Subgroups — #31817868

Order 17792.

346 conjugacy classes of subgroups.

The subgroup lattice has 346 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, complemented
2	#22581570	C₂	1	1	yes	maximal, characteristic, complemented
3	#22581242	C₂	1	1	yes	maximal, characteristic, complemented
4	#22581241	C₂	1	1	yes	maximal, characteristic, complemented
5	#22581237	C₂	1	1	yes	maximal, characteristic, complemented
6	#22581236	C₂	1	1	yes	maximal, characteristic, complemented
7	#22581209	C₂	1	1	yes	maximal, characteristic, complemented
8	#22581204	C₂	1	1	yes	maximal, characteristic
9	#22581182	C₂	1	1	yes	maximal, characteristic, complemented
10	#22581178	C₂	1	1	yes	maximal, characteristic, complemented
11	#22581173	C₂	1	1	yes	maximal, characteristic, complemented
12	#22581170	C₂	1	1	yes	maximal, characteristic, complemented
13	#22581167	C₂	1	1	yes	maximal, characteristic, complemented
14	#22580979	C₂	1	1	yes	maximal, characteristic, complemented
15	#465×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#14555775	C₂²	1	1	yes	characteristic
17	C₂×#14555774	C₂²	1	1	yes	characteristic
18	C₂×#14555770	C₂²	1	1	yes	characteristic
19	C₂×#14555769	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	#17984831	C₂²	1	1	yes	characteristic
25	#17984829	—	2	1	no	—
26	#17984829	C₂²	1	1	yes	characteristic
27	#17984828	—	2	1	no	—
28	#17984824	—	2	1	no	—
29	#17984823	—	2	1	no	—
30	#17984822	—	2	1	no	—
31	#17984821	C₂²	1	1	yes	characteristic
32	#17984819	—	2	1	no	—
33	#17984816	C₂²	1	1	yes	characteristic
34	Q₈×Dic₁₃₉	C₂²	1	1	yes	characteristic
35	#17984807	C₂²	1	1	yes	characteristic
36	D₄×Dic₁₃₉	C₂²	1	1	yes	characteristic
37	#17984804	C₂²	1	1	yes	characteristic
38	#17984801	C₂²	1	1	yes	characteristic
39	#17984774	C₂²	1	1	yes	characteristic
40	#17984773	C₂²	1	1	yes	characteristic
41	#17984773	C₂²	1	1	yes	characteristic
42	Dic₁₃₉⋊C₈	C₂²	1	1	yes	characteristic
43	Dic₁₃₉⋊C₈	C₂²	1	1	yes	characteristic
44	Dic₁₃₉⋊C₈	C₂²	1	1	yes	characteristic
45	Dic₁₃₉⋊C₈	C₂²	1	1	yes	characteristic
46	#17984767	C₂²	1	1	yes	characteristic
47	#17984766	C₂²	1	1	yes	characteristic
48	#17984765	C₂²	1	1	yes	characteristic
49	#17984764	C₂²	1	1	yes	characteristic
50	#17984763	C₂²	1	1	yes	characteristic
51	#17984762	C₂²	1	1	yes	characteristic
52	C₂×SD₁₆×C₁₃₉	C₂²	1	1	yes	characteristic
53	C₂²×C₈×C₁₃₉	C₂²	1	1	yes	characteristic
54	C₂²⋊Q₈×C₁₃₉	C₂²	1	1	yes	characteristic
55	#121×C₁₃₉	C₂²	1	1	yes	characteristic
56	#106×C₁₃₉	C₂²	1	1	yes	characteristic
57	#103×C₁₃₉	C₂²	1	1	yes	characteristic
58	#102×C₁₃₉	C₂²	1	1	yes	characteristic
59	C₂×Dic₂₇₈	—	2	1	no	—
60	C₂×Dic₂₇₈	—	2	1	no	—
61	C₂×Dic₂₇₈	C₂³	1	1	yes	characteristic
