Subgroups — #31817816

Order 17792.

358 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22581570	C₂	1	1	yes	maximal, characteristic, complemented
2	#22581556	C₂	1	1	yes	maximal, characteristic, complemented
3	#22581263	C₂	1	1	yes	maximal, characteristic, complemented
4	#22581251	C₂	1	1	yes	maximal, characteristic, complemented
5	#22581242	C₂	1	1	yes	maximal, characteristic, complemented
6	#22581231	C₂	1	1	yes	maximal, characteristic, complemented
7	#22581199	C₂	1	1	yes	maximal, characteristic, complemented
8	#22581187	C₂	1	1	yes	maximal, characteristic, complemented
9	#22581180	C₂	1	1	yes	maximal, characteristic, complemented
10	#22581167	C₂	1	1	yes	maximal, characteristic, complemented
11	#22581166	C₂	1	1	yes	maximal, characteristic, complemented
12	#22581152	C₂	1	1	yes	maximal, characteristic, complemented
13	#22581146	C₂	1	1	yes	maximal, characteristic, complemented
14	#22580981	C₂	1	1	yes	maximal, characteristic, complemented
15	#469×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#14555774	C₂²	1	1	yes	characteristic
17	C₂×#14555773	C₂²	1	1	yes	characteristic
18	C₂×#14555770	C₂²	1	1	yes	characteristic
19	C₂²×C₁₃₉⋊C₈	C₂²	1	1	yes	characteristic
20	#17984841	C₂²	1	1	yes	characteristic
21	#17984838	C₂²	1	1	yes	characteristic
22	#17984835	—	2	1	no	—
23	#17984833	C₂²	1	1	yes	characteristic
24	#17984829	—	2	1	no	—
25	#17984829	C₂²	1	1	yes	characteristic
26	#17984826	—	2	1	no	—
27	#17984824	—	2	1	no	—
28	#17984823	—	2	1	no	—
29	#17984821	C₂²	1	1	yes	characteristic
30	#17984819	—	2	1	no	—
31	#17984816	C₂²	1	1	yes	characteristic
32	#17984814	—	2	1	no	—
33	#17984813	—	2	1	no	—
34	#17984807	C₂²	1	1	yes	characteristic
35	D₄×Dic₁₃₉	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	#17984776	C₂²	1	1	yes	characteristic
40	#17984775	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	#17984765	C₂²	1	1	yes	characteristic
48	#17984764	C₂²	1	1	yes	characteristic
49	#17984763	C₂²	1	1	yes	characteristic
50	#17984763	C₂²	1	1	yes	characteristic
51	#17984762	C₂²	1	1	yes	characteristic
52	C₂×D₈×C₁₃₉	C₂²	1	1	yes	characteristic
53	C₂×M₁₆×C₁₃₉	C₂²	1	1	yes	characteristic
54	#121×C₁₃₉	C₂²	1	1	yes	characteristic
55	#121×C₁₃₉	C₂²	1	1	yes	characteristic
56	#106×C₁₃₉	C₂²	1	1	yes	characteristic
57	#102×C₁₃₉	C₂²	1	1	yes	characteristic
58	#102×C₁₃₉	C₂²	1	1	yes	characteristic
59	C₂×Dic₂₇₈	—	2	1	no	—
60	C₂×Dic₂₇₈	C₂³	1	1	yes	characteristic
61	C₂×D₅₅₆	C₂³	1	1	yes	characteristic
62	C₂×C₂²⋊D₁₃₉	—	2	1	no	—
63	C₂×C₂²⋊D₁₃₉	—	2	1	no	—
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	#14555774	—	4	1	no	—
69	#14555773	—	4	1	no	—
70	#14555770	—	4	1	no	—
71	C₂×C₁₃₉⋊C₈	—	4	1	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	D₄	1	1	yes	characteristic
