Subgroups — #31817832

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	#22581574	C₂	1	1	yes	maximal, characteristic
2	#22581565	C₂	1	1	yes	maximal, characteristic, complemented
3	#22581261	C₂	1	1	yes	maximal, characteristic, complemented
4	#22581254	C₂	1	1	yes	maximal, characteristic, complemented
5	#22581249	C₂	1	1	yes	maximal, characteristic, complemented
6	#22581239	C₂	1	1	yes	maximal, characteristic, complemented
7	#22581211	C₂	1	1	yes	maximal, characteristic, complemented
8	#22581199	C₂	1	1	yes	maximal, characteristic
9	#22581194	C₂	1	1	yes	maximal, characteristic, complemented
10	#22581182	C₂	1	1	yes	maximal, characteristic
11	#22581176	C₂	1	1	yes	maximal, characteristic, complemented
12	#22581176	C₂	1	1	yes	maximal, characteristic, complemented
13	#22581164	C₂	1	1	yes	maximal, characteristic, complemented
14	#22581159	C₂	1	1	yes	maximal, characteristic, complemented
15	#424×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×#14555774	C₂²	1	1	yes	characteristic
19	C₂×#14555774	C₂²	1	1	yes	characteristic
20	#17984842	C₂²	1	1	yes	characteristic
21	#17984842	C₂²	1	1	yes	characteristic
22	#17984837	—	2	1	no	—
23	#17984835	—	2	1	no	—
24	#17984835	—	2	1	no	—
25	#17984834	C₂²	1	1	yes	characteristic
26	#17984832	—	2	1	no	—
27	#17984832	—	2	1	no	—
28	#17984830	C₂²	1	1	yes	characteristic
29	#17984828	—	2	1	no	—
30	#17984824	—	2	1	no	—
31	#17984821	C₂²	1	1	yes	characteristic
32	#17984821	C₂²	1	1	yes	characteristic
33	#17984821	C₂²	1	1	yes	characteristic
34	#17984820	—	2	1	no	—
35	#17984816	C₂²	1	1	yes	characteristic
36	C₄×Dic₂₇₈	C₂²	1	1	yes	characteristic
37	C₄×Dic₂₇₈	C₂²	1	1	yes	characteristic
38	C₄×Dic₂₇₈	C₂²	1	1	yes	characteristic
39	C₄×D₅₅₆	C₂²	1	1	yes	characteristic
40	#17984776	C₂²	1	1	yes	characteristic
41	#17984776	C₂²	1	1	yes	characteristic
42	#17984774	C₂²	1	1	yes	characteristic
43	#17984774	C₂²	1	1	yes	characteristic
44	#17984767	C₂²	1	1	yes	characteristic
45	#17984767	C₂²	1	1	yes	characteristic
46	#17984767	C₂²	1	1	yes	characteristic
47	#17984767	C₂²	1	1	yes	characteristic
48	#17984767	C₂²	1	1	yes	characteristic
49	#17984767	C₂²	1	1	yes	characteristic
50	#17984764	C₂²	1	1	yes	characteristic
51	#17984764	C₂²	1	1	yes	characteristic
52	C₂×M₁₆×C₁₃₉	C₂²	1	1	yes	characteristic
53	C₂²×C₈×C₁₃₉	C₂²	1	1	yes	characteristic
54	#117×C₁₃₉	C₂²	1	1	yes	characteristic
55	C₄⋊C₈×C₁₃₉	C₂²	1	1	yes	characteristic
56	C₄⋊C₈×C₁₃₉	C₂²	1	1	yes	characteristic
57	C₄⋊C₈×C₁₃₉	C₂²	1	1	yes	characteristic
58	C₄⋊C₈×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×D₅₅₆	C₂³	1	1	yes	characteristic
64	C₂×C₂²⋊D₁₃₉	—	2	1	no	—
65	C₂×C₄×D₁₃₉	—	2	1	no	—
66	Dic₅₅₆	—	4	1	no	—
67	Dic₅₅₆	—	4	1	no	—
68	#14555774	—	4	1	no	—
69	#14555774	—	4	1	no	—
70	C₄⋊Dic₁₃₉	—	2	1	no	—
71	C₄⋊Dic₁₃₉	—	2	1	no	—
72	C₄⋊Dic₁₃₉	—	2	1	no	—
73	C₄⋊Dic₁₃₉	C₂³	1	1	yes	characteristic
74	C₄⋊Dic₁₃₉	C₂³	1	1	yes	characteristic
75	C₄⋊Dic₁₃₉	C₂³	1	1	yes	characteristic
76	C₄⋊Dic₁₃₉	C₂³	1	1	yes	characteristic
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	—	2	1	no	—
