Subgroups — #31816636

Order 17792.

370 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22581576	C₂	1	1	yes	maximal, characteristic
2	#22581430	C₂	1	1	yes	maximal, characteristic, complemented
3	#22581277	C₂	1	1	yes	maximal, characteristic
4	#22581247	C₂	1	1	yes	maximal, characteristic, complemented
5	#22581237	C₂	1	1	yes	maximal, characteristic, complemented
6	#22581178	C₂	1	1	yes	maximal, characteristic, complemented
7	#22581178	C₂	1	1	yes	maximal, characteristic, complemented
8	#22581108	C₂	1	1	yes	maximal, characteristic, complemented
9	#22581094	C₂	1	1	yes	maximal, characteristic
10	#22581057	C₂	1	1	yes	maximal, characteristic, complemented
11	#22581052	C₂	1	1	yes	maximal, characteristic, complemented
12	#22581045	C₂	1	1	yes	maximal, characteristic, complemented
13	#22581040	C₂	1	1	yes	maximal, characteristic, complemented
14	#22581011	C₂	1	1	yes	maximal, characteristic, complemented
15	#485×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂²×Dic₂₇₈	C₂²	1	1	yes	characteristic
17	C₂×#14555775	C₂²	1	1	yes	characteristic
18	C₂×#14555775	C₂²	1	1	yes	characteristic
19	C₂×#14555774	C₂²	1	1	yes	characteristic
20	C₂×#14555774	C₂²	1	1	yes	characteristic
21	#17984841	C₂²	1	1	yes	characteristic
22	#17984837	—	2	1	no	—
23	#17984831	C₂²	1	1	yes	characteristic
24	#17984828	—	2	1	no	—
25	#17984824	—	2	1	no	—
26	#17984822	—	2	1	no	—
27	#17984816	C₂²	1	1	yes	characteristic
28	Q₈×Dic₁₃₉	C₂²	1	1	yes	characteristic
29	C₄×Dic₂₇₈	—	2	1	no	—
30	C₄×Dic₂₇₈	—	2	1	no	—
31	#17984807	C₂²	1	1	yes	characteristic
32	#17984807	C₂²	1	1	yes	characteristic
33	#17984807	C₂²	1	1	yes	characteristic
34	#17984807	C₂²	1	1	yes	characteristic
35	#17984804	C₂²	1	1	yes	characteristic
36	#17984801	C₂²	1	1	yes	characteristic
37	#17984798	—	2	1	no	—
38	#17984798	—	2	1	no	—
39	#17984775	C₂²	1	1	yes	characteristic
40	#17984773	C₂²	1	1	yes	characteristic
41	Dic₁₃₉⋊C₈	C₂²	1	1	yes	characteristic
42	Dic₁₃₉⋊C₈	C₂²	1	1	yes	characteristic
43	#17984767	C₂²	1	1	yes	characteristic
44	#17984767	C₂²	1	1	yes	characteristic
45	#17984765	C₂²	1	1	yes	characteristic
46	#17984765	C₂²	1	1	yes	characteristic
47	#17984762	C₂²	1	1	yes	characteristic
48	#17984762	C₂²	1	1	yes	characteristic
