Subgroups — #31816766

Order 17792.

372 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22581580	C₂	1	1	yes	maximal, characteristic, complemented
2	#22581523	C₂	1	1	yes	maximal, characteristic
3	#22581322	C₂	1	1	yes	maximal, characteristic, complemented
4	#22581320	C₂	1	1	yes	maximal, characteristic, complemented
5	#22581311	C₂	1	1	yes	maximal, characteristic, complemented
6	#22581307	C₂	1	1	yes	maximal, characteristic, complemented
7	#22581306	C₂	1	1	yes	maximal, characteristic, complemented
8	#22581305	C₂	1	1	yes	maximal, characteristic, complemented
9	#22581302	C₂	1	1	yes	maximal, characteristic, complemented
10	#22580960	C₂	1	1	yes	maximal, characteristic, complemented
11	#22580951	C₂	1	1	yes	maximal, characteristic
12	#22580946	C₂	1	1	yes	maximal, characteristic
13	#103×D₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
14	#22580910	C₂	1	1	yes	maximal, characteristic, complemented
15	#489×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#14555784	C₂²	1	1	yes	characteristic
17	C₂×Q₈×D₁₃₉	C₂²	1	1	yes	characteristic
18	C₂×#14555768	C₂²	1	1	yes	characteristic
19	C₂×C₈×D₁₃₉	C₂²	1	1	yes	characteristic
20	#17984842	C₂²	1	1	yes	characteristic
21	#17984841	C₂²	1	1	yes	characteristic
22	#17984840	—	2	1	no	—
23	#17984840	C₂²	1	1	yes	characteristic
24	#17984834	C₂²	1	1	yes	characteristic
25	#17984833	C₂²	1	1	yes	characteristic
26	#17984832	—	2	1	no	—
27	#17984829	C₂²	1	1	yes	characteristic
28	#17984824	—	2	1	no	—
29	#17984824	—	2	1	no	—
30	#17984820	—	2	1	no	—
31	#17984819	—	2	1	no	—
32	#17984819	—	2	1	no	—
33	#17984818	—	2	1	no	—
34	#17984800	C₂²	1	1	yes	characteristic
35	#17984799	C₂²	1	1	yes	characteristic
36	C₄⋊C₄×D₁₃₉	C₂²	1	1	yes	characteristic
37	#17984767	C₂²	1	1	yes	characteristic
38	#17984767	C₂²	1	1	yes	characteristic
39	#17984767	C₂²	1	1	yes	characteristic
40	#17984767	C₂²	1	1	yes	characteristic
41	#17984766	C₂²	1	1	yes	characteristic
42	#17984766	C₂²	1	1	yes	characteristic
43	#17984765	C₂²	1	1	yes	characteristic
44	#17984765	C₂²	1	1	yes	characteristic
45	#17984765	C₂²	1	1	yes	characteristic
46	#17984765	C₂²	1	1	yes	characteristic
47	#17984763	C₂²	1	1	yes	characteristic
48	#17984763	C₂²	1	1	yes	characteristic
49	#17984752	C₂²	1	1	yes	characteristic
50	C₄×C₁₃₉⋊C₈	C₂²	1	1	yes	characteristic
51	C₈×Dic₁₃₉	C₂²	1	1	yes	characteristic
52	C₄⋊Q₈×C₁₃₉	C₂²	1	1	yes	characteristic
53	#124×C₁₃₉	C₂²	1	1	yes	characteristic
54	#103×C₁₃₉	C₂²	1	1	yes	characteristic
55	#103×C₁₃₉	C₂²	1	1	yes	characteristic
56	#102×C₁₃₉	C₂²	1	1	yes	characteristic
57	#102×C₁₃₉	C₂²	1	1	yes	characteristic
58	#97×C₁₃₉	C₂²	1	1	yes	characteristic
59	#14555784	—	2	2	no	—