82	Dic₁₃₉⋊C₄	—	2	1	no	—
83	Dic₁₃₉⋊C₄	—	2	1	no	—
84	Dic₁₃₉⋊C₄	—	2	1	no	—
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	—	2	1	no	—
92	C₄×Dic₁₃₉	—	2	1	no	—
93	C₄×Dic₁₃₉	—	2	1	no	—
94	C₄×Dic₁₃₉	—	2	1	no	—
95	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
96	M₁₆×C₁₃₉	—	4	1	no	—
97	C₂×C₈×C₁₃₉	—	4	1	no	—
98	C₂×C₈×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	C₂×C₈×C₁₃₉	C₂³	1	1	yes	characteristic
102	C₄⋊C₄×C₁₃₉	D₄	1	1	yes	characteristic
103	C₄⋊C₄×C₁₃₉	D₄	1	1	yes	characteristic
104	C₂²⋊C₄×C₁₃₉	D₄	1	1	yes	characteristic
105	C₂²⋊C₄×C₁₃₉	D₄	1	1	yes	characteristic
106	C₄²×C₁₃₉	C₂³	1	1	yes	characteristic
107	C₄²×C₁₃₉	C₂³	1	1	yes	characteristic
108	C₂²×D₁₃₉	—	2	1	no	—
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	D₅₅₆	—	2	1	no	—
117	D₅₅₆	—	2	1	no	—
118	C₂²⋊D₁₃₉	—	8	1	no	—
119	C₂×Dic₁₃₉	—	2	1	no	—
120	C₂×Dic₁₃₉	—	2	1	no	—
121	C₂×Dic₁₃₉	—	2	1	no	—
122	C₂×Dic₁₃₉	—	2	1	no	—
123	C₂×Dic₁₃₉	—	2	1	no	—
124	C₂×Dic₁₃₉	—	2	1	no	—
125	C₂×Dic₁₃₉	—	2	1	no	—
126	C₄×D₁₃₉	—	8	1	no	—
127	C₂³×C₁₃₉	C₂×D₄	1	1	yes	characteristic
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	C₂×D₄	1	1	yes	characteristic
133	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
134	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
135	C₂×C₄×C₁₃₉	C₂⁴	1	1	yes	characteristic
136	C₈×C₁₃₉	—	2	1	no	—
137	C₈×C₁₃₉	—	2	1	no	—
138	C₈×C₁₃₉	—	2	1	no	—
139	C₈×C₁₃₉	—	2	1	no	—
140	C₂²×C₁₃₉	—	2	1	no	—
141	C₂²×C₁₃₉	—	2	1	no	—
142	C₂²×C₁₃₉	—	2	1	no	—
143	C₂²×C₁₃₉	C₂²×D₄	1	1	yes	characteristic
144	C₂×D₁₃₉	—	4	1	no	—
145	C₂×D₁₃₉	—	4	1	no	—
146	C₂×D₁₃₉	—	4	1	no	—
147	C₄×C₁₃₉	—	2	1	no	—
148	C₄×C₁₃₉	—	2	1	no	—
149	C₄×C₁₃₉	—	2	1	no	—
150	C₄×C₁₃₉	—	2	1	no	—
151	C₄×C₁₃₉	—	2	1	no	—
152	C₄×C₁₃₉	H(5,𝔽₂)	1	1	yes	characteristic
153	C₄×C₁₃₉	2₋¹⁺⁴	1	1	yes	characteristic
154	Dic₁₃₉	—	4	1	no	—
155	Dic₁₃₉	—	4	1	no	—
156	Dic₁₃₉	—	4	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	—	4	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	—	8	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	—	139	1	no	—
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	D₁₃₉	1	1	yes	characteristic, complemented
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	C₂×D₁₃₉	1	1	yes	characteristic
188	C₂²×C₈	C₂×D₁₃₉	1	1	yes	characteristic
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	—	139	1	no	—
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	—	139	1	no	—
209	#117	C₂×D₁₃₉	1	1	yes	characteristic
210	#107	—	139	1	no	—
211	#107	—	139	1	no	—
212	#106	—	139	1	no	—
213	#106	—	139	1	no	—
214	C₄⋊C₈	C₂×D₁₃₉	1	1	yes	characteristic
215	C₄⋊C₈	C₂×D₁₃₉	1	1	yes	characteristic
216	C₄⋊C₈	C₂×D₁₃₉	1	1	yes	characteristic
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	—	139	1	no	—
225	#102	—	139	1	no	—
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	—	139	1	no	—
232	C₂²×C₄	—	278	1	no	—
233	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
234	Q₁₆	—	556	1	no	—
235	Q₁₆	—	556	1	no	—
236	SD₁₆	—	556	1	no	—
237	SD₁₆	—	556	1	no	—
