Subgroups — #31818235

Order 17792.

408 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22581638	C₂	1	1	yes	maximal, characteristic, complemented
2	#22581597	C₂	1	1	yes	maximal, characteristic
3	#22581306	C₂	1	1	yes	maximal, characteristic, complemented
4	#22581283	C₂	1	1	yes	maximal, characteristic, complemented
5	#22581275	C₂	1	1	yes	maximal, characteristic, complemented
6	#22581251	C₂	1	1	yes	maximal, characteristic, complemented
7	#22581194	C₂	1	1	yes	maximal, characteristic, complemented
8	#22581181	C₂	1	1	yes	maximal, characteristic, complemented
9	#22581174	C₂	1	1	yes	maximal, characteristic, complemented
10	#22581116	C₂	1	1	yes	maximal, characteristic, complemented
11	#22581104	C₂	1	1	yes	maximal, characteristic, complemented
12	#22581044	C₂	1	1	yes	maximal, characteristic, complemented
13	C₈×C₂²⋊D₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
14	#22580946	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂×#102×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#14555784	C₂²	1	1	yes	characteristic
17	C₂×Dic₅₅₆	C₂²	1	1	yes	characteristic, complemented
18	C₂×#14555773	C₂²	1	1	yes	characteristic
19	C₂×#14555773	C₂²	1	1	yes	characteristic
20	C₂×C₈×D₁₃₉	C₂²	1	1	yes	characteristic, complemented
21	#17984837	—	2	1	no	—
22	#17984833	C₂²	1	1	yes	characteristic, complemented
23	#17984832	—	2	1	no	—
24	#17984829	C₂²	1	1	yes	characteristic
25	#17984825	—	2	1	no	—
26	#17984823	—	2	1	no	—
27	#17984822	C₂²	1	1	yes	characteristic, complemented
28	#17984821	C₂²	1	1	yes	characteristic
29	#17984821	C₂²	1	1	yes	characteristic
30	#17984814	—	2	1	no	—
31	#17984813	C₂²	1	1	yes	characteristic
32	#17984811	—	2	1	no	—
33	#17984807	C₂²	1	1	yes	characteristic, complemented
34	D₄×Dic₁₃₉	C₂²	1	1	yes	characteristic
35	C₄×C₂²⋊D₁₃₉	C₂²	1	1	yes	characteristic
36	#17984801	—	2	1	no	—
37	#17984800	C₂²	1	1	yes	characteristic, complemented
38	C₂×C₂²⋊Dic₁₃₉	—	2	1	no	—
39	#17984776	C₂²	1	1	yes	characteristic, complemented
40	#17984773	C₂²	1	1	yes	characteristic, complemented
41	#17984773	C₂²	1	1	yes	characteristic, complemented
42	Dic₁₃₉⋊C₈	C₂²	1	1	yes	characteristic, complemented