60	#14555784	—	2	2	no	—
61	C₂×Dic₂₇₈	—	2	1	no	—
62	C₂×Dic₂₇₈	—	2	1	no	—
63	C₂×Dic₂₇₈	C₂³	1	1	yes	characteristic
64	C₂×Dic₂₇₈	C₂³	1	1	yes	characteristic
65	Q₈×D₁₃₉	—	2	2	no	—
66	Q₈×D₁₃₉	—	2	2	no	—
67	C₂×C₂²⋊D₁₃₉	—	2	1	no	—
68	C₂²×Dic₁₃₉	—	2	1	no	—
69	C₂×C₄×D₁₃₉	—	2	1	no	—
70	C₂×C₄×D₁₃₉	—	2	1	no	—
71	C₂×C₄×D₁₃₉	C₂³	1	1	yes	characteristic
72	#14555768	—	2	2	no	—
73	C₂×C₁₃₉⋊C₈	C₂³	1	1	yes	characteristic
74	C₂×C₁₃₉⋊C₈	C₂³	1	1	yes	characteristic
75	C₈×D₁₃₉	—	2	2	no	—
76	C₄⋊Dic₁₃₉	—	2	1	no	—
77	C₄⋊Dic₁₃₉	—	2	1	no	—
78	C₄⋊Dic₁₃₉	—	2	1	no	—
79	C₄⋊Dic₁₃₉	C₂³	1	1	yes	characteristic
80	C₄⋊Dic₁₃₉	C₂³	1	1	yes	characteristic
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	—	2	1	no	—
86	Dic₁₃₉⋊C₄	—	2	1	no	—
87	Dic₁₃₉⋊C₄	—	2	1	no	—
88	Dic₁₃₉⋊C₄	D₄	1	1	yes	characteristic
89	Dic₁₃₉⋊C₄	D₄	1	1	yes	characteristic
90	C₂²⋊Dic₁₃₉	—	2	1	no	—
91	C₂²⋊Dic₁₃₉	—	2	1	no	—
92	(C₂×D₁₃₉)⋊C₄	—	2	1	no	—
93	(C₂×D₁₃₉)⋊C₄	D₄	1	1	yes	characteristic
94	(C₂×D₁₃₉)⋊C₄	D₄	1	1	yes	characteristic
95	C₄×Dic₁₃₉	—	2	1	no	—
96	C₄×Dic₁₃₉	C₂³	1	1	yes	characteristic
97	C₂×Q₈×C₁₃₉	—	2	1	no	—
98	C₂×Q₈×C₁₃₉	C₂³	1	1	yes	characteristic
99	C₂×D₄×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	—	2	1	no	—
103	C₄⋊C₄×C₁₃₉	C₂³	1	1	yes	characteristic
104	C₄⋊C₄×C₁₃₉	C₂³	1	1	yes	characteristic
105	C₂²⋊C₄×C₁₃₉	—	2	1	no	—
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	—	4	2	no	—
114	Dic₂₇₈	—	8	1	no	—
115	C₂²⋊D₁₃₉	—	4	2	no	—
116	C₂×Dic₁₃₉	—	2	1	no	—
117	C₂×Dic₁₃₉	—	2	1	no	—
118	C₂×Dic₁₃₉	—	2	1	no	—
119	C₂×Dic₁₃₉	—	2	1	no	—
120	C₂×Dic₁₃₉	—	4	2	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	—	4	2	no	—
125	C₄×D₁₃₉	—	8	1	no	—
126	C₄×D₁₃₉	#41	1	2	yes	—
127	C₄×D₁₃₉	#41	1	2	yes	—
128	C₁₃₉⋊C₈	—	2	1	no	—
129	C₁₃₉⋊C₈	—	2	1	no	—
130	C₂³×C₁₃₉	—	2	1	no	—
131	Q₈×C₁₃₉	—	2	1	no	—
132	Q₈×C₁₃₉	—	2	1	no	—
133	Q₈×C₁₃₉	—	8	1	no	—
134	D₄×C₁₃₉	—	2	1	no	—
135	D₄×C₁₃₉	—	2	1	no	—
136	C₂×C₄×C₁₃₉	—	2	1	no	—
137	C₂×C₄×C₁₃₉	—	2	1	no	—
138	C₂×C₄×C₁₃₉	—	2	1	no	—
139	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
140	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
141	C₂×C₄×C₁₃₉	C₂⁴	1	1	yes	characteristic
142	C₈×C₁₃₉	—	2	1	no	—
143	C₈×C₁₃₉	—	2	1	no	—
144	C₂²×C₁₃₉	—	4	1	no	—
145	C₂²×C₁₃₉	—	4	1	no	—
146	C₂²×C₁₃₉	—	4	1	no	—
147	C₂²×C₁₃₉	C₂²×D₄	1	1	yes	characteristic
148	C₂×D₁₃₉	—	2	1	no	—
149	C₂×D₁₃₉	—	2	1	no	—
150	C₂×D₁₃₉	#124	1	2	yes	—
151	C₄×C₁₃₉	—	2	1	no	—
152	C₄×C₁₃₉	—	2	1	no	—
153	C₄×C₁₃₉	—	4	1	no	—
154	C₄×C₁₃₉	—	4	1	no	—
155	C₄×C₁₃₉	—	4	1	no	—
156	C₄×C₁₃₉	C₂×#41	1	1	yes	characteristic
157	C₄×C₁₃₉	C₂×#41	1	1	yes	characteristic
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	Dic₁₃₉	—	4	1	no	—
