Subgroups — #31819090

Order 17792.

336 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22581679	C₂	1	1	yes	maximal, characteristic, complemented
2	#22581675	C₂	1	1	yes	maximal, characteristic
3	#22581309	C₂	1	1	yes	maximal, characteristic
4	#22581308	C₂	1	1	yes	maximal, characteristic
5	#22581302	C₂	1	1	yes	maximal, characteristic, complemented
6	#22581260	C₂	1	1	yes	maximal, characteristic, complemented
7	#22581259	C₂	1	1	yes	maximal, characteristic, complemented
8	#22581232	C₂	1	1	yes	maximal, characteristic
9	#22581177	C₂	1	1	yes	maximal, characteristic
10	#22581175	C₂	1	1	yes	maximal, characteristic
11	#22581161	C₂	1	1	yes	maximal, characteristic, complemented
12	C₈×Dic₂₇₈	C₂	1	1	yes	maximal, characteristic, complemented
13	#22581045	C₂	1	1	yes	maximal, characteristic, complemented
14	#22581038	C₂	1	1	yes	maximal, characteristic
15	C₄×Q₁₆×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#14555775	C₂²	1	1	yes	characteristic
17	C₂×#14555772	C₂²	1	1	yes	characteristic
18	#17984841	—	2	1	no	—
19	#17984841	C₂²	1	1	yes	characteristic
20	#17984840	C₂²	1	1	yes	characteristic
21	#17984837	—	2	1	no	—
22	#17984836	—	2	1	no	—
23	#17984834	—	2	1	no	—
24	#17984834	C₂²	1	1	yes	characteristic
25	#17984833	—	2	1	no	—
26	#17984833	C₂²	1	1	yes	characteristic
27	#17984831	C₂²	1	1	yes	characteristic
28	#17984830	C₂²	1	1	yes	characteristic
29	#17984820	—	2	1	no	—
30	Q₈×Dic₁₃₉	C₂²	1	1	yes	characteristic
31	Q₈×Dic₁₃₉	C₂²	1	1	yes	characteristic
32	C₄×Dic₂₇₈	—	2	1	no	—
33	C₄×Dic₂₇₈	C₂²	1	1	yes	characteristic
34	#17984807	C₂²	1	1	yes	characteristic
35	#17984804	C₂²	1	1	yes	characteristic
36	#17984799	—	2	1	no	—
37	#17984775	C₂²	1	1	yes	characteristic
38	#17984773	C₂²	1	1	yes	characteristic
39	C₄⋊(C₁₃₉⋊C₈)	C₂²	1	1	yes	characteristic
40	Dic₁₃₉⋊C₈	C₂²	1	1	yes	characteristic
41	Dic₁₃₉⋊C₈	C₂²	1	1	yes	characteristic
42	#17984767	C₂²	1	1	yes	characteristic
43	#17984767	C₂²	1	1	yes	characteristic
44	#17984766	C₂²	1	1	yes	characteristic
45	#17984766	C₂²	1	1	yes	characteristic
46	#17984765	C₂²	1	1	yes	characteristic
47	#17984764	C₂²	1	1	yes	characteristic