43	#17984767	C₂²	1	1	yes	characteristic, complemented
44	#17984767	C₂²	1	1	yes	characteristic, complemented
45	#17984765	C₂²	1	1	yes	characteristic, complemented
46	#17984765	C₂²	1	1	yes	characteristic, complemented
47	#17984763	C₂²	1	1	yes	characteristic
48	#17984763	C₂²	1	1	yes	characteristic
49	C₂²⋊(C₁₃₉⋊C₈)	C₂²	1	1	yes	characteristic
50	(C₂×D₁₃₉)⋊C₈	C₂²	1	1	yes	characteristic, complemented
51	C₈×Dic₁₃₉	C₂²	1	1	yes	characteristic, complemented
52	C₂²×D₄×C₁₃₉	C₂²	1	1	yes	characteristic
53	C₂²×C₈×C₁₃₉	C₂²	1	1	yes	characteristic
54	C₂×C₄⋊C₄×C₁₃₉	C₂²	1	1	yes	characteristic
55	#102×C₁₃₉	C₂²	1	1	yes	characteristic
56	#102×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	—	4	1	no	—
61	C₂×Dic₂₇₈	C₂³	1	1	yes	characteristic
62	C₂×Dic₂₇₈	C₂³	1	1	yes	characteristic
63	C₂×C₂²⋊D₁₃₉	—	2	1	no	—
64	C₂×C₂²⋊D₁₃₉	—	2	1	no	—
65	C₂×C₂²⋊D₁₃₉	D₄	1	1	yes	characteristic
66	C₂²×Dic₁₃₉	—	2	1	no	—
67	C₂²×Dic₁₃₉	—	2	1	no	—
68	C₂×C₄×D₁₃₉	C₂³	1	1	yes	characteristic
69	Dic₅₅₆	—	4	1	no	—
70	#14555773	—	2	2	no	—
71	#14555773	—	4	1	no	—
72	C₂×C₁₃₉⋊C₈	C₂³	1	1	yes	characteristic
73	C₂×C₁₃₉⋊C₈	C₂³	1	1	yes	characteristic
74	C₈×D₁₃₉	—	4	1	no	—
75	C₄⋊Dic₁₃₉	—	2	1	no	—
76	C₄⋊Dic₁₃₉	—	2	1	no	—
77	C₄⋊Dic₁₃₉	C₂³	1	1	yes	characteristic
78	C₄⋊Dic₁₃₉	C₂³	1	1	yes	characteristic
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	D₄	1	1	yes	characteristic
84	C₂²⋊Dic₁₃₉	—	2	1	no	—
85	C₂²⋊Dic₁₃₉	—	2	1	no	—
86	C₂²⋊Dic₁₃₉	—	2	1	no	—
87	C₂²⋊Dic₁₃₉	—	2	1	no	—
88	C₂²⋊Dic₁₃₉	—	2	1	no	—
89	C₂²⋊Dic₁₃₉	—	2	1	no	—
90	C₂²⋊Dic₁₃₉	D₄	1	1	yes	characteristic
91	(C₂×D₁₃₉)⋊C₄	—	2	1	no	—
92	(C₂×D₁₃₉)⋊C₄	—	2	1	no	—
93	(C₂×D₁₃₉)⋊C₄	D₄	1	1	yes	characteristic
94	C₄×Dic₁₃₉	—	2	1	no	—
95	C₄×Dic₁₃₉	—	2	1	no	—
96	C₄×Dic₁₃₉	C₂³	1	1	yes	characteristic
97	C₂⁴×C₁₃₉	—	2	1	no	—
98	C₂×D₄×C₁₃₉	—	2	2	no	—
99	C₂×D₄×C₁₃₉	—	2	1	no	—
100	C₂×D₄×C₁₃₉	—	4	1	no	—
101	C₂×D₄×C₁₃₉	C₂³	1	1	yes	characteristic
102	C₂×D₄×C₁₃₉	C₂³	1	1	yes	characteristic
103	C₂²×C₄×C₁₃₉	—	2	1	no	—
104	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
105	C₂×C₈×C₁₃₉	—	4	1	no	—
106	C₂×C₈×C₁₃₉	C₂³	1	1	yes	characteristic