76	C₄⋊Dic₁₃₉	D₄	1	1	yes	characteristic
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	C₂²⋊Dic₁₃₉	—	2	1	no	—
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	D₄	1	1	yes	characteristic
89	C₂²⋊Dic₁₃₉	D₄	1	1	yes	characteristic
90	(C₂×D₁₃₉)⋊C₄	—	2	1	no	—
91	(C₂×D₁₃₉)⋊C₄	—	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×D₄×C₁₃₉	—	2	1	no	—
96	C₂×D₄×C₁₃₉	—	2	1	no	—
97	C₂×D₄×C₁₃₉	C₂³	1	1	yes	characteristic
98	C₂×D₄×C₁₃₉	C₂³	1	1	yes	characteristic
99	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
100	D₈×C₁₃₉	—	4	1	no	—
101	M₁₆×C₁₃₉	—	4	1	no	—
102	C₂×C₈×C₁₃₉	C₂³	1	1	yes	characteristic
103	C₂×C₈×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	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	D₅₅₆	—	2	1	no	—
113	D₅₅₆	—	2	1	no	—
114	C₂²⋊D₁₃₉	—	8	1	no	—
115	C₂²⋊D₁₃₉	—	8	1	no	—
116	C₂²⋊D₁₃₉	—	8	1	no	—
117	C₂×Dic₁₃₉	—	2	1	no	—
118	C₂×Dic₁₃₉	—	2	1	no	—
119	C₂×Dic₁₃₉	—	2	1	no	—
120	C₂×Dic₁₃₉	—	8	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	C₂×Dic₁₃₉	C₂×D₄	1	1	yes	characteristic
126	C₄×D₁₃₉	—	8	1	no	—
127	C₁₃₉⋊C₈	—	2	1	no	—
128	C₁₃₉⋊C₈	—	2	1	no	—
129	C₂³×C₁₃₉	—	2	1	no	—
130	C₂³×C₁₃₉	—	2	1	no	—
131	C₂³×C₁₃₉	C₂×D₄	1	1	yes	characteristic
132	D₄×C₁₃₉	—	2	1	no	—
133	D₄×C₁₃₉	—	2	1	no	—
134	D₄×C₁₃₉	—	2	1	no	—
135	D₄×C₁₃₉	—	2	1	no	—
136	D₄×C₁₃₉	—	8	1	no	—
137	D₄×C₁₃₉	—	8	1	no	—
138	C₂×C₄×C₁₃₉	—	2	1	no	—
139	C₂×C₄×C₁₃₉	—	2	1	no	—
140	C₂×C₄×C₁₃₉	—	2	1	no	—
141	C₂×C₄×C₁₃₉	—	2	1	no	—
142	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
143	C₂×C₄×C₁₃₉	C₂⁴	1	1	yes	characteristic
144	C₈×C₁₃₉	—	2	1	no	—
145	C₈×C₁₃₉	—	2	1	no	—
146	C₂²×C₁₃₉	—	2	1	no	—
147	C₂²×C₁₃₉	—	2	1	no	—
148	C₂²×C₁₃₉	—	2	1	no	—
149	C₂²×C₁₃₉	—	4	1	no	—
150	C₂²×C₁₃₉	—	4	1	no	—
151	C₂²×C₁₃₉	—	4	1	no	—
152	C₂²×C₁₃₉	—	4	1	no	—
153	C₂²×C₁₃₉	—	4	1	no	—
154	C₂²×C₁₃₉	—	4	1	no	—
155	C₂²×C₁₃₉	C₂²×D₄	1	1	yes	characteristic
156	C₂×D₁₃₉	—	4	1	no	—
157	C₂×D₁₃₉	—	4	1	no	—
158	C₂×D₁₃₉	—	4	1	no	—
159	C₄×C₁₃₉	—	2	1	no	—
160	C₄×C₁₃₉	—	4	1	no	—
161	C₄×C₁₃₉	—	4	1	no	—
162	C₄×C₁₃₉	H(5,𝔽₂)	1	1	yes	characteristic
163	C₄×C₁₃₉	2₋¹⁺⁴	1	1	yes	characteristic
164	Dic₁₃₉	—	2	1	no	—
165	Dic₁₃₉	—	2	1	no	—
166	Dic₁₃₉	—	2	1	no	—
167	Dic₁₃₉	—	2	1	no	—
168	Dic₁₃₉	—	4	1	no	—
169	Dic₁₃₉	—	4	1	no	—
170	Dic₁₃₉	—	4	1	no	—
171	C₂×C₁₃₉	—	4	1	no	—
172	C₂×C₁₃₉	—	8	1	no	—
173	C₂×C₁₃₉	—	8	1	no	—
174	C₂×C₁₃₉	#537	1	1	yes	characteristic
175	C₂×C₁₃₉	#576	1	1	yes	characteristic
176	C₂×C₁₃₉	#577	1	1	yes	characteristic
177	D₁₃₉	—	8	1	no	—
178	C₁₃₉	#3257	1	1	yes	characteristic, complemented, Sylow
179	#3257	—	139	1	no	maximal, Sylow
180	#542	—	139	1	no	—
181	#540	—	139	1	no	—
182	#479	—	139	1	no	—
183	#478	—	139	1	no	—
184	#476	—	139	1	no	—
