Subgroups — #34296996

Order 20224.

878 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19098960	C₂	1	1	yes	maximal, characteristic
2	#23765971	C₂	1	2	yes	maximal
3	#23764359	C₂	1	2	yes	maximal
4	#23764128	C₂	1	1	yes	maximal, characteristic
5	#23764125	C₂	1	2	yes	maximal
6	#23763555	C₂	1	1	yes	maximal, characteristic
7	#106×Dic₇₉	C₂	1	1	yes	maximal, characteristic
8	#106×Dic₇₉	C₂	1	1	yes	maximal, characteristic
9	#23763288	C₂	1	2	yes	maximal
10	C₄×C₈×Dic₇₉	C₂	1	1	yes	maximal, characteristic
11	C₇₉×#1941	C₂	1	1	yes	maximal, characteristic
12	C₂×#14579160	C₂²	1	2	yes	—
13	C₂×#14579126	—	2	2	no	—
14	#19098960	C₄	1	8	yes	complemented
15	C₂×#14579093	C₂²	1	1	yes	characteristic
16	C₂×#14579093	C₂²	1	1	yes	characteristic
17	C₂×#14579092	C₂²	1	2	yes	—
18	C₂×#14579092	C₂²	1	2	yes	—
19	C₂×C₄×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
20	C₂×C₈×Dic₇₉	C₂²	1	1	yes	characteristic
21	C₂×C₈×Dic₇₉	C₂²	1	1	yes	characteristic
22	C₂×C₈×Dic₇₉	C₂²	1	1	yes	characteristic
23	C₂×C₈×Dic₇₉	C₂²	1	1	yes	characteristic
24	#19098467	—	2	2	no	—
25	#19098461	C₂²	1	2	yes	—
26	#19098458	—	2	2	no	—
27	#19098432	C₂²	1	2	yes	—
28	C₄⋊C₄×Dic₇₉	C₂²	1	2	yes	—
29	C₄⋊C₄×Dic₇₉	C₂²	1	2	yes	—
30	#19098414	—	2	2	no	—
31	C₄²×Dic₇₉	C₂²	1	1	yes	characteristic
32	#19098326	C₂²	1	2	yes	—
33	#19098325	C₂²	1	2	yes	—
34	#19098325	C₂²	1	2	yes	—
35	C₂×#106×C₇₉	C₂²	1	1	yes	characteristic
36	C₂×#106×C₇₉	C₂²	1	1	yes	characteristic
37	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
38	#389×C₇₉	C₂²	1	2	yes	—
39	#340×C₇₉	C₂²	1	2	yes	—
40	C₂²×Dic₁₅₈	—	2	2	no	—
41	C₂²×C₇₉⋊C₈	C₂³	1	1	yes	characteristic
42	C₂²×C₇₉⋊C₈	C₂³	1	1	yes	characteristic
43	#14579160	—	2	4	no	—
44	#14579160	C₂×C₄	1	8	yes	—
45	#14579126	—	4	8	no	—
46	C₂×C₄⋊Dic₇₉	C₂³	1	2	yes	—
47	C₂×C₄⋊Dic₇₉	C₂³	1	2	yes	—
48	C₂×Dic₇₉⋊C₄	—	2	2	no	—
49	C₂×Dic₇₉⋊C₄	—	2	2	no	—
50	C₂×Dic₇₉⋊C₄	—	2	2	no	—
51	C₂×Dic₇₉⋊C₄	—	2	2	no	—
52	C₂×C₄×Dic₇₉	—	2	2	no	—