49	C₂²⋊(C₁₃₉⋊C₈)	C₂²	1	1	yes	characteristic
50	#17984752	C₂²	1	1	yes	characteristic
51	C₈×Dic₁₃₉	C₂²	1	1	yes	characteristic
52	C₂²⋊Q₈×C₁₃₉	C₂²	1	1	yes	characteristic
53	#117×C₁₃₉	C₂²	1	1	yes	characteristic
54	#107×C₁₃₉	C₂²	1	1	yes	characteristic
55	#106×C₁₃₉	C₂²	1	1	yes	characteristic
56	#103×C₁₃₉	C₂²	1	1	yes	characteristic
57	#103×C₁₃₉	C₂²	1	1	yes	characteristic
58	C₂²⋊C₈×C₁₃₉	C₂²	1	1	yes	characteristic
59	C₂×Dic₂₇₈	—	2	2	no	—
60	C₂×Dic₂₇₈	—	2	2	no	—
61	C₂×Dic₂₇₈	—	2	1	no	—
62	C₂×Dic₂₇₈	—	2	1	no	—
63	C₂×Dic₂₇₈	C₂³	1	1	yes	characteristic
64	C₂×Dic₂₇₈	C₂³	1	1	yes	characteristic
65	C₂×D₅₅₆	C₂³	1	1	yes	characteristic
66	C₂×C₂²⋊D₁₃₉	—	2	1	no	—
67	C₂²×Dic₁₃₉	—	2	1	no	—
68	C₂×C₄×D₁₃₉	—	2	1	no	—
69	#14555775	—	2	2	no	—
70	#14555775	—	2	2	no	—
71	#14555774	—	2	2	no	—
72	#14555774	—	2	2	no	—
73	C₂×C₁₃₉⋊C₈	C₂³	1	1	yes	characteristic
74	C₂×C₁₃₉⋊C₈	C₂³	1	1	yes	characteristic
75	C₄⋊Dic₁₃₉	—	2	1	no	—
76	C₄⋊Dic₁₃₉	D₄	1	1	yes	characteristic
77	C₄⋊Dic₁₃₉	D₄	1	1	yes	characteristic
78	C₄⋊Dic₁₃₉	C₂³	1	1	yes	characteristic
79	Dic₁₃₉⋊C₄	—	2	1	no	—
80	Dic₁₃₉⋊C₄	—	2	1	no	—
81	Dic₁₃₉⋊C₄	—	2	1	no	—
82	Dic₁₃₉⋊C₄	—	2	1	no	—
83	Dic₁₃₉⋊C₄	—	2	1	no	—
84	C₂²⋊Dic₁₃₉	D₄	1	1	yes	characteristic
85	C₂²⋊Dic₁₃₉	D₄	1	1	yes	characteristic
86	(C₂×D₁₃₉)⋊C₄	—	2	1	no	—
87	(C₂×D₁₃₉)⋊C₄	—	2	1	no	—
88	(C₂×D₁₃₉)⋊C₄	—	2	1	no	—
89	(C₂×D₁₃₉)⋊C₄	—	2	1	no	—
90	C₄×Dic₁₃₉	—	2	1	no	—
91	C₄×Dic₁₃₉	—	2	1	no	—
92	C₄×Dic₁₃₉	—	2	1	no	—
93	C₄×Dic₁₃₉	—	2	1	no	—
94	C₄×Dic₁₃₉	—	2	1	no	—
95	C₄×Dic₁₃₉	C₂³	1	1	yes	characteristic
96	C₄×Dic₁₃₉	C₂³	1	1	yes	characteristic
97	C₂×Q₈×C₁₃₉	C₂³	1	1	yes	characteristic
98	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
99	C₂×C₈×C₁₃₉	C₂³	1	1	yes	characteristic
100	C₂×C₈×C₁₃₉	C₂³	1	1	yes	characteristic
101	C₄⋊C₄×C₁₃₉	—	2	1	no	—
102	C₄⋊C₄×C₁₃₉	C₂³	1	1	yes	characteristic
103	C₄⋊C₄×C₁₃₉	C₂³	1	1	yes	characteristic
104	C₄⋊C₄×C₁₃₉	C₂³	1	1	yes	characteristic
105	C₂²⋊C₄×C₁₃₉	—	2	1	no	—
106	C₂²⋊C₄×C₁₃₉	—	2	1	no	—
107	C₄²×C₁₃₉	—	2	1	no	—
108	C₂²×D₁₃₉	—	2	1	no	—
109	Dic₂₇₈	—	2	4	no	—
110	Dic₂₇₈	—	2	1	no	—
111	Dic₂₇₈	—	2	1	no	—
112	Dic₂₇₈	—	8	1	no	—
113	Dic₂₇₈	#41	1	2	yes	—
114	Dic₂₇₈	#41	1	2	yes	—
115	D₅₅₆	—	2	1	no	—
116	D₅₅₆	—	2	1	no	—
117	C₂²⋊D₁₃₉	—	8	1	no	—
118	C₂×Dic₁₃₉	—	2	4	no	—
119	C₂×Dic₁₃₉	—	2	1	no	—
