Subgroups — #23777892

Order 10112.

890 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇₉⋊#564	C₂	1	4	yes	maximal, complemented
2	#19099292	C₂	1	1	yes	maximal, characteristic, complemented
3	#19099291	C₂	1	2	yes	maximal, complemented
4	#19099291	C₂	1	1	yes	maximal, characteristic, complemented
5	#19099290	C₂	1	2	yes	maximal, complemented
6	#19099289	C₂	1	2	yes	maximal, complemented
7	#19099288	C₂	1	2	yes	maximal, complemented
8	#19099288	C₂	1	1	yes	maximal, characteristic, complemented
9	#19099176	C₂	1	2	yes	maximal
10	#19099174	C₂	1	2	yes	maximal
11	#19099172	C₂	1	4	yes	maximal
12	#19099171	C₂	1	2	yes	maximal
13	#19099171	C₂	1	2	yes	maximal
14	#19099118	C₂	1	1	yes	maximal, characteristic
15	#19099117	C₂	1	1	yes	maximal, characteristic
16	#19099117	C₂	1	1	yes	maximal, characteristic
17	#557×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
18	#14579161	C₂²	1	2	yes	—
19	#14579161	C₂²	1	1	yes	characteristic
20	#14579160	C₂²	1	4	yes	—
21	#14579160	C₂²	1	1	yes	characteristic
22	#14579160	C₂²	1	1	yes	characteristic
23	#14579159	C₂²	1	2	yes	—
24	#14579159	C₂²	1	2	yes	—
25	#14579159	C₂²	1	1	yes	characteristic
26	#14579159	C₂²	1	1	yes	characteristic
27	#14579156	C₂²	1	4	yes	—
28	#14579156	C₂²	1	4	yes	—
29	#14579156	C₂²	1	4	yes	—
30	#14579155	C₂²	1	4	yes	—
31	#14579153	C₂²	1	4	yes	—
32	#14579153	C₂²	1	4	yes	—
33	#14579153	C₂²	1	2	yes	—
34	#14579152	C₂²	1	2	yes	—
35	#14579152	C₂²	1	2	yes	—
36	#14579152	C₂²	1	4	yes	—
37	#14579152	C₂²	1	4	yes	—
38	#14579152	C₂²	1	4	yes	—
39	#14579152	C₂²	1	2	yes	—
40	#14579151	C₂²	1	4	yes	—
41	#14579151	C₂²	1	4	yes	—
42	#14579151	C₂²	1	4	yes	—
43	#14579151	C₂²	1	4	yes	—
44	#14579151	C₂²	1	4	yes	—
45	#14579151	C₂²	1	4	yes	—
46	#14579151	C₂²	1	4	yes	—
47	#14579151	C₂²	1	2	yes	—
48	#14579151	C₂²	1	2	yes	—
49	#14579141	C₂²	1	1	yes	characteristic
50	#14579141	C₂²	1	2	yes	—
51	#14579141	C₂²	1	1	yes	characteristic
52	#14579139	C₂²	1	2	yes	—
53	#14579139	C₂²	1	2	yes	—