48	#17984764	C₂²	1	1	yes	characteristic
49	#17984762	C₂²	1	1	yes	characteristic
50	C₈×Dic₁₃₉	C₂²	1	1	yes	characteristic
51	C₈×Dic₁₃₉	C₂²	1	1	yes	characteristic
52	C₂×Q₁₆×C₁₃₉	C₂²	1	1	yes	characteristic
53	C₄×Q₈×C₁₃₉	C₂²	1	1	yes	characteristic
54	C₄×Q₈×C₁₃₉	C₂²	1	1	yes	characteristic
55	#107×C₁₃₉	C₂²	1	1	yes	characteristic
56	#103×C₁₃₉	C₂²	1	1	yes	characteristic
57	#103×C₁₃₉	C₂²	1	1	yes	characteristic
58	C₄×C₈×C₁₃₉	C₂²	1	1	yes	characteristic
59	C₂×Dic₂₇₈	—	2	1	no	—
60	C₂×Dic₂₇₈	D₄	1	1	yes	characteristic
61	C₂×Dic₂₇₈	C₂³	1	1	yes	characteristic
62	C₂×D₅₅₆	C₂³	1	1	yes	characteristic
63	C₂×C₄×D₁₃₉	—	2	1	no	—
64	#14555775	—	2	2	no	—
65	#14555772	—	2	2	no	—
66	C₂×C₁₃₉⋊C₈	C₂³	1	1	yes	characteristic
67	C₂×C₁₃₉⋊C₈	C₂³	1	1	yes	characteristic
68	C₄⋊Dic₁₃₉	—	2	1	no	—
69	C₄⋊Dic₁₃₉	—	2	1	no	—
70	C₄⋊Dic₁₃₉	—	2	1	no	—
71	C₄⋊Dic₁₃₉	D₄	1	1	yes	characteristic
72	C₄⋊Dic₁₃₉	C₂³	1	1	yes	characteristic
73	C₄⋊Dic₁₃₉	C₂³	1	1	yes	characteristic
74	Dic₁₃₉⋊C₄	—	2	1	no	—
75	Dic₁₃₉⋊C₄	—	2	1	no	—
76	Dic₁₃₉⋊C₄	—	2	1	no	—
77	Dic₁₃₉⋊C₄	—	2	1	no	—
78	Dic₁₃₉⋊C₄	—	2	1	no	—
79	Dic₁₃₉⋊C₄	—	2	1	no	—
80	Dic₁₃₉⋊C₄	—	2	1	no	—
81	Dic₁₃₉⋊C₄	D₄	1	1	yes	characteristic
82	Dic₁₃₉⋊C₄	D₄	1	1	yes	characteristic
83	(C₂×D₁₃₉)⋊C₄	—	2	1	no	—
84	(C₂×D₁₃₉)⋊C₄	—	2	1	no	—
85	(C₂×D₁₃₉)⋊C₄	—	2	1	no	—
86	(C₂×D₁₃₉)⋊C₄	—	2	1	no	—
87	(C₂×D₁₃₉)⋊C₄	—	2	1	no	—
88	C₄×Dic₁₃₉	—	2	1	no	—
89	C₄×Dic₁₃₉	—	2	1	no	—
90	C₄×Dic₁₃₉	—	2	1	no	—
91	C₄×Dic₁₃₉	C₂³	1	1	yes	characteristic
92	C₄×Dic₁₃₉	C₂³	1	1	yes	characteristic
93	C₂×Q₈×C₁₃₉	C₂³	1	1	yes	characteristic
94	C₂×Q₈×C₁₃₉	C₂³	1	1	yes	characteristic
95	Q₁₆×C₁₃₉	—	2	2	no	—
96	C₂×C₈×C₁₃₉	C₂³	1	1	yes	characteristic
97	C₂×C₈×C₁₃₉	C₂³	1	1	yes	characteristic
98	C₄⋊C₄×C₁₃₉	—	2	1	no	—
99	C₄⋊C₄×C₁₃₉	—	2	1	no	—
100	C₄⋊C₄×C₁₃₉	C₂³	1	1	yes	characteristic
101	C₄⋊C₄×C₁₃₉	C₂³	1	1	yes	characteristic
102	C₄²×C₁₃₉	—	2	1	no	—
103	C₄²×C₁₃₉	—	2	1	no	—
104	C₄²×C₁₃₉	C₂³	1	1	yes	characteristic
105	C₂²×D₁₃₉	—	2	1	no	—
106	Dic₂₇₈	—	2	2	no	—
107	Dic₂₇₈	—	2	1	no	—
108	Dic₂₇₈	—	2	1	no	—
109	Dic₂₇₈	—	4	2	no	—
110	D₅₅₆	—	2	1	no	—
111	D₅₅₆	—	2	1	no	—
112	C₂×Dic₁₃₉	—	2	1	no	—
113	C₂×Dic₁₃₉	—	2	1	no	—
