Subgroups — #31817834

Order 17792.

334 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22581568	C₂	1	1	yes	maximal, characteristic
2	#22581566	C₂	1	1	yes	maximal, characteristic, complemented
3	#22581257	C₂	1	1	yes	maximal, characteristic, complemented
4	#22581247	C₂	1	1	yes	maximal, characteristic, complemented
5	#22581237	C₂	1	1	yes	maximal, characteristic, complemented
6	#22581212	C₂	1	1	yes	maximal, characteristic, complemented
7	#22581198	C₂	1	1	yes	maximal, characteristic
8	#22581195	C₂	1	1	yes	maximal, characteristic, complemented
9	#22581181	C₂	1	1	yes	maximal, characteristic
10	#22581174	C₂	1	1	yes	maximal, characteristic, complemented
11	#22581174	C₂	1	1	yes	maximal, characteristic, complemented
12	#22581164	C₂	1	1	yes	maximal, characteristic, complemented
13	#22581158	C₂	1	1	yes	maximal, characteristic, complemented
14	#22580978	C₂	1	1	yes	maximal, characteristic, complemented
15	#478×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×#14555773	C₂²	1	1	yes	characteristic
19	C₂×#14555773	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	#17984841	C₂²	1	1	yes	characteristic
24	#17984837	—	2	1	no	—
25	#17984837	—	2	1	no	—
26	#17984835	—	2	1	no	—
27	#17984833	—	2	1	no	—
28	#17984833	—	2	1	no	—
29	#17984829	C₂²	1	1	yes	characteristic
30	#17984828	—	2	1	no	—
31	#17984825	—	2	1	no	—
32	#17984822	C₂²	1	1	yes	characteristic
33	#17984822	C₂²	1	1	yes	characteristic
34	#17984822	C₂²	1	1	yes	characteristic
35	#17984821	—	2	1	no	—
36	#17984813	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×D₄×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	#102×C₁₃₉	C₂²	1	1	yes	characteristic
58	#102×C₁₃₉	C₂²	1	1	yes	characteristic
59	C₂×Dic₂₇₈	—	2	1	no	—
60	C₂×Dic₂₇₈	C₂³	1	1	yes	characteristic
61	C₂×Dic₂₇₈	C₂³	1	1	yes	characteristic
62	C₂×Dic₂₇₈	C₂³	1	1	yes	characteristic
63	C₂×C₂²⋊D₁₃₉	—	2	1	no	—
64	C₂×C₄×D₁₃₉	C₂³	1	1	yes	characteristic
65	Dic₅₅₆	—	4	1	no	—
66	Dic₅₅₆	—	4	1	no	—
67	#14555773	—	4	1	no	—
68	#14555773	—	4	1	no	—
69	#14555768	—	4	1	no	—
70	C₂×C₁₃₉⋊C₈	C₂³	1	1	yes	characteristic
71	C₂×C₁₃₉⋊C₈	C₂³	1	1	yes	characteristic
72	C₈×D₁₃₉	—	4	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	—	2	1	no	—
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	D₄	1	1	yes	characteristic
84	Dic₁₃₉⋊C₄	D₄	1	1	yes	characteristic
85	C₂²⋊Dic₁₃₉	—	2	1	no	—
86	C₂²⋊Dic₁₃₉	—	2	1	no	—
87	C₂²⋊Dic₁₃₉	—	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×D₄×C₁₃₉	C₂³	1	1	yes	characteristic
98	C₂²×C₄×C₁₃₉	—	2	1	no	—
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	—	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	C₂²⋊C₄×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	Dic₂₇₈	—	8	1	no	—
116	C₂²⋊D₁₃₉	—	8	1	no	—
117	C₂×Dic₁₃₉	—	2	1	no	—
118	C₂×Dic₁₃₉	—	2	1	no	—
119	C₂×Dic₁₃₉	—	2	1	no	—
120	C₂×Dic₁₃₉	—	2	1	no	—
121	C₂×Dic₁₃₉	C₂×D₄	1	1	yes	characteristic
122	C₂×Dic₁₃₉	C₂×D₄	1	1	yes	characteristic
123	C₂×Dic₁₃₉	C₂×D₄	1	1	yes	characteristic
124	C₄×D₁₃₉	—	2	1	no	—
