#41	1	1	yes	characteristic
134	C₄×D₁₃₇	C₂⁴	1	1	yes	characteristic
135	#11764040	C₂²×C₄	1	2	yes	—
136	Q₈×C₁₃₇	—	2	1	no	—
137	Q₈×C₁₃₇	C₂²×C₄	1	2	yes	—
138	Q₈×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
139	D₄×C₁₃₇	—	2	1	no	—
140	D₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
141	C₂×C₄×C₁₃₇	—	2	1	no	—
142	C₂×C₄×C₁₃₇	—	2	1	no	—
143	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
144	C₈×C₁₃₇	C₂²×C₄	1	2	yes	—
145	C₂²×C₁₃₇	—	2	1	no	—
146	C₂²×C₁₃₇	C₄×D₄	1	1	yes	characteristic
147	C₂×D₁₃₇	—	2	1	no	—
148	C₂×D₁₃₇	—	2	1	no	—
149	C₂×D₁₃₇	—	2	1	no	—
150	C₂×D₁₃₇	—	4	1	no	—
151	C₂×D₁₃₇	—	4	1	no	—
152	C₂×D₁₃₇	C₄×D₄	1	1	yes	characteristic
153	C₂×D₁₃₇	C₂²×D₄	1	1	yes	characteristic
154	#10587739	—	2	2	no	—
155	#10587739	—	2	2	no	—
156	#10587739	—	4	1	no	—
157	#10587739	—	4	2	no	—
158	#10587739	—	4	1	no	—
159	C₄×C₁₃₇	—	2	1	no	—
160	C₄×C₁₃₇	—	2	2	no	—
161	C₄×C₁₃₇	C₄×D₄	1	1	yes	characteristic
162	C₄×C₁₃₇	C₂³×C₄	1	1	yes	characteristic
163	Dic₁₃₇	—	2	1	no	—
164	Dic₁₃₇	—	2	2	no	—
165	Dic₁₃₇	C₄×D₄	1	1	yes	characteristic
166	Dic₁₃₇	C₂×#41	1	1	yes	characteristic
167	C₂×C₁₃₇	—	2	1	no	—
168	C₂×C₁₃₇	—	4	1	no	—
169	C₂×C₁₃₇	C₂×C₄×D₄	1	1	yes	characteristic
170	D₁₃₇	—	2	1	no	—
171	D₁₃₇	—	4	1	no	—
172	D₁₃₇	#577	1	1	yes	characteristic
173	D₁₃₇	#578	1	1	yes	characteristic
174	C₁₃₇	#2947	1	1	yes	characteristic, complemented, Sylow
175	#2947	—	137	1	no	maximal, Sylow
176	C₂×#137	—	137	1	no	—
177	#520	—	137	1	no	—
178	#519	—	137	1	no	—
179	#441	—	137	2	no	—
180	#440	—	137	2	no	—
181	C₄×Q₁₆	—	137	2	no	—
182	C₄×SD₁₆	—	137	2	no	—
183	#428	—	137	1	no	—
184	#416	—	137	1	no	—
185	C₂×#103	—	137	1	no	—
186	#405	—	137	1	no	—
187	C₂×#41	—	137	1	no	—
188	C₂²×Q₈	—	137	1	no	—
189	#137	—	137	1	no	—
190	#137	—	137	1	no	—
191	#137	—	137	2	no	—
192	#137	—	137	2	no	—
193	#137	—	137	1	no	—
194	#137	#10587739	1	1	yes	characteristic, complemented
195	C₂×Q₁₆	—	137	2	no	—
196	C₂×SD₁₆	—	137	2	no	—
197	C₂×M₁₆	—	137	1	no	—
198	C₂²×C₈	—	137	1	no	—
199	C₄×Q₈	—	274	1	no	—
200	C₄×Q₈	—	274	1	no	—
201	C₄×Q₈	—	137	2	no	—
202	C₄×Q₈	—	137	2	no	—
203	C₄×Q₈	—	137	2	no	—
204	C₄×D₄	—	274	2	no	—
205	C₄×D₄	—	137	2	no	—
206	#117	—	274	1	no	—
207	#117	—	274	2	no	—
208	#117	—	137	1	no	—
209	#117	—	137	1	no	—
210	C₂×C₄⋊C₄	—	274	1	no	—
211	C₂×C₄⋊C₄	—	137	1	no	—
212	#107	—	137	2	no	—
213	#106	—	137	2	no	—
214	#103	—	137	2	no	—
215	#103	—	137	2	no	—
216	#103	—	137	1	no	—
217	#103	—	137	1	no	—
218	#102	—	137	1	no	—
219	#102	—	137	1	no	—
220	#97	—	137	2	no	—
221	C₄×C₈	—	137	2	no	—
222	#41	—	274	1	no	—
223	#41	—	274	1	no	—
224	#41	—	137	1	no	—
225	#41	—	137	1	no	—
226	#41	—	137	1	no	—
227	#41	C₂×#10587739	1	1	yes	characteristic
228	C₂×Q₈	—	274	1	no	—
229	C₂×Q₈	—	274	1	no	—
230	C₂×Q₈	—	274	1	no	—
231	C₂×Q₈	—	137	2	no	—
232	C₂×Q₈	—	137	1	no	—
233	C₂×Q₈	—	137	2	no	—
234	C₂×Q₈	—	137	1	no	—
235	C₂×Q₈	C₂×#10587739	1	1	yes	characteristic, complemented
236	C₂×D₄	—	274	1	no	—
237	C₂×D₄	—	137	1	no	—