120	C₂×Dic₁₃₉	—	2	1	no	—
121	C₂×Dic₁₃₉	—	2	1	no	—
122	C₂×Dic₁₃₉	C₂×D₄	1	1	yes	characteristic
123	C₂×Dic₁₃₉	C₂×D₄	1	1	yes	characteristic
124	C₂×Dic₁₃₉	C₂×D₄	1	1	yes	characteristic
125	C₂×Dic₁₃₉	C₂×D₄	1	1	yes	characteristic
126	C₄×D₁₃₉	—	8	1	no	—
127	C₁₃₉⋊C₈	—	2	1	no	—
128	C₁₃₉⋊C₈	—	2	1	no	—
129	C₂³×C₁₃₉	C₂×D₄	1	1	yes	characteristic
130	Q₈×C₁₃₉	—	2	1	no	—
131	Q₈×C₁₃₉	—	2	1	no	—
132	C₂×C₄×C₁₃₉	—	2	1	no	—
133	C₂×C₄×C₁₃₉	—	2	1	no	—
134	C₂×C₄×C₁₃₉	—	2	1	no	—
135	C₂×C₄×C₁₃₉	—	2	1	no	—
136	C₂×C₄×C₁₃₉	—	2	1	no	—
137	C₂×C₄×C₁₃₉	—	2	1	no	—
138	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
139	C₂×C₄×C₁₃₉	C₂⁴	1	1	yes	characteristic
140	C₈×C₁₃₉	—	2	1	no	—
141	C₈×C₁₃₉	—	2	1	no	—
142	C₂²×C₁₃₉	—	2	1	no	—
143	C₂²×C₁₃₉	—	2	1	no	—
144	C₂²×C₁₃₉	—	2	1	no	—
145	C₂²×C₁₃₉	C₂²×D₄	1	1	yes	characteristic
146	C₂×D₁₃₉	—	4	1	no	—
147	C₂×D₁₃₉	—	4	1	no	—
148	C₂×D₁₃₉	—	4	1	no	—
149	C₄×C₁₃₉	—	2	1	no	—
150	C₄×C₁₃₉	—	4	1	no	—
151	C₄×C₁₃₉	—	4	1	no	—
152	C₄×C₁₃₉	—	4	1	no	—
153	C₄×C₁₃₉	—	4	1	no	—
154	C₄×C₁₃₉	C₂×#41	1	1	yes	characteristic
155	C₄×C₁₃₉	C₂×#41	1	1	yes	characteristic
156	Dic₁₃₉	—	2	2	no	—
157	Dic₁₃₉	—	2	2	no	—
158	Dic₁₃₉	—	2	1	no	—
159	Dic₁₃₉	—	2	1	no	—
160	Dic₁₃₉	—	2	1	no	—
161	Dic₁₃₉	—	2	1	no	—
162	Dic₁₃₉	—	4	1	no	—
163	C₂×C₁₃₉	—	4	1	no	—
164	C₂×C₁₃₉	#525	1	1	yes	characteristic
165	C₂×C₁₃₉	#577	1	1	yes	characteristic
166	C₂×C₁₃₉	#578	1	1	yes	characteristic
167	D₁₃₉	—	8	1	no	—
168	C₁₃₉	#3121	1	1	yes	characteristic, complemented, Sylow
169	#3121	—	139	1	no	maximal, Sylow
170	#542	—	139	1	no	—
171	#519	—	139	1	no	—
172	#485	D₁₃₉	1	1	yes	characteristic, complemented
173	#482	—	139	1	no	—
174	#477	—	139	1	no	—
175	#475	—	139	1	no	—
176	#464	—	139	1	no	—
177	#464	—	139	1	no	—
178	#451	—	139	1	no	—
179	#448	—	139	1	no	—
180	#441	—	139	1	no	—
181	#440	—	139	1	no	—
182	C₄×Q₁₆	—	139	1	no	—
183	C₄×SD₁₆	—	139	1	no	—
184	#433	—	139	1	no	—
185	C₂²×Q₈	—	139	1	no	—
186	C₂×Q₁₆	—	139	1	no	—
187	C₂×Q₁₆	—	139	1	no	—
188	C₂×SD₁₆	—	139	1	no	—
189	C₂×SD₁₆	—	139	1	no	—
190	C₄⋊Q₈	—	139	1	no	—
191	#126	—	278	1	no	—
192	#124	—	278	1	no	—
193	#124	—	139	1	no	—
194	#123	—	278	1	no	—
195	C₂²⋊Q₈	—	278	1	no	—
196	C₂²⋊Q₈	C₂×D₁₃₉	1	1	yes	characteristic
197	#121	—	139	1	no	—
198	C₄×Q₈	—	278	1	no	—