238	M₁₆	—	4	1	no	—
239	C₂×C₈	—	4	1	no	—
240	C₂×C₈	C₂²×D₁₃₉	1	1	yes	characteristic
241	C₂×C₈	C₂²×D₁₃₉	1	1	yes	characteristic
242	C₂×C₈	C₂²×D₁₃₉	1	1	yes	characteristic
243	C₂×C₈	C₂²×D₁₃₉	1	1	yes	characteristic
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	—	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	—	139	1	no	—
259	C₄⋊C₄	D₅₅₆	1	1	yes	characteristic
260	C₄⋊C₄	D₅₅₆	1	1	yes	characteristic
261	C₂²⋊C₄	—	278	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	D₅₅₆	1	1	yes	characteristic
269	C₂²⋊C₄	D₅₅₆	1	1	yes	characteristic
270	C₄²	—	278	1	no	—
271	C₄²	—	278	1	no	—
272	C₄²	—	278	1	no	—
273	C₄²	C₂²×D₁₃₉	1	1	yes	characteristic
274	C₄²	C₂²×D₁₃₉	1	1	yes	characteristic
275	C₂³	—	278	1	no	—
276	C₂³	C₂×D₅₅₆	1	1	yes	characteristic
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	—	278	1	no	—
285	D₄	—	278	1	no	—
286	D₄	—	1112	1	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	278	1	no	—
290	C₂×C₄	—	278	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	—	1112	1	no	—
297	C₂×C₄	C₂×D₅₅₆	1	1	yes	characteristic
298	C₂×C₄	C₂×D₅₅₆	1	1	yes	characteristic
299	C₂×C₄	C₂×D₅₅₆	1	1	yes	characteristic
300	C₂×C₄	C₂×D₅₅₆	1	1	yes	characteristic
301	C₂×C₄	C₂×D₅₅₆	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	—	2	1	no	—
306	C₈	—	2	1	no	—
307	C₂²	—	2	1	no	—
308	C₂²	—	2	1	no	—
309	C₂²	—	2	1	no	—
310	C₂²	—	556	1	no	—
311	C₂²	—	556	1	no	—
312	C₂²	—	556	1	no	—
313	C₂²	C₂²×D₅₅₆	1	1	yes	characteristic
314	C₄	—	2	1	no	—
315	C₄	—	2	1	no	—
316	C₄	—	2	1	no	—
317	C₄	—	2	1	no	—
318	C₄	—	2	1	no	—
319	C₄	—	556	1	no	—
320	C₄	—	556	1	no	—
321	C₄	—	556	1	no	—
322	C₄	—	556	1	no	—
323	C₄	—	556	1	no	—
324	C₄	—	556	1	no	—
325	C₄	—	556	1	no	—
326	C₄	#17984887	1	1	yes	characteristic
327	C₄	#17984890	1	1	yes	characteristic
328	C₂	—	4	1	no	—
329	C₂	—	1112	1	no	—
330	C₂	#22581534	1	1	yes	characteristic
331	C₂	#22581765	1	1	yes	characteristic
332	C₂	#22581770	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₅₅₆
commutator, cosocle	C₂×C₄×C₁₃₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₃₉
Fitting	#424×C₁₃₉	C₂
socle	C₂²×C₁₃₉	C₂²×D₄
prime radical	#17984816	C₂²
hypercenter	#424	D₁₃₉
hypocenter	C₁₃₉	#3258
LCS	G ▷ C₂×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂² ◁ #117 ◁ #424

Complements

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

N	H ≅ G/N	type
2. #22581565	C₂	semidirect
3. #22581261	C₂	semidirect
4. #22581254	C₂	semidirect
5. #22581249	C₂	semidirect
6. #22581239	C₂	semidirect
7. #22581211	C₂	semidirect
9. #22581194	C₂	semidirect
11. #22581176	C₂	semidirect
12. #22581176	C₂	semidirect
13. #22581164	C₂	semidirect
14. #22581159	C₂	semidirect
15. #424×C₁₃₉	C₂	semidirect
166. C₁₃₉	#3258	semidirect
182. #424	D₁₃₉	semidirect

Maximal subgroups

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

subgroup	index
#22581574	2
#22581565	2
#22581261	2
#22581254	2
#22581249	2
#22581239	2
#22581211	2
#22581199	2
#22581194	2
#22581182	2
#22581176	2
#22581164	2
#22581159	2
#424×C₁₃₉	2
#3258	139