53	C₂×C₄×Dic₇₉	—	2	2	no	—
54	C₂×C₄×Dic₇₉	D₄	1	2	yes	—
55	C₂×C₄×Dic₇₉	D₄	1	2	yes	—
56	C₂×C₄×Dic₇₉	C₂³	1	1	yes	characteristic
57	C₂×C₄×Dic₇₉	C₂³	1	1	yes	characteristic
58	#14579093	—	2	2	no	—
59	#14579093	C₂×C₄	1	4	yes	—
60	#14579092	—	2	4	no	—
61	#14579092	C₂×C₄	1	8	yes	—
62	C₄×C₇₉⋊C₈	—	2	2	no	—
63	C₈×Dic₇₉	—	2	2	no	—
64	C₈×Dic₇₉	C₂×C₄	1	4	yes	—
65	C₈×Dic₇₉	Q₈	1	4	yes	complemented
66	C₈×Dic₇₉	Q₈	1	4	yes	complemented
67	C₇₉⋊#95	—	2	2	no	—
68	C₇₉⋊#95	—	2	2	no	—
69	C₇₉⋊#95	—	2	2	no	—
70	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
71	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
72	C₂×C₄⋊C₄×C₇₉	—	2	2	no	—
73	C₂×C₄⋊C₄×C₇₉	C₂³	1	2	yes	—
74	C₂×C₄⋊C₄×C₇₉	C₂³	1	2	yes	—
75	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
76	#106×C₇₉	—	2	2	no	—
77	#106×C₇₉	C₂×C₄	1	4	yes	—
78	C₄×C₈×C₇₉	—	2	2	no	—
79	#95×C₇₉	—	2	2	no	—
80	C₂×Dic₁₅₈	—	2	8	no	—
81	C₂×Dic₁₅₈	—	4	4	no	—
82	C₂×Dic₁₅₈	—	4	4	no	—
83	C₂²×Dic₇₉	—	2	2	no	—
84	C₂²×Dic₇₉	—	2	2	no	—
85	C₂²×Dic₇₉	C₂×D₄	1	1	yes	characteristic
86	C₂²×Dic₇₉	C₂×D₄	1	2	yes	—
87	C₂²×Dic₇₉	C₂×D₄	1	1	yes	characteristic
88	C₂×C₇₉⋊C₈	—	2	1	no	—
89	C₂×C₇₉⋊C₈	—	2	1	no	—
90	C₂×C₇₉⋊C₈	C₂²×C₄	1	2	yes	—
91	C₂×C₇₉⋊C₈	C₂×Q₈	1	2	yes	—
92	C₂×C₇₉⋊C₈	C₂×Q₈	1	2	yes	—
93	C₂×C₇₉⋊C₈	#41	1	2	yes	—
94	C₄⋊Dic₇₉	—	2	2	no	—
95	C₄⋊Dic₇₉	—	2	2	no	—
96	C₄⋊Dic₇₉	—	2	2	no	—
97	C₄⋊Dic₇₉	C₂²×C₄	1	4	yes	—
98	Dic₇₉⋊C₄	—	2	8	no	—
99	Dic₇₉⋊C₄	—	4	4	no	—
100	Dic₇₉⋊C₄	—	4	4	no	—
101	Dic₇₉⋊C₄	—	4	4	no	—
102	C₄×Dic₇₉	—	2	4	no	—
103	C₄×Dic₇₉	—	4	4	no	—
104	C₄×Dic₇₉	—	4	4	no	—
105	C₄×Dic₇₉	SD₁₆	1	8	yes	—
106	C₄×Dic₇₉	C₂²×C₄	1	2	yes	—
107	C₄×Dic₇₉	C₂×Q₈	1	2	yes	—
108	C₄×Dic₇₉	C₂×Q₈	1	2	yes	—
109	C₄×Dic₇₉	#41	1	2	yes	—
110	C₂²×C₄×C₇₉	—	2	2	no	—
111	C₂²×C₄×C₇₉	—	2	2	no	—
112	C₂²×C₄×C₇₉	C₂×D₄	1	2	yes	—
113	C₂²×C₄×C₇₉	C₂⁴	1	1	yes	characteristic
114	C₂×C₈×C₇₉	—	2	1	no	—
115	C₂×C₈×C₇₉	—	2	1	no	—
116	C₂×C₈×C₇₉	C₂²×C₄	1	2	yes	—
117	C₂×C₈×C₇₉	C₂×Q₈	1	2	yes	—
118	C₂×C₈×C₇₉	C₂×Q₈	1	2	yes	—
