Subgroups — #23768482

Order 10112.

492 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	C₂×#14579151	C₂	1	1	yes	maximal, characteristic, complemented
2	#19098499	C₂	1	1	yes	maximal, characteristic
3	#19098496	C₂	1	1	yes	maximal, characteristic, complemented
4	#19098486	C₂	1	1	yes	maximal, characteristic
5	#19098482	C₂	1	1	yes	maximal, characteristic, complemented
6	#19098470	C₂	1	1	yes	maximal, characteristic, complemented
7	#19098458	C₂	1	1	yes	maximal, characteristic, complemented
8	#19098458	C₂	1	1	yes	maximal, characteristic, complemented
9	#19098457	C₂	1	1	yes	maximal, characteristic
10	#19098457	C₂	1	1	yes	maximal, characteristic
11	#19098444	C₂	1	1	yes	maximal, characteristic, complemented
12	#19098436	C₂	1	1	yes	maximal, characteristic
13	C₄⋊C₄×Dic₇₉	C₂	1	1	yes	maximal, characteristic, complemented
14	C₄×C₂²⋊Dic₇₉	C₂	1	1	yes	maximal, characteristic
15	C₂×C₂²⋊Q₈×C₇₉	C₂	1	1	yes	maximal, characteristic
16	#14579151	—	2	4	no	—
17	C₂×C₄⋊Dic₇₉	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊Dic₇₉	C₂²	1	1	yes	characteristic
19	C₂×Dic₇₉⋊C₄	C₂²	1	1	yes	characteristic
20	C₂×Dic₇₉⋊C₄	C₂²	1	1	yes	characteristic
21	C₂×Dic₇₉⋊C₄	C₂²	1	1	yes	characteristic
22	C₂×Dic₇₉⋊C₄	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×C₂²⋊Dic₇₉	C₂²	1	1	yes	characteristic
28	C₂×C₂²⋊Dic₇₉	C₂²	1	1	yes	characteristic
29	C₂×C₂²⋊Dic₇₉	C₂²	1	1	yes	characteristic
30	C₂×C₂²⋊Dic₇₉	C₂²	1	1	yes	characteristic
31	C₂×C₄×Dic₇₉	C₂²	1	1	yes	characteristic
32	C₂×C₄×Dic₇₉	C₂²	1	1	yes	characteristic
33	C₂×C₄×Dic₇₉	C₂²	1	1	yes	characteristic
34	C₂×C₄×Dic₇₉	C₂²	1	1	yes	characteristic
35	C₇₉⋊#95	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	C₂²×Q₈×C₇₉	C₂²	1	1	yes	characteristic
46	C₂³×C₄×C₇₉	C₂²	1	1	yes	characteristic
47	C₂²⋊Q₈×C₇₉	—	2	4	no	—
48	C₂×C₄⋊C₄×C₇₉	C₂²	1	1	yes	characteristic
49	C₂×C₄⋊C₄×C₇₉	C₂²	1	1	yes	characteristic
50	C₂×C₄⋊C₄×C₇₉	C₂²	1	1	yes	characteristic
51	C₂×C₂²⋊C₄×C₇₉	C₂²	1	1	yes	characteristic
