Subgroups — #31819145

Order 17792.

338 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×Dic₂₇₈	C₂	1	1	yes	maximal, characteristic, complemented
2	#22581675	C₂	1	1	yes	maximal, characteristic
3	#22581314	C₂	1	1	yes	maximal, characteristic
4	#22581313	C₂	1	1	yes	maximal, characteristic
5	#22581302	C₂	1	1	yes	maximal, characteristic, complemented
6	#22581242	C₂	1	1	yes	maximal, characteristic
7	#22581238	C₂	1	1	yes	maximal, characteristic, complemented
8	#22581237	C₂	1	1	yes	maximal, characteristic, complemented
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	#22581081	C₂	1	1	yes	maximal, characteristic, complemented
13	#22581057	C₂	1	1	yes	maximal, characteristic, complemented
14	#22581038	C₂	1	1	yes	maximal, characteristic
15	#441×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	#17984842	—	2	1	no	—
19	#17984842	C₂²	1	1	yes	characteristic
20	#17984841	—	2	1	no	—
21	#17984841	—	2	1	no	—
22	#17984841	C₂²	1	1	yes	characteristic
23	#17984841	C₂²	1	1	yes	characteristic
24	#17984840	C₂²	1	1	yes	characteristic
25	#17984837	—	2	1	no	—
26	#17984836	—	2	1	no	—
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	#17984773	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	#17984752	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	#106×C₁₃₉	C₂²	1	1	yes	characteristic
56	#103×C₁₃₉	C₂²	1	1	yes	characteristic
57	#103×C₁₃₉	C₂²	1	1	yes	characteristic
58	#97×C₁₃₉	C₂²	1	1	yes	characteristic
59	C₂×Dic₂₇₈	—	2	1	no	—
60	C₂×Dic₂₇₈	—	2	1	no	—
61	C₂×Dic₂₇₈	—	2	1	no	—
62	C₂×Dic₂₇₈	D₄	1	1	yes	characteristic
63	C₂×Dic₂₇₈	C₂³	1	1	yes	characteristic
64	C₂×D₅₅₆	C₂³	1	1	yes	characteristic
65	C₂×C₄×D₁₃₉	—	2	1	no	—
66	#14555775	—	2	2	no	—
67	#14555772	—	2	2	no	—
68	C₂×C₁₃₉⋊C₈	C₂³	1	1	yes	characteristic
69	C₂×C₁₃₉⋊C₈	C₂³	1	1	yes	characteristic
70	C₄⋊Dic₁₃₉	—	2	1	no	—
71	C₄⋊Dic₁₃₉	—	2	1	no	—
72	C₄⋊Dic₁₃₉	—	2	1	no	—
73	C₄⋊Dic₁₃₉	D₄	1	1	yes	characteristic
74	C₄⋊Dic₁₃₉	C₂³	1	1	yes	characteristic
75	C₄⋊Dic₁₃₉	C₂³	1	1	yes	characteristic
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	—	8	1	no	—
110	Dic₂₇₈	—	8	1	no	—
111	Dic₂₇₈	—	8	1	no	—
112	D₅₅₆	—	2	1	no	—
113	D₅₅₆	—	2	1	no	—
114	C₂×Dic₁₃₉	—	2	1	no	—
115	C₂×Dic₁₃₉	—	2	1	no	—
116	C₂×Dic₁₃₉	—	2	1	no	—
117	C₂×Dic₁₃₉	C₂×D₄	1	1	yes	characteristic
118	C₂×Dic₁₃₉	C₂×D₄	1	1	yes	characteristic
119	C₂×Dic₁₃₉	C₂×D₄	1	1	yes	characteristic
120	C₂×Dic₁₃₉	C₂×D₄	1	1	yes	characteristic
121	C₄×D₁₃₉	—	4	2	no	—
122	C₁₃₉⋊C₈	—	2	1	no	—
123	C₁₃₉⋊C₈	—	2	1	no	—
124	Q₈×C₁₃₉	—	2	1	no	—
125	Q₈×C₁₃₉	—	2	1	no	—
126	Q₈×C₁₃₉	—	2	1	no	—
127	Q₈×C₁₃₉	#41	1	2	yes	—
128	C₂×C₄×C₁₃₉	—	2	1	no	—
129	C₂×C₄×C₁₃₉	—	2	1	no	—
130	C₂×C₄×C₁₃₉	—	2	1	no	—
131	C₂×C₄×C₁₃₉	—	2	1	no	—
132	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
133	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