119	C₂×C₈×C₇₉	#41	1	2	yes	—
120	C₄⋊C₄×C₇₉	—	2	2	no	—
121	C₄⋊C₄×C₇₉	—	2	2	no	—
122	C₄⋊C₄×C₇₉	—	2	2	no	—
123	C₄⋊C₄×C₇₉	—	4	4	no	—
124	C₄⋊C₄×C₇₉	C₂²×C₄	1	4	yes	—
125	C₄²×C₇₉	—	2	1	no	—
126	C₄²×C₇₉	—	2	1	no	—
127	C₂³×C₇₉	C₂²×D₄	1	1	yes	characteristic
128	C₂×C₄×C₇₉	—	2	2	no	—
129	C₂×C₄×C₇₉	—	2	2	no	—
130	C₂×C₄×C₇₉	—	2	4	no	—
131	C₂×C₄×C₇₉	—	4	2	no	—
132	C₂×C₄×C₇₉	—	4	2	no	—
133	C₂×C₄×C₇₉	—	4	2	no	—
134	C₂×C₄×C₇₉	—	4	2	no	—
135	C₂×C₄×C₇₉	—	4	2	no	—
136	C₂×C₄×C₇₉	C₂×SD₁₆	1	4	yes	—
137	C₂×C₄×C₇₉	C₂³×C₄	1	1	yes	characteristic
138	C₂×C₄×C₇₉	C₂²×Q₈	1	1	yes	characteristic
139	C₂×C₄×C₇₉	C₂²×Q₈	1	1	yes	characteristic
140	C₂×C₄×C₇₉	C₂×#41	1	1	yes	characteristic
141	C₂×C₄×C₇₉	C₂×#41	1	1	yes	characteristic
142	C₂×C₄×C₇₉	C₂×#41	1	1	yes	characteristic
143	C₂×Dic₇₉	—	2	2	no	—
144	C₂×Dic₇₉	—	2	4	no	—
145	C₂×Dic₇₉	—	4	2	no	—
146	C₂×Dic₇₉	—	4	2	no	—
147	C₂×Dic₇₉	—	4	2	no	—
148	C₂×Dic₇₉	—	4	2	no	—
149	C₂×Dic₇₉	—	4	2	no	—
150	C₂×Dic₇₉	C₄×D₄	1	4	yes	—
151	C₂×Dic₇₉	C₄⋊Q₈	1	2	yes	—
152	C₂×Dic₇₉	C₄⋊Q₈	1	2	yes	—
153	C₂×Dic₇₉	C₂×SD₁₆	1	4	yes	—
154	C₂×Dic₇₉	C₂×SD₁₆	1	4	yes	—
155	C₂×Dic₇₉	#135	1	4	yes	—
156	Dic₁₅₈	—	4	16	no	—
157	C₈×C₇₉	—	2	2	no	—
158	C₈×C₇₉	C₄×Q₈	1	4	yes	complemented
159	C₇₉⋊C₈	—	2	2	no	—
160	C₇₉⋊C₈	C₄×Q₈	1	4	yes	complemented
161	C₂²×C₇₉	C₂×C₄×D₄	1	1	yes	characteristic
162	C₂²×C₇₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
163	C₂²×C₇₉	#532	1	1	yes	characteristic
164	C₂²×C₇₉	C₂²×SD₁₆	1	2	yes	—
165	C₂²×C₇₉	C₂×#135	1	2	yes	—
166	C₄×C₇₉	—	2	4	no	—
167	C₄×C₇₉	—	4	4	no	—
168	C₄×C₇₉	—	8	2	no	—
169	C₄×C₇₉	C₂×C₄×Q₈	1	1	yes	characteristic
170	C₄×C₇₉	C₂×C₄×Q₈	1	1	yes	characteristic
171	C₄×C₇₉	#533	1	1	yes	characteristic
172	C₄×C₇₉	#533	1	1	yes	characteristic
173	Dic₇₉	—	2	2	no	—
174	Dic₇₉	—	4	4	no	—
175	Dic₇₉	—	8	2	no	—
176	Dic₇₉	C₄×SD₁₆	1	8	yes	—
177	Dic₇₉	#498	1	4	yes	complemented
178	C₄×#498	—	79	1	no	maximal, Sylow
179	C₂×C₇₉	C₄×C₄⋊Q₈	1	1	yes	characteristic
