Subgroups — #23768976

Order 10112.

492 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	C₂×C₄×Dic₁₅₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#14579119	C₂	1	1	yes	maximal, characteristic
3	#19098496	C₂	1	1	yes	maximal, characteristic, complemented
4	#19098492	C₂	1	1	yes	maximal, characteristic
5	#19098484	C₂	1	1	yes	maximal, characteristic, complemented
6	#19098476	C₂	1	1	yes	maximal, characteristic
7	#19098455	C₂	1	1	yes	maximal, characteristic
8	#19098454	C₂	1	1	yes	maximal, characteristic
9	#19098450	C₂	1	1	yes	maximal, characteristic, complemented
10	#19098446	C₂	1	1	yes	maximal, characteristic, complemented
11	#19098444	C₂	1	1	yes	maximal, characteristic, complemented
12	#19098437	C₂	1	1	yes	maximal, characteristic
13	#19098434	C₂	1	1	yes	maximal, characteristic
14	#19098430	C₂	1	1	yes	maximal, characteristic, complemented
15	#387×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂²×Dic₁₅₈	C₂²	1	1	yes	characteristic
17	C₂²×C₄×D₇₉	C₂²	1	1	yes	characteristic
18	C₄×Dic₁₅₈	—	2	4	no	—
19	#14579119	—	2	4	no	—
20	C₂×C₄⋊Dic₇₉	C₂²	1	1	yes	characteristic
21	C₂×C₄⋊Dic₇₉	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊Dic₇₉	C₂²	1	1	yes	characteristic
23	C₂×Dic₇₉⋊C₄	C₂²	1	1	yes	characteristic
24	C₂×Dic₇₉⋊C₄	C₂²	1	1	yes	characteristic
25	C₂×Dic₇₉⋊C₄	C₂²	1	1	yes	characteristic
26	C₂×Dic₇₉⋊C₄	C₂²	1	1	yes	characteristic
27	C₂×(C₂×D₇₉)⋊C₄	C₂²	1	1	yes	characteristic
28	C₂×(C₂×D₇₉)⋊C₄	C₂²	1	1	yes	characteristic
29	C₂×(C₂×D₇₉)⋊C₄	C₂²	1	1	yes	characteristic
30	C₂×(C₂×D₇₉)⋊C₄	C₂²	1	1	yes	characteristic
31	C₂×(C₂×D₇₉)⋊C₄	C₂²	1	1	yes	characteristic
32	C₂×(C₂×D₇₉)⋊C₄	C₂²	1	1	yes	characteristic
33	C₂×C₄×Dic₇₉	C₂²	1	1	yes	characteristic
34	C₂×C₄×Dic₇₉	C₂²	1	1	yes	characteristic
35	C₂×C₄×Dic₇₉	C₂²	1	1	yes	characteristic
36	C₇₉⋊#95	C₂²	1	1	yes	characteristic
37	C₇₉⋊#95	C₂²	1	1	yes	characteristic
38	C₇₉⋊#95	C₂²	1	1	yes	characteristic
39	C₇₉⋊#95	C₂²	1	1	yes	characteristic
40	C₇₉⋊#95	C₂²	1	1	yes	characteristic
41	C₇₉⋊#95	C₂²	1	1	yes	characteristic
42	C₇₉⋊#95	C₂²	1	1	yes	characteristic
43	C₇₉⋊#95	C₂²	1	1	yes	characteristic
44	C₇₉⋊#95	C₂²	1	1	yes	characteristic
45	C₇₉⋊#95	C₂²	1	1	yes	characteristic
46	C₂×C₄⋊C₄×C₇₉	C₂²	1	1	yes	characteristic