164	Dic₁₃₉	#124	1	2	yes	—
165	C₂×C₁₃₉	—	8	1	no	—
166	C₂×C₁₃₉	C₂×#124	1	1	yes	characteristic
167	C₂×C₁₃₉	#577	1	1	yes	characteristic
168	C₂×C₁₃₉	#578	1	1	yes	characteristic
169	D₁₃₉	—	2	2	no	—
170	C₁₃₉	#3137	1	1	yes	characteristic, complemented, Sylow
171	#3137	—	139	1	no	maximal, Sylow
172	#544	—	139	1	no	—
173	#536	—	139	1	no	—
174	#491	—	139	1	no	—
175	#491	—	139	1	no	—
176	#489	—	139	1	no	—
177	#489	D₁₃₉	1	1	yes	characteristic, complemented
178	#488	—	139	1	no	—
179	#488	—	139	1	no	—
180	#487	—	139	1	no	—
181	#487	—	139	1	no	—
182	#419	—	139	1	no	—
183	#417	—	139	1	no	—
184	#416	—	139	1	no	—
185	C₂×#103	—	139	1	no	—
186	#405	—	139	1	no	—
187	C₂×#41	—	139	1	no	—
188	C₂²×Q₈	—	139	1	no	—
189	C₂×M₁₆	—	139	1	no	—
190	C₂²×C₈	—	139	1	no	—
191	C₄⋊Q₈	—	278	1	no	—
192	C₄⋊Q₈	—	139	1	no	—
193	C₄⋊Q₈	—	139	1	no	—
194	C₄⋊Q₈	—	139	1	no	—
195	C₄⋊Q₈	C₂×D₁₃₉	1	1	yes	characteristic
196	#125	—	278	1	no	—
197	#125	—	139	1	no	—
198	#125	—	139	1	no	—
199	#124	—	139	1	no	—
200	#124	C₂×D₁₃₉	1	1	yes	characteristic
201	#123	—	278	1	no	—
202	#123	—	278	1	no	—
203	C₂²⋊Q₈	—	278	1	no	—
204	C₂²⋊Q₈	—	278	1	no	—
205	C₂²⋊Q₈	—	278	1	no	—
206	C₂²⋊Q₈	—	278	1	no	—
207	#117	—	139	1	no	—
208	#117	—	139	1	no	—
209	C₂×C₄⋊C₄	—	139	1	no	—
210	#103	—	139	1	no	—
211	#103	—	139	1	no	—
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	—	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	#102	C₂×D₁₃₉	1	1	yes	characteristic
225	#102	C₂×D₁₃₉	1	1	yes	characteristic
226	#97	—	139	1	no	—
227	#97	C₂×D₁₃₉	1	1	yes	characteristic
228	C₄×C₈	—	139	1	no	—
229	C₄×C₈	—	139	1	no	—
230	#41	—	278	2	no	—
231	#41	—	278	2	no	—
232	C₂×Q₈	—	2	1	no	—
233	C₂×Q₈	—	278	1	no	—
234	C₂×Q₈	—	278	1	no	—
235	C₂×Q₈	—	278	2	no	—
236	C₂×Q₈	—	278	2	no	—
237	C₂×Q₈	—	139	1	no	—
238	C₂×Q₈	—	139	1	no	—
239	C₂×Q₈	C₂²×D₁₃₉	1	1	yes	characteristic
240	C₂×D₄	—	278	1	no	—
241	C₂×D₄	C₂²×D₁₃₉	1	1	yes	characteristic
242	C₂²×C₄	—	278	1	no	—
243	C₂²×C₄	—	278	1	no	—
244	C₂²×C₄	—	278	1	no	—
245	C₂²×C₄	—	139	1	no	—
246	M₁₆	—	278	2	no	—
247	C₂×C₈	—	278	2	no	—
248	C₂×C₈	—	139	1	no	—
249	C₂×C₈	—	139	1	no	—
250	C₂×C₈	C₂²×D₁₃₉	1	1	yes	characteristic
251	C₂×C₈	C₂²×D₁₃₉	1	1	yes	characteristic
252	C₄⋊C₄	—	2	1	no	—
253	C₄⋊C₄	—	278	1	no	—
254	C₄⋊C₄	—	278	1	no	—
255	C₄⋊C₄	—	278	1	no	—
256	C₄⋊C₄	—	278	1	no	—
257	C₄⋊C₄	—	278	1	no	—
258	C₄⋊C₄	—	278	1	no	—
259	C₄⋊C₄	—	278	1	no	—
260	C₄⋊C₄	—	278	1	no	—
261	C₄⋊C₄	—	278	1	no	—
262	C₄⋊C₄	—	278	1	no	—
263	C₄⋊C₄	—	139	1	no	—
264	C₄⋊C₄	—	139	1	no	—
265	C₄⋊C₄	—	139	1	no	—
266	C₄⋊C₄	—	139	1	no	—
267	C₄⋊C₄	C₂²×D₁₃₉	1	1	yes	characteristic
268	C₄⋊C₄	C₂²×D₁₃₉	1	1	yes	characteristic
269	C₂²⋊C₄	—	2	1	no	—