180	C₂×C₇₉	C₂×C₄×SD₁₆	1	2	yes	—
181	C₂×C₇₉	C₂×#498	1	1	yes	characteristic
182	C₂×C₇₉	#3160	1	1	yes	characteristic
183	C₂×C₇₉	#3167	1	2	yes	—
184	C₂×#498	—	79	1	no	—
185	C₄×C₄⋊Q₈	—	79	2	no	—
186	#1990	—	79	2	no	—
187	#1941	—	79	1	no	—
188	#1941	—	79	2	no	—
189	#1941	D₇₉	1	1	yes	characteristic
190	#1844	—	79	1	no	—
191	C₄×#106	—	79	1	no	—
192	C₄×#106	—	79	1	no	—
193	C₄×#106	—	79	2	no	—
194	C₄²×C₈	—	79	1	no	—
195	C₇₉	C₄×#498	1	1	yes	characteristic, complemented, Sylow
196	C₂×C₄⋊Q₈	—	79	2	no	—
197	C₂×C₄×Q₈	—	158	2	no	—
198	#498	—	79	8	no	—
199	C₂×#106	—	79	1	no	—
200	C₂×#106	—	79	2	no	—
201	C₂×#106	—	79	1	no	—
202	C₂×#106	—	79	2	no	—
203	C₂×#106	C₂×D₇₉	1	1	yes	characteristic
204	C₂×#106	C₂×D₇₉	1	1	yes	characteristic
205	C₂×C₄×C₈	—	79	1	no	—
206	C₂×C₄×C₈	—	79	1	no	—
207	C₂×C₄×C₈	—	79	1	no	—
208	C₂×C₄×C₈	—	79	1	no	—
209	C₂×C₄×C₈	—	79	1	no	—
210	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
211	#391	—	158	2	no	—
212	#389	—	79	2	no	—
213	#389	—	158	2	no	—
214	#389	C₂×D₇₉	1	2	yes	—
215	#384	—	79	2	no	—
216	C₄×C₄⋊C₄	—	79	2	no	—
217	C₄×C₄⋊C₄	—	79	2	no	—
218	C₄×C₄⋊C₄	—	158	2	no	—
219	C₄³	—	79	1	no	—
220	#340	—	79	2	no	—
221	#340	—	79	2	no	—
222	#340	—	79	2	no	—
223	#340	C₂×D₇₉	1	2	yes	—
224	C₂²×Q₈	—	158	2	no	—
225	C₂²×C₈	—	79	1	no	—
226	C₂²×C₈	—	79	1	no	—
227	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
228	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
229	C₄⋊Q₈	—	79	8	no	—
230	C₄⋊Q₈	—	158	4	no	—
231	C₄×Q₈	—	316	8	no	—
232	C₂×C₄⋊C₄	—	2	2	no	—
233	C₂×C₄⋊C₄	—	79	2	no	—
234	C₂×C₄⋊C₄	—	79	2	no	—
235	C₂×C₄⋊C₄	—	158	2	no	—
236	C₂×C₄⋊C₄	—	158	2	no	—
237	C₂×C₄⋊C₄	—	158	2	no	—
238	C₂×C₄⋊C₄	—	158	2	no	—
239	C₂×C₄⋊C₄	C₂²×D₇₉	1	2	yes	—
240	C₂×C₄⋊C₄	C₂²×D₇₉	1	2	yes	—
241	C₂×C₄²	—	79	2	no	—
242	C₂×C₄²	—	79	1	no	—
243	C₂×C₄²	—	79	1	no	—
244	C₂×C₄²	—	79	2	no	—
245	C₂×C₄²	—	158	2	no	—
246	C₂×C₄²	—	158	2	no	—
247	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
