Subgroups — #31816634

Order 17792.

370 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22581576	C₂	1	1	yes	maximal, characteristic
2	#22581431	C₂	1	1	yes	maximal, characteristic, complemented
3	#22581291	C₂	1	1	yes	maximal, characteristic, complemented
4	#22581278	C₂	1	1	yes	maximal, characteristic
5	#22581246	C₂	1	1	yes	maximal, characteristic, complemented
6	#22581236	C₂	1	1	yes	maximal, characteristic, complemented
7	#22581177	C₂	1	1	yes	maximal, characteristic, complemented
8	#22581177	C₂	1	1	yes	maximal, characteristic, complemented
9	#22581092	C₂	1	1	yes	maximal, characteristic
10	#22581060	C₂	1	1	yes	maximal, characteristic, complemented
11	#22581056	C₂	1	1	yes	maximal, characteristic, complemented
12	Q₁₆×Dic₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
13	#22581038	C₂	1	1	yes	maximal, characteristic, complemented
14	#22581011	C₂	1	1	yes	maximal, characteristic, complemented
15	#451×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#14555772	C₂²	1	1	yes	characteristic
17	C₂×#14555772	C₂²	1	1	yes	characteristic
18	#17984841	C₂²	1	1	yes	characteristic
19	#17984837	—	2	1	no	—
20	#17984831	C₂²	1	1	yes	characteristic
21	#17984828	—	2	1	no	—
22	#17984824	—	2	1	no	—
23	#17984822	—	2	1	no	—
24	#17984816	C₂²	1	1	yes	characteristic
25	Q₈×Dic₁₃₉	—	2	1	no	—
26	Q₈×Dic₁₃₉	—	2	1	no	—
27	Q₈×Dic₁₃₉	C₂²	1	1	yes	characteristic
28	Q₈×Dic₁₃₉	C₂²	1	1	yes	characteristic
29	Q₈×Dic₁₃₉	C₂²	1	1	yes	characteristic
30	Q₈×Dic₁₃₉	C₂²	1	1	yes	characteristic
31	Q₈×Dic₁₃₉	C₂²	1	1	yes	characteristic
32	#17984804	C₂²	1	1	yes	characteristic
33	#17984801	—	2	1	no	—
34	#17984801	—	2	1	no	—
35	#17984801	C₂²	1	1	yes	characteristic
36	#17984801	C₂²	1	1	yes	characteristic
37	#17984776	C₂²	1	1	yes	characteristic
38	#17984775	C₂²	1	1	yes	characteristic
39	#17984774	C₂²	1	1	yes	characteristic
40	#17984773	C₂²	1	1	yes	characteristic
41	Dic₁₃₉⋊C₈	C₂²	1	1	yes	characteristic
42	Dic₁₃₉⋊C₈	C₂²	1	1	yes	characteristic
43	#17984766	C₂²	1	1	yes	characteristic
44	#17984766	C₂²	1	1	yes	characteristic
45	#17984766	C₂²	1	1	yes	characteristic
46	#17984766	C₂²	1	1	yes	characteristic
