Subgroups — #31819174

Order 17792.

344 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22581688	C₂	1	1	yes	maximal, characteristic
2	#22581681	C₂	1	1	yes	maximal, characteristic, complemented
3	#22581318	C₂	1	1	yes	maximal, characteristic
4	#22581313	C₂	1	1	yes	maximal, characteristic, complemented
5	#22581308	C₂	1	1	yes	maximal, characteristic, complemented
6	#22581261	C₂	1	1	yes	maximal, characteristic, complemented
7	#22581254	C₂	1	1	yes	maximal, characteristic, complemented
8	#22581244	C₂	1	1	yes	maximal, characteristic, complemented
9	#22581176	C₂	1	1	yes	maximal, characteristic, complemented
10	#22581176	C₂	1	1	yes	maximal, characteristic, complemented
11	#22581154	C₂	1	1	yes	maximal, characteristic
12	#22581061	C₂	1	1	yes	maximal, characteristic, complemented
13	#22581055	C₂	1	1	yes	maximal, characteristic, complemented
14	C₄×#14555774	C₂	1	1	yes	maximal, characteristic, complemented
15	#446×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×Dic₅₅₆	C₂²	1	1	yes	characteristic
17	C₂×#14555774	C₂²	1	1	yes	characteristic
18	C₂×#14555774	C₂²	1	1	yes	characteristic
19	#17984842	C₂²	1	1	yes	characteristic
20	#17984840	—	2	1	no	—
21	#17984839	C₂²	1	1	yes	characteristic
22	#17984834	C₂²	1	1	yes	characteristic
23	#17984834	C₂²	1	1	yes	characteristic
24	#17984832	—	2	1	no	—
25	#17984832	—	2	1	no	—
26	#17984831	—	2	1	no	—
27	#17984830	C₂²	1	1	yes	characteristic
28	#17984830	C₂²	1	1	yes	characteristic
29	#17984827	—	2	1	no	—
30	#17984827	—	2	1	no	—
31	C₄×Dic₂₇₈	C₂²	1	1	yes	characteristic
32	C₄×Dic₂₇₈	C₂²	1	1	yes	characteristic
33	C₄×Dic₂₇₈	C₂²	1	1	yes	characteristic
34	C₄×Dic₂₇₈	C₂²	1	1	yes	characteristic
35	#17984807	—	2	1	no	—
36	C₄×D₅₅₆	C₂²	1	1	yes	characteristic
37	C₄×D₅₅₆	C₂²	1	1	yes	characteristic
38	#17984804	—	2	1	no	—
39	#17984776	C₂²	1	1	yes	characteristic
40	#17984774	C₂²	1	1	yes	characteristic
41	#17984774	C₂²	1	1	yes	characteristic
42	#17984767	C₂²	1	1	yes	characteristic
43	#17984767	C₂²	1	1	yes	characteristic
44	#17984767	C₂²	1	1	yes	characteristic
45	#17984767	C₂²	1	1	yes	characteristic
46	#17984764	C₂²	1	1	yes	characteristic
47	#17984764	C₂²	1	1	yes	characteristic
48	#17984764	C₂²	1	1	yes	characteristic
49	#17984764	C₂²	1	1	yes	characteristic
50	#17984764	C₂²	1	1	yes	characteristic
51	#17984764	C₂²	1	1	yes	characteristic
52	C₄×Q₈×C₁₃₉	C₂²	1	1	yes	characteristic
53	C₄⋊C₈×C₁₃₉	C₂²	1	1	yes	characteristic
54	C₄⋊C₈×C₁₃₉	C₂²	1	1	yes	characteristic
55	C₄⋊C₈×C₁₃₉	C₂²	1	1	yes	characteristic
56	#97×C₁₃₉	C₂²	1	1	yes	characteristic