134	C₂×C₄×C₁₃₉	C₂⁴	1	1	yes	characteristic
135	C₈×C₁₃₉	—	2	1	no	—
136	C₈×C₁₃₉	—	2	1	no	—
137	C₂²×C₁₃₉	C₂²×D₄	1	1	yes	characteristic
138	C₂×D₁₃₉	—	4	1	no	—
139	C₂×D₁₃₉	—	4	1	no	—
140	C₂×D₁₃₉	—	4	1	no	—
141	C₄×C₁₃₉	—	2	2	no	—
142	C₄×C₁₃₉	—	2	1	no	—
143	C₄×C₁₃₉	—	2	1	no	—
144	C₄×C₁₃₉	—	4	1	no	—
145	C₄×C₁₃₉	—	4	1	no	—
146	C₄×C₁₃₉	—	4	1	no	—
147	C₄×C₁₃₉	C₂×#41	1	1	yes	characteristic
148	C₄×C₁₃₉	2₋¹⁺⁴	1	1	yes	characteristic
149	Dic₁₃₉	—	2	1	no	—
150	Dic₁₃₉	—	2	1	no	—
151	Dic₁₃₉	—	2	1	no	—
152	Dic₁₃₉	—	4	1	no	—
153	Dic₁₃₉	—	4	1	no	—
154	Dic₁₃₉	—	4	1	no	—
155	Dic₁₃₉	#137	1	2	yes	—
156	C₂×C₁₃₉	#548	1	1	yes	characteristic
157	C₂×C₁₃₉	C₂×#137	1	1	yes	characteristic
158	C₂×C₁₃₉	#577	1	1	yes	characteristic
159	D₁₃₉	—	8	1	no	—
160	C₁₃₉	#3369	1	1	yes	characteristic, complemented, Sylow
161	#3369	—	139	1	no	maximal, Sylow
162	Q₈²	—	139	1	no	—
163	#556	—	139	1	no	—
164	#489	—	139	1	no	—
165	#489	—	139	1	no	—
166	#487	—	139	1	no	—
167	#476	—	139	1	no	—
168	#475	—	139	1	no	—
169	#475	—	139	1	no	—
170	#464	—	139	1	no	—
171	#464	—	139	1	no	—
172	#462	—	139	1	no	—
173	#446	—	139	1	no	—
174	#441	—	139	1	no	—
175	#441	D₁₃₉	1	1	yes	characteristic, complemented
176	C₄×SD₁₆	—	139	1	no	—
177	C₂×Q₁₆	—	139	1	no	—
178	C₂×Q₁₆	C₂×D₁₃₉	1	1	yes	characteristic
179	C₂×SD₁₆	—	139	1	no	—
180	C₄⋊Q₈	—	278	1	no	—
181	C₄⋊Q₈	—	278	1	no	—
182	C₄⋊Q₈	—	278	1	no	—
183	C₄⋊Q₈	—	139	1	no	—
184	C₄⋊Q₈	—	139	1	no	—
185	C₄⋊Q₈	—	139	1	no	—
186	C₄⋊Q₈	—	139	1	no	—
187	#126	—	278	1	no	—
188	#126	—	278	1	no	—
189	#124	—	139	1	no	—
190	#124	—	139	1	no	—
191	C₂²⋊Q₈	—	278	1	no	—
192	C₄×Q₈	—	278	1	no	—
193	C₄×Q₈	—	139	1	no	—
194	C₄×Q₈	—	139	1	no	—
195	C₄×Q₈	—	139	1	no	—
196	C₄×Q₈	—	139	1	no	—
197	C₄×Q₈	C₂×D₁₃₉	1	1	yes	characteristic
198	C₄×Q₈	C₂×D₁₃₉	1	1	yes	characteristic
199	C₄×D₄	—	139	1	no	—
200	#117	—	278	1	no	—
201	#106	—	139	1	no	—
202	#106	—	139	1	no	—
203	#106	C₂×D₁₃₉	1	1	yes	characteristic
204	C₄⋊C₈	—	139	1	no	—
205	C₄⋊C₈	—	139	1	no	—
206	C₄⋊C₈	—	139	1	no	—
207	#103	—	139	1	no	—
208	#103	—	139	1	no	—
209	#103	—	139	1	no	—
210	#103	—	139	1	no	—
211	#103	—	139	1	no	—
212	#103	C₂×D₁₃₉	1	1	yes	characteristic
213	#103	C₂×D₁₃₉	1	1	yes	characteristic
214	#102	—	139	1	no	—
215	#102	—	139	1	no	—
216	#102	—	139	1	no	—
217	#97	—	139	1	no	—
218	#97	C₂×D₁₃₉	1	1	yes	characteristic
219	C₄×C₈	—	139	1	no	—
220	C₂×Q₈	—	278	1	no	—
221	C₂×Q₈	—	278	1	no	—
222	C₂×Q₈	—	278	1	no	—
223	C₂×Q₈	—	139	1	no	—
224	C₂×Q₈	—	139	1	no	—
225	C₂×Q₈	C₂²×D₁₃₉	1	1	yes	characteristic
226	C₂×Q₈	C₂²×D₁₃₉	1	1	yes	characteristic
227	C₂×D₄	—	139	1	no	—
228	C₂²×C₄	—	278	1	no	—
229	Q₁₆	—	2	2	no	—
230	Q₁₆	—	278	2	no	—
231	SD₁₆	—	278	2	no	—
232	C₂×C₈	—	139	1	no	—
233	C₂×C₈	—	139	1	no	—
234	C₂×C₈	C₂²×D₁₃₉	1	1	yes	characteristic
235	C₂×C₈	C₂²×D₁₃₉	1	1	yes	characteristic