47	#17984764	C₂²	1	1	yes	characteristic
48	#17984764	C₂²	1	1	yes	characteristic
49	C₂²⋊(C₁₃₉⋊C₈)	C₂²	1	1	yes	characteristic
50	#17984752	C₂²	1	1	yes	characteristic
51	C₈×Dic₁₃₉	C₂²	1	1	yes	characteristic
52	C₂²×Q₈×C₁₃₉	C₂²	1	1	yes	characteristic
53	C₂×Q₁₆×C₁₃₉	C₂²	1	1	yes	characteristic
54	C₂×Q₁₆×C₁₃₉	C₂²	1	1	yes	characteristic
55	C₂²⋊Q₈×C₁₃₉	C₂²	1	1	yes	characteristic
56	#103×C₁₃₉	C₂²	1	1	yes	characteristic
57	#103×C₁₃₉	C₂²	1	1	yes	characteristic
58	C₂²⋊C₈×C₁₃₉	C₂²	1	1	yes	characteristic
59	C₂×Dic₂₇₈	—	2	1	no	—
60	C₂×D₅₅₆	C₂³	1	1	yes	characteristic
61	C₂×C₂²⋊D₁₃₉	—	2	1	no	—
62	C₂×C₄×D₁₃₉	—	2	1	no	—
63	#14555772	—	2	2	no	—
64	#14555772	—	2	2	no	—
65	C₂×C₁₃₉⋊C₈	C₂³	1	1	yes	characteristic
66	C₂×C₁₃₉⋊C₈	C₂³	1	1	yes	characteristic
67	C₄⋊Dic₁₃₉	—	2	1	no	—
68	C₄⋊Dic₁₃₉	—	2	1	no	—
69	C₄⋊Dic₁₃₉	—	2	1	no	—
70	C₄⋊Dic₁₃₉	—	2	1	no	—
71	C₄⋊Dic₁₃₉	—	2	1	no	—
72	C₄⋊Dic₁₃₉	D₄	1	1	yes	characteristic
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	C₄⋊Dic₁₃₉	C₂³	1	1	yes	characteristic
77	Dic₁₃₉⋊C₄	—	2	1	no	—
78	C₂²⋊Dic₁₃₉	—	2	1	no	—
79	C₂²⋊Dic₁₃₉	D₄	1	1	yes	characteristic
80	C₂²⋊Dic₁₃₉	D₄	1	1	yes	characteristic
81	(C₂×D₁₃₉)⋊C₄	—	2	1	no	—
82	(C₂×D₁₃₉)⋊C₄	—	2	1	no	—
83	(C₂×D₁₃₉)⋊C₄	—	2	1	no	—
84	(C₂×D₁₃₉)⋊C₄	—	2	1	no	—
85	C₄×Dic₁₃₉	—	2	1	no	—
86	C₄×Dic₁₃₉	—	2	1	no	—
87	C₄×Dic₁₃₉	—	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	—	2	2	no	—
94	C₂×Q₈×C₁₃₉	—	2	2	no	—
95	C₂×Q₈×C₁₃₉	—	2	1	no	—
96	C₂×Q₈×C₁₃₉	C₂³	1	1	yes	characteristic
97	C₂×Q₈×C₁₃₉	C₂³	1	1	yes	characteristic
98	C₂×Q₈×C₁₃₉	C₂³	1	1	yes	characteristic
99	C₂²×C₄×C₁₃₉	—	2	1	no	—
100	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
101	Q₁₆×C₁₃₉	—	2	2	no	—
102	Q₁₆×C₁₃₉	—	2	2	no	—
103	C₂×C₈×C₁₃₉	C₂³	1	1	yes	characteristic
104	C₂×C₈×C₁₃₉	C₂³	1	1	yes	characteristic
105	C₄⋊C₄×C₁₃₉	—	2	1	no	—
106	C₄⋊C₄×C₁₃₉	C₂³	1	1	yes	characteristic
107	C₂²⋊C₄×C₁₃₉	—	2	1	no	—
108	C₂²×D₁₃₉	—	2	1	no	—
109	Dic₂₇₈	—	8	1	no	—
110	D₅₅₆	—	2	1	no	—
111	D₅₅₆	—	2	1	no	—
112	C₂²⋊D₁₃₉	—	8	1	no	—
113	C₂×Dic₁₃₉	—	2	1	no	—
114	C₂×Dic₁₃₉	—	2	1	no	—
115	C₂×Dic₁₃₉	—	2	1	no	—
116	C₂×Dic₁₃₉	C₂×D₄	1	1	yes	characteristic