57	#97×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×D₅₅₆	—	2	1	no	—
63	C₂×D₅₅₆	—	2	1	no	—
64	C₂×D₅₅₆	C₂³	1	1	yes	characteristic
65	C₂×D₅₅₆	C₂³	1	1	yes	characteristic
66	C₂×C₄×D₁₃₉	—	2	1	no	—
67	C₂×C₄×D₁₃₉	—	2	1	no	—
68	Dic₅₅₆	—	2	2	no	—
69	#14555774	—	2	2	no	—
70	#14555774	—	2	2	no	—
71	C₄⋊Dic₁₃₉	—	2	1	no	—
72	C₄⋊Dic₁₃₉	C₂³	1	1	yes	characteristic
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	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	—	2	1	no	—
82	Dic₁₃₉⋊C₄	—	2	1	no	—
83	Dic₁₃₉⋊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	(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	C₂×Q₈×C₁₃₉	D₄	1	1	yes	characteristic
95	C₂×C₈×C₁₃₉	C₂³	1	1	yes	characteristic
96	C₂×C₈×C₁₃₉	C₂³	1	1	yes	characteristic
97	C₂×C₈×C₁₃₉	C₂³	1	1	yes	characteristic
98	C₂×C₈×C₁₃₉	C₂³	1	1	yes	characteristic
99	C₄⋊C₄×C₁₃₉	D₄	1	1	yes	characteristic
100	C₄⋊C₄×C₁₃₉	D₄	1	1	yes	characteristic
101	C₄⋊C₄×C₁₃₉	D₄	1	1	yes	characteristic
102	C₄²×C₁₃₉	C₂³	1	1	yes	characteristic
103	C₄²×C₁₃₉	C₂³	1	1	yes	characteristic
104	C₄²×C₁₃₉	C₂³	1	1	yes	characteristic
105	C₂²×D₁₃₉	—	2	1	no	—
106	C₂²×D₁₃₉	—	2	1	no	—
107	Dic₂₇₈	—	2	1	no	—
108	Dic₂₇₈	—	2	1	no	—
109	Dic₂₇₈	—	2	1	no	—
110	Dic₂₇₈	—	4	2	no	—
111	Dic₂₇₈	#41	1	2	yes	—
112	D₅₅₆	—	2	1	no	—
113	D₅₅₆	—	2	1	no	—
114	D₅₅₆	—	2	1	no	—
115	D₅₅₆	—	2	1	no	—
116	D₅₅₆	—	4	2	no	—
117	D₅₅₆	—	8	1	no	—
118	C₂×Dic₁₃₉	—	2	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	C₄×D₁₃₉	—	8	1	no	—
125	C₄×D₁₃₉	—	8	1	no	—
126	Q₈×C₁₃₉	—	4	1	no	—
127	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
128	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
129	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
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	C₂⁴	1	1	yes	characteristic
134	C₈×C₁₃₉	—	2	1	no	—
135	C₈×C₁₃₉	—	2	1	no	—
136	C₈×C₁₃₉	—	2	1	no	—
137	C₈×C₁₃₉	—	2	1	no	—
138	C₂²×C₁₃₉	C₂²×D₄	1	1	yes	characteristic
139	C₂×D₁₃₉	—	4	1	no	—
140	C₂×D₁₃₉	—	4	1	no	—
141	C₂×D₁₃₉	—	4	1	no	—
142	C₂×D₁₃₉	—	4	1	no	—
143	C₂×D₁₃₉	—	4	1	no	—
144	C₂×D₁₃₉	—	4	1	no	—
145	C₄×C₁₃₉	—	2	1	no	—
146	C₄×C₁₃₉	—	2	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	C₂×#41	1	1	yes	characteristic
152	C₄×C₁₃₉	2₋¹⁺⁴	1	1	yes	characteristic
153	Dic₁₃₉	—	2	2	no	—
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	C₂×C₁₃₉	#548	1	1	yes	characteristic
160	C₂×C₁₃₉	#575	1	1	yes	characteristic
161	C₂×C₁₃₉	#577	1	1	yes	characteristic
162	D₁₃₉	—	8	1	no	—
163	D₁₃₉	—	8	1	no	—
164	C₁₃₉	#3371	1	1	yes	characteristic, complemented, Sylow