62	C₂×D₅₅₆	C₂³	1	1	yes	characteristic
63	C₂×C₂²⋊D₁₃₉	—	2	1	no	—
64	C₂²×Dic₁₃₉	—	2	1	no	—
65	C₂×C₄×D₁₃₉	—	2	1	no	—
66	#14555775	—	4	1	no	—
67	#14555774	—	4	1	no	—
68	#14555770	—	4	1	no	—
69	#14555769	—	4	1	no	—
70	C₂×C₁₃₉⋊C₈	C₂³	1	1	yes	characteristic
71	C₂×C₁₃₉⋊C₈	C₂³	1	1	yes	characteristic
72	C₄⋊Dic₁₃₉	—	2	1	no	—
73	C₄⋊Dic₁₃₉	D₄	1	1	yes	characteristic
74	C₄⋊Dic₁₃₉	D₄	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	Dic₁₃₉⋊C₄	—	2	1	no	—
79	Dic₁₃₉⋊C₄	—	2	1	no	—
80	Dic₁₃₉⋊C₄	—	2	1	no	—
81	C₂²⋊Dic₁₃₉	—	2	1	no	—
82	C₂²⋊Dic₁₃₉	—	2	1	no	—
83	C₂²⋊Dic₁₃₉	—	2	1	no	—
84	C₂²⋊Dic₁₃₉	—	2	1	no	—
85	C₂²⋊Dic₁₃₉	D₄	1	1	yes	characteristic
86	C₂²⋊Dic₁₃₉	D₄	1	1	yes	characteristic
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	—	2	1	no	—
91	C₄×Dic₁₃₉	—	2	1	no	—
92	C₄×Dic₁₃₉	—	2	1	no	—
93	C₄×Dic₁₃₉	C₂³	1	1	yes	characteristic
94	C₄×Dic₁₃₉	C₂³	1	1	yes	characteristic
95	C₂×Q₈×C₁₃₉	C₂³	1	1	yes	characteristic
96	C₂×D₄×C₁₃₉	—	2	1	no	—
97	C₂×D₄×C₁₃₉	C₂³	1	1	yes	characteristic
98	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
99	SD₁₆×C₁₃₉	—	4	1	no	—
100	C₂×C₈×C₁₃₉	—	4	1	no	—
101	C₂×C₈×C₁₃₉	C₂³	1	1	yes	characteristic
102	C₂×C₈×C₁₃₉	C₂³	1	1	yes	characteristic
103	C₄⋊C₄×C₁₃₉	—	2	1	no	—
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	C₂²⋊C₄×C₁₃₉	—	2	1	no	—
108	C₂²×D₁₃₉	—	2	1	no	—
109	Dic₂₇₈	—	2	1	no	—
110	Dic₂₇₈	—	2	1	no	—
111	Dic₂₇₈	—	8	1	no	—
112	Dic₂₇₈	—	8	1	no	—
113	D₅₅₆	—	2	1	no	—
114	D₅₅₆	—	2	1	no	—
115	C₂²⋊D₁₃₉	—	8	1	no	—
116	C₂×Dic₁₃₉	—	2	1	no	—
117	C₂×Dic₁₃₉	—	2	1	no	—
118	C₂×Dic₁₃₉	—	2	1	no	—
119	C₂×Dic₁₃₉	—	8	1	no	—
120	C₂×Dic₁₃₉	C₂×D₄	1	1	yes	characteristic
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	—	8	1	no	—
125	C₁₃₉⋊C₈	—	2	1	no	—
126	C₁₃₉⋊C₈	—	2	1	no	—
127	C₂³×C₁₃₉	—	2	1	no	—
128	C₂³×C₁₃₉	C₂×D₄	1	1	yes	characteristic
129	Q₈×C₁₃₉	—	2	1	no	—
130	Q₈×C₁₃₉	—	2	1	no	—
131	D₄×C₁₃₉	—	2	1	no	—
132	D₄×C₁₃₉	—	2	1	no	—
133	D₄×C₁₃₉	—	8	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	—	2	1	no	—
139	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
140	C₂×C₄×C₁₃₉	C₂⁴	1	1	yes	characteristic
141	C₈×C₁₃₉	—	2	1	no	—
142	C₈×C₁₃₉	—	2	1	no	—
143	C₂²×C₁₃₉	—	2	1	no	—
144	C₂²×C₁₃₉	—	2	1	no	—
145	C₂²×C₁₃₉	—	2	1	no	—
146	C₂²×C₁₃₉	—	4	1	no	—
147	C₂²×C₁₃₉	—	4	1	no	—
148	C₂²×C₁₃₉	—	4	1	no	—
149	C₂²×C₁₃₉	C₂²×D₄	1	1	yes	characteristic
150	C₂×D₁₃₉	—	4	1	no	—
151	C₂×D₁₃₉	—	4	1	no	—
152	C₂×D₁₃₉	—	4	1	no	—
153	C₄×C₁₃₉	—	2	1	no	—
154	C₄×C₁₃₉	—	4	1	no	—
155	C₄×C₁₃₉	—	4	1	no	—
156	C₄×C₁₃₉	—	4	1	no	—
157	C₄×C₁₃₉	H(5,𝔽₂)	1	1	yes	characteristic
158	C₄×C₁₃₉	2₋¹⁺⁴	1	1	yes	characteristic
159	Dic₁₃₉	—	2	1	no	—
160	Dic₁₃₉	—	2	1	no	—
161	Dic₁₃₉	—	2	1	no	—
162	Dic₁₃₉	—	2	1	no	—