238	C₂²×C₄	—	274	1	no	—
239	C₂²×C₄	—	274	1	no	—
240	C₂²×C₄	—	274	1	no	—
241	C₂²×C₄	—	274	2	no	—
242	C₂²×C₄	—	137	1	no	—
243	Q₁₆	—	137	2	no	—
244	Q₁₆	—	137	2	no	—
245	Q₁₆	—	137	2	no	—
246	Q₁₆	C₂×#10587739	1	2	yes	complemented
247	SD₁₆	—	137	2	no	—
248	SD₁₆	—	137	2	no	—
249	SD₁₆	—	137	2	no	—
250	SD₁₆	C₂×#10587739	1	2	yes	complemented
251	M₁₆	—	137	1	no	—
252	M₁₆	—	137	2	no	—
253	M₁₆	C₂×#10587739	1	1	yes	characteristic, complemented
254	C₂×C₈	—	274	1	no	—
255	C₂×C₈	—	274	1	no	—
256	C₂×C₈	—	137	1	no	—
257	C₂×C₈	—	137	1	no	—
258	C₂×C₈	—	137	2	no	—
259	C₄⋊C₄	—	274	1	no	—
260	C₄⋊C₄	—	274	2	no	—
261	C₄⋊C₄	—	274	2	no	—
262	C₄⋊C₄	—	274	2	no	—
263	C₄⋊C₄	—	137	1	no	—
264	C₄⋊C₄	—	137	1	no	—
265	C₄⋊C₄	—	137	2	no	—
266	C₄⋊C₄	—	137	1	no	—
267	C₄⋊C₄	—	137	1	no	—
268	C₂²⋊C₄	—	274	1	no	—
269	C₂²⋊C₄	—	274	2	no	—
270	C₂²⋊C₄	—	137	2	no	—
271	C₄²	—	274	2	no	—
272	C₄²	—	274	2	no	—
273	C₄²	—	274	2	no	—
274	C₄²	—	274	1	no	—
275	C₄²	—	137	2	no	—
276	C₂³	—	274	1	no	—
277	C₂³	—	137	1	no	—
278	Q₈	—	2	1	no	—
279	Q₈	—	274	2	no	—
280	Q₈	—	274	2	no	—
281	Q₈	—	274	1	no	—
282	Q₈	—	274	1	no	—
283	Q₈	—	137	2	no	—
284	Q₈	—	137	1	no	—
285	Q₈	C₂²×#10587739	1	2	yes	—
286	Q₈	C₂²×#10587739	1	1	yes	characteristic
287	D₄	—	2	1	no	—
288	D₄	—	274	1	no	—
289	D₄	—	274	1	no	—
290	D₄	—	274	1	no	—
291	D₄	—	274	1	no	—
292	D₄	—	137	1	no	—
293	D₄	C₂²×#10587739	1	1	yes	characteristic
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	274	1	no	—
297	C₂×C₄	—	274	2	no	—
298	C₂×C₄	—	274	2	no	—
299	C₂×C₄	—	274	1	no	—
300	C₂×C₄	—	274	1	no	—
301	C₂×C₄	—	274	1	no	—
302	C₂×C₄	—	274	1	no	—
303	C₂×C₄	—	274	1	no	—
304	C₂×C₄	—	274	1	no	—
305	C₂×C₄	—	274	2	no	—
306	C₂×C₄	—	1096	2	no	—
307	C₂×C₄	—	1096	1	no	—
308	C₂×C₄	—	137	2	no	—
309	C₂×C₄	—	137	2	no	—
310	C₂×C₄	—	137	1	no	—
311	C₂×C₄	—	137	1	no	—
312	C₂×C₄	—	137	1	no	—
313	C₂×C₄	—	137	1	no	—
314	C₂×C₄	—	137	1	no	—
315	C₂×C₄	C₂²×#10587739	1	1	yes	characteristic
316	C₈	—	274	1	no	—
317	C₈	—	274	1	no	—
318	C₈	—	137	2	no	—
319	C₈	C₂²×#10587739	1	2	yes	—
320	C₂²	—	2	1	no	—
321	C₂²	—	274	1	no	—
322	C₂²	—	274	1	no	—
323	C₂²	—	274	1	no	—
324	C₂²	—	548	1	no	—
325	C₂²	—	548	1	no	—
326	C₂²	—	137	1	no	—
327	C₂²	—	137	1	no	—
328	C₂²	D₄×#10587739	1	1	yes	characteristic
329	C₄	—	2	1	no	—
330	C₄	—	2	2	no	—
331	C₄	—	274	2	no	—
332	C₄	—	274	2	no	—
333	C₄	—	274	1	no	—
334	C₄	—	274	2	no	—
335	C₄	—	548	2	no	—
336	C₄	—	548	1	no	—
337	C₄	—	548	1	no	—
338	C₄	—	137	1	no	—
339	C₄	—	137	1	no	—
340	C₄	D₄×#10587739	1	1	yes	characteristic
341	C₄	C₂³×#10587739	1	1	yes	characteristic
342	C₂	—	2	1	no	—
343	C₂	—	4	1	no	—
344	C₂	—	274	1	no	—
345	C₂	—	548	1	no	—
346	C₂	—	137	1	no	—
347	C₂	—	137	1	no	—
348	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	#137×C₁₃₇	C₄
socle	C₂×C₁₃₇	C₂×C₄×D₄
prime radical	#41×D₁₃₇	C₂²
cosocle	C₄×D₁₃₇	C₂⁴
hypercenter	#137	#10587739
hypocenter	C₁₃₇	#2947
LCS	G ▷ C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #137