107	C₂×C₈×C₁₃₉	C₂³	1	1	yes	characteristic
108	C₄⋊C₄×C₁₃₉	—	2	1	no	—
109	C₄⋊C₄×C₁₃₉	C₂³	1	1	yes	characteristic
110	C₄⋊C₄×C₁₃₉	C₂³	1	1	yes	characteristic
111	C₂²×D₁₃₉	C₂×D₄	1	1	yes	characteristic
112	Dic₂₇₈	—	2	1	no	—
113	Dic₂₇₈	—	2	1	no	—
114	Dic₂₇₈	—	2	1	no	—
115	Dic₂₇₈	—	2	1	no	—
116	C₂²⋊D₁₃₉	—	2	2	no	—
117	C₂²⋊D₁₃₉	—	4	2	no	—
118	C₂²⋊D₁₃₉	—	8	1	no	—
119	C₂×Dic₁₃₉	—	2	1	no	—
120	C₂×Dic₁₃₉	—	2	1	no	—
121	C₂×Dic₁₃₉	—	2	1	no	—
122	C₂×Dic₁₃₉	—	2	1	no	—
123	C₂×Dic₁₃₉	—	4	2	no	—
124	C₂×Dic₁₃₉	—	8	1	no	—
125	C₂×Dic₁₃₉	C₂×D₄	1	1	yes	characteristic
126	C₂×Dic₁₃₉	C₂×D₄	1	1	yes	characteristic
127	C₂×Dic₁₃₉	C₂×D₄	1	1	yes	characteristic
128	C₄×D₁₃₉	—	2	1	no	—
129	C₄×D₁₃₉	—	2	1	no	—
130	C₁₃₉⋊C₈	—	2	1	no	—
131	C₁₃₉⋊C₈	—	2	1	no	—
132	C₂³×C₁₃₉	—	2	1	no	—
133	C₂³×C₁₃₉	—	2	1	no	—
134	C₂³×C₁₃₉	—	4	2	no	—
135	C₂³×C₁₃₉	—	4	2	no	—
136	C₂³×C₁₃₉	—	8	1	no	—
137	C₂³×C₁₃₉	C₂×D₄	1	1	yes	characteristic
138	D₄×C₁₃₉	—	2	1	no	—
139	D₄×C₁₃₉	—	2	1	no	—
140	D₄×C₁₃₉	—	2	1	no	—
141	D₄×C₁₃₉	—	4	2	no	—
142	D₄×C₁₃₉	#41	1	2	yes	—
143	C₂×C₄×C₁₃₉	—	2	1	no	—
144	C₂×C₄×C₁₃₉	—	2	1	no	—
145	C₂×C₄×C₁₃₉	—	2	1	no	—
146	C₂×C₄×C₁₃₉	—	2	1	no	—
147	C₂×C₄×C₁₃₉	—	4	2	no	—
148	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
149	C₂×C₄×C₁₃₉	C₂⁴	1	1	yes	characteristic
150	C₈×C₁₃₉	—	2	1	no	—
151	C₈×C₁₃₉	—	2	1	no	—
152	C₂²×C₁₃₉	—	2	1	no	—
153	C₂²×C₁₃₉	—	2	1	no	—
154	C₂²×C₁₃₉	—	2	2	no	—
155	C₂²×C₁₃₉	—	4	1	no	—
156	C₂²×C₁₃₉	—	4	1	no	—
157	C₂²×C₁₃₉	—	4	1	no	—
158	C₂²×C₁₃₉	—	4	1	no	—
159	C₂²×C₁₃₉	—	4	1	no	—
160	C₂²×C₁₃₉	—	8	4	no	—
161	C₂²×C₁₃₉	#135	1	2	yes	—
162	C₂²×C₁₃₉	C₂²×D₄	1	1	yes	characteristic
163	C₂×D₁₃₉	—	2	1	no	—
164	C₂×D₁₃₉	—	2	1	no	—
165	C₂×D₁₃₉	—	2	1	no	—
166	C₄×C₁₃₉	—	2	1	no	—
167	C₄×C₁₃₉	—	4	1	no	—
168	C₄×C₁₃₉	—	4	1	no	—
169	C₄×C₁₃₉	C₂×#41	1	1	yes	characteristic
170	C₄×C₁₃₉	H(5,𝔽₂)	1	1	yes	characteristic
171	Dic₁₃₉	—	2	1	no	—
172	Dic₁₃₉	—	2	1	no	—
173	Dic₁₃₉	—	2	1	no	—
174	Dic₁₃₉	—	4	1	no	—
175	Dic₁₃₉	—	4	1	no	—
176	Dic₁₃₉	—	4	1	no	—
177	Dic₁₃₉	—	4	1	no	—
178	C₂×C₁₃₉	—	2	2	no	—
179	C₂×C₁₃₉	—	4	2	no	—
180	C₂×C₁₃₉	—	8	1	no	—
181	C₂×C₁₃₉	#546	1	1	yes	characteristic
182	C₂×C₁₃₉	C₂×#135	1	1	yes	characteristic