163	Dic₁₃₉	—	4	1	no	—
164	Dic₁₃₉	—	4	1	no	—
165	Dic₁₃₉	—	4	1	no	—
166	C₂×C₁₃₉	—	4	1	no	—
167	C₂×C₁₃₉	—	8	1	no	—
168	C₂×C₁₃₉	#537	1	1	yes	characteristic
169	C₂×C₁₃₉	#577	1	1	yes	characteristic
170	C₂×C₁₃₉	#577	1	1	yes	characteristic
171	D₁₃₉	—	8	1	no	—
172	C₁₃₉	#3263	1	1	yes	characteristic, complemented, Sylow
173	#3263	—	139	1	no	maximal, Sylow
174	#542	—	139	1	no	—
175	#542	—	139	1	no	—
176	#476	—	139	1	no	—
177	#476	—	139	1	no	—
178	#475	—	139	1	no	—
179	#475	—	139	1	no	—
180	#470	—	139	1	no	—
181	#469	—	139	1	no	—
182	#465	—	139	1	no	—
183	#465	D₁₃₉	1	1	yes	characteristic, complemented
184	#464	—	139	1	no	—
185	#464	—	139	1	no	—
186	#463	—	139	1	no	—
187	#463	—	139	1	no	—
188	#424	—	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	C₂×D₈	—	139	1	no	—
193	C₂×M₁₆	—	139	1	no	—
194	C₂²×C₈	C₂×D₁₃₉	1	1	yes	characteristic
195	C₄⋊Q₈	—	139	1	no	—
196	C₄⋊Q₈	—	139	1	no	—
197	#126	—	278	1	no	—
198	#126	—	278	1	no	—
199	#124	—	278	1	no	—
200	#124	—	278	1	no	—
201	#124	—	139	1	no	—
202	#124	—	139	1	no	—
203	#123	—	278	1	no	—
204	#123	—	278	1	no	—
205	C₂²⋊Q₈	—	278	1	no	—
206	C₂²⋊Q₈	—	278	1	no	—
207	C₂²⋊Q₈	—	139	1	no	—
208	C₂²⋊Q₈	C₂×D₁₃₉	1	1	yes	characteristic
209	#121	—	139	1	no	—
210	#121	C₂×D₁₃₉	1	1	yes	characteristic
211	C₄×Q₈	—	139	1	no	—
212	C₄×Q₈	—	139	1	no	—
213	C₄×D₄	—	139	1	no	—
214	C₄×D₄	—	139	1	no	—
215	#117	—	139	1	no	—
216	#106	—	139	1	no	—
217	#106	—	139	1	no	—
218	#106	—	139	1	no	—
219	#106	C₂×D₁₃₉	1	1	yes	characteristic
220	C₄⋊C₈	—	139	1	no	—
221	C₄⋊C₈	—	139	1	no	—
222	C₄⋊C₈	—	139	1	no	—
223	C₄⋊C₈	—	139	1	no	—
224	#103	—	139	1	no	—
225	#103	—	139	1	no	—
226	#103	—	139	1	no	—
227	#103	C₂×D₁₃₉	1	1	yes	characteristic
228	#102	—	139	1	no	—
229	#102	—	139	1	no	—
230	#102	—	139	1	no	—
231	#102	C₂×D₁₃₉	1	1	yes	characteristic
232	C₂×Q₈	—	278	1	no	—
233	C₂×Q₈	—	278	1	no	—
234	C₂×Q₈	—	139	1	no	—
235	C₂×Q₈	C₂²×D₁₃₉	1	1	yes	characteristic
236	C₂×D₄	—	2	1	no	—
237	C₂×D₄	—	278	1	no	—
238	C₂×D₄	—	139	1	no	—
239	C₂×D₄	C₂²×D₁₃₉	1	1	yes	characteristic
240	C₂²×C₄	—	278	1	no	—
241	C₂²×C₄	—	278	1	no	—
242	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
243	Q₁₆	—	556	1	no	—
244	SD₁₆	—	4	1	no	—
245	SD₁₆	—	556	1	no	—
246	D₈	—	556	1	no	—
247	M₁₆	—	556	1	no	—
248	C₂×C₈	—	4	1	no	—
249	C₂×C₈	—	139	1	no	—
250	C₂×C₈	—	139	1	no	—
251	C₂×C₈	C₂²×D₁₃₉	1	1	yes	characteristic
252	C₂×C₈	C₂²×D₁₃₉	1	1	yes	characteristic
253	C₄⋊C₄	—	2	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	—	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	—	2	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	—	278	1	no	—
272	C₂²⋊C₄	—	278	1	no	—
273	C₂²⋊C₄	—	278	1	no	—
274	C₂²⋊C₄	—	278	1	no	—