52	C₂×C₂²⋊C₄×C₇₉	C₂²	1	1	yes	characteristic
53	C₂²×Dic₇₉	C₂³	1	1	yes	characteristic
54	C₂²×Dic₇₉	C₂³	1	1	yes	characteristic
55	C₂²×Dic₇₉	C₂³	1	1	yes	characteristic
56	C₂²×Dic₇₉	C₂³	1	1	yes	characteristic
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	C₄⋊Dic₇₉	—	2	2	no	—
62	C₄⋊Dic₇₉	—	2	2	no	—
63	Dic₇₉⋊C₄	—	2	2	no	—
64	Dic₇₉⋊C₄	—	2	2	no	—
65	Dic₇₉⋊C₄	—	2	2	no	—
66	Dic₇₉⋊C₄	—	2	2	no	—
67	Dic₇₉⋊C₄	—	2	2	no	—
68	Dic₇₉⋊C₄	—	2	2	no	—
69	Dic₇₉⋊C₄	—	2	2	no	—
70	Dic₇₉⋊C₄	—	4	1	no	—
71	C₂²⋊Dic₇₉	—	2	2	no	—
72	C₂²⋊Dic₇₉	—	2	2	no	—
73	C₂²⋊Dic₇₉	—	4	1	no	—
74	C₂²⋊Dic₇₉	Q₈	1	4	yes	complemented
75	C₄×Dic₇₉	—	2	2	no	—
76	C₄×Dic₇₉	—	2	2	no	—
77	C₄×Dic₇₉	—	4	1	no	—
78	C₄×Dic₇₉	—	4	1	no	—
79	C₂⁴×C₇₉	C₂³	1	1	yes	characteristic
80	C₂×Q₈×C₇₉	—	2	2	no	—
81	C₂×Q₈×C₇₉	—	2	2	no	—
82	C₂×Q₈×C₇₉	—	4	1	no	—
83	C₂²×C₄×C₇₉	—	2	2	no	—
84	C₂²×C₄×C₇₉	—	2	2	no	—
85	C₂²×C₄×C₇₉	—	4	1	no	—
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	C₂²×C₄×C₇₉	C₂³	1	1	yes	characteristic
91	C₂²×C₄×C₇₉	C₂³	1	1	yes	characteristic
92	C₄⋊C₄×C₇₉	—	2	2	no	—
93	C₄⋊C₄×C₇₉	—	2	2	no	—
94	C₄⋊C₄×C₇₉	D₄	1	4	yes	—
95	C₂²⋊C₄×C₇₉	—	2	2	no	—
96	C₂²⋊C₄×C₇₉	—	2	2	no	—
97	C₂³×C₇₉	—	2	1	no	—
98	C₂³×C₇₉	—	2	1	no	—
99	C₂³×C₇₉	—	2	1	no	—
100	C₂³×C₇₉	—	2	1	no	—
101	C₂³×C₇₉	—	2	1	no	—
102	C₂³×C₇₉	—	2	1	no	—
103	C₂³×C₇₉	C₂×Q₈	1	2	yes	—
104	C₂³×C₇₉	C₂⁴	1	1	yes	characteristic
105	Q₈×C₇₉	—	4	4	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	—	2	1	no	—
120	C₂×C₄×C₇₉	—	4	4	no	—
121	C₂×C₄×C₇₉	C₂×D₄	1	2	yes	—
122	C₂×C₄×C₇₉	C₂×D₄	1	2	yes	—
123	C₂×C₄×C₇₉	C₂×D₄	1	2	yes	—
124	C₂×C₄×C₇₉	#41	1	2	yes	—
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	—	2	1	no	—
141	C₂×Dic₇₉	—	2	1	no	—
142	C₂×Dic₇₉	—	2	1	no	—
143	C₂×Dic₇₉	—	2	1	no	—
144	C₂×Dic₇₉	—	2	1	no	—
145	C₂×Dic₇₉	C₂×Q₈	1	2	yes	—
146	C₂×Dic₇₉	C₂×Q₈	1	2	yes	—
147	C₂×Dic₇₉	#41	1	2	yes	—
148	C₂×Dic₇₉	#41	1	2	yes	—
149	C₂²×C₇₉	—	2	2	no	—
150	C₂²×C₇₉	—	2	2	no	—
151	C₂²×C₇₉	—	2	2	no	—
152	C₂²×C₇₉	—	4	1	no	—
153	C₂²×C₇₉	—	4	1	no	—