185	#474	—	139	1	no	—
186	#469	D₁₃₉	1	1	yes	characteristic, complemented
187	#468	—	139	1	no	—
188	#466	—	139	1	no	—
189	#465	—	139	1	no	—
190	#463	—	139	1	no	—
191	#463	—	139	1	no	—
192	#460	—	139	1	no	—
193	#459	—	139	1	no	—
194	#424	—	139	1	no	—
195	C₂×SD₁₆	—	139	1	no	—
196	C₂×SD₁₆	—	139	1	no	—
197	C₂×D₈	—	139	1	no	—
198	C₂×D₈	C₂×D₁₃₉	1	1	yes	characteristic
199	C₂×M₁₆	C₂×D₁₃₉	1	1	yes	characteristic
200	C₂²×C₈	—	139	1	no	—
201	C₄⋊Q₈	—	139	1	no	—
202	#127	—	139	1	no	—
203	#126	—	278	1	no	—
204	#125	—	139	1	no	—
205	#124	—	278	1	no	—
206	#124	—	139	1	no	—
207	#123	—	278	1	no	—
208	#123	—	278	1	no	—
209	#123	—	278	1	no	—
210	C₂²⋊Q₈	—	278	1	no	—
211	C₂²⋊Q₈	—	139	1	no	—
212	#121	—	278	1	no	—
213	#121	—	278	1	no	—
214	#121	—	139	1	no	—
215	#121	C₂×D₁₃₉	1	1	yes	characteristic
216	#121	C₂×D₁₃₉	1	1	yes	characteristic
217	C₄×Q₈	—	139	1	no	—
218	C₄×D₄	—	139	1	no	—
219	C₄×D₄	—	139	1	no	—
220	C₄×D₄	—	139	1	no	—
221	#117	—	139	1	no	—
222	#107	—	139	1	no	—
223	#107	—	139	1	no	—
224	#106	—	139	1	no	—
225	#106	C₂×D₁₃₉	1	1	yes	characteristic
226	C₄⋊C₈	—	139	1	no	—
227	C₄⋊C₈	—	139	1	no	—
228	C₄⋊C₈	—	139	1	no	—
229	C₄⋊C₈	—	139	1	no	—
230	#103	—	139	1	no	—
231	#103	—	139	1	no	—
232	#102	—	139	1	no	—
233	#102	—	139	1	no	—
234	#102	—	139	1	no	—
235	#102	—	139	1	no	—
236	#102	C₂×D₁₃₉	1	1	yes	characteristic
237	#102	C₂×D₁₃₉	1	1	yes	characteristic
238	C₂×Q₈	—	278	1	no	—
239	C₂×Q₈	—	139	1	no	—
240	C₂×D₄	—	2	1	no	—
241	C₂×D₄	—	2	1	no	—
242	C₂×D₄	—	278	1	no	—
243	C₂×D₄	—	278	1	no	—
244	C₂×D₄	—	278	1	no	—
245	C₂×D₄	—	139	1	no	—
246	C₂×D₄	C₂²×D₁₃₉	1	1	yes	characteristic
247	C₂×D₄	C₂²×D₁₃₉	1	1	yes	characteristic
248	C₂²×C₄	—	278	1	no	—
249	C₂²×C₄	—	278	1	no	—
250	C₂²×C₄	—	278	1	no	—
251	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
252	SD₁₆	—	556	1	no	—
253	SD₁₆	—	556	1	no	—
254	D₈	—	4	1	no	—
255	D₈	—	556	1	no	—
256	M₁₆	—	4	1	no	—
257	C₂×C₈	—	556	1	no	—
258	C₂×C₈	—	139	1	no	—
259	C₂×C₈	—	139	1	no	—
260	C₂×C₈	C₂²×D₁₃₉	1	1	yes	characteristic
261	C₂×C₈	C₂²×D₁₃₉	1	1	yes	characteristic
262	C₄⋊C₄	—	278	1	no	—
263	C₄⋊C₄	—	278	1	no	—
264	C₄⋊C₄	—	278	1	no	—
265	C₄⋊C₄	—	278	1	no	—
266	C₄⋊C₄	—	278	1	no	—
267	C₄⋊C₄	—	139	1	no	—
268	C₄⋊C₄	—	139	1	no	—
269	C₄⋊C₄	—	139	1	no	—
270	C₄⋊C₄	—	139	1	no	—
271	C₄⋊C₄	C₂²×D₁₃₉	1	1	yes	characteristic
272	C₄⋊C₄	C₂²×D₁₃₉	1	1	yes	characteristic
273	C₂²⋊C₄	—	2	1	no	—
274	C₂²⋊C₄	—	2	1	no	—
275	C₂²⋊C₄	—	278	1	no	—
276	C₂²⋊C₄	—	278	1	no	—
277	C₂²⋊C₄	—	278	1	no	—
278	C₂²⋊C₄	—	278	1	no	—
279	C₂²⋊C₄	—	278	1	no	—
280	C₂²⋊C₄	—	278	1	no	—
281	C₂²⋊C₄	—	278	1	no	—
282	C₂²⋊C₄	—	139	1	no	—
283	C₂²⋊C₄	—	139	1	no	—
284	C₄²	—	278	1	no	—
285	C₄²	—	139	1	no	—
286	C₄²	—	139	1	no	—
287	C₂³	—	2	1	no	—
288	C₂³	—	2	1	no	—
