Subgroups — #31816562

Order 17792.

386 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	#22581507	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×#14555784	C₂	1	1	yes	maximal, characteristic
3	#22581296	C₂	1	1	yes	maximal, characteristic, complemented
4	#22581292	C₂	1	1	yes	maximal, characteristic, complemented
5	#22581247	C₂	1	1	yes	maximal, characteristic, complemented
6	#22581241	C₂	1	1	yes	maximal, characteristic, complemented
7	#22581162	C₂	1	1	yes	maximal, characteristic, complemented
8	#22581156	C₂	1	1	yes	maximal, characteristic, complemented
9	#22581116	C₂	1	1	yes	maximal, characteristic
10	#22581112	C₂	1	1	yes	maximal, characteristic
11	#22581061	C₂	1	1	yes	maximal, characteristic, complemented
12	#22581058	C₂	1	1	yes	maximal, characteristic, complemented
13	#22581053	C₂	1	1	yes	maximal, characteristic, complemented
14	#22581008	C₂	1	1	yes	maximal, characteristic, complemented
15	#440×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×#14555775	C₂²	1	1	yes	characteristic
19	C₂×#14555773	C₂²	1	1	yes	characteristic
20	#17984841	C₂²	1	1	yes	characteristic
21	#17984840	C₂²	1	1	yes	characteristic
22	#17984832	—	2	1	no	—
23	#17984822	C₂²	1	1	yes	characteristic
24	#17984820	—	2	1	no	—
25	#17984818	C₂²	1	1	yes	characteristic
26	C₄×Dic₂₇₈	—	2	1	no	—
27	C₄×Dic₂₇₈	C₂²	1	1	yes	characteristic
28	C₄×Dic₂₇₈	C₂²	1	1	yes	characteristic
29	#17984807	C₂²	1	1	yes	characteristic
30	D₄×Dic₁₃₉	C₂²	1	1	yes	characteristic
31	C₄×C₂²⋊D₁₃₉	—	2	1	no	—
32	(C₂²⋊D₁₃₉)⋊C₄	—	2	1	no	—
33	#17984800	C₂²	1	1	yes	characteristic
34	#17984799	—	2	1	no	—
35	#17984798	—	2	1	no	—
36	C₂×C₄×Dic₁₃₉	—	2	1	no	—
37	C₄²×D₁₃₉	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	#17984767	C₂²	1	1	yes	characteristic
44	#17984767	C₂²	1	1	yes	characteristic
45	#17984766	C₂²	1	1	yes	characteristic
46	#17984765	C₂²	1	1	yes	characteristic
47	#17984765	C₂²	1	1	yes	characteristic
48	#17984763	C₂²	1	1	yes	characteristic
49	(C₂×D₁₃₉)⋊C₈	C₂²	1	1	yes	characteristic
50	(C₂×D₁₃₉)⋊C₈	C₂²	1	1	yes	characteristic
51	#17984753	C₂²	1	1	yes	characteristic
52	C₂×SD₁₆×C₁₃₉	C₂²	1	1	yes	characteristic
53	C₄×Q₈×C₁₃₉	C₂²	1	1	yes	characteristic
54	C₄×D₄×C₁₃₉	C₂²	1	1	yes	characteristic
55	#107×C₁₃₉	C₂²	1	1	yes	characteristic
56	#103×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	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	D₄	1	1	yes	characteristic
68	C₂×C₄×D₁₃₉	D₄	1	1	yes	characteristic
69	C₂×C₄×D₁₃₉	C₂³	1	1	yes	characteristic
70	Dic₅₅₆	—	2	2	no	—
71	#14555775	—	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	C₂³	1	1	yes	characteristic
77	C₄⋊Dic₁₃₉	C₂³	1	1	yes	characteristic
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	(C₂×D₁₃₉)⋊C₄	—	2	1	no	—
87	(C₂×D₁₃₉)⋊C₄	—	2	1	no	—
88	(C₂×D₁₃₉)⋊C₄	—	2	1	no	—
89	C₄×Dic₁₃₉	—	2	1	no	—
90	C₄×Dic₁₃₉	—	2	1	no	—