54	#14579137	C₂²	1	1	yes	characteristic
55	#14579137	C₂²	1	2	yes	—
56	#14579137	C₂²	1	1	yes	characteristic
57	Q₈×Dic₇₉	C₂²	1	2	yes	—
58	Q₈×Dic₇₉	C₂²	1	2	yes	—
59	Q₈×Dic₇₉	C₂²	1	2	yes	—
60	C₄×Dic₁₅₈	C₂²	1	2	yes	—
61	C₄×Dic₁₅₈	C₂²	1	2	yes	—
62	C₄×Dic₁₅₈	C₂²	1	4	yes	—
63	C₄×Dic₁₅₈	C₂²	1	2	yes	—
64	C₄×Dic₁₅₈	C₂²	1	1	yes	characteristic
65	C₄×Dic₁₅₈	C₂²	1	1	yes	characteristic
66	C₄×Dic₁₅₈	C₂²	1	2	yes	—
67	#14579126	C₂²	1	4	yes	—
68	#14579126	C₂²	1	2	yes	—
69	#14579126	C₂²	1	2	yes	—
70	#14579126	C₂²	1	2	yes	—
71	#14579119	C₂²	1	2	yes	—
72	#14579118	C₂²	1	2	yes	—
73	#14579118	C₂²	1	2	yes	—
74	C₄²×D₇₉	C₂²	1	1	yes	characteristic
75	C₄⋊Q₈×C₇₉	C₂²	1	2	yes	—
76	C₄⋊Q₈×C₇₉	C₂²	1	1	yes	characteristic
77	#125×C₇₉	C₂²	1	4	yes	—
78	#125×C₇₉	C₂²	1	2	yes	—
79	C₄×Q₈×C₇₉	C₂²	1	2	yes	—
80	C₄×Q₈×C₇₉	C₂²	1	2	yes	—
81	C₄×Q₈×C₇₉	C₂²	1	1	yes	characteristic
82	C₄×Q₈×C₇₉	C₂²	1	1	yes	characteristic
83	C₂×Dic₁₅₈	C₂³	1	2	yes	—
84	C₂×Dic₁₅₈	C₂³	1	1	yes	characteristic
85	C₂×Dic₁₅₈	C₂³	1	2	yes	—
86	C₂×Dic₁₅₈	C₂³	1	2	yes	—
87	C₂×Dic₁₅₈	C₂³	1	1	yes	characteristic
88	C₂×C₄×D₇₉	C₂³	1	1	yes	characteristic
89	C₂×C₄×D₇₉	C₂³	1	1	yes	characteristic
90	C₂×C₄×D₇₉	C₂³	1	1	yes	characteristic
91	C₄⋊Dic₇₉	C₂³	1	2	yes	—
92	C₄⋊Dic₇₉	C₂³	1	1	yes	characteristic
93	C₄⋊Dic₇₉	C₂³	1	2	yes	—
94	C₄⋊Dic₇₉	C₂³	1	1	yes	characteristic
95	C₄⋊Dic₇₉	C₂³	1	2	yes	—
96	C₄⋊Dic₇₉	C₂³	1	2	yes	—
97	C₄⋊Dic₇₉	C₂³	1	4	yes	—
98	C₄⋊Dic₇₉	C₂³	1	4	yes	—
99	C₄⋊Dic₇₉	C₂³	1	4	yes	—
100	C₄⋊Dic₇₉	C₂³	1	4	yes	—
101	Dic₇₉⋊C₄	C₂³	1	2	yes	—
102	Dic₇₉⋊C₄	C₂³	1	2	yes	—
103	Dic₇₉⋊C₄	C₂³	1	2	yes	—
104	Dic₇₉⋊C₄	C₂³	1	2	yes	—
105	Dic₇₉⋊C₄	C₂³	1	2	yes	—
106	Dic₇₉⋊C₄	C₂³	1	4	yes	—
107	Dic₇₉⋊C₄	C₂³	1	4	yes	—
108	Dic₇₉⋊C₄	C₂³	1	4	yes	—
109	Dic₇₉⋊C₄	C₂³	1	4	yes	—
110	Dic₇₉⋊C₄	C₂³	1	2	yes	—
111	Dic₇₉⋊C₄	C₂³	1	2	yes	—
112	Dic₇₉⋊C₄	C₂³	1	2	yes	—
113	Dic₇₉⋊C₄	C₂³	1	2	yes	—
114	Dic₇₉⋊C₄	C₂³	1	2	yes	—
115	Dic₇₉⋊C₄	C₂³	1	4	yes	—
116	Dic₇₉⋊C₄	C₂³	1	4	yes	—
117	Dic₇₉⋊C₄	C₂³	1	4	yes	—
118	(C₂×D₇₉)⋊C₄	C₂³	1	2	yes	—
