Subgroups — #31816623

Order 17792.

372 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	#22581566	C₂	1	1	yes	maximal, characteristic, complemented
2	#22581432	C₂	1	1	yes	maximal, characteristic
3	#22581290	C₂	1	1	yes	maximal, characteristic, complemented
4	#22581285	C₂	1	1	yes	maximal, characteristic, complemented
5	#22581257	C₂	1	1	yes	maximal, characteristic, complemented
6	#22581164	C₂	1	1	yes	maximal, characteristic, complemented
7	#22581158	C₂	1	1	yes	maximal, characteristic, complemented
8	#22581116	C₂	1	1	yes	maximal, characteristic
9	#22581116	C₂	1	1	yes	maximal, characteristic
10	#22581059	C₂	1	1	yes	maximal, characteristic, complemented
11	#22581050	C₂	1	1	yes	maximal, characteristic, complemented
12	#22581047	C₂	1	1	yes	maximal, characteristic, complemented
13	#22581044	C₂	1	1	yes	maximal, characteristic, complemented
14	#22581013	C₂	1	1	yes	maximal, characteristic, complemented
15	#478×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#14555784	C₂²	1	1	yes	characteristic
17	C₂×Dic₅₅₆	C₂²	1	1	yes	characteristic
18	C₂×Dic₅₅₆	C₂²	1	1	yes	characteristic
19	C₂×#14555773	C₂²	1	1	yes	characteristic
20	C₂×#14555773	C₂²	1	1	yes	characteristic
21	#17984842	C₂²	1	1	yes	characteristic
22	#17984837	—	2	1	no	—
23	#17984837	—	2	1	no	—
24	#17984833	—	2	1	no	—
25	#17984833	—	2	1	no	—
26	#17984822	C₂²	1	1	yes	characteristic
27	#17984822	C₂²	1	1	yes	characteristic
28	C₄×Dic₂₇₈	C₂²	1	1	yes	characteristic
29	#17984807	C₂²	1	1	yes	characteristic
30	#17984807	C₂²	1	1	yes	characteristic
31	#17984807	C₂²	1	1	yes	characteristic
32	D₄×Dic₁₃₉	C₂²	1	1	yes	characteristic
33	C₄×C₂²⋊D₁₃₉	—	2	1	no	—
34	(C₂²⋊D₁₃₉)⋊C₄	—	2	1	no	—
35	#17984799	C₂²	1	1	yes	characteristic
36	#17984798	—	2	1	no	—
37	C₂×Dic₁₃₉⋊C₄	—	2	1	no	—
38	C₄⋊C₄×D₁₃₉	C₂²	1	1	yes	characteristic
39	#17984775	C₂²	1	1	yes	characteristic
40	#17984773	C₂²	1	1	yes	characteristic
41	C₄⋊(C₁₃₉⋊C₈)	C₂²	1	1	yes	characteristic
42	#17984767	C₂²	1	1	yes	characteristic
43	#17984767	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	#17984765	C₂²	1	1	yes	characteristic
48	(C₂×D₁₃₉)⋊C₈	C₂²	1	1	yes	characteristic
49	(C₂×D₁₃₉)⋊C₈	C₂²	1	1	yes	characteristic
50	#17984752	C₂²	1	1	yes	characteristic
51	C₈×Dic₁₃₉	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	#14555784	—	2	2	no	—
60	#14555784	—	2	2	no	—
61	C₂×Dic₂₇₈	C₂³	1	1	yes	characteristic
62	C₂×Dic₂₇₈	C₂³	1	1	yes	characteristic
63	C₂×Dic₂₇₈	C₂³	1	1	yes	characteristic
64	C₂×C₂²⋊D₁₃₉	—	2	1	no	—
65	C₂²×Dic₁₃₉	—	2	1	no	—
66	C₂²×Dic₁₃₉	—	2	1	no	—
67	C₂×C₄×D₁₃₉	—	2	1	no	—
68	C₂×C₄×D₁₃₉	C₂³	1	1	yes	characteristic
69	Dic₅₅₆	—	2	2	no	—
70	Dic₅₅₆	—	2	2	no	—
71	#14555773	—	2	2	no	—
72	#14555773	—	2	2	no	—
73	C₂×C₁₃₉⋊C₈	C₂³	1	1	yes	characteristic
74	C₂×C₁₃₉⋊C₈	C₂³	1	1	yes	characteristic