91	C₄×Dic₁₃₉	—	2	1	no	—
92	C₄×Dic₁₃₉	—	2	1	no	—
93	C₄×Dic₁₃₉	D₄	1	1	yes	characteristic
94	C₄×Dic₁₃₉	D₄	1	1	yes	characteristic
95	C₄×Dic₁₃₉	C₂³	1	1	yes	characteristic
96	C₂×Q₈×C₁₃₉	C₂³	1	1	yes	characteristic
97	C₂×D₄×C₁₃₉	C₂³	1	1	yes	characteristic
98	C₂²×C₄×C₁₃₉	—	2	1	no	—
99	SD₁₆×C₁₃₉	—	2	2	no	—
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	C₄²×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	—	4	2	no	—
113	Dic₂₇₈	#41	1	2	yes	—
114	C₂²⋊D₁₃₉	—	2	4	no	—
115	C₂×Dic₁₃₉	—	2	4	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	—	2	1	no	—
125	C₄×D₁₃₉	—	2	1	no	—
126	C₄×D₁₃₉	—	2	2	no	—
127	C₄×D₁₃₉	—	4	1	no	—
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	D₄×C₁₃₉	—	2	1	no	—
134	D₄×C₁₃₉	#41	1	2	yes	—
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	—	4	2	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	—	2	2	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	—	2	1	no	—
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	2	no	—
160	Dic₁₃₉	—	2	1	no	—
161	Dic₁₃₉	—	4	1	no	—
162	Dic₁₃₉	—	4	1	no	—
163	Dic₁₃₉	—	4	1	no	—
164	Dic₁₃₉	#137	1	2	yes	—
165	C₂×C₁₃₉	—	4	2	no	—
166	C₂×C₁₃₉	#525	1	1	yes	characteristic
167	C₂×C₁₃₉	C₂×#137	1	1	yes	characteristic
168	C₂×C₁₃₉	#575	1	1	yes	characteristic
169	D₁₃₉	—	4	1	no	—
170	C₁₃₉	#3116	1	1	yes	characteristic, complemented, Sylow
171	#3116	—	139	1	no	maximal, Sylow
172	#533	—	139	1	no	—
173	C₄×#41	—	139	1	no	—
174	#485	—	139	1	no	—
175	#485	—	139	1	no	—
176	#477	—	139	1	no	—
177	#476	—	139	1	no	—
178	#462	—	139	1	no	—
179	#461	—	139	1	no	—
180	#452	—	139	1	no	—
181	#452	—	139	1	no	—
182	#441	—	139	1	no	—
183	#441	—	139	1	no	—
184	#440	—	139	1	no	—
185	#440	D₁₃₉	1	1	yes	characteristic, complemented
186	#432	—	139	1	no	—
187	C₂×#41	—	139	1	no	—
188	C₂×Q₁₆	—	139	1	no	—
189	C₂×Q₁₆	—	139	1	no	—
190	C₂×SD₁₆	—	139	1	no	—
191	C₂×SD₁₆	C₂×D₁₃₉	1	1	yes	characteristic
192	C₄⋊Q₈	—	139	1	no	—
193	C₄⋊Q₈	—	139	1	no	—
194	#125	—	278	1	no	—
195	C₂²⋊Q₈	—	278	1	no	—
196	C₂²⋊Q₈	—	139	1	no	—
197	C₂²⋊Q₈	—	139	1	no	—
198	C₄×Q₈	—	278	1	no	—
199	C₄×Q₈	—	139	1	no	—
200	C₄×Q₈	—	139	1	no	—
201	C₄×Q₈	—	139	1	no	—
202	C₄×Q₈	C₂×D₁₃₉	1	1	yes	characteristic
203	C₄×D₄	—	278	1	no	—
204	C₄×D₄	—	278	1	no	—
205	C₄×D₄	—	139	1	no	—
206	C₄×D₄	C₂×D₁₃₉	1	1	yes	characteristic
207	#117	—	278	1	no	—
208	#117	—	278	1	no	—
209	#117	—	139	1	no	—
210	C₂×C₄²	—	278	1	no	—
211	C₂×C₄²	—	139	1	no	—
212	#107	—	139	1	no	—
213	#107	C₂×D₁₃₉	1	1	yes	characteristic
214	#106	—	139	1	no	—
215	#106	—	139	1	no	—
216	C₄⋊C₈	—	139	1	no	—
217	C₄⋊C₈	—	139	1	no	—
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	C₂×D₁₃₉	1	1	yes	characteristic
224	#102	—	139	1	no	—
225	#102	C₂×D₁₃₉	1	1	yes	characteristic
226	C₂²⋊C₈	—	139	1	no	—
227	C₂²⋊C₈	—	139	1	no	—
228	#97	—	139	1	no	—
229	#97	C₂×D₁₃₉	1	1	yes	characteristic
