Subgroups — #31816581

Order 17792.

406 conjugacy classes of subgroups.

The subgroup lattice has 364 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	#22581428	C₂	1	1	yes	maximal, characteristic, complemented
3	#22581295	C₂	1	1	yes	maximal, characteristic, complemented
4	#22581274	C₂	1	1	yes	maximal, characteristic, complemented
5	#22581241	C₂	1	1	yes	maximal, characteristic, complemented
6	#22581231	C₂	1	1	yes	maximal, characteristic, complemented
7	#22581166	C₂	1	1	yes	maximal, characteristic, complemented
8	#22581166	C₂	1	1	yes	maximal, characteristic, complemented
9	#22581104	C₂	1	1	yes	maximal, characteristic, complemented
10	#22581065	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	D₈×Dic₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
14	#22581011	C₂	1	1	yes	maximal, characteristic, complemented
15	#447×C₁₃₉	C₂	1	1	yes	maximal, characteristic
16	C₂×#14555773	C₂²	1	1	yes	characteristic
17	C₂×#14555773	C₂²	1	1	yes	characteristic
18	#17984841	C₂²	1	1	yes	characteristic, complemented
19	#17984835	—	2	1	no	—
20	#17984829	—	2	1	no	—
21	#17984829	C₂²	1	1	yes	characteristic, complemented
22	#17984823	—	2	1	no	—
23	#17984821	C₂²	1	1	yes	characteristic
24	#17984819	—	2	1	no	—
25	#17984807	C₂²	1	1	yes	characteristic, complemented
26	D₄×Dic₁₃₉	—	2	1	no	—
27	D₄×Dic₁₃₉	—	2	1	no	—
28	D₄×Dic₁₃₉	C₂²	1	1	yes	characteristic
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, complemented
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	#17984763	C₂²	1	1	yes	characteristic
46	#17984763	C₂²	1	1	yes	characteristic
47	#17984763	C₂²	1	1	yes	characteristic, complemented
48	#17984763	C₂²	1	1	yes	characteristic, complemented
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×D₈×C₁₃₉	C₂²	1	1	yes	characteristic
54	C₂×D₈×C₁₃₉	C₂²	1	1	yes	characteristic
55	#121×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	—	2	1	no	—
60	C₂×Dic₂₇₈	C₂³	1	1	yes	characteristic
61	C₂²×Dic₁₃₉	—	2	1	no	—
62	C₂²×Dic₁₃₉	—	2	1	no	—
63	C₂²×Dic₁₃₉	—	2	1	no	—
64	C₂²×Dic₁₃₉	—	2	1	no	—
65	C₂²×Dic₁₃₉	—	2	1	no	—
66	#14555773	—	2	2	no	—
67	#14555773	—	2	2	no	—
68	C₂×C₁₃₉⋊C₈	C₂³	1	1	yes	characteristic
69	C₂×C₁₃₉⋊C₈	C₂³	1	1	yes	characteristic
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	C₂²⋊Dic₁₃₉	—	2	1	no	—
79	C₂²⋊Dic₁₃₉	—	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	—	2	1	no	—
87	C₂²⋊Dic₁₃₉	D₄	1	1	yes	characteristic
88	C₂²⋊Dic₁₃₉	D₄	1	1	yes	characteristic
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×D₄×C₁₃₉	—	2	2	no	—
95	C₂×D₄×C₁₃₉	—	2	2	no	—
96	C₂×D₄×C₁₃₉	—	2	1	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	C₂×D₄×C₁₃₉	C₂³	1	1	yes	characteristic
101	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
102	D₈×C₁₃₉	—	2	2	no	—
103	D₈×C₁₃₉	—	2	2	no	—
104	C₂×C₈×C₁₃₉	C₂³	1	1	yes	characteristic
105	C₂×C₈×C₁₃₉	C₂³	1	1	yes	characteristic
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	Dic₂₇₈	—	8	1	no	—
111	C₂×Dic₁₃₉	—	2	1	no	—
112	C₂×Dic₁₃₉	—	2	1	no	—
113	C₂×Dic₁₃₉	—	2	1	no	—
114	C₂×Dic₁₃₉	—	2	1	no	—
115	C₂×Dic₁₃₉	—	8	1	no	—
116	C₂×Dic₁₃₉	—	8	1	no	—
