Subgroups — #31818981

Order 17792.

374 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	#22581674	C₂	1	1	yes	maximal, characteristic, complemented
2	#22581604	C₂	1	1	yes	maximal, characteristic
3	#22581357	C₂	1	1	yes	maximal, characteristic, complemented
4	#22581287	C₂	1	1	yes	maximal, characteristic, complemented
5	#22581280	C₂	1	1	yes	maximal, characteristic, complemented
6	#22581278	C₂	1	1	yes	maximal, characteristic, complemented
7	#22581236	C₂	1	1	yes	maximal, characteristic, complemented
8	#22581232	C₂	1	1	yes	maximal, characteristic, complemented
9	#22581202	C₂	1	1	yes	maximal, characteristic, complemented
10	#22581201	C₂	1	1	yes	maximal, characteristic, complemented
11	#22581038	C₂	1	1	yes	maximal, characteristic, complemented
12	#22581028	C₂	1	1	yes	maximal, characteristic, complemented
13	#22580994	C₂	1	1	yes	maximal, characteristic, complemented
14	#22580993	C₂	1	1	yes	maximal, characteristic, complemented
15	#484×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#14555785	—	2	1	no	—
17	C₂×#14555772	C₂²	1	1	yes	characteristic
18	C₂×#14555769	C₂²	1	1	yes	characteristic
19	C₂×#14555768	C₂²	1	1	yes	characteristic
20	#17984841	C₂²	1	1	yes	characteristic
21	#17984841	C₂²	1	1	yes	characteristic
22	#17984837	—	2	1	no	—
23	#17984835	—	2	1	no	—
24	#17984828	—	2	1	no	—
25	#17984827	—	2	1	no	—
26	#17984825	—	2	1	no	—
27	#17984820	—	2	1	no	—
28	#17984818	C₂²	1	1	yes	characteristic
29	#17984817	C₂²	1	1	yes	characteristic
30	#17984816	C₂²	1	1	yes	characteristic
31	Q₈×Dic₁₃₉	C₂²	1	1	yes	characteristic
32	#17984807	—	2	1	no	—
33	#17984804	C₂²	1	1	yes	characteristic
34	C₄×C₂²⋊D₁₃₉	C₂²	1	1	yes	characteristic
35	#17984801	C₂²	1	1	yes	characteristic
36	#17984800	C₂²	1	1	yes	characteristic
37	C₄⋊C₄×D₁₃₉	C₂²	1	1	yes	characteristic
38	#17984776	C₂²	1	1	yes	characteristic
39	#17984775	C₂²	1	1	yes	characteristic
40	#17984775	C₂²	1	1	yes	characteristic
41	#17984775	C₂²	1	1	yes	characteristic
42	#17984774	C₂²	1	1	yes	characteristic
43	#17984773	C₂²	1	1	yes	characteristic
44	#17984773	C₂²	1	1	yes	characteristic
45	Dic₁₃₉⋊C₈	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	#17984762	C₂²	1	1	yes	characteristic
50	(C₂×D₁₃₉)⋊C₈	C₂²	1	1	yes	characteristic
51	C₈×Dic₁₃₉	C₂²	1	1	yes	characteristic
52	C₂²⋊Q₈×C₁₃₉	C₂²	1	1	yes	characteristic
53	C₂×C₄⋊C₄×C₁₃₉	C₂²	1	1	yes	characteristic
54	#107×C₁₃₉	C₂²	1	1	yes	characteristic
55	#107×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	#14555785	—	4	4	no	—
60	C₂×Dic₂₇₈	—	2	1	no	—
61	C₂×Dic₂₇₈	—	2	1	no	—
62	C₂×D₅₅₆	—	2	1	no	—