199	C₄×Q₈	—	278	1	no	—
200	C₄×Q₈	—	139	1	no	—
201	C₄×Q₈	—	139	1	no	—
202	C₄×Q₈	—	139	1	no	—
203	C₄×Q₈	—	139	1	no	—
204	C₄×Q₈	—	139	1	no	—
205	C₄×D₄	—	139	1	no	—
206	#117	—	278	1	no	—
207	#117	—	278	1	no	—
208	#117	—	139	1	no	—
209	#117	C₂×D₁₃₉	1	1	yes	characteristic
210	#107	—	139	1	no	—
211	#107	C₂×D₁₃₉	1	1	yes	characteristic
212	#106	—	139	1	no	—
213	#106	C₂×D₁₃₉	1	1	yes	characteristic
214	C₄⋊C₈	—	139	1	no	—
215	C₄⋊C₈	—	139	1	no	—
216	#103	—	139	1	no	—
217	#103	—	139	1	no	—
218	#103	—	139	1	no	—
219	#103	—	139	1	no	—
220	#103	C₂×D₁₃₉	1	1	yes	characteristic
221	#103	C₂×D₁₃₉	1	1	yes	characteristic
222	#102	—	139	1	no	—
223	#102	—	139	1	no	—
224	C₂²⋊C₈	—	139	1	no	—
225	C₂²⋊C₈	C₂×D₁₃₉	1	1	yes	characteristic
226	#97	—	139	1	no	—
227	C₄×C₈	—	139	1	no	—
228	C₂×Q₈	—	278	2	no	—
229	C₂×Q₈	—	278	2	no	—
230	C₂×Q₈	—	278	1	no	—
231	C₂×Q₈	—	278	1	no	—
232	C₂×Q₈	—	139	1	no	—
233	C₂×Q₈	—	139	1	no	—
234	C₂×Q₈	C₂²×D₁₃₉	1	1	yes	characteristic
235	C₂×D₄	—	278	1	no	—
236	C₂×D₄	—	139	1	no	—
237	C₂²×C₄	—	278	1	no	—
238	C₂²×C₄	—	278	1	no	—
239	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
240	Q₁₆	—	278	2	no	—
241	Q₁₆	—	278	2	no	—
242	SD₁₆	—	278	2	no	—
243	SD₁₆	—	278	2	no	—
244	C₂×C₈	—	139	1	no	—
245	C₂×C₈	—	139	1	no	—
246	C₂×C₈	C₂²×D₁₃₉	1	1	yes	characteristic
247	C₂×C₈	C₂²×D₁₃₉	1	1	yes	characteristic
248	C₄⋊C₄	—	2	1	no	—
249	C₄⋊C₄	—	278	1	no	—
250	C₄⋊C₄	—	278	1	no	—
251	C₄⋊C₄	—	278	1	no	—
252	C₄⋊C₄	—	278	1	no	—
253	C₄⋊C₄	—	278	1	no	—
254	C₄⋊C₄	—	278	1	no	—
255	C₄⋊C₄	—	139	1	no	—
256	C₄⋊C₄	—	139	1	no	—
257	C₄⋊C₄	—	139	1	no	—
258	C₄⋊C₄	C₂²×D₁₃₉	1	1	yes	characteristic
259	C₄⋊C₄	C₂²×D₁₃₉	1	1	yes	characteristic
260	C₄⋊C₄	C₂²×D₁₃₉	1	1	yes	characteristic
261	C₂²⋊C₄	—	2	1	no	—
262	C₂²⋊C₄	—	2	1	no	—
263	C₂²⋊C₄	—	278	1	no	—
264	C₂²⋊C₄	—	278	1	no	—
265	C₂²⋊C₄	—	278	1	no	—
266	C₂²⋊C₄	—	278	1	no	—
267	C₂²⋊C₄	—	139	1	no	—
268	C₂²⋊C₄	—	139	1	no	—
269	C₄²	—	2	1	no	—
270	C₄²	—	278	1	no	—
271	C₄²	—	278	1	no	—
272	C₄²	—	278	1	no	—
273	C₄²	—	278	1	no	—
274	C₄²	—	278	1	no	—
275	C₄²	—	139	1	no	—
276	C₄²	—	139	1	no	—
277	C₂³	—	278	1	no	—
278	C₂³	D₄×D₁₃₉	1	1	yes	characteristic
279	Q₈	—	2	1	no	—
280	Q₈	—	2	1	no	—
281	Q₈	—	278	4	no	—
282	Q₈	—	278	1	no	—
283	Q₈	—	278	1	no	—