154	C₂²×C₇₉	—	4	1	no	—
155	C₂²×C₇₉	—	4	1	no	—
156	C₂²×C₇₉	C₂²×D₄	1	1	yes	characteristic
157	C₂²×C₇₉	C₂²×Q₈	1	1	yes	characteristic
158	C₂²×C₇₉	C₂×#41	1	1	yes	characteristic
159	C₂²×C₇₉	C₂×#41	1	1	yes	characteristic
160	C₂²×C₇₉	C₂×#41	1	1	yes	characteristic
161	C₂²×C₇₉	H(5,𝔽₂)	1	1	yes	characteristic
162	C₂²×C₇₉	2₋¹⁺⁴	1	1	yes	characteristic
163	C₄×C₇₉	—	2	2	no	—
164	C₄×C₇₉	—	2	2	no	—
165	C₄×C₇₉	—	2	2	no	—
166	C₄×C₇₉	—	2	2	no	—
167	C₄×C₇₉	—	4	1	no	—
168	C₄×C₇₉	—	4	1	no	—
169	Dic₇₉	—	2	2	no	—
170	Dic₇₉	—	2	2	no	—
171	Dic₇₉	—	2	2	no	—
172	Dic₇₉	—	2	2	no	—
173	Dic₇₉	—	4	1	no	—
174	Dic₇₉	—	4	1	no	—
175	Dic₇₉	—	4	1	no	—
176	Dic₇₉	—	4	1	no	—
177	C₂×C₇₉	—	4	2	no	—
178	C₂×C₇₉	#532	1	1	yes	characteristic
179	C₂×C₇₉	#533	1	1	yes	characteristic
180	C₂×C₇₉	#539	1	1	yes	characteristic
181	C₂×C₇₉	#546	1	1	yes	characteristic
182	C₂×C₇₉	D₄×Q₈	1	1	yes	characteristic
183	C₂×C₇₉	#552	1	1	yes	characteristic
184	C₂×C₇₉	#554	1	1	yes	characteristic
185	#2556	—	79	1	no	maximal, Sylow
186	C₇₉	#2556	1	1	yes	characteristic, complemented, Sylow
187	C₂×#125	—	79	1	no	—
188	C₂×C₂²⋊Q₈	D₇₉	1	1	yes	characteristic
189	#399	—	79	1	no	—
190	#398	—	79	1	no	—
191	#396	—	79	1	no	—
192	#395	—	79	1	no	—
193	#391	—	79	1	no	—
194	#389	—	79	1	no	—
195	#389	—	79	1	no	—
196	#388	—	79	1	no	—
197	#388	—	79	1	no	—
198	#387	—	79	1	no	—
199	#385	—	79	1	no	—
200	C₄×C₄⋊C₄	—	79	1	no	—
201	C₄×C₂²⋊C₄	—	79	1	no	—
202	C₂²×Q₈	C₂×D₇₉	1	1	yes	characteristic
203	C₂³×C₄	C₂×D₇₉	1	1	yes	characteristic
204	#125	—	158	4	no	—
205	C₂²⋊Q₈	—	2	4	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	—	79	1	no	—
210	C₂×C₄⋊C₄	—	79	1	no	—
211	C₂×C₄⋊C₄	—	79	1	no	—
212	C₂×C₄⋊C₄	—	79	1	no	—
213	C₂×C₄⋊C₄	—	79	1	no	—
214	C₂×C₄⋊C₄	—	79	1	no	—
215	C₂×C₄⋊C₄	—	79	1	no	—
216	C₂×C₄⋊C₄	C₂×D₇₉	1	1	yes	characteristic
217	C₂×C₄⋊C₄	C₂×D₇₉	1	1	yes	characteristic
218	C₂×C₄⋊C₄	C₂×D₇₉	1	1	yes	characteristic
219	C₂×C₂²⋊C₄	—	79	1	no	—
220	C₂×C₂²⋊C₄	—	79	1	no	—
221	C₂×C₂²⋊C₄	—	79	1	no	—