117	C₂×Dic₁₃₉	—	8	1	no	—
118	C₂×Dic₁₃₉	—	8	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	C₁₃₉⋊C₈	—	2	1	no	—
125	C₁₃₉⋊C₈	—	2	1	no	—
126	C₂³×C₁₃₉	—	2	4	no	—
127	C₂³×C₁₃₉	—	2	1	no	—
128	C₂³×C₁₃₉	—	2	1	no	—
129	C₂³×C₁₃₉	—	2	1	no	—
130	C₂³×C₁₃₉	—	8	1	no	—
131	C₂³×C₁₃₉	—	8	1	no	—
132	C₂³×C₁₃₉	C₂×D₄	1	1	yes	characteristic
133	D₄×C₁₃₉	—	2	4	no	—
134	D₄×C₁₃₉	—	2	1	no	—
135	D₄×C₁₃₉	—	2	1	no	—
136	D₄×C₁₃₉	—	2	1	no	—
137	D₄×C₁₃₉	—	2	1	no	—
138	D₄×C₁₃₉	—	8	1	no	—
139	D₄×C₁₃₉	#41	1	2	yes	—
140	D₄×C₁₃₉	#41	1	2	yes	—
141	C₂×C₄×C₁₃₉	—	2	1	no	—
142	C₂×C₄×C₁₃₉	—	2	1	no	—
143	C₂×C₄×C₁₃₉	—	2	1	no	—
144	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
145	C₂×C₄×C₁₃₉	C₂⁴	1	1	yes	characteristic
146	C₈×C₁₃₉	—	2	1	no	—
147	C₈×C₁₃₉	—	2	1	no	—
148	C₂²×C₁₃₉	—	2	1	no	—
149	C₂²×C₁₃₉	—	2	2	no	—
150	C₂²×C₁₃₉	—	2	2	no	—
151	C₂²×C₁₃₉	—	2	1	no	—
152	C₂²×C₁₃₉	—	2	1	no	—
153	C₂²×C₁₃₉	—	4	1	no	—
154	C₂²×C₁₃₉	—	4	1	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	C₂²×D₄	1	1	yes	characteristic
162	C₄×C₁₃₉	—	2	1	no	—
163	C₄×C₁₃₉	—	4	1	no	—
164	C₄×C₁₃₉	C₂×#41	1	1	yes	characteristic
165	C₄×C₁₃₉	C₂×#41	1	1	yes	characteristic
166	Dic₁₃₉	—	2	1	no	—
167	Dic₁₃₉	—	2	1	no	—
168	Dic₁₃₉	—	2	1	no	—
169	Dic₁₃₉	—	2	1	no	—
170	Dic₁₃₉	—	4	1	no	—
171	Dic₁₃₉	—	4	1	no	—
172	Dic₁₃₉	—	4	1	no	—
173	Dic₁₃₉	—	4	1	no	—
174	C₂×C₁₃₉	—	4	1	no	—
175	C₂×C₁₃₉	—	4	2	no	—
176	C₂×C₁₃₉	—	4	2	no	—
177	C₂×C₁₃₉	—	8	1	no	—
178	C₂×C₁₃₉	#525	1	1	yes	characteristic
179	C₂×C₁₃₉	#576	1	1	yes	characteristic
180	C₂×C₁₃₉	#577	1	1	yes	characteristic
181	C₁₃₉	#3118	1	1	yes	characteristic, complemented, Sylow
182	#3118	—	139	1	no	maximal, Sylow
183	#542	—	139	1	no	—
184	#518	—	139	1	no	—
185	#485	—	139	1	no	—
186	#482	—	139	1	no	—
187	#476	—	139	1	no	—
188	#474	—	139	1	no	—
189	#463	—	139	1	no	—
190	#463	—	139	1	no	—
191	#450	—	139	1	no	—
192	#447	D₁₃₉	1	1	yes	characteristic
193	#442	—	139	1	no	—
194	#440	—	139	1	no	—
195	C₄×SD₁₆	—	139	1	no	—
196	C₄×D₈	—	139	1	no	—
197	#433	—	139	1	no	—
198	C₂²×D₄	C₂×D₁₃₉	1	1	yes	characteristic
199	C₂×SD₁₆	—	139	1	no	—
200	C₂×SD₁₆	—	139	1	no	—
201	C₂×D₈	C₂×D₁₃₉	1	1	yes	characteristic
202	C₂×D₈	C₂×D₁₃₉	1	1	yes	characteristic
203	C₄⋊Q₈	—	139	1	no	—
204	#126	—	278	1	no	—
205	#124	—	278	1	no	—
206	#124	—	139	1	no	—
207	#123	—	278	1	no	—
208	C₂²⋊Q₈	—	278	1	no	—
209	C₂²⋊Q₈	—	139	1	no	—
210	#121	C₂×D₁₃₉	1	1	yes	characteristic
211	C₄×Q₈	—	139	1	no	—
212	C₄×D₄	—	278	1	no	—
213	C₄×D₄	—	278	1	no	—
214	C₄×D₄	—	139	1	no	—
215	C₄×D₄	—	139	1	no	—
216	C₄×D₄	—	139	1	no	—
217	C₄×D₄	—	139	1	no	—
218	C₄×D₄	—	139	1	no	—
219	#117	—	139	1	no	—
220	#117	—	139	1	no	—
221	C₂×C₂²⋊C₄	—	278	1	no	—
222	C₂×C₂²⋊C₄	—	278	1	no	—
223	#107	—	139	1	no	—
224	#107	—	139	1	no	—
225	#106	—	139	1	no	—
226	#106	—	139	1	no	—
227	C₄⋊C₈	—	139	1	no	—
228	C₄⋊C₈	—	139	1	no	—
229	#103	—	139	1	no	—
230	#103	—	139	1	no	—
231	#102	—	139	1	no	—
232	#102	—	139	1	no	—
233	#102	—	139	1	no	—
234	#102	—	139	1	no	—
235	#102	C₂×D₁₃₉	1	1	yes	characteristic
236	#102	C₂×D₁₃₉	1	1	yes	characteristic
237	C₂²⋊C₈	—	139	1	no	—
238	C₂²⋊C₈	C₂×D₁₃₉	1	1	yes	characteristic