75	C₄⋊Dic₁₃₉	—	2	1	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	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	Dic₁₃₉⋊C₄	D₄	1	1	yes	characteristic
86	Dic₁₃₉⋊C₄	D₄	1	1	yes	characteristic
87	C₂²⋊Dic₁₃₉	—	2	1	no	—
88	(C₂×D₁₃₉)⋊C₄	—	2	1	no	—
89	(C₂×D₁₃₉)⋊C₄	—	2	1	no	—
90	(C₂×D₁₃₉)⋊C₄	D₄	1	1	yes	characteristic
91	(C₂×D₁₃₉)⋊C₄	D₄	1	1	yes	characteristic
92	C₄×Dic₁₃₉	—	2	1	no	—
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	#41	1	2	yes	—
115	C₂²⋊D₁₃₉	—	2	4	no	—
116	C₂×Dic₁₃₉	—	2	4	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	—	8	1	no	—
122	C₂×Dic₁₃₉	C₂×D₄	1	1	yes	characteristic
123	C₂×Dic₁₃₉	C₂×D₄	1	1	yes	characteristic
124	C₂×Dic₁₃₉	C₂×D₄	1	1	yes	characteristic
125	C₄×D₁₃₉	—	2	1	no	—
126	C₄×D₁₃₉	—	2	1	no	—
127	C₄×D₁₃₉	—	8	1	no	—
128	C₁₃₉⋊C₈	—	2	1	no	—
129	C₁₃₉⋊C₈	—	2	1	no	—
130	C₂³×C₁₃₉	—	2	1	no	—
131	D₄×C₁₃₉	—	2	1	no	—
132	D₄×C₁₃₉	#41	1	2	yes	—
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	—	8	1	no	—
137	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
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	2	no	—
143	C₂²×C₁₃₉	—	4	1	no	—
144	C₂²×C₁₃₉	—	4	1	no	—
145	C₂²×C₁₃₉	C₂²×D₄	1	1	yes	characteristic
146	C₂×D₁₃₉	—	2	1	no	—
147	C₂×D₁₃₉	—	2	1	no	—
148	C₂×D₁₃₉	—	2	1	no	—
149	C₄×C₁₃₉	—	2	1	no	—
150	C₄×C₁₃₉	—	2	1	no	—
151	C₄×C₁₃₉	—	4	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	2	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	2	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	—	4	1	no	—
168	C₁₃₉	#3120	1	1	yes	characteristic, complemented, Sylow
169	#3120	—	139	1	no	maximal, Sylow
170	#541	—	139	1	no	—
171	#520	—	139	1	no	—
172	#484	—	139	1	no	—
173	#483	—	139	1	no	—
174	#479	—	139	1	no	—
175	#478	D₁₃₉	1	1	yes	characteristic, complemented
176	#462	—	139	1	no	—
177	#461	—	139	1	no	—
178	#452	—	139	1	no	—
179	#452	—	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	C₂×#41	—	139	1	no	—
186	C₂×Q₁₆	—	139	1	no	—
187	C₂×Q₁₆	—	139	1	no	—
188	C₂×SD₁₆	—	139	1	no	—
189	C₂×SD₁₆	—	139	1	no	—
190	C₄⋊Q₈	—	139	1	no	—
191	#126	—	278	1	no	—
192	#126	—	278	1	no	—
193	#125	—	278	1	no	—
194	#125	—	278	1	no	—
195	#125	C₂×D₁₃₉	1	1	yes	characteristic
196	C₂²⋊Q₈	—	139	1	no	—
197	C₂²⋊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	—	139	1	no	—
202	C₄×D₄	—	278	1	no	—
203	C₄×D₄	—	278	1	no	—
204	C₄×D₄	—	139	1	no	—
205	C₄×D₄	C₂×D₁₃₉	1	1	yes	characteristic
206	#117	—	278	1	no	—
207	#117	—	139	1	no	—
208	C₂×C₄⋊C₄	—	278	1	no	—
209	C₂×C₄⋊C₄	—	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	C₂×D₁₃₉	1	1	yes	characteristic
216	#103	—	139	1	no	—
217	#103	—	139	1	no	—
218	#103	—	139	1	no	—
219	#103	—	139	1	no	—
220	#103	—	139	1	no	—
221	#103	—	139	1	no	—
222	#102	C₂×D₁₃₉	1	1	yes	characteristic
223	#102	C₂×D₁₃₉	1	1	yes	characteristic
224	C₂²⋊C₈	—	139	1	no	—
225	C₂²⋊C₈	—	139	1	no	—
226	#97	—	139	1	no	—
227	C₄×C₈	—	139	1	no	—
228	#41	—	278	2	no	—
229	#41	—	278	2	no	—