125	C₄×D₁₃₉	—	2	1	no	—
126	C₁₃₉⋊C₈	—	2	1	no	—
127	C₁₃₉⋊C₈	—	2	1	no	—
128	C₂³×C₁₃₉	—	2	1	no	—
129	D₄×C₁₃₉	—	2	1	no	—
130	D₄×C₁₃₉	—	2	1	no	—
131	C₂×C₄×C₁₃₉	—	2	1	no	—
132	C₂×C₄×C₁₃₉	—	2	1	no	—
133	C₂×C₄×C₁₃₉	—	2	1	no	—
134	C₂×C₄×C₁₃₉	—	8	1	no	—
135	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
136	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
137	C₂×C₄×C₁₃₉	C₂⁴	1	1	yes	characteristic
138	C₈×C₁₃₉	—	2	1	no	—
139	C₈×C₁₃₉	—	2	1	no	—
140	C₂²×C₁₃₉	—	4	1	no	—
141	C₂²×C₁₃₉	—	4	1	no	—
142	C₂²×C₁₃₉	—	4	1	no	—
143	C₂²×C₁₃₉	C₂²×D₄	1	1	yes	characteristic
144	C₂×D₁₃₉	—	2	1	no	—
145	C₂×D₁₃₉	—	2	1	no	—
146	C₂×D₁₃₉	—	2	1	no	—
147	C₄×C₁₃₉	—	2	1	no	—
148	C₄×C₁₃₉	—	2	1	no	—
149	C₄×C₁₃₉	—	4	1	no	—
150	C₄×C₁₃₉	—	4	1	no	—
151	C₄×C₁₃₉	—	4	1	no	—
152	C₄×C₁₃₉	H(5,𝔽₂)	1	1	yes	characteristic
153	C₄×C₁₃₉	2₋¹⁺⁴	1	1	yes	characteristic
154	Dic₁₃₉	—	2	1	no	—
155	Dic₁₃₉	—	2	1	no	—
156	Dic₁₃₉	—	2	1	no	—
157	Dic₁₃₉	—	4	1	no	—
158	Dic₁₃₉	—	4	1	no	—
159	Dic₁₃₉	—	4	1	no	—
160	Dic₁₃₉	—	4	1	no	—
161	C₂×C₁₃₉	—	8	1	no	—
162	C₂×C₁₃₉	#537	1	1	yes	characteristic
163	C₂×C₁₃₉	#577	1	1	yes	characteristic
164	C₂×C₁₃₉	#578	1	1	yes	characteristic
165	D₁₃₉	—	4	1	no	—
166	C₁₃₉	#3258	1	1	yes	characteristic, complemented, Sylow
167	#3258	—	139	1	no	maximal, Sylow
168	#542	—	139	1	no	—
169	#541	—	139	1	no	—
170	#479	—	139	1	no	—
171	#478	D₁₃₉	1	1	yes	characteristic, complemented
172	#477	—	139	1	no	—
173	#475	—	139	1	no	—
174	#470	—	139	1	no	—
175	#468	—	139	1	no	—
176	#467	—	139	1	no	—
177	#465	—	139	1	no	—
178	#464	—	139	1	no	—
179	#464	—	139	1	no	—
180	#462	—	139	1	no	—
181	#461	—	139	1	no	—
182	#424	—	139	1	no	—
183	C₂×Q₁₆	—	139	1	no	—
184	C₂×Q₁₆	—	139	1	no	—
185	C₂×SD₁₆	—	139	1	no	—
186	C₂×SD₁₆	—	139	1	no	—
187	C₂×M₁₆	—	139	1	no	—
188	C₂²×C₈	—	139	1	no	—
189	C₄⋊Q₈	—	139	1	no	—
190	C₄⋊Q₈	—	139	1	no	—
191	#126	—	278	1	no	—
192	#126	—	278	1	no	—
193	#126	—	278	1	no	—
194	#125	—	278	1	no	—
195	#125	—	278	1	no	—
196	#125	C₂×D₁₃₉	1	1	yes	characteristic
197	#124	—	278	1	no	—
198	#124	—	139	1	no	—
199	#123	—	278	1	no	—
200	C₂²⋊Q₈	—	278	1	no	—
201	C₂²⋊Q₈	—	139	1	no	—
202	C₂²⋊Q₈	—	139	1	no	—
203	C₂²⋊Q₈	—	139	1	no	—
204	#121	—	139	1	no	—
205	C₄×Q₈	—	139	1	no	—
206	C₄×Q₈	—	139	1	no	—
207	C₄×Q₈	—	139	1	no	—
208	C₄×D₄	C₂×D₁₃₉	1	1	yes	characteristic
209	#117	—	139	1	no	—
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	C₄⋊C₈	—	139	1	no	—
217	C₄⋊C₈	C₂×D₁₃₉	1	1	yes	characteristic
218	#103	—	139	1	no	—
219	#103	—	139	1	no	—
220	#103	—	139	1	no	—
221	#103	—	139	1	no	—
222	#103	—	139	1	no	—
223	#103	—	139	1	no	—
224	#102	C₂×D₁₃₉	1	1	yes	characteristic
225	#102	C₂×D₁₃₉	1	1	yes	characteristic
226	C₂×Q₈	—	278	1	no	—
227	C₂×Q₈	—	139	1	no	—
228	C₂×Q₈	—	139	1	no	—
229	C₂×Q₈	—	139	1	no	—
230	C₂×D₄	—	278	1	no	—
231	C₂×D₄	C₂²×D₁₃₉	1	1	yes	characteristic
232	C₂²×C₄	—	2	1	no	—
233	C₂²×C₄	—	139	1	no	—
234	Q₁₆	—	556	1	no	—
235	Q₁₆	—	556	1	no	—
236	SD₁₆	—	556	1	no	—
237	SD₁₆	—	556	1	no	—
238	M₁₆	—	556	1	no	—