119	(C₂×D₇₉)⋊C₄	C₂³	1	2	yes	—
120	(C₂×D₇₉)⋊C₄	C₂³	1	2	yes	—
121	(C₂×D₇₉)⋊C₄	C₂³	1	2	yes	—
122	(C₂×D₇₉)⋊C₄	C₂³	1	4	yes	—
123	C₄×Dic₇₉	C₂³	1	1	yes	characteristic
124	C₄×Dic₇₉	C₂³	1	1	yes	characteristic
125	C₄×Dic₇₉	C₂³	1	1	yes	characteristic
126	C₄×Dic₇₉	C₂³	1	2	yes	—
127	C₄×Dic₇₉	C₂³	1	2	yes	—
128	C₄×Dic₇₉	C₂³	1	4	yes	—
129	C₄×Dic₇₉	C₂³	1	4	yes	—
130	C₄×Dic₇₉	C₂³	1	4	yes	—
131	C₄×Dic₇₉	C₂³	1	4	yes	—
132	C₂×Q₈×C₇₉	C₂³	1	1	yes	characteristic
133	C₂×Q₈×C₇₉	C₂³	1	2	yes	—
134	C₂×Q₈×C₇₉	C₂³	1	1	yes	characteristic
135	C₄⋊C₄×C₇₉	C₂³	1	1	yes	characteristic
136	C₄⋊C₄×C₇₉	C₂³	1	1	yes	characteristic
137	C₄⋊C₄×C₇₉	C₂³	1	2	yes	—
138	C₄⋊C₄×C₇₉	C₂³	1	2	yes	—
139	C₄⋊C₄×C₇₉	C₂³	1	4	yes	—
140	C₄⋊C₄×C₇₉	C₂³	1	2	yes	—
141	C₄⋊C₄×C₇₉	C₂³	1	4	yes	—
142	C₄⋊C₄×C₇₉	C₂³	1	2	yes	—
143	C₄⋊C₄×C₇₉	C₂³	1	4	yes	—
144	C₄²×C₇₉	C₂³	1	2	yes	—
145	C₄²×C₇₉	C₂³	1	4	yes	—
146	C₄²×C₇₉	C₂³	1	1	yes	characteristic
147	C₄²×C₇₉	C₂³	1	2	yes	—
148	C₂²×D₇₉	C₂⁴	1	1	yes	characteristic
149	Q₈×C₇₉	—	2	4	no	—
150	Q₈×C₇₉	—	2	2	no	—
151	Q₈×C₇₉	—	2	2	no	—
152	C₂×C₄×C₇₉	C₂⁴	1	4	yes	—
153	C₂×C₄×C₇₉	C₂⁴	1	2	yes	—
154	C₂×C₄×C₇₉	C₂⁴	1	2	yes	—
155	C₂×C₄×C₇₉	C₂⁴	1	1	yes	characteristic
156	C₂×C₄×C₇₉	C₂⁴	1	2	yes	—
157	C₂×C₄×C₇₉	C₂⁴	1	2	yes	—
158	C₂×C₄×C₇₉	C₂⁴	1	1	yes	characteristic
159	C₂×C₄×C₇₉	C₂⁴	1	1	yes	characteristic
160	C₂×Dic₇₉	C₂⁴	1	2	yes	—
161	C₂×Dic₇₉	C₂⁴	1	2	yes	—
162	C₂×Dic₇₉	C₂⁴	1	2	yes	—
163	C₂×Dic₇₉	C₂⁴	1	2	yes	—
164	C₂×Dic₇₉	C₂⁴	1	4	yes	—
165	C₂×Dic₇₉	C₂⁴	1	1	yes	characteristic
166	C₂×Dic₇₉	C₂⁴	1	1	yes	characteristic
167	C₂×Dic₇₉	C₂⁴	1	1	yes	characteristic
168	C₄×D₇₉	—	2	2	no	—
169	C₄×D₇₉	—	2	2	no	—
170	C₄×D₇₉	—	2	2	no	—
171	Dic₁₅₈	—	2	2	no	—
172	Dic₁₅₈	—	2	2	no	—
173	Dic₁₅₈	—	2	4	no	—
174	Dic₁₅₈	—	2	4	no	—
175	Dic₁₅₈	—	2	4	no	—
176	C₂²×C₇₉	C₂⁵	1	1	yes	characteristic
177	C₂×D₇₉	—	2	1	no	—
178	C₂×D₇₉	—	2	1	no	—
179	C₂×D₇₉	—	2	1	no	—
180	C₄×C₇₉	—	2	4	no	—
181	C₄×C₇₉	—	2	2	no	—
182	C₄×C₇₉	—	2	2	no	—
183	C₄×C₇₉	—	2	2	no	—
184	C₄×C₇₉	—	2	2	no	—