63	C₂×D₅₅₆	C₂³	1	1	yes	characteristic
64	C₂×C₂²⋊D₁₃₉	—	2	1	no	—
65	C₂×C₂²⋊D₁₃₉	D₄	1	1	yes	characteristic
66	C₂×C₄×D₁₃₉	—	2	1	no	—
67	C₂×C₄×D₁₃₉	—	2	1	no	—
68	C₂×C₄×D₁₃₉	C₂³	1	1	yes	characteristic
69	#14555772	—	2	2	no	—
70	#14555769	—	2	2	no	—
71	#14555768	—	2	2	no	—
72	C₂×C₁₃₉⋊C₈	C₂³	1	1	yes	characteristic
73	C₂×C₁₃₉⋊C₈	C₂³	1	1	yes	characteristic
74	C₄⋊Dic₁₃₉	—	2	1	no	—
75	C₄⋊Dic₁₃₉	C₂³	1	1	yes	characteristic
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	D₄	1	1	yes	characteristic
83	C₂²⋊Dic₁₃₉	—	2	1	no	—
84	C₂²⋊Dic₁₃₉	D₄	1	1	yes	characteristic
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	(C₂×D₁₃₉)⋊C₄	D₄	1	1	yes	characteristic
91	C₄×Dic₁₃₉	—	2	1	no	—
92	C₄×Dic₁₃₉	—	2	1	no	—
93	C₄×Dic₁₃₉	—	2	1	no	—
94	C₄×Dic₁₃₉	C₂³	1	1	yes	characteristic
95	C₂×Q₈×C₁₃₉	C₂³	1	1	yes	characteristic
96	C₂²×C₄×C₁₃₉	—	2	1	no	—
97	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
98	C₂×C₈×C₁₃₉	C₂³	1	1	yes	characteristic
99	C₂×C₈×C₁₃₉	C₂³	1	1	yes	characteristic
100	C₄⋊C₄×C₁₃₉	—	2	1	no	—
101	C₄⋊C₄×C₁₃₉	—	2	1	no	—
102	C₄⋊C₄×C₁₃₉	C₂³	1	1	yes	characteristic
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×D₁₃₉	—	2	1	no	—
107	C₂²×D₁₃₉	C₂×D₄	1	1	yes	characteristic
108	Dic₂₇₈	—	4	2	no	—
109	Dic₂₇₈	—	4	2	no	—
110	D₅₅₆	—	2	1	no	—
111	D₅₅₆	—	2	1	no	—
112	D₅₅₆	—	4	2	no	—
113	C₂²⋊D₁₃₉	—	2	2	no	—
114	C₂²⋊D₁₃₉	—	4	2	no	—
115	C₂×Dic₁₃₉	—	2	1	no	—
116	C₂×Dic₁₃₉	—	2	1	no	—
117	C₂×Dic₁₃₉	—	2	1	no	—
118	C₂×Dic₁₃₉	C₂×D₄	1	1	yes	characteristic
119	C₂×Dic₁₃₉	C₂×D₄	1	1	yes	characteristic
120	C₂×Dic₁₃₉	C₂×D₄	1	1	yes	characteristic
121	C₄×D₁₃₉	—	2	1	no	—
122	C₄×D₁₃₉	—	2	1	no	—
123	C₄×D₁₃₉	—	4	2	no	—
124	C₄×D₁₃₉	—	4	2	no	—
125	C₁₃₉⋊C₈	—	2	1	no	—
126	C₁₃₉⋊C₈	#41	1	2	yes	complemented
127	C₂³×C₁₃₉	C₂×D₄	1	1	yes	characteristic
128	Q₈×C₁₃₉	—	2	1	no	—
129	Q₈×C₁₃₉	—	2	1	no	—
130	C₂×C₄×C₁₃₉	—	2	1	no	—
131	C₂×C₄×C₁₃₉	—	2	1	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	—	4	2	no	—
137	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
138	C₂×C₄×C₁₃₉	C₂⁴	1	1	yes	characteristic
139	C₈×C₁₃₉	—	2	1	no	—
140	C₈×C₁₃₉	—	2	1	no	—
141	C₂²×C₁₃₉	—	2	1	no	—
142	C₂²×C₁₃₉	—	2	1	no	—
143	C₂²×C₁₃₉	—	2	1	no	—
144	C₂²×C₁₃₉	C₂²×D₄	1	1	yes	characteristic
145	C₂×D₁₃₉	—	2	1	no	—
146	C₂×D₁₃₉	—	2	1	no	—
147	C₂×D₁₃₉	—	2	1	no	—
148	C₂×D₁₃₉	—	4	1	no	—
149	C₂×D₁₃₉	—	4	1	no	—
150	C₂×D₁₃₉	—	4	1	no	—
151	C₄×C₁₃₉	—	2	1	no	—
152	C₄×C₁₃₉	—	4	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	2₋¹⁺⁴	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	AGL(1,ℤ/8)	1	2	yes	—
164	C₂×C₁₃₉	—	4	1	no	—
165	C₂×C₁₃₉	#547	1	1	yes	characteristic