248	#106	—	2	2	no	—
249	#106	—	79	4	no	—
250	#106	—	79	8	no	—
251	#106	—	158	4	no	—
252	#106	—	158	2	no	—
253	#106	C₄×D₇₉	1	4	yes	—
254	C₄×C₈	—	2	2	no	—
255	C₄×C₈	—	79	4	no	—
256	C₄×C₈	—	79	4	no	—
257	C₄×C₈	—	79	4	no	—
258	C₄×C₈	—	158	2	no	—
259	C₄×C₈	—	158	2	no	—
260	#95	—	2	2	no	—
261	#95	—	158	2	no	—
262	#95	—	158	2	no	—
263	#95	—	158	2	no	—
264	C₂×Q₈	—	316	4	no	—
265	C₂×Q₈	—	316	4	no	—
266	C₂×Q₈	—	158	8	no	—
267	C₂²×C₄	—	2	2	no	—
268	C₂²×C₄	—	2	2	no	—
269	C₂²×C₄	—	79	1	no	—
270	C₂²×C₄	—	79	2	no	—
271	C₂²×C₄	—	79	1	no	—
272	C₂²×C₄	—	158	2	no	—
273	C₂²×C₄	—	158	2	no	—
274	C₂²×C₄	D₄×D₇₉	1	2	yes	—
275	C₂²×C₄	C₂³×D₇₉	1	1	yes	characteristic
276	C₂×C₈	—	2	1	no	—
277	C₂×C₈	—	2	1	no	—
278	C₂×C₈	—	79	2	no	—
279	C₂×C₈	—	79	2	no	—
280	C₂×C₈	—	79	2	no	—
281	C₂×C₈	—	79	2	no	—
282	C₂×C₈	—	158	1	no	—
283	C₂×C₈	—	158	1	no	—
284	C₂×C₈	C₂×C₄×D₇₉	1	2	yes	—
285	C₂×C₈	Q₈×D₇₉	1	2	yes	—
286	C₂×C₈	Q₈×D₇₉	1	2	yes	—
287	C₂×C₈	#11929623	1	2	yes	—
288	C₄⋊C₄	—	2	2	no	—
289	C₄⋊C₄	—	2	2	no	—
290	C₄⋊C₄	—	2	2	no	—
291	C₄⋊C₄	—	4	4	no	—
292	C₄⋊C₄	—	316	4	no	—
293	C₄⋊C₄	—	316	4	no	—
294	C₄⋊C₄	—	316	4	no	—
295	C₄⋊C₄	—	79	4	no	—
296	C₄⋊C₄	—	158	8	no	—
297	C₄⋊C₄	—	158	2	no	—
298	C₄⋊C₄	—	158	2	no	—
299	C₄⋊C₄	—	158	2	no	—
300	C₄⋊C₄	C₂×C₄×D₇₉	1	4	yes	—
301	C₄²	—	2	1	no	—
302	C₄²	—	2	1	no	—
303	C₄²	—	316	4	no	—
304	C₄²	—	316	4	no	—
305	C₄²	—	79	2	no	—
306	C₄²	—	79	2	no	—
307	C₄²	—	79	2	no	—
308	C₄²	—	79	8	no	—
309	C₄²	—	79	2	no	—
310	C₄²	—	158	4	no	—
311	C₂³	C₂×D₄×D₇₉	1	1	yes	characteristic
312	Q₈	—	316	16	no	—
313	C₂×C₄	—	2	2	no	—
314	C₂×C₄	—	2	2	no	—
315	C₂×C₄	—	2	4	no	—
316	C₂×C₄	—	4	2	no	—
317	C₂×C₄	—	4	2	no	—
318	C₂×C₄	—	4	2	no	—
319	C₂×C₄	—	4	2	no	—
320	C₂×C₄	—	4	2	no	—
321	C₂×C₄	—	316	2	no	—
322	C₂×C₄	—	316	2	no	—
323	C₂×C₄	—	316	2	no	—
324	C₂×C₄	—	316	2	no	—
325	C₂×C₄	—	316	2	no	—
326	C₂×C₄	—	79	4	no	—
327	C₂×C₄	—	79	4	no	—
328	C₂×C₄	—	79	4	no	—
329	C₂×C₄	—	79	4	no	—
330	C₂×C₄	—	79	2	no	—
331	C₂×C₄	—	79	2	no	—