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	C₄×D₁₃₉	—	8	1	no	—
121	C₁₃₉⋊C₈	—	2	1	no	—
122	C₁₃₉⋊C₈	—	2	1	no	—
123	C₂³×C₁₃₉	C₂×D₄	1	1	yes	characteristic
124	Q₈×C₁₃₉	—	2	4	no	—
125	Q₈×C₁₃₉	—	2	1	no	—
126	Q₈×C₁₃₉	—	2	1	no	—
127	Q₈×C₁₃₉	—	2	1	no	—
128	Q₈×C₁₃₉	—	2	1	no	—
129	Q₈×C₁₃₉	#41	1	2	yes	—
130	Q₈×C₁₃₉	#41	1	2	yes	—
131	C₂×C₄×C₁₃₉	—	2	4	no	—
132	C₂×C₄×C₁₃₉	—	2	1	no	—
133	C₂×C₄×C₁₃₉	—	2	1	no	—
134	C₂×C₄×C₁₃₉	—	2	1	no	—
135	C₂×C₄×C₁₃₉	—	2	1	no	—
136	C₂×C₄×C₁₃₉	—	2	1	no	—
137	C₂×C₄×C₁₃₉	—	2	1	no	—
138	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
139	C₂×C₄×C₁₃₉	C₂⁴	1	1	yes	characteristic
140	C₈×C₁₃₉	—	2	1	no	—
141	C₈×C₁₃₉	—	2	1	no	—
142	C₂²×C₁₃₉	—	2	1	no	—
143	C₂²×C₁₃₉	—	2	1	no	—
144	C₂²×C₁₃₉	—	2	1	no	—
145	C₂²×C₁₃₉	C₂²×D₄	1	1	yes	characteristic
146	C₂×D₁₃₉	—	4	1	no	—
147	C₂×D₁₃₉	—	4	1	no	—
148	C₂×D₁₃₉	—	4	1	no	—
149	C₄×C₁₃₉	—	2	2	no	—
150	C₄×C₁₃₉	—	2	2	no	—
151	C₄×C₁₃₉	—	2	1	no	—
152	C₄×C₁₃₉	—	4	1	no	—
153	C₄×C₁₃₉	—	4	1	no	—
154	C₄×C₁₃₉	C₂×#41	1	1	yes	characteristic
155	C₄×C₁₃₉	C₂×#41	1	1	yes	characteristic
156	Dic₁₃₉	—	2	1	no	—
157	Dic₁₃₉	—	2	1	no	—
158	Dic₁₃₉	—	2	1	no	—
159	Dic₁₃₉	—	2	1	no	—
160	Dic₁₃₉	—	4	1	no	—
161	Dic₁₃₉	—	4	1	no	—
162	Dic₁₃₉	—	4	1	no	—
163	C₂×C₁₃₉	—	4	1	no	—
164	C₂×C₁₃₉	#525	1	1	yes	characteristic
165	C₂×C₁₃₉	#577	1	1	yes	characteristic
166	C₂×C₁₃₉	#578	1	1	yes	characteristic
167	D₁₃₉	—	8	1	no	—
168	C₁₃₉	#3121	1	1	yes	characteristic, complemented, Sylow
169	#3121	—	139	1	no	maximal, Sylow
170	#542	—	139	1	no	—
171	#519	—	139	1	no	—
172	#485	—	139	1	no	—
173	#482	—	139	1	no	—
174	#477	—	139	1	no	—
175	#475	—	139	1	no	—
176	#464	—	139	1	no	—
177	#464	—	139	1	no	—
178	#451	D₁₃₉	1	1	yes	characteristic, complemented
179	#448	—	139	1	no	—
180	#441	—	139	1	no	—
181	#440	—	139	1	no	—
182	C₄×Q₁₆	—	139	1	no	—
183	C₄×SD₁₆	—	139	1	no	—
184	#433	—	139	1	no	—
185	C₂²×Q₈	C₂×D₁₃₉	1	1	yes	characteristic
186	C₂×Q₁₆	C₂×D₁₃₉	1	1	yes	characteristic
187	C₂×Q₁₆	C₂×D₁₃₉	1	1	yes	characteristic
188	C₂×SD₁₆	—	139	1	no	—
189	C₂×SD₁₆	—	139	1	no	—
190	C₄⋊Q₈	—	139	1	no	—
191	#126	—	278	1	no	—
192	#124	—	278	1	no	—