185	C₄×C₇₉	—	2	1	no	—
186	C₄×C₇₉	2₋¹⁺⁴	1	2	yes	—
187	C₄×C₇₉	2₋¹⁺⁴	1	2	yes	—
188	Dic₇₉	—	2	2	no	—
189	Dic₇₉	—	2	2	no	—
190	Dic₇₉	—	2	2	no	—
191	Dic₇₉	—	2	2	no	—
192	Dic₇₉	—	2	4	no	—
193	Dic₇₉	—	2	1	no	—
194	Dic₇₉	—	2	1	no	—
195	Dic₇₉	2₋¹⁺⁴	1	2	yes	—
196	C₂×C₇₉	C₂×2₋¹⁺⁴	1	1	yes	characteristic
197	C₂×C₇₉	C₂×2₋¹⁺⁴	1	1	yes	characteristic
198	C₂×C₇₉	C₂×2₋¹⁺⁴	1	1	yes	characteristic
199	D₇₉	—	4	1	no	—
200	#3556	—	79	1	no	maximal, Sylow
201	C₇₉	#3556	1	1	yes	characteristic, complemented, Sylow
202	#564	—	79	4	no	—
203	#557	—	79	2	no	—
204	#557	—	79	1	no	—
205	#557	—	79	2	no	—
206	#557	—	79	2	no	—
207	#557	—	79	2	no	—
208	#557	—	79	1	no	—
209	#557	—	79	1	no	—
210	#557	D₇₉	1	1	yes	characteristic, complemented
211	#541	—	79	2	no	—
212	#541	—	79	4	no	—
213	#541	—	79	2	no	—
214	#541	—	79	2	no	—
215	#541	—	79	2	no	—
216	#533	—	79	1	no	—
217	#533	—	79	1	no	—
218	#533	—	79	1	no	—
219	C₄⋊Q₈	—	79	1	no	—
220	C₄⋊Q₈	—	79	1	no	—
221	C₄⋊Q₈	—	79	2	no	—
222	C₄⋊Q₈	—	79	2	no	—
223	C₄⋊Q₈	—	79	1	no	—
224	C₄⋊Q₈	—	79	1	no	—
225	C₄⋊Q₈	—	79	4	no	—
226	C₄⋊Q₈	—	79	1	no	—
227	C₄⋊Q₈	—	79	2	no	—
228	C₄⋊Q₈	C₂×D₇₉	1	1	yes	characteristic
229	C₄⋊Q₈	C₂×D₇₉	1	2	yes	—
230	#126	—	79	4	no	—
231	#126	—	79	4	no	—
232	#126	—	79	4	no	—
233	#126	—	79	4	no	—
234	#125	—	79	2	no	—
235	#125	—	79	2	no	—
236	#125	—	79	2	no	—
237	#125	—	79	4	no	—
238	#125	—	79	2	no	—
239	#125	—	79	2	no	—
240	#125	—	79	4	no	—
241	#125	—	79	4	no	—
242	#125	—	79	4	no	—
243	#125	—	79	4	no	—
244	#125	—	79	4	no	—
245	#125	—	79	4	no	—
246	#125	—	79	4	no	—
247	#125	—	79	4	no	—
248	#125	—	79	4	no	—
249	#125	—	79	2	no	—
250	#125	—	79	4	no	—
251	#125	—	79	4	no	—
252	#125	C₂×D₇₉	1	2	yes	—
253	#125	C₂×D₇₉	1	4	yes	—
254	C₂²⋊Q₈	—	79	2	no	—
255	C₂²⋊Q₈	—	79	1	no	—
256	C₂²⋊Q₈	—	79	1	no	—
257	C₂²⋊Q₈	—	79	2	no	—
258	C₂²⋊Q₈	—	79	2	no	—
259	C₂²⋊Q₈	—	79	2	no	—
260	C₂²⋊Q₈	—	79	1	no	—
261	C₂²⋊Q₈	—	79	1	no	—
262	C₄×Q₈	—	79	2	no	—
263	C₄×Q₈	—	79	2	no	—
264	C₄×Q₈	—	79	2	no	—
265	C₄×Q₈	—	79	2	no	—