183	C₂×C₁₃₉	#577	1	1	yes	characteristic
184	D₁₃₉	—	4	1	no	—
185	C₁₃₉	#3296	1	1	yes	characteristic, complemented, Sylow
186	#3296	—	139	1	no	maximal, Sylow
187	#552	—	139	1	no	—
188	#546	—	139	1	no	—
189	#488	—	139	1	no	—
190	#483	—	139	1	no	—
191	#482	—	139	1	no	—
192	#478	—	139	1	no	—
193	#467	—	139	1	no	—
194	#465	—	139	1	no	—
195	#464	—	139	1	no	—
196	#452	—	139	1	no	—
197	#450	—	139	1	no	—
198	C₄×SD₁₆	—	139	1	no	—
199	C₈×D₄	—	139	1	no	—
200	#416	—	139	1	no	—
201	C₂×#102	D₁₃₉	1	1	yes	characteristic, complemented
202	C₂×#41	—	139	1	no	—
203	C₂²×D₄	C₂×D₁₃₉	1	1	yes	characteristic
204	C₂×Q₁₆	—	139	1	no	—
205	C₂×SD₁₆	—	139	1	no	—
206	C₂×SD₁₆	—	139	1	no	—
207	C₂²×C₈	—	139	1	no	—
208	C₂²×C₈	C₂×D₁₃₉	1	1	yes	characteristic
209	#126	—	278	1	no	—
210	#125	—	278	1	no	—
211	#125	—	139	1	no	—
212	#124	—	139	1	no	—
213	#123	—	278	1	no	—
214	#123	—	278	1	no	—
215	C₂²⋊Q₈	—	139	1	no	—
216	C₂²⋊Q₈	—	139	1	no	—
217	C₂²⋊Q₈	—	139	1	no	—
218	#121	—	278	1	no	—
219	#121	—	139	1	no	—
220	#120	—	278	1	no	—
221	C₄×Q₈	—	139	1	no	—
222	C₄×D₄	—	139	1	no	—
223	C₄×D₄	—	139	1	no	—
224	#117	—	278	1	no	—
225	#117	—	139	1	no	—
226	C₂×C₄⋊C₄	C₂×D₁₃₉	1	1	yes	characteristic
227	C₂×C₂²⋊C₄	—	278	1	no	—
228	#107	—	139	1	no	—
229	#106	—	139	1	no	—
230	#106	—	139	1	no	—
231	C₄⋊C₈	—	139	1	no	—
232	#103	—	139	1	no	—
233	#103	—	139	1	no	—
234	#103	—	139	1	no	—
235	#103	—	139	1	no	—
236	#102	—	139	1	no	—
237	#102	—	139	1	no	—
238	#102	C₂×D₁₃₉	1	1	yes	characteristic
239	#102	C₂×D₁₃₉	1	1	yes	characteristic
240	#102	C₂×D₁₃₉	1	1	yes	characteristic
241	#102	C₂×D₁₃₉	1	1	yes	characteristic
242	C₂²⋊C₈	—	139	1	no	—
243	C₂²⋊C₈	—	139	1	no	—
244	C₄×C₈	—	139	1	no	—
245	C₂⁴	—	2	1	no	—
246	#41	—	278	2	no	—
247	#41	—	556	1	no	—
248	C₂×Q₈	—	139	1	no	—
249	C₂×Q₈	—	139	1	no	—
250	C₂×D₄	—	2	2	no	—
251	C₂×D₄	—	2	1	no	—
252	C₂×D₄	—	4	1	no	—
253	C₂×D₄	—	278	1	no	—
254	C₂×D₄	—	278	1	no	—
255	C₂×D₄	—	139	1	no	—
256	C₂×D₄	C₂²×D₁₃₉	1	1	yes	characteristic
257	C₂×D₄	C₂²×D₁₃₉	1	1	yes	characteristic
258	C₂²×C₄	—	2	1	no	—
259	C₂²×C₄	—	278	1	no	—
260	C₂²×C₄	—	278	1	no	—
261	C₂²×C₄	—	139	1	no	—