47	C₂×C₄⋊C₄×C₇₉	C₂²	1	1	yes	characteristic
48	C₂×C₄²×C₇₉	C₂²	1	1	yes	characteristic
49	#95×C₇₉	C₂²	1	1	yes	characteristic
50	#95×C₇₉	C₂²	1	1	yes	characteristic
51	#95×C₇₉	C₂²	1	1	yes	characteristic
52	#95×C₇₉	C₂²	1	1	yes	characteristic
53	C₂³×D₇₉	C₂³	1	1	yes	characteristic
54	C₂×Dic₁₅₈	—	2	2	no	—
55	C₂×Dic₁₅₈	—	2	2	no	—
56	C₂×Dic₁₅₈	—	2	2	no	—
57	C₂²×Dic₇₉	C₂³	1	1	yes	characteristic
58	C₂²×Dic₇₉	C₂³	1	1	yes	characteristic
59	C₂²×Dic₇₉	C₂³	1	1	yes	characteristic
60	C₂²×Dic₇₉	C₂³	1	1	yes	characteristic
61	C₂²×Dic₇₉	C₂³	1	1	yes	characteristic
62	C₂²×Dic₇₉	C₂³	1	1	yes	characteristic
63	C₂²×Dic₇₉	C₂³	1	1	yes	characteristic
64	C₂×C₄×D₇₉	—	2	2	no	—
65	C₂×C₄×D₇₉	—	2	2	no	—
66	C₂×C₄×D₇₉	—	2	2	no	—
67	C₄⋊Dic₇₉	—	2	2	no	—
68	C₄⋊Dic₇₉	—	2	2	no	—
69	C₄⋊Dic₇₉	—	2	2	no	—
70	Dic₇₉⋊C₄	—	2	2	no	—
71	Dic₇₉⋊C₄	—	2	2	no	—
72	Dic₇₉⋊C₄	—	2	2	no	—
73	Dic₇₉⋊C₄	—	2	2	no	—
74	(C₂×D₇₉)⋊C₄	—	2	2	no	—
75	(C₂×D₇₉)⋊C₄	—	2	2	no	—
76	(C₂×D₇₉)⋊C₄	—	2	2	no	—
77	(C₂×D₇₉)⋊C₄	—	2	2	no	—
78	(C₂×D₇₉)⋊C₄	—	2	2	no	—
79	(C₂×D₇₉)⋊C₄	—	2	2	no	—
80	C₄×Dic₇₉	—	2	2	no	—
81	C₄×Dic₇₉	—	2	2	no	—
82	C₄×Dic₇₉	D₄	1	4	yes	—
83	C₂²×C₄×C₇₉	C₂³	1	1	yes	characteristic
84	C₂²×C₄×C₇₉	C₂³	1	1	yes	characteristic
85	C₂²×C₄×C₇₉	C₂³	1	1	yes	characteristic
86	C₂²×C₄×C₇₉	C₂³	1	1	yes	characteristic
87	C₂²×C₄×C₇₉	C₂³	1	1	yes	characteristic
88	C₂²×C₄×C₇₉	C₂³	1	1	yes	characteristic
89	C₂²×C₄×C₇₉	C₂³	1	1	yes	characteristic
90	C₄⋊C₄×C₇₉	—	2	2	no	—
91	C₄⋊C₄×C₇₉	—	2	2	no	—
92	C₄²×C₇₉	—	2	2	no	—
93	C₂³×C₇₉	C₂⁴	1	1	yes	characteristic
94	C₂²×D₇₉	—	2	1	no	—
95	C₂²×D₇₉	—	2	1	no	—
96	C₂²×D₇₉	—	2	1	no	—
97	C₂²×D₇₉	—	2	1	no	—
98	C₂²×D₇₉	—	2	1	no	—
99	C₂²×D₇₉	—	2	1	no	—
100	C₂²×D₇₉	#41	1	2	yes	—
101	C₂×C₄×C₇₉	—	2	1	no	—
102	C₂×C₄×C₇₉	—	2	1	no	—
103	C₂×C₄×C₇₉	—	2	1	no	—
104	C₂×C₄×C₇₉	—	2	1	no	—
105	C₂×C₄×C₇₉	—	2	1	no	—
106	C₂×C₄×C₇₉	—	2	1	no	—
107	C₂×C₄×C₇₉	—	2	1	no	—
108	C₂×C₄×C₇₉	—	2	1	no	—
109	C₂×C₄×C₇₉	—	2	1	no	—
110	C₂×C₄×C₇₉	—	2	1	no	—
111	C₂×C₄×C₇₉	—	2	1	no	—
112	C₂×C₄×C₇₉	—	2	1	no	—
113	C₂×C₄×C₇₉	—	2	1	no	—
114	C₂×C₄×C₇₉	—	2	1	no	—
115	C₂×C₄×C₇₉	—	2	1	no	—
116	C₂×C₄×C₇₉	—	2	1	no	—
117	C₂×C₄×C₇₉	—	2	1	no	—
118	C₂×C₄×C₇₉	—	2	1	no	—
119	C₂×C₄×C₇₉	C₂×D₄	1	2	yes	—
120	C₂×C₄×C₇₉	#41	1	2	yes	—
121	C₂×C₄×C₇₉	#41	1	2	yes	—