284	Q₈	—	1112	1	no	—
285	Q₈	—	139	2	no	—
286	Q₈	—	139	2	no	—
287	D₄	—	278	1	no	—
288	D₄	—	278	1	no	—
289	D₄	—	1112	1	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	278	4	no	—
297	C₂×C₄	—	278	1	no	—
298	C₂×C₄	—	278	1	no	—
299	C₂×C₄	—	278	1	no	—
300	C₂×C₄	—	1112	1	no	—
301	C₂×C₄	—	139	1	no	—
302	C₂×C₄	—	139	1	no	—
303	C₂×C₄	—	139	1	no	—
304	C₂×C₄	—	139	1	no	—
305	C₂×C₄	D₄×D₁₃₉	1	1	yes	characteristic
306	C₂×C₄	C₂³×D₁₃₉	1	1	yes	characteristic
307	C₈	—	2	1	no	—
308	C₈	—	2	1	no	—
309	C₈	—	278	1	no	—
310	C₈	—	278	1	no	—
311	C₂²	—	2	1	no	—
312	C₂²	—	2	1	no	—
313	C₂²	—	2	1	no	—
314	C₂²	—	556	1	no	—
315	C₂²	—	556	1	no	—
316	C₂²	—	556	1	no	—
317	C₂²	C₂×D₄×D₁₃₉	1	1	yes	characteristic
318	C₄	—	2	1	no	—
319	C₄	—	4	1	no	—
320	C₄	—	4	1	no	—
321	C₄	—	4	1	no	—
322	C₄	—	4	1	no	—
323	C₄	—	278	1	no	—
324	C₄	—	278	1	no	—
325	C₄	—	278	1	no	—
326	C₄	—	278	1	no	—
327	C₄	—	278	2	no	—
328	C₄	—	278	2	no	—
329	C₄	—	556	1	no	—
330	C₄	#41×D₁₃₉	1	1	yes	characteristic
331	C₄	#41×D₁₃₉	1	1	yes	characteristic
332	C₂	—	4	1	no	—
333	C₂	—	1112	1	no	—
334	C₂	#22581461	1	1	yes	characteristic
335	C₂	#22581760	1	1	yes	characteristic
336	C₂	#22581769	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₃₉
commutator, cosocle	C₂×C₄×C₁₃₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₃₉
Fitting	#485×C₁₃₉	C₂
socle	C₂²×C₁₃₉	C₂²×D₄
prime radical	#17984816	C₂²
hypercenter	#485	D₁₃₉
hypocenter	C₁₃₉	#3121
LCS	G ▷ C₂×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #485

Complements

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

N	H ≅ G/N	type
2. #22581430	C₂	semidirect
4. #22581247	C₂	semidirect
5. #22581237	C₂	semidirect
6. #22581178	C₂	semidirect
7. #22581178	C₂	semidirect
8. #22581108	C₂	semidirect
10. #22581057	C₂	semidirect
11. #22581052	C₂	semidirect
12. #22581045	C₂	semidirect
13. #22581040	C₂	semidirect
14. #22581011	C₂	semidirect
15. #485×C₁₃₉	C₂	semidirect
168. C₁₃₉	#3121	semidirect
172. #485	D₁₃₉	semidirect

Maximal subgroups

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

subgroup	index
#22581576	2
#22581430	2
#22581277	2
#22581247	2
#22581237	2
#22581178	2
#22581108	2
#22581094	2
#22581057	2
#22581052	2
#22581045	2
#22581040	2
#22581011	2
#485×C₁₃₉	2
#3121	139