Subgroups — C₇₉×#3258 #23760880

Order 10112.

334 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#542×C₇₉	C₂	1	1	yes	maximal, characteristic
2	#541×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
3	#479×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
4	#478×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
5	#477×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
6	#475×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
7	#470×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
8	#468×C₇₉	C₂	1	1	yes	maximal, characteristic
9	#467×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
10	#465×C₇₉	C₂	1	1	yes	maximal, characteristic
11	#464×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
12	#464×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
13	#462×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
14	#461×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
15	#424×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×Q₁₆×C₇₉	C₂²	1	1	yes	characteristic
17	C₂×Q₁₆×C₇₉	C₂²	1	1	yes	characteristic
18	C₂×SD₁₆×C₇₉	C₂²	1	1	yes	characteristic
19	C₂×SD₁₆×C₇₉	C₂²	1	1	yes	characteristic
20	C₂×M₁₆×C₇₉	C₂²	1	1	yes	characteristic
21	C₂²×C₈×C₇₉	C₂²	1	1	yes	characteristic
22	C₄⋊Q₈×C₇₉	C₂²	1	1	yes	characteristic
23	C₄⋊Q₈×C₇₉	C₂²	1	1	yes	characteristic
24	#126×C₇₉	—	2	1	no	—
25	#126×C₇₉	—	2	1	no	—
26	#126×C₇₉	—	2	1	no	—
27	#125×C₇₉	—	2	1	no	—
28	#125×C₇₉	—	2	1	no	—
29	#125×C₇₉	C₂²	1	1	yes	characteristic
30	#124×C₇₉	—	2	1	no	—
31	#124×C₇₉	C₂²	1	1	yes	characteristic
32	#123×C₇₉	—	2	1	no	—
33	C₂²⋊Q₈×C₇₉	—	2	1	no	—
34	C₂²⋊Q₈×C₇₉	C₂²	1	1	yes	characteristic
35	C₂²⋊Q₈×C₇₉	C₂²	1	1	yes	characteristic
36	C₂²⋊Q₈×C₇₉	C₂²	1	1	yes	characteristic
37	#121×C₇₉	C₂²	1	1	yes	characteristic
38	C₄×Q₈×C₇₉	C₂²	1	1	yes	characteristic
39	C₄×Q₈×C₇₉	C₂²	1	1	yes	characteristic
40	C₄×Q₈×C₇₉	C₂²	1	1	yes	characteristic
41	C₄×D₄×C₇₉	C₂²	1	1	yes	characteristic
42	#117×C₇₉	C₂²	1	1	yes	characteristic
43	#107×C₇₉	C₂²	1	1	yes	characteristic
44	#107×C₇₉	C₂²	1	1	yes	characteristic
45	#106×C₇₉	C₂²	1	1	yes	characteristic
46	#106×C₇₉	C₂²	1	1	yes	characteristic
47	C₄⋊C₈×C₇₉	C₂²	1	1	yes	characteristic
48	C₄⋊C₈×C₇₉	C₂²	1	1	yes	characteristic
49	C₄⋊C₈×C₇₉	C₂²	1	1	yes	characteristic
50	C₄⋊C₈×C₇₉	C₂²	1	1	yes	characteristic
51	#103×C₇₉	C₂²	1	1	yes	characteristic
52	#103×C₇₉	C₂²	1	1	yes	characteristic
53	#103×C₇₉	C₂²	1	1	yes	characteristic
54	#103×C₇₉	C₂²	1	1	yes	characteristic
