Subgroups — #31816650

Order 17792.

394 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22581563	C₂	1	1	yes	maximal, characteristic, complemented
2	#22581428	C₂	1	1	yes	maximal, characteristic
3	#22581295	C₂	1	1	yes	maximal, characteristic, complemented
4	#22581291	C₂	1	1	yes	maximal, characteristic, complemented
5	#22581255	C₂	1	1	yes	maximal, characteristic, complemented
6	#22581251	C₂	1	1	yes	maximal, characteristic, complemented
7	#22581156	C₂	1	1	yes	maximal, characteristic, complemented
8	#22581156	C₂	1	1	yes	maximal, characteristic, complemented
9	#22581104	C₂	1	1	yes	maximal, characteristic
10	#22581054	C₂	1	1	yes	maximal, characteristic, complemented
11	#22581050	C₂	1	1	yes	maximal, characteristic, complemented
12	#22581044	C₂	1	1	yes	maximal, characteristic, complemented
13	SD₁₆×Dic₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
14	#22581011	C₂	1	1	yes	maximal, characteristic, complemented
15	#450×C₁₃₉	C₂	1	1	yes	maximal, characteristic
16	C₂×#14555773	C₂²	1	1	yes	characteristic
17	C₂×#14555773	C₂²	1	1	yes	characteristic
18	#17984840	C₂²	1	1	yes	characteristic, complemented
19	#17984835	—	2	1	no	—
20	#17984835	—	2	1	no	—
21	#17984833	—	2	1	no	—
22	#17984833	C₂²	1	1	yes	characteristic, complemented
23	#17984832	—	2	1	no	—
24	#17984821	C₂²	1	1	yes	characteristic
25	Q₈×Dic₁₃₉	C₂²	1	1	yes	characteristic, complemented
26	#17984807	C₂²	1	1	yes	characteristic, complemented
27	D₄×Dic₁₃₉	—	2	1	no	—
28	D₄×Dic₁₃₉	—	2	1	no	—
29	D₄×Dic₁₃₉	C₂²	1	1	yes	characteristic
30	D₄×Dic₁₃₉	C₂²	1	1	yes	characteristic
31	D₄×Dic₁₃₉	C₂²	1	1	yes	characteristic
32	D₄×Dic₁₃₉	C₂²	1	1	yes	characteristic
33	#17984801	C₂²	1	1	yes	characteristic
34	#17984801	C₂²	1	1	yes	characteristic
35	C₂×C₂²⋊Dic₁₃₉	—	2	1	no	—
36	C₂×C₂²⋊Dic₁₃₉	—	2	1	no	—
37	#17984776	C₂²	1	1	yes	characteristic, complemented
38	#17984775	C₂²	1	1	yes	characteristic, complemented
39	#17984774	C₂²	1	1	yes	characteristic, complemented
40	#17984773	C₂²	1	1	yes	characteristic, complemented
41	Dic₁₃₉⋊C₈	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	#17984766	C₂²	1	1	yes	characteristic, complemented
46	#17984766	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	#17984752	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	C₂×SD₁₆×C₁₃₉	C₂²	1	1	yes	characteristic