230	C₂×Q₈	—	139	1	no	—
231	C₂×Q₈	—	139	1	no	—
232	C₂×Q₈	—	139	1	no	—
233	C₂×D₄	—	278	1	no	—
234	C₂×D₄	C₂²×D₁₃₉	1	1	yes	characteristic
235	C₂²×C₄	—	2	1	no	—
236	C₂²×C₄	—	278	1	no	—
237	C₂²×C₄	—	278	1	no	—
238	C₂²×C₄	—	278	1	no	—
239	C₂²×C₄	—	139	1	no	—
240	Q₁₆	—	278	2	no	—
241	Q₁₆	—	278	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	—	2	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	—	278	1	no	—
256	C₄⋊C₄	—	278	1	no	—
257	C₄⋊C₄	—	278	1	no	—
258	C₄⋊C₄	—	278	1	no	—
259	C₄⋊C₄	—	139	1	no	—
260	C₄⋊C₄	—	139	1	no	—
261	C₄⋊C₄	—	139	1	no	—
262	C₄⋊C₄	C₂²×D₁₃₉	1	1	yes	characteristic
263	C₄⋊C₄	C₂²×D₁₃₉	1	1	yes	characteristic
264	C₄⋊C₄	C₂²×D₁₃₉	1	1	yes	characteristic
265	C₂²⋊C₄	—	2	1	no	—
266	C₂²⋊C₄	—	278	1	no	—
267	C₂²⋊C₄	—	278	1	no	—
268	C₂²⋊C₄	—	278	1	no	—
269	C₂²⋊C₄	—	139	1	no	—
270	C₂²⋊C₄	—	139	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	C₂²×D₁₃₉	1	1	yes	characteristic
277	C₂³	—	2	1	no	—
278	C₂³	—	139	1	no	—
279	Q₈	—	278	1	no	—
280	Q₈	—	278	1	no	—
281	Q₈	—	278	1	no	—
282	Q₈	—	278	1	no	—
283	Q₈	—	278	1	no	—
284	Q₈	—	139	2	no	—
285	D₄	—	2	1	no	—
286	D₄	—	278	4	no	—
287	D₄	#14555786	1	2	yes	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	8	1	no	—
292	C₂×C₄	—	278	4	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	—	278	1	no	—
298	C₂×C₄	—	278	1	no	—
299	C₂×C₄	—	1112	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	D₄×D₁₃₉	1	1	yes	characteristic
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	2	no	—
312	C₂²	—	4	1	no	—
313	C₂²	—	4	1	no	—
314	C₂²	—	278	1	no	—
315	C₂²	—	278	1	no	—
316	C₂²	—	278	1	no	—
317	C₂²	C₂×D₄×D₁₃₉	1	1	yes	characteristic
318	C₄	—	2	1	no	—
319	C₄	—	2	1	no	—
320	C₄	—	4	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	2	no	—
327	C₄	—	556	1	no	—
328	C₄	—	556	1	no	—
329	C₄	—	556	1	no	—
330	C₄	#41×D₁₃₉	1	1	yes	characteristic
331	C₄	C₂×#14555786	1	1	yes	characteristic
332	C₂	—	4	2	no	—
333	C₂	—	556	1	no	—
334	C₂	#22581466	1	1	yes	characteristic
335	C₂	#22581763	1	1	yes	characteristic
336	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	C₂×#14555784	C₂²
hypercenter	#478	D₁₃₉
hypocenter	C₁₃₉	#3120
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
1. #22581566	C₂	semidirect
3. #22581290	C₂	semidirect
4. #22581285	C₂	semidirect
5. #22581257	C₂	semidirect
6. #22581164	C₂	semidirect
7. #22581158	C₂	semidirect
10. #22581059	C₂	semidirect
11. #22581050	C₂	semidirect
12. #22581047	C₂	semidirect
13. #22581044	C₂	semidirect
14. #22581013	C₂	semidirect
15. #478×C₁₃₉	C₂	semidirect
168. C₁₃₉	#3120	semidirect
175. #478	D₁₃₉	semidirect

Maximal subgroups

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

subgroup	index
#22581566	2
#22581432	2
#22581290	2
#22581285	2
#22581257	2
#22581164	2
#22581158	2
#22581116	2
#22581059	2
#22581050	2
#22581047	2
#22581044	2
#22581013	2
#478×C₁₃₉	2
#3120	139