55	#103×C₇₉	C₂²	1	1	yes	characteristic
56	#103×C₇₉	C₂²	1	1	yes	characteristic
57	#102×C₇₉	C₂²	1	1	yes	characteristic
58	#102×C₇₉	C₂²	1	1	yes	characteristic
59	C₂×Q₈×C₇₉	—	2	1	no	—
60	C₂×Q₈×C₇₉	C₂³	1	1	yes	characteristic
61	C₂×Q₈×C₇₉	C₂³	1	1	yes	characteristic
62	C₂×Q₈×C₇₉	C₂³	1	1	yes	characteristic
63	C₂×D₄×C₇₉	—	2	1	no	—
64	C₂×D₄×C₇₉	C₂³	1	1	yes	characteristic
65	C₂²×C₄×C₇₉	—	2	1	no	—
66	C₂²×C₄×C₇₉	C₂³	1	1	yes	characteristic
67	Q₁₆×C₇₉	—	4	1	no	—
68	Q₁₆×C₇₉	—	4	1	no	—
69	SD₁₆×C₇₉	—	4	1	no	—
70	SD₁₆×C₇₉	—	4	1	no	—
71	M₁₆×C₇₉	—	4	1	no	—
72	C₂×C₈×C₇₉	—	4	1	no	—
73	C₂×C₈×C₇₉	C₂³	1	1	yes	characteristic
74	C₂×C₈×C₇₉	C₂³	1	1	yes	characteristic
75	C₂×C₈×C₇₉	C₂³	1	1	yes	characteristic
76	C₂×C₈×C₇₉	C₂³	1	1	yes	characteristic
77	C₄⋊C₄×C₇₉	—	2	1	no	—
78	C₄⋊C₄×C₇₉	—	2	1	no	—
79	C₄⋊C₄×C₇₉	—	2	1	no	—
80	C₄⋊C₄×C₇₉	—	2	1	no	—
81	C₄⋊C₄×C₇₉	—	2	1	no	—
82	C₄⋊C₄×C₇₉	—	2	1	no	—
83	C₄⋊C₄×C₇₉	—	2	1	no	—
84	C₄⋊C₄×C₇₉	—	2	1	no	—
85	C₄⋊C₄×C₇₉	—	2	1	no	—
86	C₄⋊C₄×C₇₉	—	2	1	no	—
87	C₄⋊C₄×C₇₉	—	2	1	no	—
88	C₄⋊C₄×C₇₉	D₄	1	1	yes	characteristic
89	C₄⋊C₄×C₇₉	D₄	1	1	yes	characteristic
90	C₄⋊C₄×C₇₉	C₂³	1	1	yes	characteristic
91	C₄⋊C₄×C₇₉	C₂³	1	1	yes	characteristic
92	C₄⋊C₄×C₇₉	C₂³	1	1	yes	characteristic
93	C₄⋊C₄×C₇₉	C₂³	1	1	yes	characteristic
94	C₂²⋊C₄×C₇₉	—	2	1	no	—
95	C₂²⋊C₄×C₇₉	—	2	1	no	—
96	C₂²⋊C₄×C₇₉	—	2	1	no	—
97	C₂²⋊C₄×C₇₉	—	2	1	no	—
98	C₂²⋊C₄×C₇₉	—	2	1	no	—
99	C₂²⋊C₄×C₇₉	—	2	1	no	—
100	C₂²⋊C₄×C₇₉	—	2	1	no	—
101	C₂²⋊C₄×C₇₉	D₄	1	1	yes	characteristic
102	C₂²⋊C₄×C₇₉	D₄	1	1	yes	characteristic
103	C₄²×C₇₉	—	2	1	no	—
104	C₄²×C₇₉	—	2	1	no	—
105	C₄²×C₇₉	—	2	1	no	—
106	C₄²×C₇₉	C₂³	1	1	yes	characteristic
107	C₄²×C₇₉	C₂³	1	1	yes	characteristic
108	C₂³×C₇₉	—	2	1	no	—
109	C₂³×C₇₉	C₂×D₄	1	1	yes	characteristic
110	Q₈×C₇₉	—	2	1	no	—
111	Q₈×C₇₉	—	2	1	no	—
112	Q₈×C₇₉	—	2	1	no	—
113	Q₈×C₇₉	—	2	1	no	—
114	Q₈×C₇₉	—	2	1	no	—
115	Q₈×C₇₉	—	2	1	no	—
116	Q₈×C₇₉	—	8	1	no	—
117	D₄×C₇₉	—	2	1	no	—
118	D₄×C₇₉	—	2	1	no	—
119	D₄×C₇₉	—	8	1	no	—
120	C₂×C₄×C₇₉	—	2	1	no	—
121	C₂×C₄×C₇₉	—	2	1	no	—
122	C₂×C₄×C₇₉	—	2	1	no	—
123	C₂×C₄×C₇₉	—	2	1	no	—
124	C₂×C₄×C₇₉	—	2	1	no	—
125	C₂×C₄×C₇₉	—	2	1	no	—
126	C₂×C₄×C₇₉	—	2	1	no	—
127	C₂×C₄×C₇₉	—	2	1	no	—
128	C₂×C₄×C₇₉	—	2	1	no	—
129	C₂×C₄×C₇₉	—	8	1	no	—
130	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
131	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
132	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
133	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
134	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