54	C₂×SD₁₆×C₁₃₉	C₂²	1	1	yes	characteristic
55	C₂²⋊Q₈×C₁₃₉	C₂²	1	1	yes	characteristic
56	#102×C₁₃₉	C₂²	1	1	yes	characteristic
57	#102×C₁₃₉	C₂²	1	1	yes	characteristic
58	C₂²⋊C₈×C₁₃₉	C₂²	1	1	yes	characteristic
59	C₂×Dic₂₇₈	C₂³	1	1	yes	characteristic
60	C₂²×Dic₁₃₉	—	2	1	no	—
61	C₂²×Dic₁₃₉	—	2	1	no	—
62	C₂²×Dic₁₃₉	—	2	1	no	—
63	C₂²×Dic₁₃₉	—	2	1	no	—
64	#14555773	—	2	2	no	—
65	#14555773	—	2	2	no	—
66	C₂×C₁₃₉⋊C₈	C₂³	1	1	yes	characteristic
67	C₂×C₁₃₉⋊C₈	C₂³	1	1	yes	characteristic
68	C₄⋊Dic₁₃₉	—	2	1	no	—
69	C₄⋊Dic₁₃₉	—	2	1	no	—
70	C₄⋊Dic₁₃₉	D₄	1	1	yes	characteristic
71	C₄⋊Dic₁₃₉	D₄	1	1	yes	characteristic
72	C₄⋊Dic₁₃₉	C₂³	1	1	yes	characteristic
73	C₄⋊Dic₁₃₉	C₂³	1	1	yes	characteristic
74	C₄⋊Dic₁₃₉	C₂³	1	1	yes	characteristic
75	Dic₁₃₉⋊C₄	—	2	1	no	—
76	Dic₁₃₉⋊C₄	—	2	1	no	—
77	Dic₁₃₉⋊C₄	—	2	1	no	—
78	Dic₁₃₉⋊C₄	—	2	1	no	—
79	Dic₁₃₉⋊C₄	—	2	1	no	—
80	C₂²⋊Dic₁₃₉	—	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	—	2	1	no	—
86	C₂²⋊Dic₁₃₉	D₄	1	1	yes	characteristic
87	C₂²⋊Dic₁₃₉	D₄	1	1	yes	characteristic
88	C₄×Dic₁₃₉	—	2	1	no	—
89	C₄×Dic₁₃₉	—	2	1	no	—
90	C₄×Dic₁₃₉	—	2	1	no	—
91	C₄×Dic₁₃₉	C₂³	1	1	yes	characteristic
92	C₄×Dic₁₃₉	C₂³	1	1	yes	characteristic
93	C₂⁴×C₁₃₉	—	2	1	no	—
94	C₂×Q₈×C₁₃₉	C₂³	1	1	yes	characteristic
95	C₂×D₄×C₁₃₉	—	2	2	no	—
96	C₂×D₄×C₁₃₉	—	2	2	no	—
97	C₂×D₄×C₁₃₉	—	2	1	no	—
98	C₂×D₄×C₁₃₉	C₂³	1	1	yes	characteristic
99	C₂×D₄×C₁₃₉	C₂³	1	1	yes	characteristic
100	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
101	SD₁₆×C₁₃₉	—	2	2	no	—
102	SD₁₆×C₁₃₉	—	2	2	no	—
103	C₂×C₈×C₁₃₉	C₂³	1	1	yes	characteristic
104	C₂×C₈×C₁₃₉	C₂³	1	1	yes	characteristic
105	C₄⋊C₄×C₁₃₉	—	2	1	no	—
106	C₄⋊C₄×C₁₃₉	C₂³	1	1	yes	characteristic
107	C₂²⋊C₄×C₁₃₉	—	2	1	no	—
108	Dic₂₇₈	—	2	1	no	—
109	Dic₂₇₈	—	2	1	no	—
110	C₂×Dic₁₃₉	—	2	1	no	—
111	C₂×Dic₁₃₉	—	2	1	no	—
112	C₂×Dic₁₃₉	—	2	1	no	—
113	C₂×Dic₁₃₉	—	2	1	no	—
114	C₂×Dic₁₃₉	—	8	1	no	—
115	C₂×Dic₁₃₉	—	8	1	no	—
116	C₂×Dic₁₃₉	—	8	1	no	—
117	C₂×Dic₁₃₉	—	8	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	C₂×Dic₁₃₉	C₂×D₄	1	1	yes	characteristic
122	C₁₃₉⋊C₈	—	2	1	no	—
123	C₁₃₉⋊C₈	—	2	1	no	—
124	C₂³×C₁₃₉	—	2	4	no	—
125	C₂³×C₁₃₉	—	2	1	no	—
126	C₂³×C₁₃₉	—	2	1	no	—
127	C₂³×C₁₃₉	—	8	1	no	—
128	C₂³×C₁₃₉	—	8	1	no	—
129	C₂³×C₁₃₉	C₂×D₄	1	1	yes	characteristic
130	Q₈×C₁₃₉	—	2	1	no	—