193	#124	—	139	1	no	—
194	#123	—	278	1	no	—
195	C₂²⋊Q₈	—	278	1	no	—
196	C₂²⋊Q₈	C₂×D₁₃₉	1	1	yes	characteristic
197	#121	—	139	1	no	—
198	C₄×Q₈	—	278	1	no	—
199	C₄×Q₈	—	278	1	no	—
200	C₄×Q₈	—	139	1	no	—
201	C₄×Q₈	—	139	1	no	—
202	C₄×Q₈	—	139	1	no	—
203	C₄×Q₈	—	139	1	no	—
204	C₄×Q₈	—	139	1	no	—
205	C₄×D₄	—	139	1	no	—
206	#117	—	278	1	no	—
207	#117	—	278	1	no	—
208	#117	—	139	1	no	—
209	#117	—	139	1	no	—
210	#107	—	139	1	no	—
211	#107	—	139	1	no	—
212	#106	—	139	1	no	—
213	#106	—	139	1	no	—
214	C₄⋊C₈	—	139	1	no	—
215	C₄⋊C₈	—	139	1	no	—
216	#103	—	139	1	no	—
217	#103	—	139	1	no	—
218	#103	—	139	1	no	—
219	#103	—	139	1	no	—
220	#103	C₂×D₁₃₉	1	1	yes	characteristic
221	#103	C₂×D₁₃₉	1	1	yes	characteristic
222	#102	—	139	1	no	—
223	#102	—	139	1	no	—
224	C₂²⋊C₈	—	139	1	no	—
225	C₂²⋊C₈	C₂×D₁₃₉	1	1	yes	characteristic
226	#97	—	139	1	no	—
227	C₄×C₈	—	139	1	no	—
228	C₂×Q₈	—	2	2	no	—
229	C₂×Q₈	—	2	2	no	—
230	C₂×Q₈	—	2	1	no	—
231	C₂×Q₈	—	278	1	no	—
232	C₂×Q₈	C₂²×D₁₃₉	1	1	yes	characteristic
233	C₂×Q₈	C₂²×D₁₃₉	1	1	yes	characteristic
234	C₂×Q₈	C₂²×D₁₃₉	1	1	yes	characteristic
235	C₂×D₄	—	278	1	no	—
236	C₂×D₄	—	139	1	no	—
237	C₂²×C₄	—	2	1	no	—
238	C₂²×C₄	—	278	1	no	—
239	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
240	Q₁₆	—	2	2	no	—
241	Q₁₆	—	2	2	no	—
242	SD₁₆	—	278	2	no	—
243	SD₁₆	—	278	2	no	—
244	C₂×C₈	—	139	1	no	—
245	C₂×C₈	—	139	1	no	—
246	C₂×C₈	C₂²×D₁₃₉	1	1	yes	characteristic
247	C₂×C₈	C₂²×D₁₃₉	1	1	yes	characteristic
248	C₄⋊C₄	—	2	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	—	139	1	no	—
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	—	139	1	no	—
268	C₂²⋊C₄	—	139	1	no	—
269	C₄²	—	278	1	no	—
270	C₄²	—	278	1	no	—
271	C₄²	—	278	1	no	—
272	C₄²	—	278	1	no	—
273	C₄²	—	278	1	no	—
274	C₄²	—	278	1	no	—
275	C₄²	—	139	1	no	—
276	C₄²	—	139	1	no	—
277	C₂³	—	278	1	no	—
278	C₂³	D₄×D₁₃₉	1	1	yes	characteristic
279	Q₈	—	2	4	no	—
280	Q₈	—	2	1	no	—
281	Q₈	—	2	1	no	—
282	Q₈	—	2	1	no	—
283	Q₈	—	2	1	no	—
284	Q₈	—	1112	1	no	—
285	Q₈	#14555784	1	2	yes	—
286	Q₈	#14555784	1	2	yes	—
287	D₄	—	278	1	no	—
288	D₄	—	278	1	no	—
289	D₄	—	1112	1	no	—