266	C₄×Q₈	—	79	4	no	—
267	C₄×Q₈	—	79	2	no	—
268	C₄×Q₈	—	79	2	no	—
269	C₄×Q₈	—	79	2	no	—
270	C₄×Q₈	—	79	2	no	—
271	C₄×Q₈	—	79	1	no	—
272	C₄×Q₈	—	79	1	no	—
273	C₄×Q₈	—	79	4	no	—
274	C₄×Q₈	—	79	2	no	—
275	C₄×Q₈	—	79	2	no	—
276	C₄×Q₈	C₂×D₇₉	1	1	yes	characteristic
277	C₄×Q₈	C₂×D₇₉	1	1	yes	characteristic
278	C₄×Q₈	C₂×D₇₉	1	2	yes	—
279	C₄×Q₈	C₂×D₇₉	1	2	yes	—
280	#117	—	79	2	no	—
281	#117	—	79	2	no	—
282	#117	—	79	2	no	—
283	C₂×C₄²	—	79	1	no	—
284	C₂×Q₈	—	79	2	no	—
285	C₂×Q₈	—	79	2	no	—
286	C₂×Q₈	—	79	1	no	—
287	C₂×Q₈	—	79	2	no	—
288	C₂×Q₈	—	79	1	no	—
289	C₂×Q₈	C₂²×D₇₉	1	2	yes	—
290	C₂×Q₈	C₂²×D₇₉	1	1	yes	characteristic
291	C₂×Q₈	C₂²×D₇₉	1	1	yes	characteristic
292	C₂²×C₄	—	79	1	no	—
293	C₂²×C₄	—	79	1	no	—
294	C₂²×C₄	—	79	1	no	—
295	C₄⋊C₄	—	79	2	no	—
296	C₄⋊C₄	—	79	2	no	—
297	C₄⋊C₄	—	79	2	no	—
298	C₄⋊C₄	—	79	2	no	—
299	C₄⋊C₄	—	79	2	no	—
300	C₄⋊C₄	—	79	2	no	—
301	C₄⋊C₄	—	79	2	no	—
302	C₄⋊C₄	—	79	2	no	—
303	C₄⋊C₄	—	79	4	no	—
304	C₄⋊C₄	—	79	2	no	—
305	C₄⋊C₄	—	79	2	no	—
306	C₄⋊C₄	—	79	4	no	—
307	C₄⋊C₄	—	79	4	no	—
308	C₄⋊C₄	—	79	4	no	—
309	C₄⋊C₄	—	79	4	no	—
310	C₄⋊C₄	—	79	4	no	—
311	C₄⋊C₄	—	79	4	no	—
312	C₄⋊C₄	—	79	2	no	—
313	C₄⋊C₄	—	79	2	no	—
314	C₄⋊C₄	—	79	1	no	—
315	C₄⋊C₄	—	79	1	no	—
316	C₄⋊C₄	—	79	4	no	—
317	C₄⋊C₄	—	79	2	no	—
318	C₄⋊C₄	—	79	2	no	—
319	C₄⋊C₄	—	79	4	no	—
320	C₄⋊C₄	—	79	4	no	—
321	C₄⋊C₄	—	79	4	no	—
322	C₄⋊C₄	C₂²×D₇₉	1	2	yes	—
323	C₄⋊C₄	C₂²×D₇₉	1	2	yes	—
324	C₄⋊C₄	C₂²×D₇₉	1	1	yes	characteristic
325	C₄⋊C₄	C₂²×D₇₉	1	4	yes	—
326	C₄⋊C₄	C₂²×D₇₉	1	1	yes	characteristic
327	C₄⋊C₄	C₂²×D₇₉	1	2	yes	—
328	C₄⋊C₄	C₂²×D₇₉	1	2	yes	—
329	C₄⋊C₄	C₂²×D₇₉	1	4	yes	—
330	C₄⋊C₄	C₂²×D₇₉	1	4	yes	—
331	C₂²⋊C₄	—	79	2	no	—
332	C₂²⋊C₄	—	79	4	no	—
333	C₂²⋊C₄	—	79	2	no	—
334	C₂²⋊C₄	—	79	2	no	—
335	C₂²⋊C₄	—	79	2	no	—
336	C₄²	—	79	1	no	—
337	C₄²	—	79	1	no	—
338	C₄²	—	79	1	no	—
339	C₄²	—	79	4	no	—
340	C₄²	—	79	2	no	—
341	C₄²	—	79	2	no	—
342	C₄²	—	79	4	no	—
343	C₄²	—	79	4	no	—
344	C₄²	—	79	4	no	—
345	C₄²	C₂²×D₇₉	1	1	yes	characteristic