131	Q₈×C₁₃₉	—	2	1	no	—
132	D₄×C₁₃₉	—	2	4	no	—
133	D₄×C₁₃₉	—	2	1	no	—
134	D₄×C₁₃₉	—	2	1	no	—
135	D₄×C₁₃₉	#41	1	2	yes	—
136	D₄×C₁₃₉	#41	1	2	yes	—
137	C₂×C₄×C₁₃₉	—	2	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	C₂×D₄	1	1	yes	characteristic
142	C₂×C₄×C₁₃₉	C₂⁴	1	1	yes	characteristic
143	C₈×C₁₃₉	—	2	1	no	—
144	C₈×C₁₃₉	—	2	1	no	—
145	C₂²×C₁₃₉	—	2	1	no	—
146	C₂²×C₁₃₉	—	2	2	no	—
147	C₂²×C₁₃₉	—	2	2	no	—
148	C₂²×C₁₃₉	—	2	1	no	—
149	C₂²×C₁₃₉	—	2	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	—	8	4	no	—
155	C₂²×C₁₃₉	C₂²×D₄	1	1	yes	characteristic
156	C₄×C₁₃₉	—	2	1	no	—
157	C₄×C₁₃₉	—	4	1	no	—
158	C₄×C₁₃₉	—	4	1	no	—
159	C₄×C₁₃₉	C₂×#41	1	1	yes	characteristic
160	C₄×C₁₃₉	C₂×#41	1	1	yes	characteristic
161	Dic₁₃₉	—	2	1	no	—
162	Dic₁₃₉	—	2	1	no	—
163	Dic₁₃₉	—	2	1	no	—
164	Dic₁₃₉	—	2	1	no	—
165	Dic₁₃₉	—	4	1	no	—
166	Dic₁₃₉	—	4	1	no	—
167	Dic₁₃₉	—	4	1	no	—
168	Dic₁₃₉	—	4	1	no	—
169	C₂×C₁₃₉	—	4	1	no	—
170	C₂×C₁₃₉	—	4	2	no	—
171	C₂×C₁₃₉	—	4	2	no	—
172	C₂×C₁₃₉	#525	1	1	yes	characteristic
173	C₂×C₁₃₉	#577	1	1	yes	characteristic
174	C₂×C₁₃₉	#577	1	1	yes	characteristic
175	C₁₃₉	#3123	1	1	yes	characteristic, complemented, Sylow
176	#3123	—	139	1	no	maximal, Sylow
177	#541	—	139	1	no	—
178	#518	—	139	1	no	—
179	#485	—	139	1	no	—
180	#485	—	139	1	no	—
181	#478	—	139	1	no	—
182	#478	—	139	1	no	—
183	#461	—	139	1	no	—
184	#461	—	139	1	no	—
185	#450	—	139	1	no	—
186	#450	D₁₃₉	1	1	yes	characteristic
187	#440	—	139	1	no	—
188	#440	—	139	1	no	—
189	C₄×SD₁₆	—	139	1	no	—
190	C₄×SD₁₆	—	139	1	no	—
191	#433	—	139	1	no	—
192	C₂²×D₄	C₂×D₁₃₉	1	1	yes	characteristic
193	C₂×SD₁₆	—	139	1	no	—
194	C₂×SD₁₆	—	139	1	no	—
195	C₂×SD₁₆	C₂×D₁₃₉	1	1	yes	characteristic
196	C₂×SD₁₆	C₂×D₁₃₉	1	1	yes	characteristic
197	C₄⋊Q₈	—	139	1	no	—
198	#126	—	278	1	no	—
199	#126	—	278	1	no	—
200	#125	—	278	1	no	—
201	#125	—	278	1	no	—
202	#125	—	139	1	no	—
203	C₂²⋊Q₈	—	139	1	no	—
204	C₂²⋊Q₈	C₂×D₁₃₉	1	1	yes	characteristic
205	C₄×Q₈	—	139	1	no	—
206	C₄×Q₈	—	139	1	no	—
207	C₄×D₄	—	278	1	no	—
208	C₄×D₄	—	278	1	no	—
209	C₄×D₄	—	139	1	no	—
210	C₄×D₄	—	139	1	no	—
211	C₄×D₄	—	139	1	no	—
212	C₄×D₄	—	139	1	no	—
213	#117	—	139	1	no	—
214	#117	—	139	1	no	—
215	C₂×C₂²⋊C₄	—	278	1	no	—
216	C₂×C₂²⋊C₄	—	278	1	no	—
217	#107	—	139	1	no	—
218	#107	—	139	1	no	—