222	C₂×C₂²⋊C₄	—	79	1	no	—
223	C₂×C₂²⋊C₄	C₂×D₇₉	1	1	yes	characteristic
224	C₂×C₂²⋊C₄	C₂×D₇₉	1	1	yes	characteristic
225	C₂×C₄²	—	79	1	no	—
226	C₂×C₄²	—	79	1	no	—
227	C₂×C₄²	—	79	1	no	—
228	C₂×C₄²	—	79	1	no	—
229	#95	—	79	1	no	—
230	#95	—	79	1	no	—
231	#95	—	79	1	no	—
232	#95	—	79	1	no	—
233	#95	—	79	1	no	—
234	#95	—	79	1	no	—
235	#95	—	79	1	no	—
236	#95	—	79	1	no	—
237	#95	—	79	1	no	—
238	#95	—	79	1	no	—
239	C₂⁴	C₂²×D₇₉	1	1	yes	characteristic
240	C₂×Q₈	—	2	2	no	—
241	C₂×Q₈	—	2	2	no	—
242	C₂×Q₈	—	4	1	no	—
243	C₂²×C₄	—	2	2	no	—
244	C₂²×C₄	—	2	2	no	—
245	C₂²×C₄	—	4	1	no	—
246	C₂²×C₄	—	79	1	no	—
247	C₂²×C₄	—	79	1	no	—
248	C₂²×C₄	—	79	1	no	—
249	C₂²×C₄	—	79	1	no	—
250	C₂²×C₄	—	79	1	no	—
251	C₂²×C₄	—	79	1	no	—
252	C₂²×C₄	—	79	1	no	—
253	C₂²×C₄	—	79	1	no	—
254	C₂²×C₄	C₂²×D₇₉	1	1	yes	characteristic
255	C₂²×C₄	C₂²×D₇₉	1	1	yes	characteristic
256	C₂²×C₄	C₂²×D₇₉	1	1	yes	characteristic
257	C₂²×C₄	C₂²×D₇₉	1	1	yes	characteristic
258	C₂²×C₄	C₂²×D₇₉	1	1	yes	characteristic
259	C₂²×C₄	C₂²×D₇₉	1	1	yes	characteristic
260	C₄⋊C₄	—	2	2	no	—
261	C₄⋊C₄	—	2	2	no	—
262	C₄⋊C₄	—	316	1	no	—
263	C₄⋊C₄	—	158	2	no	—
264	C₄⋊C₄	—	158	2	no	—
265	C₄⋊C₄	—	158	2	no	—
266	C₄⋊C₄	—	158	2	no	—
267	C₄⋊C₄	—	158	2	no	—
268	C₄⋊C₄	—	158	2	no	—
269	C₄⋊C₄	—	158	2	no	—
270	C₄⋊C₄	—	158	2	no	—
271	C₄⋊C₄	—	158	2	no	—
272	C₄⋊C₄	C₂²⋊D₇₉	1	4	yes	—
273	C₂²⋊C₄	—	2	2	no	—
274	C₂²⋊C₄	—	2	2	no	—
275	C₂²⋊C₄	—	316	1	no	—
276	C₂²⋊C₄	—	79	4	no	—
277	C₂²⋊C₄	—	158	2	no	—
278	C₂²⋊C₄	—	158	2	no	—
279	C₄²	—	316	1	no	—
280	C₄²	—	316	1	no	—
281	C₄²	—	158	2	no	—
282	C₄²	—	158	2	no	—
283	C₂³	—	2	1	no	—
284	C₂³	—	2	1	no	—
285	C₂³	—	2	1	no	—
286	C₂³	—	2	1	no	—
287	C₂³	—	2	1	no	—
288	C₂³	—	2	1	no	—
289	C₂³	Q₈×D₇₉	1	2	yes	—
290	C₂³	C₂³×D₇₉	1	1	yes	characteristic
291	Q₈	—	4	4	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	—	2	1	no	—
303	C₂×C₄	—	2	1	no	—
304	C₂×C₄	—	2	1	no	—
305	C₂×C₄	—	2	1	no	—
306	C₂×C₄	—	4	4	no	—
307	C₂×C₄	—	79	2	no	—
308	C₂×C₄	—	79	2	no	—
309	C₂×C₄	—	79	2	no	—
