Subgroups — #31817857

Order 17792.

322 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	#22581566	C₂	1	1	yes	maximal, characteristic
2	#22581561	C₂	1	1	yes	maximal, characteristic
3	#22581263	C₂	1	1	yes	maximal, characteristic, complemented
4	#22581259	C₂	1	1	yes	maximal, characteristic, complemented
5	#22581257	C₂	1	1	yes	maximal, characteristic, complemented
6	#22581212	C₂	1	1	yes	maximal, characteristic
7	#22581210	C₂	1	1	yes	maximal, characteristic
8	#22581195	C₂	1	1	yes	maximal, characteristic
9	#22581193	C₂	1	1	yes	maximal, characteristic
10	#22581164	C₂	1	1	yes	maximal, characteristic, complemented
11	#22581163	C₂	1	1	yes	maximal, characteristic, complemented
12	#22581162	C₂	1	1	yes	maximal, characteristic, complemented
13	#22581162	C₂	1	1	yes	maximal, characteristic, complemented
14	#22580978	C₂	1	1	yes	maximal, characteristic
15	#479×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×Dic₅₅₆	C₂²	1	1	yes	characteristic
17	C₂×Dic₅₅₆	C₂²	1	1	yes	characteristic
18	C₂×#14555775	C₂²	1	1	yes	characteristic
19	C₂×#14555775	C₂²	1	1	yes	characteristic
20	C₂×#14555768	C₂²	1	1	yes	characteristic
21	C₂×C₈×D₁₃₉	C₂²	1	1	yes	characteristic
22	#17984842	C₂²	1	1	yes	characteristic
23	#17984840	C₂²	1	1	yes	characteristic
24	#17984837	—	2	1	no	—
25	#17984837	—	2	1	no	—
26	#17984837	—	2	1	no	—
27	#17984836	—	2	1	no	—
28	#17984834	—	2	1	no	—
29	#17984833	—	2	1	no	—
30	#17984833	—	2	1	no	—
31	#17984833	C₂²	1	1	yes	characteristic
32	#17984832	—	2	1	no	—
33	#17984822	C₂²	1	1	yes	characteristic
34	#17984822	C₂²	1	1	yes	characteristic
35	#17984822	C₂²	1	1	yes	characteristic
36	#17984818	C₂²	1	1	yes	characteristic
37	C₄×Dic₂₇₈	C₂²	1	1	yes	characteristic
38	#17984807	C₂²	1	1	yes	characteristic
39	#17984807	C₂²	1	1	yes	characteristic
40	#17984799	C₂²	1	1	yes	characteristic
41	#17984775	C₂²	1	1	yes	characteristic
42	#17984773	C₂²	1	1	yes	characteristic
43	C₄⋊(C₁₃₉⋊C₈)	C₂²	1	1	yes	characteristic
44	Dic₁₃₉⋊C₈	C₂²	1	1	yes	characteristic
45	Dic₁₃₉⋊C₈	C₂²	1	1	yes	characteristic
46	#17984767	C₂²	1	1	yes	characteristic
47	#17984767	C₂²	1	1	yes	characteristic
48	#17984765	C₂²	1	1	yes	characteristic
49	#17984765	C₂²	1	1	yes	characteristic
50	#17984765	C₂²	1	1	yes	characteristic
51	#17984765	C₂²	1	1	yes	characteristic
52	#125×C₁₃₉	C₂²	1	1	yes	characteristic