165	#3371	—	139	1	no	maximal, Sylow
166	#559	—	139	1	no	—
167	#557	—	139	1	no	—
168	#490	—	139	1	no	—
169	#489	—	139	1	no	—
170	#488	—	139	1	no	—
171	#479	—	139	1	no	—
172	#478	—	139	1	no	—
173	#476	—	139	1	no	—
174	#464	—	139	1	no	—
175	#464	—	139	1	no	—
176	#460	—	139	1	no	—
177	#446	D₁₃₉	1	1	yes	characteristic, complemented
178	#441	—	139	1	no	—
179	#440	—	139	1	no	—
180	C₄×SD₁₆	—	139	1	no	—
181	C₂×Q₁₆	—	139	1	no	—
182	C₂×SD₁₆	—	139	1	no	—
183	C₂×SD₁₆	—	139	1	no	—
184	C₄⋊Q₈	—	278	1	no	—
185	C₄⋊Q₈	—	139	1	no	—
186	#127	—	139	1	no	—
187	#125	—	278	1	no	—
188	#125	—	278	1	no	—
189	#125	—	139	1	no	—
190	#125	—	139	1	no	—
191	#124	—	278	1	no	—
192	#124	—	139	1	no	—
193	#124	—	139	1	no	—
194	#123	—	278	1	no	—
195	#123	—	278	1	no	—
196	C₄×Q₈	—	278	1	no	—
197	C₄×Q₈	—	139	1	no	—
198	C₄×Q₈	—	139	1	no	—
199	C₄×Q₈	—	139	1	no	—
200	C₄×Q₈	—	139	1	no	—
201	C₄×Q₈	C₂×D₁₃₉	1	1	yes	characteristic
202	C₄×D₄	—	278	1	no	—
203	C₄×D₄	—	139	1	no	—
204	C₄×D₄	—	139	1	no	—
205	#107	—	139	1	no	—
206	#106	—	139	1	no	—
207	#106	—	139	1	no	—
208	C₄⋊C₈	C₂×D₁₃₉	1	1	yes	characteristic
209	C₄⋊C₈	C₂×D₁₃₉	1	1	yes	characteristic
210	C₄⋊C₈	C₂×D₁₃₉	1	1	yes	characteristic
211	#103	—	139	1	no	—
212	#103	—	139	1	no	—
213	#103	—	139	1	no	—
214	#103	—	139	1	no	—
215	#102	—	139	1	no	—
216	#102	—	139	1	no	—
217	#102	—	139	1	no	—
218	#102	—	139	1	no	—
219	#102	—	139	1	no	—
220	#102	—	139	1	no	—
221	#97	C₂×D₁₃₉	1	1	yes	characteristic
222	#97	C₂×D₁₃₉	1	1	yes	characteristic
223	C₄×C₈	C₂×D₁₃₉	1	1	yes	characteristic
224	C₂×Q₈	—	278	1	no	—
225	C₂×Q₈	—	139	1	no	—
226	C₂×Q₈	—	139	1	no	—
227	C₂×Q₈	D₅₅₆	1	1	yes	characteristic
228	C₂×D₄	—	278	1	no	—
229	C₂×D₄	—	278	1	no	—
230	C₂×D₄	—	139	1	no	—
231	C₂×D₄	—	139	1	no	—
232	C₂²×C₄	—	278	1	no	—
233	C₂²×C₄	—	278	1	no	—
234	Q₁₆	—	278	2	no	—
235	SD₁₆	—	278	2	no	—
236	SD₁₆	—	278	2	no	—
237	C₂×C₈	C₂²×D₁₃₉	1	1	yes	characteristic
238	C₂×C₈	C₂²×D₁₃₉	1	1	yes	characteristic
239	C₂×C₈	C₂²×D₁₃₉	1	1	yes	characteristic
240	C₂×C₈	C₂²×D₁₃₉	1	1	yes	characteristic
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	—	139	1	no	—
251	C₄⋊C₄	—	139	1	no	—
252	C₄⋊C₄	—	139	1	no	—
253	C₄⋊C₄	—	139	1	no	—
254	C₄⋊C₄	D₅₅₆	1	1	yes	characteristic
255	C₄⋊C₄	D₅₅₆	1	1	yes	characteristic
256	C₄⋊C₄	D₅₅₆	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	—	278	1	no	—
262	C₂²⋊C₄	—	278	1	no	—
263	C₄²	—	278	1	no	—
264	C₄²	—	278	1	no	—
265	C₄²	—	278	1	no	—
266	C₄²	—	278	1	no	—
267	C₄²	C₂²×D₁₃₉	1	1	yes	characteristic
268	C₄²	C₂²×D₁₃₉	1	1	yes	characteristic