289	C₂³	—	278	1	no	—
290	C₂³	D₄×D₁₃₉	1	1	yes	characteristic
291	Q₈	—	278	1	no	—
292	Q₈	—	278	1	no	—
293	Q₈	—	1112	1	no	—
294	D₄	—	2	1	no	—
295	D₄	—	2	1	no	—
296	D₄	—	2	1	no	—
297	D₄	—	2	1	no	—
298	D₄	—	8	1	no	—
299	D₄	—	8	1	no	—
300	D₄	—	278	1	no	—
301	D₄	—	278	1	no	—
302	D₄	—	1112	1	no	—
303	D₄	—	1112	1	no	—
304	D₄	—	1112	1	no	—
305	C₂×C₄	—	2	1	no	—
306	C₂×C₄	—	2	1	no	—
307	C₂×C₄	—	2	1	no	—
308	C₂×C₄	—	2	1	no	—
309	C₂×C₄	—	278	1	no	—
310	C₂×C₄	—	278	1	no	—
311	C₂×C₄	—	278	1	no	—
312	C₂×C₄	—	1112	1	no	—
313	C₂×C₄	—	1112	1	no	—
314	C₂×C₄	—	1112	1	no	—
315	C₂×C₄	—	139	1	no	—
316	C₂×C₄	—	139	1	no	—
317	C₂×C₄	—	139	1	no	—
318	C₂×C₄	—	139	1	no	—
319	C₂×C₄	D₄×D₁₃₉	1	1	yes	characteristic
320	C₂×C₄	C₂³×D₁₃₉	1	1	yes	characteristic
321	C₈	—	2	1	no	—
322	C₈	—	2	1	no	—
323	C₈	—	278	1	no	—
324	C₈	—	278	1	no	—
325	C₂²	—	2	1	no	—
326	C₂²	—	2	1	no	—
327	C₂²	—	2	1	no	—
328	C₂²	—	4	1	no	—
329	C₂²	—	4	1	no	—
330	C₂²	—	4	1	no	—
331	C₂²	—	4	1	no	—
332	C₂²	—	4	1	no	—
333	C₂²	—	4	1	no	—
334	C₂²	—	556	1	no	—
335	C₂²	—	556	1	no	—
336	C₂²	—	556	1	no	—
337	C₂²	C₂×D₄×D₁₃₉	1	1	yes	characteristic
338	C₄	—	2	1	no	—
339	C₄	—	4	1	no	—
340	C₄	—	4	1	no	—
341	C₄	—	278	1	no	—
342	C₄	—	278	1	no	—
343	C₄	—	278	1	no	—
344	C₄	—	278	1	no	—
345	C₄	—	556	1	no	—
346	C₄	—	556	1	no	—
347	C₄	—	556	1	no	—
348	C₄	#17984887	1	1	yes	characteristic
349	C₄	#17984889	1	1	yes	characteristic
350	C₂	—	4	1	no	—
351	C₂	—	8	1	no	—
352	C₂	—	8	1	no	—
353	C₂	—	1112	1	no	—
354	C₂	#22581530	1	1	yes	characteristic
355	C₂	#22581757	1	1	yes	characteristic
356	C₂	#22581763	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₃₉
commutator, cosocle	C₂×C₄×C₁₃₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₃₉
Fitting	#469×C₁₃₉	C₂
socle	C₂²×C₁₃₉	C₂²×D₄
hypercenter	#469	D₁₃₉
hypocenter	C₁₃₉	#3257
LCS	G ▷ C₂×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #469

Complements

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

N	H ≅ G/N	type
1. #22581570	C₂	semidirect
2. #22581556	C₂	semidirect
3. #22581263	C₂	semidirect
4. #22581251	C₂	semidirect
5. #22581242	C₂	semidirect
6. #22581231	C₂	semidirect
7. #22581199	C₂	semidirect
8. #22581187	C₂	semidirect
9. #22581180	C₂	semidirect
10. #22581167	C₂	semidirect
11. #22581166	C₂	semidirect
12. #22581152	C₂	semidirect
13. #22581146	C₂	semidirect
14. #22580981	C₂	semidirect
15. #469×C₁₃₉	C₂	semidirect
178. C₁₃₉	#3257	semidirect
186. #469	D₁₃₉	semidirect

Maximal subgroups

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

subgroup	index
#22581570	2
#22581556	2
#22581263	2
#22581251	2
#22581242	2
#22581231	2
#22581199	2
#22581187	2
#22581180	2
#22581167	2
#22581166	2
#22581152	2
#22581146	2
#22580981	2
#469×C₁₃₉	2
#3257	139