219	#106	—	139	1	no	—
220	#106	—	139	1	no	—
221	C₄⋊C₈	—	139	1	no	—
222	C₄⋊C₈	—	139	1	no	—
223	#103	—	139	1	no	—
224	#103	—	139	1	no	—
225	#103	—	139	1	no	—
226	#103	—	139	1	no	—
227	#102	—	139	1	no	—
228	#102	—	139	1	no	—
229	#102	C₂×D₁₃₉	1	1	yes	characteristic
230	#102	C₂×D₁₃₉	1	1	yes	characteristic
231	C₂²⋊C₈	—	139	1	no	—
232	C₂²⋊C₈	C₂×D₁₃₉	1	1	yes	characteristic
233	#97	—	139	1	no	—
234	C₄×C₈	—	139	1	no	—
235	C₂⁴	—	2	1	no	—
236	C₂×Q₈	—	139	1	no	—
237	C₂×Q₈	C₂²×D₁₃₉	1	1	yes	characteristic
238	C₂×D₄	—	2	2	no	—
239	C₂×D₄	—	2	2	no	—
240	C₂×D₄	—	2	1	no	—
241	C₂×D₄	C₂²×D₁₃₉	1	1	yes	characteristic
242	C₂×D₄	C₂²×D₁₃₉	1	1	yes	characteristic
243	C₂²×C₄	—	278	1	no	—
244	C₂²×C₄	—	278	1	no	—
245	C₂²×C₄	—	278	1	no	—
246	C₂²×C₄	—	278	1	no	—
247	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
248	SD₁₆	—	2	2	no	—
249	SD₁₆	—	2	2	no	—
250	SD₁₆	—	278	2	no	—
251	SD₁₆	—	278	2	no	—
252	C₂×C₈	—	139	1	no	—
253	C₂×C₈	—	139	1	no	—
254	C₂×C₈	C₂²×D₁₃₉	1	1	yes	characteristic
255	C₂×C₈	C₂²×D₁₃₉	1	1	yes	characteristic
256	C₄⋊C₄	—	2	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	—	278	1	no	—
262	C₄⋊C₄	—	278	1	no	—
263	C₄⋊C₄	—	278	1	no	—
264	C₄⋊C₄	—	139	1	no	—
265	C₄⋊C₄	—	139	1	no	—
266	C₄⋊C₄	—	139	1	no	—
267	C₄⋊C₄	—	139	1	no	—
268	C₄⋊C₄	—	139	1	no	—
269	C₄⋊C₄	C₂²×D₁₃₉	1	1	yes	characteristic
270	C₂²⋊C₄	—	2	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	—	278	1	no	—
276	C₂²⋊C₄	—	278	1	no	—
277	C₂²⋊C₄	—	139	1	no	—
278	C₂²⋊C₄	—	139	1	no	—
279	C₄²	—	278	1	no	—
280	C₄²	—	278	1	no	—
281	C₄²	—	278	1	no	—
282	C₄²	—	139	1	no	—
283	C₄²	—	139	1	no	—
284	C₂³	—	2	4	no	—
285	C₂³	—	2	1	no	—
286	C₂³	—	2	1	no	—
287	C₂³	—	8	1	no	—
288	C₂³	—	8	1	no	—
289	C₂³	D₄×D₁₃₉	1	1	yes	characteristic
290	Q₈	—	2	1	no	—
291	Q₈	—	2	1	no	—
292	Q₈	—	278	1	no	—
293	Q₈	—	278	1	no	—
294	D₄	—	2	4	no	—
295	D₄	—	2	1	no	—
296	D₄	—	2	1	no	—
297	D₄	#14555784	1	2	yes	—
298	D₄	#14555784	1	2	yes	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	1	no	—
301	C₂×C₄	—	2	1	no	—
302	C₂×C₄	—	2	1	no	—
303	C₂×C₄	—	278	1	no	—
304	C₂×C₄	—	278	1	no	—
305	C₂×C₄	—	278	1	no	—
306	C₂×C₄	—	278	1	no	—
307	C₂×C₄	—	1112	1	no	—
308	C₂×C₄	—	1112	1	no	—
309	C₂×C₄	—	1112	1	no	—
310	C₂×C₄	—	1112	1	no	—
311	C₂×C₄	—	139	1	no	—
312	C₂×C₄	—	139	1	no	—
313	C₂×C₄	—	139	1	no	—
314	C₂×C₄	—	139	1	no	—