239	C₂×C₈	—	556	1	no	—
240	C₂×C₈	—	139	1	no	—
241	C₂×C₈	—	139	1	no	—
242	C₂×C₈	C₂²×D₁₃₉	1	1	yes	characteristic
243	C₂×C₈	C₂²×D₁₃₉	1	1	yes	characteristic
244	C₄⋊C₄	—	2	1	no	—
245	C₄⋊C₄	—	2	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	—	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	—	278	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	—	278	1	no	—
268	C₂²⋊C₄	—	139	1	no	—
269	C₂²⋊C₄	—	139	1	no	—
270	C₄²	—	278	1	no	—
271	C₄²	—	278	1	no	—
272	C₄²	—	278	1	no	—
273	C₄²	—	139	1	no	—
274	C₄²	C₂²×D₁₃₉	1	1	yes	characteristic
275	C₂³	—	2	1	no	—
276	C₂³	—	139	1	no	—
277	Q₈	—	278	1	no	—
278	Q₈	—	278	1	no	—
279	Q₈	—	278	1	no	—
280	Q₈	—	278	1	no	—
281	Q₈	—	278	1	no	—
282	Q₈	—	278	1	no	—
283	Q₈	—	1112	1	no	—
284	D₄	—	2	1	no	—
285	D₄	—	2	1	no	—
286	D₄	—	1112	1	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	8	1	no	—
291	C₂×C₄	—	278	1	no	—
292	C₂×C₄	—	278	1	no	—
293	C₂×C₄	—	278	1	no	—
294	C₂×C₄	—	278	1	no	—
295	C₂×C₄	—	278	1	no	—
296	C₂×C₄	—	278	1	no	—
297	C₂×C₄	—	139	1	no	—
298	C₂×C₄	—	139	1	no	—
299	C₂×C₄	—	139	1	no	—
300	C₂×C₄	D₄×D₁₃₉	1	1	yes	characteristic
301	C₂×C₄	D₄×D₁₃₉	1	1	yes	characteristic
302	C₂×C₄	C₂³×D₁₃₉	1	1	yes	characteristic
303	C₈	—	2	1	no	—
304	C₈	—	2	1	no	—
305	C₈	—	278	1	no	—
306	C₈	—	278	1	no	—
307	C₂²	—	4	1	no	—
308	C₂²	—	4	1	no	—
309	C₂²	—	4	1	no	—
310	C₂²	—	278	1	no	—
311	C₂²	—	278	1	no	—
312	C₂²	—	278	1	no	—
313	C₂²	C₂×D₄×D₁₃₉	1	1	yes	characteristic
314	C₄	—	2	1	no	—
315	C₄	—	2	1	no	—
316	C₄	—	4	1	no	—
317	C₄	—	4	1	no	—
318	C₄	—	4	1	no	—
319	C₄	—	278	1	no	—
320	C₄	—	278	1	no	—
321	C₄	—	278	1	no	—
322	C₄	—	556	1	no	—
323	C₄	—	556	1	no	—
324	C₄	—	556	1	no	—
325	C₄	—	556	1	no	—
326	C₄	#17984888	1	1	yes	characteristic
327	C₄	#17984890	1	1	yes	characteristic
328	C₂	—	8	1	no	—
329	C₂	—	556	1	no	—
330	C₂	#22581533	1	1	yes	characteristic
331	C₂	#22581763	1	1	yes	characteristic
332	C₂	#22581767	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₃₉
commutator, cosocle	C₂×C₄×C₁₃₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₃₉
Fitting	#478×C₁₃₉	C₂
socle	C₂²×C₁₃₉	C₂²×D₄
prime radical	#17984813	C₂²
hypercenter	#478	D₁₃₉
hypocenter	C₁₃₉	#3258
LCS	G ▷ C₂×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #478

Complements

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

N	H ≅ G/N	type
2. #22581566	C₂	semidirect
3. #22581257	C₂	semidirect
4. #22581247	C₂	semidirect
5. #22581237	C₂	semidirect
6. #22581212	C₂	semidirect
8. #22581195	C₂	semidirect
10. #22581174	C₂	semidirect
11. #22581174	C₂	semidirect
12. #22581164	C₂	semidirect
13. #22581158	C₂	semidirect
14. #22580978	C₂	semidirect
15. #478×C₁₃₉	C₂	semidirect
166. C₁₃₉	#3258	semidirect
171. #478	D₁₃₉	semidirect

Maximal subgroups

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

subgroup	index
#22581568	2
#22581566	2
#22581257	2
#22581247	2
#22581237	2
#22581212	2
#22581198	2
#22581195	2
#22581181	2
#22581174	2
#22581164	2
#22581158	2
#22580978	2
#478×C₁₃₉	2
#3258	139