239	#97	—	139	1	no	—
240	C₄×C₈	—	139	1	no	—
241	C₂⁴	—	2	1	no	—
242	C₂×Q₈	—	278	1	no	—
243	C₂×Q₈	—	139	1	no	—
244	C₂×D₄	—	2	2	no	—
245	C₂×D₄	—	2	2	no	—
246	C₂×D₄	—	2	1	no	—
247	C₂×D₄	—	2	1	no	—
248	C₂×D₄	C₂²×D₁₃₉	1	1	yes	characteristic
249	C₂×D₄	C₂²×D₁₃₉	1	1	yes	characteristic
250	C₂×D₄	C₂²×D₁₃₉	1	1	yes	characteristic
251	C₂²×C₄	—	278	1	no	—
252	C₂²×C₄	—	278	1	no	—
253	C₂²×C₄	—	278	1	no	—
254	C₂²×C₄	—	278	1	no	—
255	C₂²×C₄	—	278	1	no	—
256	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
257	SD₁₆	—	278	2	no	—
258	SD₁₆	—	278	2	no	—
259	D₈	—	2	2	no	—
260	D₈	—	2	2	no	—
261	C₂×C₈	—	139	1	no	—
262	C₂×C₈	—	139	1	no	—
263	C₂×C₈	C₂²×D₁₃₉	1	1	yes	characteristic
264	C₂×C₈	C₂²×D₁₃₉	1	1	yes	characteristic
265	C₄⋊C₄	—	278	1	no	—
266	C₄⋊C₄	—	278	1	no	—
267	C₄⋊C₄	—	278	1	no	—
268	C₄⋊C₄	—	139	1	no	—
269	C₄⋊C₄	—	139	1	no	—
270	C₄⋊C₄	—	139	1	no	—
271	C₄⋊C₄	—	139	1	no	—
272	C₄⋊C₄	—	139	1	no	—
273	C₄⋊C₄	C₂²×D₁₃₉	1	1	yes	characteristic
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	—	278	1	no	—
283	C₂²⋊C₄	—	278	1	no	—
284	C₂²⋊C₄	—	139	1	no	—
285	C₂²⋊C₄	—	139	1	no	—
286	C₄²	—	278	1	no	—
287	C₄²	—	278	1	no	—
288	C₄²	—	139	1	no	—
289	C₄²	—	139	1	no	—
290	C₂³	—	2	4	no	—
291	C₂³	—	2	1	no	—
292	C₂³	—	2	1	no	—
293	C₂³	—	2	1	no	—
294	C₂³	—	8	1	no	—
295	C₂³	—	8	1	no	—
296	C₂³	D₄×D₁₃₉	1	1	yes	characteristic
297	Q₈	—	278	1	no	—
298	Q₈	—	278	1	no	—
299	Q₈	—	1112	1	no	—
300	D₄	—	2	4	no	—
301	D₄	—	2	1	no	—
302	D₄	—	2	1	no	—
303	D₄	—	2	1	no	—
304	D₄	—	2	1	no	—
305	D₄	—	8	1	no	—
306	D₄	#14555784	1	2	yes	—
307	D₄	#14555784	1	2	yes	—
308	C₂×C₄	—	2	1	no	—
309	C₂×C₄	—	2	1	no	—
310	C₂×C₄	—	2	1	no	—
311	C₂×C₄	—	278	1	no	—
312	C₂×C₄	—	278	1	no	—
313	C₂×C₄	—	278	1	no	—
314	C₂×C₄	—	278	1	no	—
315	C₂×C₄	—	1112	1	no	—
316	C₂×C₄	—	1112	1	no	—
317	C₂×C₄	—	1112	1	no	—
318	C₂×C₄	—	1112	1	no	—
319	C₂×C₄	—	1112	1	no	—
320	C₂×C₄	—	139	1	no	—
321	C₂×C₄	—	139	1	no	—
322	C₂×C₄	—	139	1	no	—
323	C₂×C₄	—	139	1	no	—
324	C₂×C₄	D₄×D₁₃₉	1	1	yes	characteristic
325	C₂×C₄	C₂³×D₁₃₉	1	1	yes	characteristic
326	C₈	—	2	1	no	—
327	C₈	—	2	1	no	—
328	C₈	—	278	1	no	—
329	C₈	—	278	1	no	—
330	C₂²	—	2	1	no	—
331	C₂²	—	2	1	no	—
332	C₂²	—	2	2	no	—
333	C₂²	—	2	2	no	—
334	C₂²	—	2	1	no	—
335	C₂²	—	4	1	no	—
336	C₂²	—	4	1	no	—
337	C₂²	—	4	1	no	—
338	C₂²	—	4	1	no	—
339	C₂²	—	4	1	no	—
340	C₂²	—	4	1	no	—
341	C₂²	—	4	1	no	—
342	C₂²	—	8	4	no	—
343	C₂²	C₂×D₄×D₁₃₉	1	1	yes	characteristic
344	C₄	—	2	1	no	—
345	C₄	—	4	1	no	—
346	C₄	—	278	1	no	—
347	C₄	—	278	1	no	—
348	C₄	—	278	1	no	—
349	C₄	—	278	1	no	—
350	C₄	—	556	1	no	—
351	C₄	—	556	1	no	—
352	C₄	—	556	1	no	—
353	C₄	—	556	1	no	—
354	C₄	C₂×#14555784	1	1	yes	characteristic
355	C₄	C₂×#14555784	1	1	yes	characteristic
356	C₂	—	4	1	no	—
357	C₂	—	4	2	no	—
358	C₂	—	4	2	no	—
359	C₂	—	8	1	no	—
360	C₂	#22581459	1	1	yes	characteristic
361	C₂	#22581754	1	1	yes	characteristic
362	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, prime radical	#447×C₁₃₉	C₂
socle	C₂²×C₁₃₉	C₂²×D₄
hypercenter	#447	D₁₃₉
hypocenter	C₁₃₉	#3118
LCS	G ▷ C₂×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #447