53	C₄×Q₈×C₁₃₉	C₂²	1	1	yes	characteristic
54	#107×C₁₃₉	C₂²	1	1	yes	characteristic
55	#106×C₁₃₉	C₂²	1	1	yes	characteristic
56	C₄⋊C₈×C₁₃₉	C₂²	1	1	yes	characteristic
57	#103×C₁₃₉	C₂²	1	1	yes	characteristic
58	#103×C₁₃₉	C₂²	1	1	yes	characteristic
59	C₂×Dic₂₇₈	C₂³	1	1	yes	characteristic
60	C₂×Dic₂₇₈	C₂³	1	1	yes	characteristic
61	C₂×Dic₂₇₈	C₂³	1	1	yes	characteristic
62	C₂×C₄×D₁₃₉	C₂³	1	1	yes	characteristic
63	Dic₅₅₆	—	4	1	no	—
64	Dic₅₅₆	—	4	1	no	—
65	#14555775	—	4	1	no	—
66	#14555775	—	4	1	no	—
67	#14555768	—	4	1	no	—
68	C₂×C₁₃₉⋊C₈	C₂³	1	1	yes	characteristic
69	C₂×C₁₃₉⋊C₈	C₂³	1	1	yes	characteristic
70	C₈×D₁₃₉	—	4	1	no	—
71	C₄⋊Dic₁₃₉	—	2	1	no	—
72	C₄⋊Dic₁₃₉	—	2	1	no	—
73	C₄⋊Dic₁₃₉	—	2	1	no	—
74	C₄⋊Dic₁₃₉	—	2	1	no	—
75	C₄⋊Dic₁₃₉	—	2	1	no	—
76	C₄⋊Dic₁₃₉	—	2	1	no	—
77	C₄⋊Dic₁₃₉	C₂³	1	1	yes	characteristic
78	Dic₁₃₉⋊C₄	—	2	1	no	—
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	Dic₁₃₉⋊C₄	—	2	1	no	—
85	Dic₁₃₉⋊C₄	D₄	1	1	yes	characteristic
86	Dic₁₃₉⋊C₄	D₄	1	1	yes	characteristic
87	(C₂×D₁₃₉)⋊C₄	—	2	1	no	—
88	(C₂×D₁₃₉)⋊C₄	—	2	1	no	—
89	(C₂×D₁₃₉)⋊C₄	—	2	1	no	—
90	(C₂×D₁₃₉)⋊C₄	—	2	1	no	—
91	(C₂×D₁₃₉)⋊C₄	D₄	1	1	yes	characteristic
92	(C₂×D₁₃₉)⋊C₄	D₄	1	1	yes	characteristic
93	C₄×Dic₁₃₉	—	2	1	no	—
94	C₄×Dic₁₃₉	—	2	1	no	—
95	C₄×Dic₁₃₉	—	2	1	no	—
96	C₄×Dic₁₃₉	C₂³	1	1	yes	characteristic
97	C₂×Q₈×C₁₃₉	C₂³	1	1	yes	characteristic
98	C₂×C₈×C₁₃₉	C₂³	1	1	yes	characteristic
99	C₂×C₈×C₁₃₉	C₂³	1	1	yes	characteristic
100	C₄⋊C₄×C₁₃₉	—	2	1	no	—
101	C₄⋊C₄×C₁₃₉	—	2	1	no	—
102	C₄⋊C₄×C₁₃₉	—	2	1	no	—
103	C₄⋊C₄×C₁₃₉	C₂³	1	1	yes	characteristic
104	C₄⋊C₄×C₁₃₉	C₂³	1	1	yes	characteristic
105	C₄⋊C₄×C₁₃₉	C₂³	1	1	yes	characteristic
106	C₄²×C₁₃₉	—	2	1	no	—
107	C₄²×C₁₃₉	C₂³	1	1	yes	characteristic
108	C₂²×D₁₃₉	C₂×D₄	1	1	yes	characteristic
109	Dic₂₇₈	—	2	1	no	—
110	Dic₂₇₈	—	2	1	no	—
111	Dic₂₇₈	—	2	1	no	—
112	Dic₂₇₈	—	2	1	no	—
113	Dic₂₇₈	—	2	1	no	—
114	Dic₂₇₈	—	2	1	no	—
115	C₂×Dic₁₃₉	—	2	1	no	—
116	C₂×Dic₁₃₉	—	2	1	no	—
117	C₂×Dic₁₃₉	—	2	1	no	—
118	C₂×Dic₁₃₉	—	2	1	no	—
119	C₂×Dic₁₃₉	C₂×D₄	1	1	yes	characteristic
120	C₂×Dic₁₃₉	C₂×D₄	1	1	yes	characteristic
121	C₂×Dic₁₃₉	C₂×D₄	1	1	yes	characteristic
122	C₄×D₁₃₉	—	2	1	no	—
123	C₄×D₁₃₉	—	2	1	no	—
124	C₁₃₉⋊C₈	—	2	1	no	—
125	C₁₃₉⋊C₈	—	2	1	no	—
126	Q₈×C₁₃₉	—	2	1	no	—
127	Q₈×C₁₃₉	—	2	1	no	—
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	—	2	1	no	—
139	C₂×D₁₃₉	—	2	1	no	—