Complements

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

N	H ≅ G/N	type
2. #22578471	C₂	semidirect
3. #22578443	C₂	semidirect
4. #22578442	C₂	semidirect
5. Q₁₆×#10587739	C₂	semidirect
6. SD₁₆×#10587739	C₂	semidirect
7. #22578414	C₂	semidirect
9. C₂×#17984151	C₂	semidirect
10. #22578369	C₂	semidirect
38. #17984121	C₄	semidirect
39. #17984120	C₄	semidirect
40. #17984119	C₄	semidirect
41. #17984118	C₄	semidirect
42. #17984117	C₄	semidirect
46. #137×C₁₃₇	C₄	semidirect
76. C₂×Dic₂₇₄	C₂×C₄	semidirect
79. Q₈×D₁₃₇	C₂×C₄	semidirect
87. Dic₅₄₈	C₂×C₄	semidirect
88. #14555500	C₂×C₄	semidirect
89. #14555500	C₂×C₄	semidirect
90. #14555499	C₂×C₄	semidirect
91. #14555498	C₂×C₄	semidirect
92. #14555497	C₂×C₄	semidirect
93. #14555494	C₂×C₄	semidirect
94. #14555493	C₂×C₄	semidirect
97. C₂×Q₈×C₁₃₇	C₂×C₄	semidirect
98. Q₁₆×C₁₃₇	C₂×C₄	semidirect
99. SD₁₆×C₁₃₇	C₂×C₄	semidirect
100. M₁₆×C₁₃₇	C₂×C₄	semidirect
174. C₁₃₇	#2947	semidirect
194. #137	#10587739	semidirect
235. C₂×Q₈	C₂×#10587739	semidirect
246. Q₁₆	C₂×#10587739	semidirect
250. SD₁₆	C₂×#10587739	semidirect
253. M₁₆	C₂×#10587739	semidirect

Maximal subgroups

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

subgroup	index
#22578472	2
#22578471	2
#22578443	2
#22578442	2
Q₁₆×#10587739	2
SD₁₆×#10587739	2
#22578414	2
#22578393	2
C₂×#17984151	2
#22578369	2
#137×D₁₃₇	2
#2947	137