275	C₂²⋊C₄	—	139	1	no	—
276	C₂²⋊C₄	—	139	1	no	—
277	C₄²	—	278	1	no	—
278	C₄²	—	278	1	no	—
279	C₄²	—	139	1	no	—
280	C₄²	—	139	1	no	—
281	C₂³	—	2	1	no	—
282	C₂³	—	278	1	no	—
283	C₂³	D₄×D₁₃₉	1	1	yes	characteristic
284	Q₈	—	2	1	no	—
285	Q₈	—	2	1	no	—
286	Q₈	—	278	1	no	—
287	Q₈	—	278	1	no	—
288	Q₈	—	1112	1	no	—
289	Q₈	—	1112	1	no	—
290	D₄	—	2	1	no	—
291	D₄	—	2	1	no	—
292	D₄	—	8	1	no	—
293	D₄	—	278	1	no	—
294	D₄	—	278	1	no	—
295	D₄	—	1112	1	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	1	no	—
301	C₂×C₄	—	278	1	no	—
302	C₂×C₄	—	278	1	no	—
303	C₂×C₄	—	278	1	no	—
304	C₂×C₄	—	1112	1	no	—
305	C₂×C₄	—	1112	1	no	—
306	C₂×C₄	—	139	1	no	—
307	C₂×C₄	—	139	1	no	—
308	C₂×C₄	—	139	1	no	—
309	C₂×C₄	—	139	1	no	—
310	C₂×C₄	D₄×D₁₃₉	1	1	yes	characteristic
311	C₂×C₄	C₂³×D₁₃₉	1	1	yes	characteristic
312	C₈	—	2	1	no	—
313	C₈	—	2	1	no	—
314	C₈	—	278	1	no	—
315	C₈	—	278	1	no	—
316	C₂²	—	2	1	no	—
317	C₂²	—	2	1	no	—
318	C₂²	—	2	1	no	—
319	C₂²	—	4	1	no	—
320	C₂²	—	4	1	no	—
321	C₂²	—	4	1	no	—
322	C₂²	—	556	1	no	—
323	C₂²	—	556	1	no	—
324	C₂²	—	556	1	no	—
325	C₂²	C₂×D₄×D₁₃₉	1	1	yes	characteristic
326	C₄	—	2	1	no	—
327	C₄	—	4	1	no	—
328	C₄	—	4	1	no	—
329	C₄	—	4	1	no	—
330	C₄	—	278	1	no	—
331	C₄	—	278	1	no	—
332	C₄	—	278	1	no	—
333	C₄	—	278	1	no	—
334	C₄	—	556	1	no	—
335	C₄	—	556	1	no	—
336	C₄	—	556	1	no	—
337	C₄	#17984887	1	1	yes	characteristic
338	C₄	#17984889	1	1	yes	characteristic
339	C₂	—	4	1	no	—
340	C₂	—	8	1	no	—
341	C₂	—	1112	1	no	—
342	C₂	#22581530	1	1	yes	characteristic
343	C₂	#22581759	1	1	yes	characteristic
344	C₂	#22581760	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₃₉
commutator, cosocle	C₂×C₄×C₁₃₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₃₉
Fitting	#465×C₁₃₉	C₂
socle	C₂²×C₁₃₉	C₂²×D₄
prime radical	#22581204	C₂
hypercenter	#465	D₁₃₉
hypocenter	C₁₃₉	#3263
LCS	G ▷ C₂×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #465

Complements

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

N	H ≅ G/N	type
1. #22581576	C₂	semidirect
2. #22581570	C₂	semidirect
3. #22581242	C₂	semidirect
4. #22581241	C₂	semidirect
5. #22581237	C₂	semidirect
6. #22581236	C₂	semidirect
7. #22581209	C₂	semidirect
9. #22581182	C₂	semidirect
10. #22581178	C₂	semidirect
11. #22581173	C₂	semidirect
12. #22581170	C₂	semidirect
13. #22581167	C₂	semidirect
14. #22580979	C₂	semidirect
15. #465×C₁₃₉	C₂	semidirect
172. C₁₃₉	#3263	semidirect
183. #465	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
#22581570	2
#22581242	2
#22581241	2
#22581237	2
#22581236	2
#22581209	2
#22581204	2
#22581182	2
#22581178	2
#22581173	2
#22581170	2
#22581167	2
#22580979	2
#465×C₁₃₉	2
#3263	139