262	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
263	Q₁₆	—	556	1	no	—
264	SD₁₆	—	278	2	no	—
265	SD₁₆	—	556	1	no	—
266	C₂×C₈	—	4	1	no	—
267	C₂×C₈	—	556	1	no	—
268	C₂×C₈	—	139	1	no	—
269	C₂×C₈	—	139	1	no	—
270	C₂×C₈	C₂²×D₁₃₉	1	1	yes	characteristic
271	C₂×C₈	C₂²×D₁₃₉	1	1	yes	characteristic
272	C₄⋊C₄	—	2	1	no	—
273	C₄⋊C₄	—	278	1	no	—
274	C₄⋊C₄	—	278	1	no	—
275	C₄⋊C₄	—	278	1	no	—
276	C₄⋊C₄	—	278	1	no	—
277	C₄⋊C₄	—	278	1	no	—
278	C₄⋊C₄	—	278	1	no	—
279	C₄⋊C₄	—	139	1	no	—
280	C₄⋊C₄	—	139	1	no	—
281	C₄⋊C₄	—	139	1	no	—
282	C₄⋊C₄	C₂²×D₁₃₉	1	1	yes	characteristic
283	C₄⋊C₄	C₂²×D₁₃₉	1	1	yes	characteristic
284	C₂²⋊C₄	—	278	1	no	—
285	C₂²⋊C₄	—	278	1	no	—
286	C₂²⋊C₄	—	278	1	no	—
287	C₂²⋊C₄	—	278	1	no	—
288	C₂²⋊C₄	—	278	1	no	—
289	C₂²⋊C₄	—	278	1	no	—
290	C₂²⋊C₄	—	278	1	no	—
291	C₂²⋊C₄	—	278	1	no	—
292	C₂²⋊C₄	—	139	1	no	—
293	C₂²⋊C₄	—	139	1	no	—
294	C₄²	—	278	1	no	—
295	C₄²	—	278	1	no	—
296	C₄²	—	139	1	no	—
297	C₂³	—	2	1	no	—
298	C₂³	—	2	1	no	—
299	C₂³	—	4	2	no	—
300	C₂³	—	4	2	no	—
301	C₂³	—	8	1	no	—
302	C₂³	—	139	1	no	—
303	C₂³	D₄×D₁₃₉	1	1	yes	characteristic
304	Q₈	—	278	1	no	—
305	Q₈	—	278	1	no	—
306	Q₈	—	278	1	no	—
307	Q₈	—	278	1	no	—
308	D₄	—	2	1	no	—
309	D₄	—	2	1	no	—
310	D₄	—	2	1	no	—
311	D₄	—	4	2	no	—
312	D₄	—	278	2	no	—
313	D₄	—	556	2	no	—
314	D₄	—	1112	1	no	—
315	D₄	#14555785	1	2	yes	—
316	C₂×C₄	—	2	1	no	—
317	C₂×C₄	—	2	1	no	—
318	C₂×C₄	—	2	1	no	—
319	C₂×C₄	—	2	1	no	—
320	C₂×C₄	—	4	2	no	—
321	C₂×C₄	—	278	1	no	—
322	C₂×C₄	—	278	1	no	—
323	C₂×C₄	—	278	1	no	—
324	C₂×C₄	—	278	1	no	—
325	C₂×C₄	—	278	1	no	—
326	C₂×C₄	—	278	1	no	—
327	C₂×C₄	—	556	2	no	—
328	C₂×C₄	—	1112	1	no	—
329	C₂×C₄	—	139	1	no	—
330	C₂×C₄	—	139	1	no	—
331	C₂×C₄	—	139	1	no	—
332	C₂×C₄	D₄×D₁₃₉	1	1	yes	characteristic
333	C₂×C₄	C₂³×D₁₃₉	1	1	yes	characteristic
334	C₈	—	2	1	no	—
335	C₈	—	2	1	no	—
336	C₈	—	278	1	no	—
337	C₈	—	278	1	no	—
338	C₂²	—	2	1	no	—
339	C₂²	—	2	1	no	—
340	C₂²	—	2	2	no	—
341	C₂²	—	4	1	no	—
342	C₂²	—	4	1	no	—
343	C₂²	—	4	1	no	—
344	C₂²	—	4	1	no	—
345	C₂²	—	4	1	no	—
346	C₂²	—	8	4	no	—
347	C₂²	—	278	1	no	—
348	C₂²	—	278	1	no	—
349	C₂²	—	278	1	no	—
350	C₂²	#17984861	1	2	yes	—
351	C₂²	C₂×D₄×D₁₃₉	1	1	yes	characteristic