140	C₂×D₁₃₉	—	2	1	no	—
141	C₄×C₁₃₉	—	2	1	no	—
142	C₄×C₁₃₉	—	2	1	no	—
143	C₄×C₁₃₉	—	4	1	no	—
144	C₄×C₁₃₉	—	4	1	no	—
145	C₄×C₁₃₉	—	4	1	no	—
146	C₄×C₁₃₉	—	4	1	no	—
147	C₄×C₁₃₉	H(5,𝔽₂)	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	—	4	1	no	—
156	C₂×C₁₃₉	#537	1	1	yes	characteristic
157	C₂×C₁₃₉	#578	1	1	yes	characteristic
158	C₂×C₁₃₉	#578	1	1	yes	characteristic
159	D₁₃₉	—	4	1	no	—
160	C₁₃₉	#3261	1	1	yes	characteristic, complemented, Sylow
161	#3261	—	139	1	no	maximal, Sylow
162	#541	—	139	1	no	—
163	#541	—	139	1	no	—
164	#479	—	139	1	no	—
165	#479	—	139	1	no	—
166	#479	—	139	1	no	—
167	#479	D₁₃₉	1	1	yes	characteristic, complemented
168	#470	—	139	1	no	—
169	#470	—	139	1	no	—
170	#467	—	139	1	no	—
171	#467	—	139	1	no	—
172	#462	—	139	1	no	—
173	#462	—	139	1	no	—
174	#462	—	139	1	no	—
175	#462	—	139	1	no	—
176	#424	—	139	1	no	—
177	C₂×Q₁₆	—	139	1	no	—
178	C₂×Q₁₆	—	139	1	no	—
179	C₂×Q₁₆	—	139	1	no	—
180	C₂×Q₁₆	—	139	1	no	—
181	C₂×M₁₆	—	139	1	no	—
182	C₂²×C₈	—	139	1	no	—
183	C₄⋊Q₈	—	139	1	no	—
184	C₄⋊Q₈	—	139	1	no	—
185	#126	—	278	1	no	—
186	#126	—	278	1	no	—
187	#126	—	278	1	no	—
188	#126	—	278	1	no	—
189	#125	—	278	1	no	—
190	#125	—	278	1	no	—
191	#125	—	278	1	no	—
192	#125	—	278	1	no	—
193	#125	—	139	1	no	—
194	#125	C₂×D₁₃₉	1	1	yes	characteristic
195	C₂²⋊Q₈	—	139	1	no	—
196	C₂²⋊Q₈	—	139	1	no	—
197	C₂²⋊Q₈	—	139	1	no	—
198	C₂²⋊Q₈	—	139	1	no	—
199	C₄×Q₈	—	139	1	no	—
200	C₄×Q₈	—	139	1	no	—
201	C₄×Q₈	—	139	1	no	—
202	C₄×Q₈	C₂×D₁₃₉	1	1	yes	characteristic
203	#117	—	139	1	no	—
204	#107	—	139	1	no	—
205	#107	C₂×D₁₃₉	1	1	yes	characteristic
206	#106	—	139	1	no	—
207	#106	C₂×D₁₃₉	1	1	yes	characteristic
208	C₄⋊C₈	—	139	1	no	—
209	C₄⋊C₈	—	139	1	no	—
210	C₄⋊C₈	—	139	1	no	—
211	C₄⋊C₈	C₂×D₁₃₉	1	1	yes	characteristic
212	#103	—	139	1	no	—
213	#103	—	139	1	no	—
214	#103	—	139	1	no	—
215	#103	—	139	1	no	—
216	#103	—	139	1	no	—
217	#103	—	139	1	no	—
218	#103	C₂×D₁₃₉	1	1	yes	characteristic
219	#103	C₂×D₁₃₉	1	1	yes	characteristic
220	C₂×Q₈	—	139	1	no	—
221	C₂×Q₈	—	139	1	no	—
222	C₂×Q₈	—	139	1	no	—
223	C₂×Q₈	C₂²×D₁₃₉	1	1	yes	characteristic
224	C₂²×C₄	—	139	1	no	—
225	Q₁₆	—	556	1	no	—
226	Q₁₆	—	556	1	no	—
227	Q₁₆	—	556	1	no	—
228	Q₁₆	—	556	1	no	—
229	M₁₆	—	556	1	no	—
230	C₂×C₈	—	556	1	no	—
231	C₂×C₈	—	139	1	no	—
232	C₂×C₈	—	139	1	no	—
233	C₂×C₈	C₂²×D₁₃₉	1	1	yes	characteristic