290	C₂×C₄	—	2	4	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	278	1	no	—
298	C₂×C₄	—	278	1	no	—
299	C₂×C₄	—	278	1	no	—
300	C₂×C₄	—	1112	1	no	—
301	C₂×C₄	—	139	1	no	—
302	C₂×C₄	—	139	1	no	—
303	C₂×C₄	—	139	1	no	—
304	C₂×C₄	—	139	1	no	—
305	C₂×C₄	D₄×D₁₃₉	1	1	yes	characteristic
306	C₂×C₄	C₂³×D₁₃₉	1	1	yes	characteristic
307	C₈	—	2	1	no	—
308	C₈	—	2	1	no	—
309	C₈	—	278	1	no	—
310	C₈	—	278	1	no	—
311	C₂²	—	2	1	no	—
312	C₂²	—	2	1	no	—
313	C₂²	—	2	1	no	—
314	C₂²	—	556	1	no	—
315	C₂²	—	556	1	no	—
316	C₂²	—	556	1	no	—
317	C₂²	C₂×D₄×D₁₃₉	1	1	yes	characteristic
318	C₄	—	2	2	no	—
319	C₄	—	2	2	no	—
320	C₄	—	2	1	no	—
321	C₄	—	4	1	no	—
322	C₄	—	4	1	no	—
323	C₄	—	278	1	no	—
324	C₄	—	278	1	no	—
325	C₄	—	278	1	no	—
326	C₄	—	278	1	no	—
327	C₄	—	556	1	no	—
328	C₄	—	556	1	no	—
329	C₄	—	556	1	no	—
330	C₄	C₂×#14555784	1	1	yes	characteristic
331	C₄	C₂×#14555784	1	1	yes	characteristic
332	C₂	—	4	1	no	—
333	C₂	—	1112	1	no	—
334	C₂	#22581459	1	1	yes	characteristic
335	C₂	#22581762	1	1	yes	characteristic
336	C₂	#22581766	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₃₉
commutator, cosocle	C₂×C₄×C₁₃₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₃₉
Fitting	#451×C₁₃₉	C₂
socle	C₂²×C₁₃₉	C₂²×D₄
prime radical	#17984816	C₂²
hypercenter	#451	D₁₃₉
hypocenter	C₁₃₉	#3121
LCS	G ▷ C₂×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #451

Complements

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

N	H ≅ G/N	type
2. #22581431	C₂	semidirect
3. #22581291	C₂	semidirect
5. #22581246	C₂	semidirect
6. #22581236	C₂	semidirect
7. #22581177	C₂	semidirect
8. #22581177	C₂	semidirect
10. #22581060	C₂	semidirect
11. #22581056	C₂	semidirect
12. Q₁₆×Dic₁₃₉	C₂	semidirect
13. #22581038	C₂	semidirect
14. #22581011	C₂	semidirect
15. #451×C₁₃₉	C₂	semidirect
168. C₁₃₉	#3121	semidirect
178. #451	D₁₃₉	semidirect

Maximal subgroups

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

subgroup	index
#22581576	2
#22581431	2
#22581291	2
#22581278	2
#22581246	2
#22581236	2
#22581177	2
#22581092	2
#22581060	2
#22581056	2
Q₁₆×Dic₁₃₉	2
#22581038	2
#22581011	2
#451×C₁₃₉	2
#3121	139