346	C₄²	C₂²×D₇₉	1	2	yes	—
347	C₄²	C₂²×D₇₉	1	2	yes	—
348	C₄²	C₂²×D₇₉	1	4	yes	—
349	C₂³	—	79	1	no	—
350	Q₈	—	2	4	no	—
351	Q₈	—	2	2	no	—
352	Q₈	—	2	2	no	—
353	Q₈	—	158	4	no	—
354	Q₈	—	158	4	no	—
355	Q₈	—	158	2	no	—
356	Q₈	—	158	4	no	—
357	Q₈	—	158	2	no	—
358	C₂×C₄	—	79	1	no	—
359	C₂×C₄	—	79	2	no	—
360	C₂×C₄	—	79	2	no	—
361	C₂×C₄	—	79	2	no	—
362	C₂×C₄	—	79	2	no	—
363	C₂×C₄	—	79	4	no	—
364	C₂×C₄	—	79	1	no	—
365	C₂×C₄	—	79	1	no	—
366	C₂×C₄	—	158	2	no	—
367	C₂×C₄	—	158	2	no	—
368	C₂×C₄	—	158	2	no	—
369	C₂×C₄	C₂³×D₇₉	1	4	yes	—
370	C₂×C₄	C₂³×D₇₉	1	2	yes	—
371	C₂×C₄	C₂³×D₇₉	1	2	yes	—
372	C₂×C₄	C₂³×D₇₉	1	1	yes	characteristic
373	C₂×C₄	C₂³×D₇₉	1	2	yes	—
374	C₂×C₄	C₂³×D₇₉	1	2	yes	—
375	C₂×C₄	C₂³×D₇₉	1	1	yes	characteristic
376	C₂×C₄	C₂³×D₇₉	1	1	yes	characteristic
377	C₂²	—	158	1	no	—
378	C₂²	—	158	1	no	—
379	C₂²	—	158	1	no	—
380	C₂²	C₂⁴×D₇₉	1	1	yes	characteristic
381	C₄	—	2	1	no	—
382	C₄	—	2	2	no	—
383	C₄	—	2	2	no	—
384	C₄	—	2	4	no	—
385	C₄	—	2	2	no	—
386	C₄	—	2	2	no	—
387	C₄	—	79	2	no	—
388	C₄	—	158	2	no	—
389	C₄	—	158	2	no	—
390	C₄	—	158	1	no	—
391	C₄	—	158	2	no	—
392	C₄	—	158	1	no	—
393	C₄	—	158	2	no	—
394	C₄	—	158	4	no	—
395	C₄	#14579208	1	2	yes	—
396	C₄	#14579209	1	2	yes	—
397	C₂	—	316	1	no	—
398	C₂	2₋¹⁺⁴×D₇₉	1	1	yes	characteristic
399	C₂	C₂×#14579208	1	1	yes	characteristic
400	C₂	C₂×#14579209	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₇₉
commutator, socle, cosocle	C₂²×C₇₉	C₂⁵
Fitting	#557×C₇₉	C₂
prime radical	C₂²×D₇₉	C₂⁴
hypercenter	#557	D₇₉
hypocenter	C₇₉	#3556
LCS	G ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ #557

Complements

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

N	H ≅ G/N	type
1. C₇₉⋊#564	C₂	semidirect
2. #19099292	C₂	semidirect
3. #19099291	C₂	semidirect
4. #19099291	C₂	semidirect
5. #19099290	C₂	semidirect
6. #19099289	C₂	semidirect
7. #19099288	C₂	semidirect
8. #19099288	C₂	semidirect
17. #557×C₇₉	C₂	semidirect
201. C₇₉	#3556	semidirect
210. #557	D₇₉	semidirect