270	C₂²⋊C₄	—	2	1	no	—
271	C₂²⋊C₄	—	278	1	no	—
272	C₂²⋊C₄	—	278	1	no	—
273	C₂²⋊C₄	—	278	1	no	—
274	C₂²⋊C₄	—	139	1	no	—
275	C₂²⋊C₄	—	139	1	no	—
276	C₄²	—	278	1	no	—
277	C₄²	—	139	1	no	—
278	C₄²	C₂²×D₁₃₉	1	1	yes	characteristic
279	C₂³	—	2	1	no	—
280	C₂³	—	139	1	no	—
281	Q₈	—	2	1	no	—
282	Q₈	—	2	1	no	—
283	Q₈	—	8	1	no	—
284	Q₈	—	278	1	no	—
285	Q₈	—	278	1	no	—
286	Q₈	—	278	1	no	—
287	Q₈	—	278	1	no	—
288	Q₈	—	556	2	no	—
289	Q₈	—	1112	1	no	—
290	D₄	—	2	1	no	—
291	D₄	—	2	1	no	—
292	D₄	—	556	2	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	278	1	no	—
297	C₂×C₄	—	278	1	no	—
298	C₂×C₄	—	278	1	no	—
299	C₂×C₄	—	278	1	no	—
300	C₂×C₄	—	556	2	no	—
301	C₂×C₄	—	556	2	no	—
302	C₂×C₄	—	1112	1	no	—
303	C₂×C₄	—	139	1	no	—
304	C₂×C₄	—	139	1	no	—
305	C₂×C₄	—	139	1	no	—
306	C₂×C₄	—	139	2	no	—
307	C₂×C₄	—	139	2	no	—
308	C₂×C₄	D₄×D₁₃₉	1	1	yes	characteristic
309	C₂×C₄	D₄×D₁₃₉	1	1	yes	characteristic
310	C₂×C₄	C₂³×D₁₃₉	1	1	yes	characteristic
311	C₈	—	2	1	no	—
312	C₈	—	2	1	no	—
313	C₈	—	278	1	no	—
314	C₈	—	278	1	no	—
315	C₂²	—	4	1	no	—
316	C₂²	—	4	1	no	—
317	C₂²	—	4	1	no	—
318	C₂²	—	278	1	no	—
319	C₂²	—	278	1	no	—
320	C₂²	—	139	2	no	—
321	C₂²	C₂×D₄×D₁₃₉	1	1	yes	characteristic
322	C₄	—	2	1	no	—
323	C₄	—	2	1	no	—
324	C₄	—	4	1	no	—
325	C₄	—	4	1	no	—
326	C₄	—	4	1	no	—
327	C₄	—	278	1	no	—
328	C₄	—	278	1	no	—
329	C₄	—	556	1	no	—
330	C₄	—	556	1	no	—
331	C₄	—	556	1	no	—
332	C₄	—	556	1	no	—
333	C₄	—	139	2	no	—
334	C₄	#41×D₁₃₉	1	1	yes	characteristic
335	C₄	#41×D₁₃₉	1	1	yes	characteristic
336	C₂	—	8	1	no	—
337	C₂	—	278	2	no	—
338	C₂	#124×D₁₃₉	1	1	yes	characteristic
339	C₂	#22581763	1	1	yes	characteristic
340	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	#489×C₁₃₉	C₂
socle	C₂²×C₁₃₉	C₂²×D₄
prime radical	C₂×#14555784	C₂²
hypercenter	#489	D₁₃₉
hypocenter	C₁₃₉	#3137
LCS	G ▷ C₂×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #489

Complements

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

N	H ≅ G/N	type
1. #22581580	C₂	semidirect
3. #22581322	C₂	semidirect
4. #22581320	C₂	semidirect
5. #22581311	C₂	semidirect
6. #22581307	C₂	semidirect
7. #22581306	C₂	semidirect
8. #22581305	C₂	semidirect
9. #22581302	C₂	semidirect
10. #22580960	C₂	semidirect
13. #103×D₁₃₉	C₂	semidirect
14. #22580910	C₂	semidirect
15. #489×C₁₃₉	C₂	semidirect
170. C₁₃₉	#3137	semidirect
177. #489	D₁₃₉	semidirect

Maximal subgroups

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

subgroup	index
#22581580	2
#22581523	2
#22581322	2
#22581320	2
#22581311	2
#22581307	2
#22581306	2
#22581305	2
#22581302	2
#22580960	2
#22580951	2
#22580946	2
#103×D₁₃₉	2
#22580910	2
#489×C₁₃₉	2
#3137	139