135	C₂×C₄×C₇₉	C₂⁴	1	1	yes	characteristic
136	C₈×C₇₉	—	2	1	no	—
137	C₈×C₇₉	—	2	1	no	—
138	C₈×C₇₉	—	2	1	no	—
139	C₈×C₇₉	—	2	1	no	—
140	C₂²×C₇₉	—	2	1	no	—
141	C₂²×C₇₉	—	2	1	no	—
142	C₂²×C₇₉	—	2	1	no	—
143	C₂²×C₇₉	—	4	1	no	—
144	C₂²×C₇₉	—	4	1	no	—
145	C₂²×C₇₉	—	4	1	no	—
146	C₂²×C₇₉	C₂²×D₄	1	1	yes	characteristic
147	C₄×C₇₉	—	2	1	no	—
148	C₄×C₇₉	—	2	1	no	—
149	C₄×C₇₉	—	2	1	no	—
150	C₄×C₇₉	—	2	1	no	—
151	C₄×C₇₉	—	2	1	no	—
152	C₄×C₇₉	—	4	1	no	—
153	C₄×C₇₉	—	4	1	no	—
154	C₄×C₇₉	—	4	1	no	—
155	C₄×C₇₉	—	4	1	no	—
156	C₄×C₇₉	—	4	1	no	—
157	C₄×C₇₉	—	4	1	no	—
158	C₄×C₇₉	—	4	1	no	—
159	C₄×C₇₉	H(5,𝔽₂)	1	1	yes	characteristic
160	C₄×C₇₉	2₋¹⁺⁴	1	1	yes	characteristic
161	C₂×C₇₉	—	4	1	no	—
162	C₂×C₇₉	—	8	1	no	—
163	C₂×C₇₉	#537	1	1	yes	characteristic
164	C₂×C₇₉	#577	1	1	yes	characteristic
165	C₂×C₇₉	#578	1	1	yes	characteristic
166	#3258	C₇₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
167	C₇₉	#3258	1	1	yes	characteristic, direct product factor, Sylow
168	#542	C₂×C₇₉	1	1	yes	characteristic
169	#541	C₂×C₇₉	1	1	yes	characteristic, complemented
170	#479	C₂×C₇₉	1	1	yes	characteristic, complemented
171	#478	C₂×C₇₉	1	1	yes	characteristic, complemented
172	#477	C₂×C₇₉	1	1	yes	characteristic, complemented
173	#475	C₂×C₇₉	1	1	yes	characteristic, complemented
174	#470	C₂×C₇₉	1	1	yes	characteristic, complemented
175	#468	C₂×C₇₉	1	1	yes	characteristic
176	#467	C₂×C₇₉	1	1	yes	characteristic, complemented
177	#465	C₂×C₇₉	1	1	yes	characteristic
178	#464	C₂×C₇₉	1	1	yes	characteristic, complemented
179	#464	C₂×C₇₉	1	1	yes	characteristic, complemented
180	#462	C₂×C₇₉	1	1	yes	characteristic, complemented
181	#461	C₂×C₇₉	1	1	yes	characteristic, complemented
182	#424	C₂×C₇₉	1	1	yes	characteristic, complemented
183	C₂×Q₁₆	C₂²×C₇₉	1	1	yes	characteristic
184	C₂×Q₁₆	C₂²×C₇₉	1	1	yes	characteristic
185	C₂×SD₁₆	C₂²×C₇₉	1	1	yes	characteristic
186	C₂×SD₁₆	C₂²×C₇₉	1	1	yes	characteristic
187	C₂×M₁₆	C₂²×C₇₉	1	1	yes	characteristic
188	C₂²×C₈	C₂²×C₇₉	1	1	yes	characteristic
189	C₄⋊Q₈	C₂²×C₇₉	1	1	yes	characteristic
190	C₄⋊Q₈	C₂²×C₇₉	1	1	yes	characteristic
191	#126	—	2	1	no	—
192	#126	—	2	1	no	—
193	#126	—	2	1	no	—
194	#125	—	2	1	no	—
195	#125	—	2	1	no	—
196	#125	C₂²×C₇₉	1	1	yes	characteristic
197	#124	—	2	1	no	—
198	#124	C₂²×C₇₉	1	1	yes	characteristic
199	#123	—	2	1	no	—
200	C₂²⋊Q₈	—	2	1	no	—
201	C₂²⋊Q₈	C₂²×C₇₉	1	1	yes	characteristic
202	C₂²⋊Q₈	C₂²×C₇₉	1	1	yes	characteristic
203	C₂²⋊Q₈	C₂²×C₇₉	1	1	yes	characteristic