122	C₂×Dic₇₉	—	2	1	no	—
123	C₂×Dic₇₉	—	2	1	no	—
124	C₂×Dic₇₉	—	2	1	no	—
125	C₂×Dic₇₉	—	2	1	no	—
126	C₂×Dic₇₉	—	2	1	no	—
127	C₂×Dic₇₉	—	2	1	no	—
128	C₂×Dic₇₉	—	2	1	no	—
129	C₂×Dic₇₉	—	2	1	no	—
130	C₂×Dic₇₉	—	2	1	no	—
131	C₂×Dic₇₉	—	2	1	no	—
132	C₂×Dic₇₉	—	2	1	no	—
133	C₂×Dic₇₉	—	2	1	no	—
134	C₂×Dic₇₉	—	2	1	no	—
135	C₂×Dic₇₉	—	2	1	no	—
136	C₂×Dic₇₉	—	2	1	no	—
137	C₂×Dic₇₉	—	2	1	no	—
138	C₂×Dic₇₉	—	2	1	no	—
139	C₂×Dic₇₉	—	2	1	no	—
140	C₂×Dic₇₉	C₂×D₄	1	2	yes	—
141	C₂×Dic₇₉	C₂×D₄	1	2	yes	—
142	C₂×Dic₇₉	#41	1	2	yes	—
143	C₄×D₇₉	—	2	8	no	—
144	Dic₁₅₈	—	2	8	no	—
145	C₂²×C₇₉	C₂²×D₄	1	1	yes	characteristic
146	C₂²×C₇₉	C₂×#41	1	1	yes	characteristic
147	C₂²×C₇₉	C₂×#41	1	1	yes	characteristic
148	C₂²×C₇₉	C₂×#41	1	1	yes	characteristic
149	C₂²×C₇₉	C₂×#41	1	1	yes	characteristic
150	C₂²×C₇₉	2₋¹⁺⁴	1	1	yes	characteristic
151	C₂²×C₇₉	2₋¹⁺⁴	1	1	yes	characteristic
152	C₂×D₇₉	—	2	2	no	—
153	C₂×D₇₉	—	2	2	no	—
154	C₂×D₇₉	—	2	2	no	—
155	C₂×D₇₉	—	4	1	no	—
156	C₂×D₇₉	—	4	1	no	—
157	C₂×D₇₉	—	4	1	no	—
158	C₂×D₇₉	—	4	1	no	—
159	C₄×C₇₉	—	2	2	no	—
160	C₄×C₇₉	—	2	2	no	—
161	C₄×C₇₉	—	2	2	no	—
162	C₄×C₇₉	—	4	1	no	—
163	C₄×C₇₉	—	4	1	no	—
164	C₄×C₇₉	—	4	1	no	—
165	C₄×C₇₉	—	4	1	no	—
166	Dic₇₉	—	2	2	no	—
167	Dic₇₉	—	2	2	no	—
168	Dic₇₉	—	2	2	no	—
169	Dic₇₉	—	4	1	no	—
170	Dic₇₉	—	4	1	no	—
171	Dic₇₉	—	4	1	no	—
172	Dic₇₉	—	4	1	no	—
173	C₂×C₇₉	#525	1	1	yes	characteristic
174	C₂×C₇₉	#525	1	1	yes	characteristic
175	C₂×C₇₉	#536	1	1	yes	characteristic
176	C₂×C₇₉	#552	1	1	yes	characteristic
177	C₂×C₇₉	#552	1	1	yes	characteristic
178	C₂×C₇₉	#556	1	1	yes	characteristic
179	C₂×C₇₉	#556	1	1	yes	characteristic
180	D₇₉	—	4	2	no	—
181	#2599	—	79	1	no	maximal, Sylow
182	C₇₉	#2599	1	1	yes	characteristic, complemented, Sylow
183	C₂×C₄×Q₈	—	79	1	no	—
184	C₂×#117	—	79	1	no	—
185	#398	—	79	1	no	—
186	#397	—	79	1	no	—
187	#395	—	79	1	no	—
188	#393	—	79	1	no	—
189	#388	—	79	1	no	—
190	#388	—	79	1	no	—
191	#387	—	79	1	no	—
192	#387	—	79	1	no	—
193	#387	—	79	1	no	—
194	#387	D₇₉	1	1	yes	characteristic, complemented
195	#385	—	79	1	no	—
196	#385	—	79	1	no	—
197	#383	—	79	1	no	—
198	C₂²×Q₈	—	79	1	no	—
199	C₂³×C₄	—	79	1	no	—
200	C₄×Q₈	—	158	4	no	—
201	#117	—	158	4	no	—
202	C₂×C₄⋊C₄	—	79	1	no	—
203	C₂×C₄⋊C₄	—	79	1	no	—