236	C₄⋊C₄	—	2	1	no	—
237	C₄⋊C₄	—	2	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	—	278	1	no	—
245	C₄⋊C₄	—	278	1	no	—
246	C₄⋊C₄	—	139	1	no	—
247	C₄⋊C₄	—	139	1	no	—
248	C₄⋊C₄	—	139	1	no	—
249	C₄⋊C₄	—	139	1	no	—
250	C₄⋊C₄	—	139	1	no	—
251	C₄⋊C₄	C₂²×D₁₃₉	1	1	yes	characteristic
252	C₄⋊C₄	C₂²×D₁₃₉	1	1	yes	characteristic
253	C₂²⋊C₄	—	278	1	no	—
254	C₂²⋊C₄	—	278	1	no	—
255	C₂²⋊C₄	—	278	1	no	—
256	C₂²⋊C₄	—	278	1	no	—
257	C₂²⋊C₄	—	278	1	no	—
258	C₄²	—	2	1	no	—
259	C₄²	—	2	1	no	—
260	C₄²	—	278	1	no	—
261	C₄²	—	278	1	no	—
262	C₄²	—	278	1	no	—
263	C₄²	—	139	1	no	—
264	C₄²	—	139	1	no	—
265	C₄²	C₂²×D₁₃₉	1	1	yes	characteristic
266	C₂³	—	278	1	no	—
267	Q₈	—	2	1	no	—
268	Q₈	—	2	1	no	—
269	Q₈	—	2	1	no	—
270	Q₈	—	278	2	no	—
271	Q₈	—	278	1	no	—
272	Q₈	—	278	1	no	—
273	Q₈	—	1112	1	no	—
274	Q₈	—	1112	1	no	—
275	Q₈	—	1112	1	no	—
276	Q₈	#14555785	1	2	yes	—
277	D₄	—	278	1	no	—
278	D₄	—	278	1	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	278	1	no	—
284	C₂×C₄	—	278	1	no	—
285	C₂×C₄	—	278	1	no	—
286	C₂×C₄	—	556	2	no	—
287	C₂×C₄	—	139	1	no	—
288	C₂×C₄	—	139	1	no	—
289	C₂×C₄	—	139	1	no	—
290	C₂×C₄	—	139	1	no	—
291	C₂×C₄	D₄×D₁₃₉	1	1	yes	characteristic
292	C₂×C₄	D₄×D₁₃₉	1	1	yes	characteristic
293	C₂×C₄	C₂³×D₁₃₉	1	1	yes	characteristic
294	C₈	—	2	1	no	—
295	C₈	—	2	1	no	—
296	C₈	—	278	1	no	—
297	C₈	—	278	1	no	—
298	C₂²	—	556	1	no	—
299	C₂²	—	556	1	no	—
300	C₂²	—	556	1	no	—
301	C₂²	C₂×D₄×D₁₃₉	1	1	yes	characteristic
302	C₄	—	2	1	no	—
303	C₄	—	2	2	no	—
304	C₄	—	2	1	no	—
305	C₄	—	4	1	no	—
306	C₄	—	4	1	no	—
307	C₄	—	4	1	no	—
308	C₄	—	278	1	no	—
309	C₄	—	278	1	no	—
310	C₄	—	278	1	no	—
311	C₄	—	556	1	no	—
312	C₄	—	556	1	no	—
313	C₄	—	556	1	no	—
314	C₄	—	139	2	no	—
315	C₄	C₂×#14555785	1	1	yes	characteristic
316	C₄	#17984889	1	1	yes	characteristic
317	C₂	—	1112	1	no	—
318	C₂	#22581612	1	1	yes	characteristic
319	C₂	#137×D₁₃₉	1	1	yes	characteristic
320	C₂	#22581762	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₃₉
commutator, cosocle	C₂×C₄×C₁₃₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₃₉
Fitting	#441×C₁₃₉	C₂
socle	C₂²×C₁₃₉	C₂²×D₄
prime radical	C₂×D₅₅₆	C₂³
hypercenter	#441	D₁₃₉
hypocenter	C₁₃₉	#3369
LCS	G ▷ C₂×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #441

Complements

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

N	H ≅ G/N	type
1. Q₈×Dic₂₇₈	C₂	semidirect
5. #22581302	C₂	semidirect
7. #22581238	C₂	semidirect
8. #22581237	C₂	semidirect
11. #22581161	C₂	semidirect
12. #22581081	C₂	semidirect
13. #22581057	C₂	semidirect
15. #441×C₁₃₉	C₂	semidirect
160. C₁₃₉	#3369	semidirect
175. #441	D₁₃₉	semidirect

Maximal subgroups

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

subgroup	index
Q₈×Dic₂₇₈	2
#22581675	2
#22581314	2
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