Subgroups — #23766918

Order 10112.

500 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×#14579138	C₂	1	1	yes	maximal, characteristic
2	C₂×#14579126	C₂	1	1	yes	maximal, characteristic, complemented
3	#19098499	C₂	1	1	yes	maximal, characteristic
4	#19098496	C₂	1	1	yes	maximal, characteristic, complemented
5	#19098494	C₂	1	1	yes	maximal, characteristic, complemented
6	#19098467	C₂	1	1	yes	maximal, characteristic, complemented
7	#19098458	C₂	1	1	yes	maximal, characteristic, complemented
8	#19098457	C₂	1	1	yes	maximal, characteristic
9	#19098451	C₂	1	1	yes	maximal, characteristic
10	#19098449	C₂	1	1	yes	maximal, characteristic, complemented
11	#19098448	C₂	1	1	yes	maximal, characteristic, complemented
12	#19098432	C₂	1	1	yes	maximal, characteristic, complemented
13	#19098421	C₂	1	1	yes	maximal, characteristic
14	C₂²⋊C₄×Dic₇₉	C₂	1	1	yes	maximal, characteristic
15	#399×C₇₉	C₂	1	1	yes	maximal, characteristic
16	C₂²×Dic₁₅₈	C₂²	1	1	yes	characteristic
17	C₂³×Dic₇₉	C₂²	1	1	yes	characteristic
18	#14579138	—	2	4	no	—
19	#14579126	—	2	4	no	—
20	C₂×C₄⋊Dic₇₉	C₂²	1	1	yes	characteristic
21	C₂×C₄⋊Dic₇₉	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×Dic₇₉⋊C₄	C₂²	1	1	yes	characteristic
28	C₂×Dic₇₉⋊C₄	C₂²	1	1	yes	characteristic
29	C₂×Dic₇₉⋊C₄	C₂²	1	1	yes	characteristic
30	C₂×C₂²⋊Dic₇₉	C₂²	1	1	yes	characteristic
31	C₂×C₂²⋊Dic₇₉	C₂²	1	1	yes	characteristic
32	C₂×C₂²⋊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	C₂×C₄×Dic₇₉	C₂²	1	1	yes	characteristic
36	C₂×C₄×Dic₇₉	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×C₄⋊C₄×C₇₉	C₂²	1	1	yes	characteristic
49	C₂×C₂²⋊C₄×C₇₉	C₂²	1	1	yes	characteristic
50	C₂×C₂²⋊C₄×C₇₉	C₂²	1	1	yes	characteristic
51	C₂×C₂²⋊C₄×C₇₉	C₂²	1	1	yes	characteristic
52	#95×C₇₉	C₂²	1	1	yes	characteristic
53	C₂×Dic₁₅₈	—	2	2	no	—
54	C₂×Dic₁₅₈	—	2	2	no	—
55	C₂×Dic₁₅₈	D₄	1	4	yes	—
56	C₂²×Dic₇₉	—	2	2	no	—
57	C₂²×Dic₇₉	—	2	2	no	—
58	C₂²×Dic₇₉	—	2	2	no	—
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	C₂²×Dic₇₉	C₂³	1	1	yes	characteristic
65	C₂²×Dic₇₉	C₂³	1	1	yes	characteristic
66	C₂²×Dic₇₉	C₂³	1	1	yes	characteristic
67	C₄⋊Dic₇₉	—	2	2	no	—
68	C₄⋊Dic₇₉	—	2	2	no	—
69	Dic₇₉⋊C₄	—	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	Dic₇₉⋊C₄	—	4	1	no	—
75	Dic₇₉⋊C₄	—	4	1	no	—
76	Dic₇₉⋊C₄	—	4	1	no	—
77	C₂²⋊Dic₇₉	—	2	2	no	—
78	C₂²⋊Dic₇₉	—	4	1	no	—
79	C₂²⋊Dic₇₉	Q₈	1	4	yes	complemented
80	C₄×Dic₇₉	—	2	2	no	—
81	C₄×Dic₇₉	—	2	2	no	—
82	C₄×Dic₇₉	—	2	2	no	—
83	C₄×Dic₇₉	—	2	2	no	—
84	C₂⁴×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	C₂²×C₄×C₇₉	C₂³	1	1	yes	characteristic
91	C₄⋊C₄×C₇₉	—	2	2	no	—
92	C₄⋊C₄×C₇₉	—	2	2	no	—
93	C₄⋊C₄×C₇₉	—	4	1	no	—
94	C₂²⋊C₄×C₇₉	—	2	2	no	—
95	C₂²⋊C₄×C₇₉	—	2	2	no	—
96	C₂²⋊C₄×C₇₉	—	4	1	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	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	—	2	1	no	—
120	C₂×C₄×C₇₉	—	2	1	no	—
121	C₂×C₄×C₇₉	C₂×D₄	1	2	yes	—
122	C₂×C₄×C₇₉	#41	1	2	yes	—
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	—	2	1	no	—
141	C₂×Dic₇₉	—	4	4	no	—
142	C₂×Dic₇₉	C₂×D₄	1	2	yes	—
143	C₂×Dic₇₉	C₂×D₄	1	2	yes	—
144	C₂×Dic₇₉	C₂×Q₈	1	2	yes	—
145	C₂×Dic₇₉	C₂×Q₈	1	2	yes	—
146	C₂×Dic₇₉	#41	1	2	yes	—
147	C₂×Dic₇₉	#41	1	2	yes	—
148	Dic₁₅₈	—	2	8	no	—
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	—	4	1	no	—
166	C₄×C₇₉	—	4	1	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	—	4	1	no	—