114	C₂×Dic₁₃₉	—	2	1	no	—
115	C₂×Dic₁₃₉	C₂×D₄	1	1	yes	characteristic
116	C₂×Dic₁₃₉	C₂×D₄	1	1	yes	characteristic
117	C₂×Dic₁₃₉	C₂×D₄	1	1	yes	characteristic
118	C₂×Dic₁₃₉	C₂×D₄	1	1	yes	characteristic
119	C₄×D₁₃₉	—	4	2	no	—
120	C₁₃₉⋊C₈	—	2	1	no	—
121	C₁₃₉⋊C₈	—	2	1	no	—
122	Q₈×C₁₃₉	—	2	1	no	—
123	Q₈×C₁₃₉	—	2	1	no	—
124	Q₈×C₁₃₉	—	2	1	no	—
125	Q₈×C₁₃₉	#41	1	2	yes	—
126	C₂×C₄×C₁₃₉	—	2	1	no	—
127	C₂×C₄×C₁₃₉	—	2	1	no	—
128	C₂×C₄×C₁₃₉	—	2	1	no	—
129	C₂×C₄×C₁₃₉	—	2	1	no	—
130	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
131	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
132	C₂×C₄×C₁₃₉	C₂⁴	1	1	yes	characteristic
133	C₈×C₁₃₉	—	2	1	no	—
134	C₈×C₁₃₉	—	2	1	no	—
135	C₂²×C₁₃₉	C₂²×D₄	1	1	yes	characteristic
136	C₂×D₁₃₉	—	4	1	no	—
137	C₂×D₁₃₉	—	4	1	no	—
138	C₂×D₁₃₉	—	4	1	no	—
139	C₄×C₁₃₉	—	2	2	no	—
140	C₄×C₁₃₉	—	2	1	no	—
141	C₄×C₁₃₉	—	2	1	no	—
142	C₄×C₁₃₉	—	4	1	no	—
143	C₄×C₁₃₉	—	4	1	no	—
144	C₄×C₁₃₉	—	4	1	no	—
145	C₄×C₁₃₉	C₂×#41	1	1	yes	characteristic
146	C₄×C₁₃₉	2₋¹⁺⁴	1	1	yes	characteristic
147	Dic₁₃₉	—	2	1	no	—
148	Dic₁₃₉	—	2	1	no	—
149	Dic₁₃₉	—	2	1	no	—
150	Dic₁₃₉	—	4	1	no	—
151	Dic₁₃₉	—	4	1	no	—
152	Dic₁₃₉	—	4	1	no	—
153	Dic₁₃₉	#135	1	2	yes	—
154	C₂×C₁₃₉	#548	1	1	yes	characteristic
155	C₂×C₁₃₉	C₂×#135	1	1	yes	characteristic
156	C₂×C₁₃₉	#578	1	1	yes	characteristic
157	D₁₃₉	—	8	1	no	—
158	C₁₃₉	#3364	1	1	yes	characteristic, complemented, Sylow
159	#3364	—	139	1	no	maximal, Sylow
160	#557	—	139	1	no	—
161	#556	—	139	1	no	—
162	#488	—	139	1	no	—
163	#488	—	139	1	no	—
164	#487	—	139	1	no	—
165	#479	—	139	1	no	—
166	#479	—	139	1	no	—
167	#474	—	139	1	no	—
168	#464	—	139	1	no	—
169	#464	—	139	1	no	—
170	#462	—	139	1	no	—
171	C₈×Q₈	—	139	1	no	—
172	C₄×Q₁₆	—	139	1	no	—
173	C₄×Q₁₆	D₁₃₉	1	1	yes	characteristic, complemented
174	C₄×SD₁₆	—	139	1	no	—
175	C₂×Q₁₆	—	139	1	no	—
176	C₂×Q₁₆	C₂×D₁₃₉	1	1	yes	characteristic
177	C₂×SD₁₆	—	139	1	no	—
178	C₄⋊Q₈	—	278	1	no	—
179	C₄⋊Q₈	—	139	1	no	—
180	C₄⋊Q₈	—	139	1	no	—
181	#126	—	278	1	no	—
182	#126	—	278	1	no	—
183	#125	—	278	1	no	—
184	#125	—	278	1	no	—
185	#125	—	139	1	no	—
186	#125	—	139	1	no	—