204	#121	C₂²×C₇₉	1	1	yes	characteristic
205	C₄×Q₈	C₂²×C₇₉	1	1	yes	characteristic
206	C₄×Q₈	C₂²×C₇₉	1	1	yes	characteristic
207	C₄×Q₈	C₂²×C₇₉	1	1	yes	characteristic
208	C₄×D₄	C₂²×C₇₉	1	1	yes	characteristic
209	#117	C₂²×C₇₉	1	1	yes	characteristic
210	#107	C₂²×C₇₉	1	1	yes	characteristic
211	#107	C₂²×C₇₉	1	1	yes	characteristic
212	#106	C₂²×C₇₉	1	1	yes	characteristic
213	#106	C₂²×C₇₉	1	1	yes	characteristic
214	C₄⋊C₈	C₂²×C₇₉	1	1	yes	characteristic
215	C₄⋊C₈	C₂²×C₇₉	1	1	yes	characteristic
216	C₄⋊C₈	C₂²×C₇₉	1	1	yes	characteristic
217	C₄⋊C₈	C₂²×C₇₉	1	1	yes	characteristic
218	#103	C₂²×C₇₉	1	1	yes	characteristic
219	#103	C₂²×C₇₉	1	1	yes	characteristic
220	#103	C₂²×C₇₉	1	1	yes	characteristic
221	#103	C₂²×C₇₉	1	1	yes	characteristic
222	#103	C₂²×C₇₉	1	1	yes	characteristic
223	#103	C₂²×C₇₉	1	1	yes	characteristic
224	#102	C₂²×C₇₉	1	1	yes	characteristic
225	#102	C₂²×C₇₉	1	1	yes	characteristic
226	C₂×Q₈	—	2	1	no	—
227	C₂×Q₈	C₂³×C₇₉	1	1	yes	characteristic
228	C₂×Q₈	C₂³×C₇₉	1	1	yes	characteristic
229	C₂×Q₈	C₂³×C₇₉	1	1	yes	characteristic
230	C₂×D₄	—	2	1	no	—
231	C₂×D₄	C₂³×C₇₉	1	1	yes	characteristic
232	C₂²×C₄	—	2	1	no	—
233	C₂²×C₄	C₂³×C₇₉	1	1	yes	characteristic
234	Q₁₆	—	4	1	no	—
235	Q₁₆	—	4	1	no	—
236	SD₁₆	—	4	1	no	—
237	SD₁₆	—	4	1	no	—
238	M₁₆	—	4	1	no	—
239	C₂×C₈	—	4	1	no	—
240	C₂×C₈	C₂³×C₇₉	1	1	yes	characteristic
241	C₂×C₈	C₂³×C₇₉	1	1	yes	characteristic
242	C₂×C₈	C₂³×C₇₉	1	1	yes	characteristic
243	C₂×C₈	C₂³×C₇₉	1	1	yes	characteristic
244	C₄⋊C₄	—	2	1	no	—
245	C₄⋊C₄	—	2	1	no	—
246	C₄⋊C₄	—	2	1	no	—
247	C₄⋊C₄	—	2	1	no	—
248	C₄⋊C₄	—	2	1	no	—
249	C₄⋊C₄	—	2	1	no	—
250	C₄⋊C₄	—	2	1	no	—
251	C₄⋊C₄	—	2	1	no	—
252	C₄⋊C₄	—	2	1	no	—
253	C₄⋊C₄	—	2	1	no	—
254	C₄⋊C₄	—	2	1	no	—
255	C₄⋊C₄	D₄×C₇₉	1	1	yes	characteristic
256	C₄⋊C₄	D₄×C₇₉	1	1	yes	characteristic
257	C₄⋊C₄	C₂³×C₇₉	1	1	yes	characteristic
258	C₄⋊C₄	C₂³×C₇₉	1	1	yes	characteristic
259	C₄⋊C₄	C₂³×C₇₉	1	1	yes	characteristic
260	C₄⋊C₄	C₂³×C₇₉	1	1	yes	characteristic
261	C₂²⋊C₄	—	2	1	no	—
262	C₂²⋊C₄	—	2	1	no	—
263	C₂²⋊C₄	—	2	1	no	—
264	C₂²⋊C₄	—	2	1	no	—
265	C₂²⋊C₄	—	2	1	no	—
266	C₂²⋊C₄	—	2	1	no	—
267	C₂²⋊C₄	—	2	1	no	—
268	C₂²⋊C₄	D₄×C₇₉	1	1	yes	characteristic
269	C₂²⋊C₄	D₄×C₇₉	1	1	yes	characteristic
270	C₄²	—	2	1	no	—
271	C₄²	—	2	1	no	—
272	C₄²	—	2	1	no	—
273	C₄²	C₂³×C₇₉	1	1	yes	characteristic
274	C₄²	C₂³×C₇₉	1	1	yes	characteristic
275	C₂³	—	2	1	no	—
276	C₂³	C₂×D₄×C₇₉	1	1	yes	characteristic
277	Q₈	—	2	1	no	—
278	Q₈	—	2	1	no	—
279	Q₈	—	2	1	no	—