166	C₂×C₁₃₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
167	C₂×C₁₃₉	#578	1	1	yes	characteristic
168	D₁₃₉	—	4	1	no	—
169	D₁₃₉	—	8	1	no	—
170	C₁₃₉	#3353	1	1	yes	characteristic, complemented, Sylow
171	#3353	—	139	1	no	maximal, Sylow
172	#556	—	139	1	no	—
173	#547	—	139	1	no	—
174	#500	—	139	1	no	—
175	#484	—	139	1	no	—
176	#484	D₁₃₉	1	1	yes	characteristic, complemented
177	#482	—	139	1	no	—
178	#482	—	139	1	no	—
179	#475	—	139	1	no	—
180	#474	—	139	1	no	—
181	#468	—	139	1	no	—
182	#468	—	139	1	no	—
183	C₄×SD₁₆	—	139	1	no	—
184	#436	—	139	1	no	—
185	#428	—	139	1	no	—
186	#428	—	139	1	no	—
187	C₂×#41	—	278	1	no	—
188	C₂×SD₁₆	—	139	1	no	—
189	C₂×M₁₆	—	139	1	no	—
190	C₂×M₁₆	—	139	1	no	—
191	C₄⋊Q₈	—	139	1	no	—
192	C₄⋊Q₈	—	139	1	no	—
193	#126	—	278	1	no	—
194	#126	—	278	1	no	—
195	#124	—	278	1	no	—
196	#123	—	278	1	no	—
197	#123	—	278	1	no	—
198	C₂²⋊Q₈	—	278	1	no	—
199	C₂²⋊Q₈	—	139	1	no	—
200	C₂²⋊Q₈	C₂×D₁₃₉	1	1	yes	characteristic
201	#121	—	139	1	no	—
202	#121	—	139	1	no	—
203	C₄×Q₈	—	278	1	no	—
204	C₄×Q₈	—	139	1	no	—
205	C₄×D₄	—	139	1	no	—
206	C₄×D₄	—	139	1	no	—
207	#117	—	139	1	no	—
208	#117	—	139	1	no	—
209	C₂×C₄⋊C₄	—	139	1	no	—
210	C₂×C₄⋊C₄	C₂×D₁₃₉	1	1	yes	characteristic
211	#107	—	139	1	no	—
212	#107	—	139	1	no	—
213	#107	—	139	1	no	—
214	#107	—	139	1	no	—
215	#107	C₂×D₁₃₉	1	1	yes	characteristic
216	#107	C₂×D₁₃₉	1	1	yes	characteristic
217	#106	—	139	1	no	—
218	#106	—	139	1	no	—
219	#106	—	139	1	no	—
220	C₄⋊C₈	—	139	1	no	—
221	#103	—	139	1	no	—
222	#103	C₂×D₁₃₉	1	1	yes	characteristic
223	#103	C₂×D₁₃₉	1	1	yes	characteristic
224	#102	—	139	1	no	—
225	#102	—	139	1	no	—
226	#102	—	139	1	no	—
227	C₂²⋊C₈	—	139	1	no	—
228	C₂²⋊C₈	C₂×D₁₃₉	1	1	yes	characteristic
229	C₄×C₈	—	139	1	no	—
230	#41	—	556	4	no	—
231	C₂×Q₈	—	278	1	no	—
232	C₂×Q₈	—	278	1	no	—
233	C₂×Q₈	C₂²×D₁₃₉	1	1	yes	characteristic
234	C₂×D₄	—	278	1	no	—
235	C₂×D₄	—	278	1	no	—
236	C₂×D₄	—	139	1	no	—
237	C₂×D₄	—	139	1	no	—
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	C₂²×D₁₃₉	1	1	yes	characteristic
243	SD₁₆	—	278	2	no	—
244	M₁₆	—	278	2	no	—
245	M₁₆	—	278	2	no	—
246	C₂×C₈	—	139	1	no	—
247	C₂×C₈	—	139	1	no	—
248	C₂×C₈	C₂²×D₁₃₉	1	1	yes	characteristic
249	C₂×C₈	C₂²×D₁₃₉	1	1	yes	characteristic
250	C₄⋊C₄	—	2	1	no	—
251	C₄⋊C₄	—	2	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	—	139	1	no	—
258	C₄⋊C₄	—	139	1	no	—
259	C₄⋊C₄	—	139	1	no	—
260	C₄⋊C₄	—	139	1	no	—
261	C₄⋊C₄	C₂²×D₁₃₉	1	1	yes	characteristic
262	C₄⋊C₄	C₂²×D₁₃₉	1	1	yes	characteristic
263	C₄⋊C₄	C₂²×D₁₃₉	1	1	yes	characteristic
264	C₂²⋊C₄	—	2	1	no	—