187	#124	—	139	1	no	—
188	#124	—	139	1	no	—
189	C₂²⋊Q₈	—	278	1	no	—
190	C₄×Q₈	—	278	1	no	—
191	C₄×Q₈	—	139	1	no	—
192	C₄×Q₈	—	139	1	no	—
193	C₄×Q₈	—	139	1	no	—
194	C₄×Q₈	—	139	1	no	—
195	C₄×Q₈	C₂×D₁₃₉	1	1	yes	characteristic
196	C₄×Q₈	C₂×D₁₃₉	1	1	yes	characteristic
197	C₄×D₄	—	139	1	no	—
198	#117	—	278	1	no	—
199	#107	—	139	1	no	—
200	#107	C₂×D₁₃₉	1	1	yes	characteristic
201	#106	—	139	1	no	—
202	C₄⋊C₈	—	139	1	no	—
203	C₄⋊C₈	—	139	1	no	—
204	C₄⋊C₈	—	139	1	no	—
205	#103	—	139	1	no	—
206	#103	—	139	1	no	—
207	#103	—	139	1	no	—
208	#103	—	139	1	no	—
209	#103	—	139	1	no	—
210	#103	C₂×D₁₃₉	1	1	yes	characteristic
211	#103	C₂×D₁₃₉	1	1	yes	characteristic
212	#102	—	139	1	no	—
213	#102	—	139	1	no	—
214	#102	—	139	1	no	—
215	C₄×C₈	—	139	1	no	—
216	C₄×C₈	—	139	1	no	—
217	C₄×C₈	C₂×D₁₃₉	1	1	yes	characteristic
218	C₂×Q₈	—	278	1	no	—
219	C₂×Q₈	—	139	1	no	—
220	C₂×Q₈	—	139	1	no	—
221	C₂×Q₈	C₂²×D₁₃₉	1	1	yes	characteristic
222	C₂×Q₈	C₂²×D₁₃₉	1	1	yes	characteristic
223	C₂×D₄	—	139	1	no	—
224	C₂²×C₄	—	278	1	no	—
225	Q₁₆	—	2	2	no	—
226	Q₁₆	—	278	2	no	—
227	SD₁₆	—	278	2	no	—
228	C₂×C₈	—	139	1	no	—
229	C₂×C₈	—	139	1	no	—
230	C₂×C₈	C₂²×D₁₃₉	1	1	yes	characteristic
231	C₂×C₈	C₂²×D₁₃₉	1	1	yes	characteristic
232	C₄⋊C₄	—	2	1	no	—
233	C₄⋊C₄	—	2	1	no	—
234	C₄⋊C₄	—	278	1	no	—
235	C₄⋊C₄	—	278	1	no	—
236	C₄⋊C₄	—	278	1	no	—
237	C₄⋊C₄	—	278	1	no	—
238	C₄⋊C₄	—	278	1	no	—
239	C₄⋊C₄	—	278	1	no	—
240	C₄⋊C₄	—	278	1	no	—
241	C₄⋊C₄	—	278	1	no	—
242	C₄⋊C₄	—	278	1	no	—
243	C₄⋊C₄	—	278	1	no	—
244	C₄⋊C₄	—	139	1	no	—
245	C₄⋊C₄	—	139	1	no	—
246	C₄⋊C₄	—	139	1	no	—
247	C₄⋊C₄	—	139	1	no	—
248	C₄⋊C₄	—	139	1	no	—
249	C₄⋊C₄	C₂²×D₁₃₉	1	1	yes	characteristic
250	C₄⋊C₄	C₂²×D₁₃₉	1	1	yes	characteristic
251	C₂²⋊C₄	—	278	1	no	—
252	C₂²⋊C₄	—	278	1	no	—
253	C₂²⋊C₄	—	278	1	no	—
254	C₂²⋊C₄	—	278	1	no	—
255	C₂²⋊C₄	—	278	1	no	—
256	C₄²	—	2	1	no	—
257	C₄²	—	2	1	no	—
258	C₄²	—	278	1	no	—
259	C₄²	—	278	1	no	—
260	C₄²	—	278	1	no	—
261	C₄²	—	139	1	no	—
262	C₄²	—	139	1	no	—
263	C₄²	C₂²×D₁₃₉	1	1	yes	characteristic
264	C₂³	—	278	1	no	—
265	Q₈	—	2	1	no	—
266	Q₈	—	2	1	no	—
267	Q₈	—	2	1	no	—
268	Q₈	—	278	2	no	—