280	Q₈	—	2	1	no	—
281	Q₈	—	2	1	no	—
282	Q₈	—	2	1	no	—
283	Q₈	—	8	1	no	—
284	D₄	—	2	1	no	—
285	D₄	—	2	1	no	—
286	D₄	—	8	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	—	8	1	no	—
297	C₂×C₄	C₂×D₄×C₇₉	1	1	yes	characteristic
298	C₂×C₄	C₂×D₄×C₇₉	1	1	yes	characteristic
299	C₂×C₄	C₂×D₄×C₇₉	1	1	yes	characteristic
300	C₂×C₄	C₂×D₄×C₇₉	1	1	yes	characteristic
301	C₂×C₄	C₂×D₄×C₇₉	1	1	yes	characteristic
302	C₂×C₄	C₂⁴×C₇₉	1	1	yes	characteristic
303	C₈	—	2	1	no	—
304	C₈	—	2	1	no	—
305	C₈	—	2	1	no	—
306	C₈	—	2	1	no	—
307	C₂²	—	2	1	no	—
308	C₂²	—	2	1	no	—
309	C₂²	—	2	1	no	—
310	C₂²	—	4	1	no	—
311	C₂²	—	4	1	no	—
312	C₂²	—	4	1	no	—
313	C₂²	C₂²×D₄×C₇₉	1	1	yes	characteristic
314	C₄	—	2	1	no	—
315	C₄	—	2	1	no	—
316	C₄	—	2	1	no	—
317	C₄	—	2	1	no	—
318	C₄	—	2	1	no	—
319	C₄	—	4	1	no	—
320	C₄	—	4	1	no	—
321	C₄	—	4	1	no	—
322	C₄	—	4	1	no	—
323	C₄	—	4	1	no	—
324	C₄	—	4	1	no	—
325	C₄	—	4	1	no	—
326	C₄	H(5,𝔽₂)×C₇₉	1	1	yes	characteristic
327	C₄	2₋¹⁺⁴×C₇₉	1	1	yes	characteristic
328	C₂	—	4	1	no	—
329	C₂	—	8	1	no	—
330	C₂	#537×C₇₉	1	1	yes	characteristic
331	C₂	#577×C₇₉	1	1	yes	characteristic
332	C₂	#578×C₇₉	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₇₉	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₇₉
prime radical	#121×C₇₉	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇₉ ◁ #117×C₇₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. #541×C₇₉	C₂	semidirect
3. #479×C₇₉	C₂	semidirect
4. #478×C₇₉	C₂	semidirect
5. #477×C₇₉	C₂	semidirect
6. #475×C₇₉	C₂	semidirect
7. #470×C₇₉	C₂	semidirect
9. #467×C₇₉	C₂	semidirect
11. #464×C₇₉	C₂	semidirect
12. #464×C₇₉	C₂	semidirect
13. #462×C₇₉	C₂	semidirect
14. #461×C₇₉	C₂	semidirect
15. #424×C₇₉	C₂	semidirect
166. #3258	C₇₉	direct (N × H)
167. C₇₉	#3258	direct (N × H)
169. #541	C₂×C₇₉	semidirect
170. #479	C₂×C₇₉	semidirect
171. #478	C₂×C₇₉	semidirect
172. #477	C₂×C₇₉	semidirect
173. #475	C₂×C₇₉	semidirect
174. #470	C₂×C₇₉	semidirect
176. #467	C₂×C₇₉	semidirect
178. #464	C₂×C₇₉	semidirect
179. #464	C₂×C₇₉	semidirect
180. #462	C₂×C₇₉	semidirect
181. #461	C₂×C₇₉	semidirect
182. #424	C₂×C₇₉	semidirect

Maximal subgroups

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

subgroup	index
#542×C₇₉	2
#541×C₇₉	2
#479×C₇₉	2
#478×C₇₉	2
#477×C₇₉	2
#475×C₇₉	2
#470×C₇₉	2
#468×C₇₉	2
#467×C₇₉	2
#465×C₇₉	2
#464×C₇₉	2
#462×C₇₉	2
#461×C₇₉	2
#424×C₇₉	2
#3258	79

Subgroups — C79×#3258 #23760880

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₉×#3258 #23760880