234	C₂×C₈	C₂²×D₁₃₉	1	1	yes	characteristic
235	C₄⋊C₄	—	2	1	no	—
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	—	278	1	no	—
247	C₄⋊C₄	—	278	1	no	—
248	C₄⋊C₄	—	278	1	no	—
249	C₄⋊C₄	—	278	1	no	—
250	C₄⋊C₄	—	278	1	no	—
251	C₄⋊C₄	—	139	1	no	—
252	C₄⋊C₄	—	139	1	no	—
253	C₄⋊C₄	—	139	1	no	—
254	C₄⋊C₄	C₂²×D₁₃₉	1	1	yes	characteristic
255	C₄⋊C₄	C₂²×D₁₃₉	1	1	yes	characteristic
256	C₄⋊C₄	C₂²×D₁₃₉	1	1	yes	characteristic
257	C₂²⋊C₄	—	278	1	no	—
258	C₂²⋊C₄	—	278	1	no	—
259	C₂²⋊C₄	—	278	1	no	—
260	C₂²⋊C₄	—	278	1	no	—
261	C₂²⋊C₄	—	139	1	no	—
262	C₂²⋊C₄	—	139	1	no	—
263	C₄²	—	2	1	no	—
264	C₄²	—	278	1	no	—
265	C₄²	—	278	1	no	—
266	C₄²	—	278	1	no	—
267	C₄²	—	139	1	no	—
268	C₄²	C₂²×D₁₃₉	1	1	yes	characteristic
269	C₂³	—	139	1	no	—
270	Q₈	—	2	1	no	—
271	Q₈	—	2	1	no	—
272	Q₈	—	278	1	no	—
273	Q₈	—	278	1	no	—
274	Q₈	—	278	1	no	—
275	Q₈	—	278	1	no	—
276	Q₈	—	278	1	no	—
277	Q₈	—	278	1	no	—
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	278	1	no	—
283	C₂×C₄	—	278	1	no	—
284	C₂×C₄	—	278	1	no	—
285	C₂×C₄	—	278	1	no	—
286	C₂×C₄	—	278	1	no	—
287	C₂×C₄	—	278	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	—	278	1	no	—
299	C₂²	—	278	1	no	—
300	C₂²	—	278	1	no	—
301	C₂²	C₂×D₄×D₁₃₉	1	1	yes	characteristic
302	C₄	—	2	1	no	—
303	C₄	—	2	1	no	—
304	C₄	—	4	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	—	556	1	no	—
315	C₄	#17984888	1	1	yes	characteristic
316	C₄	#17984890	1	1	yes	characteristic
317	C₂	—	556	1	no	—
318	C₂	#22581533	1	1	yes	characteristic
319	C₂	#22581767	1	1	yes	characteristic
320	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	#479×C₁₃₉	C₂
socle	C₂²×C₁₃₉	C₂²×D₄
prime radical	C₂²×D₁₃₉	C₂×D₄
hypercenter	#479	D₁₃₉
hypocenter	C₁₃₉	#3261
LCS	G ▷ C₂×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #479

Complements

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

N	H ≅ G/N	type
3. #22581263	C₂	semidirect
4. #22581259	C₂	semidirect
5. #22581257	C₂	semidirect
10. #22581164	C₂	semidirect
11. #22581163	C₂	semidirect
12. #22581162	C₂	semidirect
13. #22581162	C₂	semidirect
15. #479×C₁₃₉	C₂	semidirect
160. C₁₃₉	#3261	semidirect
167. #479	D₁₃₉	semidirect

Maximal subgroups

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

subgroup	index
#22581566	2
#22581561	2
#22581263	2
#22581259	2
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