310	C₂×C₄	—	79	2	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	—	158	1	no	—
325	C₂×C₄	—	158	1	no	—
326	C₂×C₄	—	158	1	no	—
327	C₂×C₄	—	158	1	no	—
328	C₂×C₄	—	158	1	no	—
329	C₂×C₄	—	158	1	no	—
330	C₂×C₄	—	158	1	no	—
331	C₂×C₄	C₂×C₂²⋊D₇₉	1	2	yes	—
332	C₂×C₄	C₂×C₂²⋊D₇₉	1	2	yes	—
333	C₂×C₄	C₂×C₂²⋊D₇₉	1	2	yes	—
334	C₂×C₄	#11929623	1	2	yes	—
335	C₂²	—	2	2	no	—
336	C₂²	—	2	2	no	—
337	C₂²	—	2	2	no	—
338	C₂²	—	4	1	no	—
339	C₂²	—	4	1	no	—
340	C₂²	—	4	1	no	—
341	C₂²	—	4	1	no	—
342	C₂²	C₂²×C₂²⋊D₇₉	1	1	yes	characteristic
343	C₂²	C₂×Q₈×D₇₉	1	1	yes	characteristic
344	C₂²	#41×D₇₉	1	1	yes	characteristic
345	C₂²	#41×D₇₉	1	1	yes	characteristic
346	C₂²	C₂×#11929623	1	1	yes	characteristic
347	C₂²	#14579206	1	1	yes	characteristic
348	C₂²	#14579209	1	1	yes	characteristic
349	C₄	—	2	2	no	—
350	C₄	—	2	2	no	—
351	C₄	—	2	2	no	—
352	C₄	—	2	2	no	—
353	C₄	—	4	1	no	—
354	C₄	—	4	1	no	—
355	C₄	—	316	1	no	—
356	C₄	—	316	1	no	—
357	C₄	—	316	1	no	—
358	C₄	—	316	1	no	—
359	C₄	—	158	2	no	—
360	C₄	—	158	2	no	—
361	C₄	—	158	2	no	—
362	C₄	—	158	2	no	—
363	C₂	—	4	2	no	—
364	C₂	#19099109	1	1	yes	characteristic
365	C₂	#19099115	1	1	yes	characteristic
366	C₂	#19099159	1	1	yes	characteristic
367	C₂	#19099207	1	1	yes	characteristic
368	C₂	Q₈×C₂²⋊D₇₉	1	1	yes	characteristic
369	C₂	#19099247	1	1	yes	characteristic
370	C₂	#19099267	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₇₉
commutator, socle, cosocle	C₂³×C₇₉	C₂⁴
Fitting	C₂×C₂²⋊Q₈×C₇₉	C₂
prime radical	C₂⁴×C₇₉	C₂³
hypercenter	C₂×C₂²⋊Q₈	D₇₉
hypocenter	C₇₉	#2556
LCS	G ▷ C₂³×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₂²⋊Q₈

Complements

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

N	H ≅ G/N	type
1. C₂×#14579151	C₂	semidirect
3. #19098496	C₂	semidirect
5. #19098482	C₂	semidirect
6. #19098470	C₂	semidirect
7. #19098458	C₂	semidirect
8. #19098458	C₂	semidirect
11. #19098444	C₂	semidirect
13. C₄⋊C₄×Dic₇₉	C₂	semidirect
74. C₂²⋊Dic₇₉	Q₈	semidirect
186. C₇₉	#2556	semidirect