204	C₂×C₄⋊C₄	—	79	1	no	—
205	C₂×C₄⋊C₄	—	79	1	no	—
206	C₂×C₄⋊C₄	—	79	1	no	—
207	C₂×C₄⋊C₄	—	79	1	no	—
208	C₂×C₄⋊C₄	—	79	1	no	—
209	C₂×C₄⋊C₄	C₂×D₇₉	1	1	yes	characteristic
210	C₂×C₄⋊C₄	C₂×D₇₉	1	1	yes	characteristic
211	C₂×C₂²⋊C₄	—	79	1	no	—
212	C₂×C₂²⋊C₄	—	79	1	no	—
213	C₂×C₂²⋊C₄	—	79	1	no	—
214	C₂×C₂²⋊C₄	—	79	1	no	—
215	C₂×C₂²⋊C₄	—	79	1	no	—
216	C₂×C₂²⋊C₄	—	79	1	no	—
217	C₂×C₄²	—	79	1	no	—
218	C₂×C₄²	—	79	1	no	—
219	C₂×C₄²	—	79	1	no	—
220	C₂×C₄²	C₂×D₇₉	1	1	yes	characteristic
221	#95	—	79	1	no	—
222	#95	—	79	1	no	—
223	#95	—	79	1	no	—
224	#95	—	79	1	no	—
225	#95	—	79	1	no	—
226	#95	—	79	1	no	—
227	#95	—	79	1	no	—
228	#95	—	79	1	no	—
229	#95	—	79	1	no	—
230	#95	—	79	1	no	—
231	#95	C₂×D₇₉	1	1	yes	characteristic
232	#95	C₂×D₇₉	1	1	yes	characteristic
233	#95	C₂×D₇₉	1	1	yes	characteristic
234	#95	C₂×D₇₉	1	1	yes	characteristic
235	C₂⁴	—	79	1	no	—
236	C₂×Q₈	—	158	2	no	—
237	C₂×Q₈	—	158	2	no	—
238	C₂×Q₈	—	158	2	no	—
239	C₂²×C₄	—	79	1	no	—
240	C₂²×C₄	—	79	1	no	—
241	C₂²×C₄	—	79	1	no	—
242	C₂²×C₄	—	79	1	no	—
243	C₂²×C₄	—	79	1	no	—
244	C₂²×C₄	—	79	1	no	—
245	C₂²×C₄	—	79	1	no	—
246	C₂²×C₄	—	158	2	no	—
247	C₂²×C₄	—	158	2	no	—
248	C₂²×C₄	—	158	2	no	—
249	C₂²×C₄	C₂²×D₇₉	1	1	yes	characteristic
250	C₂²×C₄	C₂²×D₇₉	1	1	yes	characteristic
251	C₂²×C₄	C₂²×D₇₉	1	1	yes	characteristic
252	C₂²×C₄	C₂²×D₇₉	1	1	yes	characteristic
253	C₂²×C₄	C₂²×D₇₉	1	1	yes	characteristic
254	C₂²×C₄	C₂²×D₇₉	1	1	yes	characteristic
255	C₂²×C₄	C₂²×D₇₉	1	1	yes	characteristic
256	C₄⋊C₄	—	2	2	no	—
257	C₄⋊C₄	—	2	2	no	—
258	C₄⋊C₄	—	158	2	no	—
259	C₄⋊C₄	—	158	2	no	—
260	C₄⋊C₄	—	158	2	no	—
261	C₄⋊C₄	—	158	2	no	—
262	C₄⋊C₄	—	158	2	no	—
263	C₄⋊C₄	—	158	2	no	—
264	C₄⋊C₄	—	158	2	no	—
265	C₂²⋊C₄	—	158	2	no	—
266	C₂²⋊C₄	—	158	2	no	—
267	C₂²⋊C₄	—	158	2	no	—
268	C₂²⋊C₄	—	158	2	no	—
269	C₂²⋊C₄	—	158	2	no	—
270	C₂²⋊C₄	—	158	2	no	—
271	C₄²	—	2	2	no	—
272	C₄²	—	79	4	no	—
273	C₄²	—	158	2	no	—
274	C₄²	—	158	2	no	—
275	C₂³	—	79	2	no	—
276	C₂³	—	158	1	no	—
277	C₂³	—	158	1	no	—
278	C₂³	—	158	1	no	—
279	C₂³	—	158	1	no	—
280	C₂³	—	158	1	no	—
281	C₂³	—	158	1	no	—
282	C₂³	C₂³×D₇₉	1	1	yes	characteristic
283	Q₈	—	158	8	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	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	—	2	1	no	—