Complements

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

N	H ≅ G/N	type
1. #22581570	C₂	semidirect
2. #22581428	C₂	semidirect
3. #22581295	C₂	semidirect
4. #22581274	C₂	semidirect
5. #22581241	C₂	semidirect
6. #22581231	C₂	semidirect
7. #22581166	C₂	semidirect
8. #22581166	C₂	semidirect
9. #22581104	C₂	semidirect
10. #22581065	C₂	semidirect
11. #22581050	C₂	semidirect
12. #22581044	C₂	semidirect
13. D₈×Dic₁₃₉	C₂	semidirect
14. #22581011	C₂	semidirect
18. #17984841	C₂²	semidirect
21. #17984829	C₂²	semidirect
25. #17984807	C₂²	semidirect
32. D₄×Dic₁₃₉	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
47. #17984763	C₂²	semidirect
48. #17984763	C₂²	semidirect
50. #17984752	C₂²	semidirect
51. C₈×Dic₁₃₉	C₂²	semidirect
181. C₁₃₉	#3118	semidirect

Maximal subgroups

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

subgroup	index
#22581570	2
#22581428	2
#22581295	2
#22581274	2
#22581241	2
#22581231	2
#22581166	2
#22581104	2
#22581065	2
#22581050	2
#22581044	2
D₈×Dic₁₃₉	2
#22581011	2
#447×C₁₃₉	2
#3118	139