173	Dic₇₉	—	4	1	no	—
174	Dic₇₉	—	4	1	no	—
175	Dic₇₉	—	4	1	no	—
176	Dic₇₉	#123	1	4	yes	—
177	C₂×C₇₉	—	4	2	no	—
178	C₂×C₇₉	C₂×#123	1	1	yes	characteristic
179	C₂×C₇₉	#533	1	1	yes	characteristic
180	C₂×C₇₉	#542	1	1	yes	characteristic
181	C₂×C₇₉	D₄×Q₈	1	1	yes	characteristic
182	C₂×C₇₉	#550	1	1	yes	characteristic
183	C₂×C₇₉	#552	1	1	yes	characteristic
184	C₂×C₇₉	#554	1	1	yes	characteristic
185	#2423	—	79	1	no	maximal, Sylow
186	C₇₉	#2423	1	1	yes	characteristic, complemented, Sylow
187	C₂×C₂²⋊Q₈	—	79	1	no	—
188	C₂×C₄×Q₈	—	79	1	no	—
189	#399	—	79	1	no	—
190	#399	D₇₉	1	1	yes	characteristic
191	#398	—	79	1	no	—
192	#398	—	79	1	no	—
193	#391	—	79	1	no	—
194	#389	—	79	1	no	—
195	#388	—	79	1	no	—
196	#388	—	79	1	no	—
197	#387	—	79	1	no	—
198	#387	—	79	1	no	—
199	#384	—	79	1	no	—
200	#380	—	79	1	no	—
201	C₄×C₂²⋊C₄	—	79	1	no	—
202	C₂²×Q₈	—	79	1	no	—
203	C₂³×C₄	—	79	1	no	—
204	C₂²⋊Q₈	—	158	4	no	—
205	C₄×Q₈	—	158	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	C₂×D₇₉	1	1	yes	characteristic
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	C₂×D₇₉	1	1	yes	characteristic
239	C₂⁴	C₂²×D₇₉	1	1	yes	characteristic
240	C₂×Q₈	—	79	4	no	—
241	C₂×Q₈	—	158	2	no	—
242	C₂×Q₈	—	158	2	no	—
243	C₂²×C₄	—	79	1	no	—
244	C₂²×C₄	—	79	1	no	—
245	C₂²×C₄	—	79	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	—	158	2	no	—
252	C₂²×C₄	—	158	2	no	—
253	C₂²×C₄	—	158	2	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	—	4	1	no	—
263	C₄⋊C₄	—	316	1	no	—
264	C₄⋊C₄	—	316	1	no	—
265	C₄⋊C₄	—	316	1	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	—	158	2	no	—
273	C₂²⋊C₄	—	2	2	no	—
274	C₂²⋊C₄	—	2	2	no	—
275	C₂²⋊C₄	—	4	1	no	—
276	C₂²⋊C₄	—	316	1	no	—
277	C₂²⋊C₄	—	79	4	no	—
278	C₂²⋊C₄	—	158	2	no	—
279	C₄²	—	158	2	no	—
280	C₄²	—	158	2	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	—	158	8	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	—	2	1	no	—
307	C₂×C₄	—	2	1	no	—
308	C₂×C₄	—	316	4	no	—
309	C₂×C₄	—	79	2	no	—
310	C₂×C₄	—	79	2	no	—
311	C₂×C₄	—	79	2	no	—
312	C₂×C₄	—	79	2	no	—
313	C₂×C₄	—	79	2	no	—
314	C₂×C₄	—	79	2	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	—	158	1	no	—
332	C₂×C₄	—	158	1	no	—
333	C₂×C₄	D₄×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	C₂×D₄×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	#14579208	1	1	yes	characteristic
349	C₄	—	2	2	no	—
350	C₄	—	2	2	no	—
351	C₄	—	4	1	no	—
352	C₄	—	4	1	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	—	79	4	no	—
360	C₄	—	158	2	no	—
361	C₄	—	158	2	no	—
362	C₄	—	158	2	no	—
363	C₂	—	4	2	no	—
364	C₂	#123×D₇₉	1	1	yes	characteristic
365	C₂	#19099115	1	1	yes	characteristic
366	C₂	#19099181	1	1	yes	characteristic
367	C₂	#19099226	1	1	yes	characteristic
368	C₂	#19099233	1	1	yes	characteristic
369	C₂	#19099250	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	#399×C₇₉	C₂
prime radical	C₂⁴×C₇₉	C₂³
hypercenter	#399	D₇₉
hypocenter	C₇₉	#2423
LCS	G ▷ C₂³×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ #399

Complements

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

N	H ≅ G/N	type
2. C₂×#14579126	C₂	semidirect
4. #19098496	C₂	semidirect
5. #19098494	C₂	semidirect
6. #19098467	C₂	semidirect
7. #19098458	C₂	semidirect
10. #19098449	C₂	semidirect
11. #19098448	C₂	semidirect
12. #19098432	C₂	semidirect
79. C₂²⋊Dic₇₉	Q₈	semidirect
186. C₇₉	#2423	semidirect