269	C₄²	C₂²×D₁₃₉	1	1	yes	characteristic
270	C₂³	—	278	1	no	—
271	C₂³	—	278	1	no	—
272	Q₈	—	4	1	no	—
273	Q₈	—	278	1	no	—
274	Q₈	—	278	1	no	—
275	Q₈	—	278	1	no	—
276	Q₈	—	556	2	no	—
277	Q₈	—	139	2	no	—
278	D₄	—	278	1	no	—
279	D₄	—	278	1	no	—
280	D₄	—	278	1	no	—
281	D₄	—	278	1	no	—
282	D₄	—	556	2	no	—
283	D₄	—	1112	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	—	278	1	no	—
289	C₂×C₄	—	278	1	no	—
290	C₂×C₄	—	1112	1	no	—
291	C₂×C₄	—	1112	1	no	—
292	C₂×C₄	C₂×D₅₅₆	1	1	yes	characteristic
293	C₂×C₄	C₂×D₅₅₆	1	1	yes	characteristic
294	C₂×C₄	C₂×D₅₅₆	1	1	yes	characteristic
295	C₂×C₄	C₂×D₅₅₆	1	1	yes	characteristic
296	C₂×C₄	C₂×D₅₅₆	1	1	yes	characteristic
297	C₂×C₄	C₂×D₅₅₆	1	1	yes	characteristic
298	C₂×C₄	C₂³×D₁₃₉	1	1	yes	characteristic
299	C₈	—	2	1	no	—
300	C₈	—	2	1	no	—
301	C₈	—	2	1	no	—
302	C₈	—	2	1	no	—
303	C₂²	—	556	1	no	—
304	C₂²	—	556	1	no	—
305	C₂²	—	556	1	no	—
306	C₂²	—	556	1	no	—
307	C₂²	—	556	1	no	—
308	C₂²	—	556	1	no	—
309	C₂²	C₂²×D₅₅₆	1	1	yes	characteristic
310	C₄	—	2	1	no	—
311	C₄	—	2	1	no	—
312	C₄	—	2	1	no	—
313	C₄	—	2	1	no	—
314	C₄	—	2	1	no	—
315	C₄	—	2	1	no	—
316	C₄	—	278	2	no	—
317	C₄	—	556	1	no	—
318	C₄	—	556	1	no	—
319	C₄	—	556	1	no	—
320	C₄	—	556	1	no	—
321	C₄	—	556	1	no	—
322	C₄	#41×D₁₃₉	1	1	yes	characteristic
323	C₄	#17984890	1	1	yes	characteristic
324	C₂	—	1112	1	no	—
325	C₂	—	1112	1	no	—
326	C₂	#22581615	1	1	yes	characteristic
327	C₂	#22581753	1	1	yes	characteristic
328	C₂	#22581765	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₅₅₆
commutator, cosocle	C₂×C₄×C₁₃₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₃₉
Fitting	#446×C₁₃₉	C₂
socle	C₂²×C₁₃₉	C₂²×D₄
prime radical	#17984839	C₂²
hypercenter	#446	D₁₃₉
hypocenter	C₁₃₉	#3371
LCS	G ▷ C₂×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ #446

Complements

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

N	H ≅ G/N	type
2. #22581681	C₂	semidirect
4. #22581313	C₂	semidirect
5. #22581308	C₂	semidirect
6. #22581261	C₂	semidirect
7. #22581254	C₂	semidirect
8. #22581244	C₂	semidirect
9. #22581176	C₂	semidirect
10. #22581176	C₂	semidirect
12. #22581061	C₂	semidirect
13. #22581055	C₂	semidirect
14. C₄×#14555774	C₂	semidirect
15. #446×C₁₃₉	C₂	semidirect
164. C₁₃₉	#3371	semidirect
177. #446	D₁₃₉	semidirect

Maximal subgroups

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

subgroup	index
#22581688	2
#22581681	2
#22581318	2
#22581313	2
#22581308	2
#22581261	2
#22581254	2
#22581244	2
#22581176	2
#22581154	2
#22581061	2
#22581055	2
C₄×#14555774	2
#446×C₁₃₉	2
#3371	139