265	C₂²⋊C₄	—	278	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	C₂²⋊C₄	—	278	1	no	—
271	C₂²⋊C₄	—	139	1	no	—
272	C₂²⋊C₄	—	139	1	no	—
273	C₄²	—	278	1	no	—
274	C₄²	—	278	1	no	—
275	C₄²	—	278	1	no	—
276	C₄²	—	139	1	no	—
277	C₂³	—	278	1	no	—
278	C₂³	—	139	1	no	—
279	C₂³	D₄×D₁₃₉	1	1	yes	characteristic
280	Q₈	—	2	1	no	—
281	Q₈	—	2	1	no	—
282	Q₈	—	556	2	no	—
283	Q₈	—	556	2	no	—
284	D₄	—	278	2	no	—
285	D₄	—	278	1	no	—
286	D₄	—	278	1	no	—
287	D₄	—	556	2	no	—
288	D₄	—	556	2	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	1	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	—	4	2	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	—	278	1	no	—
301	C₂×C₄	—	556	2	no	—
302	C₂×C₄	—	556	2	no	—
303	C₂×C₄	—	139	1	no	—
304	C₂×C₄	—	139	1	no	—
305	C₂×C₄	—	139	1	no	—
306	C₂×C₄	D₄×D₁₃₉	1	1	yes	characteristic
307	C₂×C₄	C₂³×D₁₃₉	1	1	yes	characteristic
308	C₈	—	2	1	no	—
309	C₈	—	2	1	no	—
310	C₈	—	278	1	no	—
311	C₈	—	139	2	no	—
312	C₂²	—	2	1	no	—
313	C₂²	—	2	1	no	—
314	C₂²	—	2	1	no	—
315	C₂²	—	278	1	no	—
316	C₂²	—	278	1	no	—
317	C₂²	—	278	1	no	—
318	C₂²	—	556	1	no	—
319	C₂²	—	556	1	no	—
320	C₂²	—	556	1	no	—
321	C₂²	C₂×D₄×D₁₃₉	1	1	yes	characteristic
322	C₄	—	2	1	no	—
323	C₄	—	4	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	—	139	2	no	—
333	C₄	#41×D₁₃₉	1	1	yes	characteristic
334	C₄	#17984889	1	1	yes	characteristic
335	C₂	—	4	1	no	—
336	C₂	—	556	1	no	—
337	C₂	—	1112	1	no	—
338	C₂	#22581601	1	1	yes	characteristic
339	C₂	AGL(1,ℤ/8)×D₁₃₉	1	1	yes	characteristic
340	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	#484×C₁₃₉	C₂
socle	C₂²×C₁₃₉	C₂²×D₄
prime radical	#22581604	C₂
hypercenter	#484	D₁₃₉
hypocenter	C₁₃₉	#3353
LCS	G ▷ C₂×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #484

Complements

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

N	H ≅ G/N	type
1. #22581674	C₂	semidirect
3. #22581357	C₂	semidirect
4. #22581287	C₂	semidirect
5. #22581280	C₂	semidirect
6. #22581278	C₂	semidirect
7. #22581236	C₂	semidirect
8. #22581232	C₂	semidirect
9. #22581202	C₂	semidirect
10. #22581201	C₂	semidirect
11. #22581038	C₂	semidirect
12. #22581028	C₂	semidirect
13. #22580994	C₂	semidirect
14. #22580993	C₂	semidirect
15. #484×C₁₃₉	C₂	semidirect
126. C₁₃₉⋊C₈	#41	semidirect
170. C₁₃₉	#3353	semidirect
176. #484	D₁₃₉	semidirect

Maximal subgroups

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

subgroup	index
#22581674	2
#22581604	2
#22581357	2
#22581287	2
#22581280	2
#22581278	2
#22581236	2
#22581232	2
#22581202	2
#22581201	2
#22581038	2
#22581028	2
#22580994	2
#22580993	2
#484×C₁₃₉	2
#3353	139