352	C₄	—	2	1	no	—
353	C₄	—	4	1	no	—
354	C₄	—	4	1	no	—
355	C₄	—	278	1	no	—
356	C₄	—	278	1	no	—
357	C₄	—	278	1	no	—
358	C₄	—	556	1	no	—
359	C₄	—	556	1	no	—
360	C₄	—	556	1	no	—
361	C₄	—	556	1	no	—
362	C₄	C₂×#14555785	1	1	yes	characteristic
363	C₄	#17984887	1	1	yes	characteristic
364	C₂	—	2	2	no	—
365	C₂	—	4	2	no	—
366	C₂	—	8	1	no	—
367	C₂	—	556	1	no	—
368	C₂	#22581593	1	1	yes	characteristic
369	C₂	C₂×#17984861	1	1	yes	characteristic
370	C₂	#22581759	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₃₉
commutator, cosocle	C₂×C₄×C₁₃₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₃₉
Fitting	C₂×#102×C₁₃₉	C₂
socle	C₂²×C₁₃₉	C₂²×D₄
prime radical	#22581597	C₂
hypercenter	C₂×#102	D₁₃₉
hypocenter	C₁₃₉	#3296
LCS	G ▷ C₂×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#102

Complements

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

N	H ≅ G/N	type
1. #22581638	C₂	semidirect
3. #22581306	C₂	semidirect
4. #22581283	C₂	semidirect
5. #22581275	C₂	semidirect
6. #22581251	C₂	semidirect
7. #22581194	C₂	semidirect
8. #22581181	C₂	semidirect
9. #22581174	C₂	semidirect
10. #22581116	C₂	semidirect
11. #22581104	C₂	semidirect
12. #22581044	C₂	semidirect
13. C₈×C₂²⋊D₁₃₉	C₂	semidirect
14. #22580946	C₂	semidirect
15. C₂×#102×C₁₃₉	C₂	semidirect
17. C₂×Dic₅₅₆	C₂²	semidirect
20. C₂×C₈×D₁₃₉	C₂²	semidirect
22. #17984833	C₂²	semidirect
27. #17984822	C₂²	semidirect
33. #17984807	C₂²	semidirect
37. #17984800	C₂²	semidirect
39. #17984776	C₂²	semidirect
40. #17984773	C₂²	semidirect
41. #17984773	C₂²	semidirect
42. Dic₁₃₉⋊C₈	C₂²	semidirect
43. #17984767	C₂²	semidirect
44. #17984767	C₂²	semidirect
45. #17984765	C₂²	semidirect
46. #17984765	C₂²	semidirect
50. (C₂×D₁₃₉)⋊C₈	C₂²	semidirect
51. C₈×Dic₁₃₉	C₂²	semidirect
185. C₁₃₉	#3296	semidirect
201. C₂×#102	D₁₃₉	semidirect

Maximal subgroups

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

subgroup	index
#22581638	2
#22581597	2
#22581306	2
#22581283	2
#22581275	2
#22581251	2
#22581194	2
#22581181	2
#22581174	2
#22581116	2
#22581104	2
#22581044	2
C₈×C₂²⋊D₁₃₉	2
#22580946	2
C₂×#102×C₁₃₉	2
#3296	139