230	#41	—	278	2	no	—
231	#41	—	278	2	no	—
232	C₂×Q₈	—	278	1	no	—
233	C₂×Q₈	—	139	1	no	—
234	C₂×Q₈	—	139	1	no	—
235	C₂×Q₈	C₂²×D₁₃₉	1	1	yes	characteristic
236	C₂×D₄	—	278	1	no	—
237	C₂×D₄	C₂²×D₁₃₉	1	1	yes	characteristic
238	C₂²×C₄	—	2	1	no	—
239	C₂²×C₄	—	278	1	no	—
240	C₂²×C₄	—	278	1	no	—
241	C₂²×C₄	—	139	1	no	—
242	C₂²×C₄	—	139	1	no	—
243	C₂²×C₄	—	139	1	no	—
244	Q₁₆	—	278	2	no	—
245	Q₁₆	—	278	2	no	—
246	SD₁₆	—	2	2	no	—
247	SD₁₆	—	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	—	139	1	no	—
262	C₄⋊C₄	—	139	1	no	—
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	—	278	1	no	—
270	C₄²	—	2	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	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	—	278	1	no	—
284	Q₈	—	278	1	no	—
285	Q₈	—	278	1	no	—
286	Q₈	—	556	2	no	—
287	Q₈	—	139	2	no	—
288	D₄	—	2	1	no	—
289	D₄	—	278	4	no	—
290	D₄	#14555785	1	2	yes	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	4	2	no	—
295	C₂×C₄	—	278	2	no	—
296	C₂×C₄	—	278	4	no	—
297	C₂×C₄	—	278	1	no	—
298	C₂×C₄	—	278	1	no	—
299	C₂×C₄	—	278	1	no	—
300	C₂×C₄	—	278	1	no	—
301	C₂×C₄	—	278	1	no	—
302	C₂×C₄	—	278	1	no	—
303	C₂×C₄	—	556	1	no	—
304	C₂×C₄	—	556	2	no	—
305	C₂×C₄	—	139	1	no	—
306	C₂×C₄	—	139	1	no	—
307	C₂×C₄	—	139	1	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	—	2	2	no	—
316	C₂²	—	4	1	no	—
317	C₂²	—	4	1	no	—
318	C₂²	—	278	1	no	—
319	C₂²	—	278	1	no	—
320	C₂²	—	278	1	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	2	no	—
329	C₄	—	278	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	C₂×#14555785	1	1	yes	characteristic
336	C₂	—	4	2	no	—
337	C₂	—	556	1	no	—
338	C₂	#22581462	1	1	yes	characteristic
339	C₂	#137×D₁₃₉	1	1	yes	characteristic
340	C₂	#22581749	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₃₉
commutator, cosocle	C₂×C₄×C₁₃₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₃₉
Fitting	#440×C₁₃₉	C₂
socle	C₂²×C₁₃₉	C₂²×D₄
prime radical	C₂×#14555784	C₂²
hypercenter	#440	D₁₃₉
hypocenter	C₁₃₉	#3116
LCS	G ▷ C₂×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #440

Complements

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

N	H ≅ G/N	type
1. #22581507	C₂	semidirect
3. #22581296	C₂	semidirect
4. #22581292	C₂	semidirect
5. #22581247	C₂	semidirect
6. #22581241	C₂	semidirect
7. #22581162	C₂	semidirect
8. #22581156	C₂	semidirect
11. #22581061	C₂	semidirect
12. #22581058	C₂	semidirect
13. #22581053	C₂	semidirect
14. #22581008	C₂	semidirect
15. #440×C₁₃₉	C₂	semidirect
170. C₁₃₉	#3116	semidirect
185. #440	D₁₃₉	semidirect

Maximal subgroups

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

subgroup	index
#22581507	2
C₄×#14555784	2
#22581296	2
#22581292	2
#22581247	2
#22581241	2
#22581162	2
#22581156	2
#22581116	2
#22581112	2
#22581061	2
#22581058	2
#22581053	2
#22581008	2
#440×C₁₃₉	2
#3116	139