332	C₂×C₄	—	158	4	no	—
333	C₂×C₄	—	158	2	no	—
334	C₂×C₄	SD₁₆×D₇₉	1	4	yes	—
335	C₂×C₄	C₂²×C₄×D₇₉	1	1	yes	characteristic
336	C₂×C₄	C₂×Q₈×D₇₉	1	1	yes	characteristic
337	C₂×C₄	C₂×Q₈×D₇₉	1	1	yes	characteristic
338	C₂×C₄	#41×D₇₉	1	1	yes	characteristic
339	C₂×C₄	#41×D₇₉	1	1	yes	characteristic
340	C₂×C₄	C₂×#11929623	1	1	yes	characteristic
341	C₈	—	2	2	no	—
342	C₈	—	79	4	no	—
343	C₈	—	158	2	no	—
344	C₈	#14579126	1	4	yes	complemented
345	C₂²	C₄×D₄×D₇₉	1	1	yes	characteristic
346	C₂²	C₄⋊Q₈×D₇₉	1	1	yes	characteristic
347	C₂²	#19099108	1	1	yes	characteristic
348	C₂²	C₂×SD₁₆×D₇₉	1	2	yes	—
349	C₂²	#135×D₇₉	1	2	yes	—
350	C₄	—	2	4	no	—
351	C₄	—	4	4	no	—
352	C₄	—	8	2	no	—
353	C₄	—	316	4	no	—
354	C₄	—	79	4	no	—
355	C₄	—	79	8	no	—
356	C₄	—	632	2	no	—
357	C₄	—	158	2	no	—
358	C₄	C₄×Q₈×D₇₉	1	1	yes	characteristic
359	C₄	C₂×#14579126	1	1	yes	characteristic
360	C₄	#19099115	1	1	yes	characteristic
361	C₄	#19099119	1	1	yes	characteristic
362	C₂	#23765971	1	1	yes	characteristic
363	C₂	C₄×SD₁₆×D₇₉	1	2	yes	—
364	C₂	#498×D₇₉	1	1	yes	characteristic
365	C₂	#23773913	1	1	yes	characteristic
366	C₂	#23773954	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄×D₇₉
commutator	C₂×C₄×C₇₉	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₇₉
Fitting	C₇₉×#1941	C₂
socle, prime radical	C₂³×C₇₉	C₂²×D₄
cosocle	C₂²×C₄×C₇₉	C₂⁴
hypercenter	#1941	D₇₉
hypocenter	C₇₉	C₄×#498
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1941

Complements

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

N	H ≅ G/N	type
14. #19098960	C₄	semidirect
65. C₈×Dic₇₉	Q₈	semidirect
66. C₈×Dic₇₉	Q₈	semidirect
158. C₈×C₇₉	C₄×Q₈	semidirect
160. C₇₉⋊C₈	C₄×Q₈	semidirect
177. Dic₇₉	#498	semidirect
195. C₇₉	C₄×#498	semidirect
344. C₈	#14579126	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19098960	2
#23765971	2
#23764359	2
#23764128	2
#23764125	2
#23763555	2
#106×Dic₇₉	2
#23763288	2
C₄×C₈×Dic₇₉	2
C₇₉×#1941	2
C₄×#498	79