269	Q₈	—	278	1	no	—
270	Q₈	—	278	1	no	—
271	Q₈	—	556	2	no	—
272	Q₈	#14555785	1	2	yes	—
273	D₄	—	278	1	no	—
274	D₄	—	278	1	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	—	278	1	no	—
280	C₂×C₄	—	278	1	no	—
281	C₂×C₄	—	278	1	no	—
282	C₂×C₄	—	556	2	no	—
283	C₂×C₄	—	139	1	no	—
284	C₂×C₄	—	139	1	no	—
285	C₂×C₄	—	139	1	no	—
286	C₂×C₄	—	139	1	no	—
287	C₂×C₄	D₄×D₁₃₉	1	1	yes	characteristic
288	C₂×C₄	D₄×D₁₃₉	1	1	yes	characteristic
289	C₂×C₄	C₂³×D₁₃₉	1	1	yes	characteristic
290	C₈	—	2	1	no	—
291	C₈	—	2	1	no	—
292	C₈	—	278	1	no	—
293	C₈	—	278	1	no	—
294	C₂²	—	556	1	no	—
295	C₂²	—	556	1	no	—
296	C₂²	—	556	1	no	—
297	C₂²	C₂×D₄×D₁₃₉	1	1	yes	characteristic
298	C₄	—	2	1	no	—
299	C₄	—	2	2	no	—
300	C₄	—	2	1	no	—
301	C₄	—	4	1	no	—
302	C₄	—	4	1	no	—
303	C₄	—	4	1	no	—
304	C₄	—	278	1	no	—
305	C₄	—	278	1	no	—
306	C₄	—	278	1	no	—
307	C₄	—	556	1	no	—
308	C₄	—	556	1	no	—
309	C₄	—	556	1	no	—
310	C₄	—	139	2	no	—
311	C₄	C₂×#14555785	1	1	yes	characteristic
312	C₄	#17984889	1	1	yes	characteristic
313	C₂	—	1112	1	no	—
314	C₂	#22581612	1	1	yes	characteristic
315	C₂	#135×D₁₃₉	1	1	yes	characteristic
316	C₂	#22581766	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₃₉
commutator, cosocle	C₂×C₄×C₁₃₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₃₉
Fitting	C₄×Q₁₆×C₁₃₉	C₂
socle	C₂²×C₁₃₉	C₂²×D₄
prime radical	C₂×D₅₅₆	C₂³
hypercenter	C₄×Q₁₆	D₁₃₉
hypocenter	C₁₃₉	#3364
LCS	G ▷ C₂×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×Q₁₆

Complements

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

N	H ≅ G/N	type
1. #22581679	C₂	semidirect
5. #22581302	C₂	semidirect
6. #22581260	C₂	semidirect
7. #22581259	C₂	semidirect
11. #22581161	C₂	semidirect
12. C₈×Dic₂₇₈	C₂	semidirect
13. #22581045	C₂	semidirect
15. C₄×Q₁₆×C₁₃₉	C₂	semidirect
158. C₁₃₉	#3364	semidirect
173. C₄×Q₁₆	D₁₃₉	semidirect

Maximal subgroups

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

subgroup	index
#22581679	2
#22581675	2
#22581309	2
#22581308	2
#22581302	2
#22581260	2
#22581259	2
#22581232	2
#22581177	2
#22581175	2
#22581161	2
C₈×Dic₂₇₈	2
#22581045	2
#22581038	2
C₄×Q₁₆×C₁₃₉	2
#3364	139