302	C₂×C₄	—	79	2	no	—
303	C₂×C₄	—	79	2	no	—
304	C₂×C₄	—	79	2	no	—
305	C₂×C₄	—	158	8	no	—
306	C₂×C₄	—	158	1	no	—
307	C₂×C₄	—	158	1	no	—
308	C₂×C₄	—	158	1	no	—
309	C₂×C₄	—	158	1	no	—
310	C₂×C₄	—	158	1	no	—
311	C₂×C₄	—	158	1	no	—
312	C₂×C₄	—	158	1	no	—
313	C₂×C₄	—	158	1	no	—
314	C₂×C₄	—	158	1	no	—
315	C₂×C₄	—	158	1	no	—
316	C₂×C₄	—	158	1	no	—
317	C₂×C₄	—	158	1	no	—
318	C₂×C₄	—	158	1	no	—
319	C₂×C₄	—	158	1	no	—
320	C₂×C₄	—	158	1	no	—
321	C₂×C₄	—	158	1	no	—
322	C₂×C₄	—	158	1	no	—
323	C₂×C₄	—	158	1	no	—
324	C₂×C₄	D₄×D₇₉	1	2	yes	—
325	C₂×C₄	#11929624	1	2	yes	—
326	C₂×C₄	#11929625	1	2	yes	—
327	C₂²	—	316	1	no	—
328	C₂²	—	316	1	no	—
329	C₂²	—	316	1	no	—
330	C₂²	—	316	1	no	—
331	C₂²	—	158	2	no	—
332	C₂²	—	158	2	no	—
333	C₂²	—	158	2	no	—
334	C₂²	C₂×D₄×D₇₉	1	1	yes	characteristic
335	C₂²	#41×D₇₉	1	1	yes	characteristic
336	C₂²	#41×D₇₉	1	1	yes	characteristic
337	C₂²	C₂×#11929624	1	1	yes	characteristic
338	C₂²	C₂×#11929625	1	1	yes	characteristic
339	C₂²	#14579208	1	1	yes	characteristic
340	C₂²	#14579208	1	1	yes	characteristic
341	C₄	—	2	2	no	—
342	C₄	—	2	2	no	—
343	C₄	—	2	2	no	—
344	C₄	—	4	1	no	—
345	C₄	—	4	1	no	—
346	C₄	—	4	1	no	—
347	C₄	—	4	1	no	—
348	C₄	—	316	1	no	—
349	C₄	—	316	1	no	—
350	C₄	—	316	1	no	—
351	C₄	—	316	1	no	—
352	C₄	—	158	2	no	—
353	C₄	—	158	2	no	—
354	C₄	—	158	2	no	—
355	C₂	—	316	2	no	—
356	C₂	#19099072	1	1	yes	characteristic
357	C₂	#19099076	1	1	yes	characteristic
358	C₂	#19099135	1	1	yes	characteristic
359	C₂	#19099249	1	1	yes	characteristic
360	C₂	#19099251	1	1	yes	characteristic
361	C₂	#19099281	1	1	yes	characteristic
362	C₂	#19099286	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₇₉
commutator, socle, cosocle	C₂³×C₇₉	C₂⁴
Fitting	#387×C₇₉	C₂
prime radical	C₂³×D₇₉	C₂³
hypercenter	#387	D₇₉
hypocenter	C₇₉	#2599
LCS	G ▷ C₂³×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ #387

Complements

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

N	H ≅ G/N	type
1. C₂×C₄×Dic₁₅₈	C₂	semidirect
3. #19098496	C₂	semidirect
5. #19098484	C₂	semidirect
9. #19098450	C₂	semidirect
10. #19098446	C₂	semidirect
11. #19098444	C₂	semidirect
14. #19098430	C₂	semidirect
15. #387×C₇₉	C₂	semidirect
182. C₇₉	#2599	semidirect
194. #387	D₇₉	semidirect