315	C₂×C₄	D₄×D₁₃₉	1	1	yes	characteristic
316	C₂×C₄	C₂³×D₁₃₉	1	1	yes	characteristic
317	C₈	—	2	1	no	—
318	C₈	—	2	1	no	—
319	C₈	—	278	1	no	—
320	C₈	—	278	1	no	—
321	C₂²	—	2	1	no	—
322	C₂²	—	2	1	no	—
323	C₂²	—	2	2	no	—
324	C₂²	—	2	2	no	—
325	C₂²	—	2	1	no	—
326	C₂²	—	4	1	no	—
327	C₂²	—	4	1	no	—
328	C₂²	—	4	1	no	—
329	C₂²	—	4	1	no	—
330	C₂²	—	8	4	no	—
331	C₂²	C₂×D₄×D₁₃₉	1	1	yes	characteristic
332	C₄	—	2	1	no	—
333	C₄	—	4	1	no	—
334	C₄	—	4	1	no	—
335	C₄	—	278	1	no	—
336	C₄	—	278	1	no	—
337	C₄	—	278	1	no	—
338	C₄	—	278	1	no	—
339	C₄	—	556	1	no	—
340	C₄	—	556	1	no	—
341	C₄	—	556	1	no	—
342	C₄	—	556	1	no	—
343	C₄	C₂×#14555784	1	1	yes	characteristic
344	C₄	C₂×#14555784	1	1	yes	characteristic
345	C₂	—	4	1	no	—
346	C₂	—	4	2	no	—
347	C₂	—	4	2	no	—
348	C₂	#22581459	1	1	yes	characteristic
349	C₂	#22581759	1	1	yes	characteristic
350	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	#450×C₁₃₉	C₂
socle	C₂²×C₁₃₉	C₂²×D₄
prime radical	C₂²×D₄×C₁₃₉	C₂²
hypercenter	#450	D₁₃₉
hypocenter	C₁₃₉	#3123
LCS	G ▷ C₂×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #450

Complements

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

N	H ≅ G/N	type
1. #22581563	C₂	semidirect
3. #22581295	C₂	semidirect
4. #22581291	C₂	semidirect
5. #22581255	C₂	semidirect
6. #22581251	C₂	semidirect
7. #22581156	C₂	semidirect
8. #22581156	C₂	semidirect
10. #22581054	C₂	semidirect
11. #22581050	C₂	semidirect
12. #22581044	C₂	semidirect
13. SD₁₆×Dic₁₃₉	C₂	semidirect
14. #22581011	C₂	semidirect
18. #17984840	C₂²	semidirect
22. #17984833	C₂²	semidirect
25. Q₈×Dic₁₃₉	C₂²	semidirect
26. #17984807	C₂²	semidirect
37. #17984776	C₂²	semidirect
38. #17984775	C₂²	semidirect
39. #17984774	C₂²	semidirect
40. #17984773	C₂²	semidirect
41. Dic₁₃₉⋊C₈	C₂²	semidirect
42. Dic₁₃₉⋊C₈	C₂²	semidirect
43. #17984767	C₂²	semidirect
44. #17984767	C₂²	semidirect
45. #17984766	C₂²	semidirect
46. #17984766	C₂²	semidirect
50. #17984752	C₂²	semidirect
51. C₈×Dic₁₃₉	C₂²	semidirect
175. C₁₃₉	#3123	semidirect

Maximal subgroups

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

subgroup	index
#22581563	2
#22581428	2
#22581295	2
#22581291	2
#22581255	2
#22581251	2
#22581156	2
#22581104	2
#22581054	2
#22581050	2
#22581044	2
SD₁₆×Dic₁₃₉	2
#22581011	2
#450×C₁₃₉	2
#3123	139