Subgroups — #34286927

Order 20224.

866 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₈×C₇₉⋊C₈	C₂	1	1	yes	maximal, characteristic
2	#23765878	C₂	1	1	yes	maximal, characteristic
3	#23763272	C₂	1	2	yes	maximal
4	#23763270	C₂	1	1	yes	maximal, characteristic
5	#23763270	C₂	1	2	yes	maximal
6	C₈×C₄⋊Dic₇₉	C₂	1	1	yes	maximal, characteristic
7	#23763259	C₂	1	2	yes	maximal
8	C₄⋊C₈×Dic₇₉	C₂	1	1	yes	maximal, characteristic
9	#23763088	C₂	1	2	yes	maximal
10	C₄×#14579071	C₂	1	1	yes	maximal, characteristic
11	C₂×#446×C₇₉	C₂	1	1	yes	maximal, characteristic
12	Q₈×C₇₉⋊C₈	C₄	1	8	yes	complemented
13	C₂×C₄⋊(C₇₉⋊C₈)	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊(C₇₉⋊C₈)	C₂²	1	2	yes	—
15	C₂×#14579071	C₂²	1	1	yes	characteristic
16	C₂×#14579071	C₂²	1	1	yes	characteristic
17	C₂×#14579071	C₂²	1	2	yes	—
18	C₂×#14579071	C₂²	1	1	yes	characteristic
19	C₂×#14579071	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₇₉⋊C₈	C₂²	1	2	yes	—
22	C₂×C₈×Dic₇₉	C₂²	1	2	yes	—
23	C₂×C₈×Dic₇₉	C₂²	1	2	yes	—
24	#19098470	C₄	1	2	yes	—
25	#19098462	C₄	1	2	yes	—
26	#19098449	C₄	1	4	yes	—
27	C₄×C₄⋊Dic₇₉	C₂²	1	1	yes	characteristic
28	C₄×C₄⋊Dic₇₉	C₂²	1	2	yes	—
29	#19098409	C₂²	1	2	yes	—
30	C₄²×Dic₇₉	C₂²	1	1	yes	characteristic
31	#19098317	C₂²	1	2	yes	—
32	#19098317	C₂²	1	2	yes	—
33	#19098317	C₂²	1	2	yes	—
34	C₂×C₄×Q₈×C₇₉	C₂²	1	1	yes	characteristic
35	#446×C₇₉	C₄	1	8	yes	complemented
36	C₂×C₄⋊C₈×C₇₉	C₂²	1	1	yes	characteristic
37	C₂×C₄⋊C₈×C₇₉	C₂²	1	2	yes	—
38	C₂×#97×C₇₉	C₂²	1	2	yes	—
39	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
40	C₂²×C₇₉⋊C₈	C₂³	1	2	yes	—
41	C₂²×C₇₉⋊C₈	C₂³	1	1	yes	characteristic
42	C₂²×C₇₉⋊C₈	C₂³	1	1	yes	characteristic
43	C₂×C₄⋊Dic₇₉	C₂×C₄	1	2	yes	—
44	C₂×C₄⋊Dic₇₉	C₂×C₄	1	4	yes	—
45	C₂×C₄×Dic₇₉	C₂×C₄	1	2	yes	—
46	C₂×C₄×Dic₇₉	C₂×C₄	1	2	yes	—
47	C₂×C₄×Dic₇₉	C₂³	1	1	yes	characteristic
48	C₂×C₄×Dic₇₉	C₂³	1	2	yes	—
49	C₂×C₄×Dic₇₉	C₂³	1	1	yes	characteristic
50	C₄⋊(C₇₉⋊C₈)	C₂×C₄	1	4	yes	—
51	C₄⋊(C₇₉⋊C₈)	C₂×C₄	1	8	yes	—
52	#14579071	—	2	2	no	—
53	#14579071	—	2	2	no	—
54	#14579071	—	2	4	no	—
55	#14579071	—	2	2	no	—
56	#14579071	Q₈	1	4	yes	complemented
57	C₄×C₇₉⋊C₈	C₂×C₄	1	4	yes	—
58	C₄×C₇₉⋊C₈	C₂×C₄	1	8	yes	—
59	C₈×Dic₇₉	—	2	4	no	—
60	C₈×Dic₇₉	—	2	4	no	—
61	C₇₉⋊#95	C₂×C₄	1	4	yes	—
62	C₇₉⋊#95	C₂×C₄	1	4	yes	—
63	C₇₉⋊#95	C₂×C₄	1	2	yes	—
64	C₇₉⋊#95	C₂×C₄	1	4	yes	—
65	C₂²×Q₈×C₇₉	C₂×C₄	1	1	yes	characteristic
66	C₂²×C₈×C₇₉	C₂³	1	2	yes	—
67	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
68	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
69	C₄×Q₈×C₇₉	C₂×C₄	1	4	yes	—
70	C₄×Q₈×C₇₉	C₂×C₄	1	4	yes	—
71	C₂×C₄⋊C₄×C₇₉	C₂×C₄	1	2	yes	—
72	C₂×C₄⋊C₄×C₇₉	C₂×C₄	1	1	yes	characteristic
73	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
74	C₂×C₄²×C₇₉	C₂³	1	2	yes	—
75	C₄⋊C₈×C₇₉	C₂×C₄	1	4	yes	—
76	C₄⋊C₈×C₇₉	C₂×C₄	1	8	yes	—
77	#97×C₇₉	C₂×C₄	1	8	yes	—
78	C₄×C₈×C₇₉	C₂×C₄	1	4	yes	—
79	C₂²×Dic₇₉	C₂²×C₄	1	4	yes	—
80	C₂²×Dic₇₉	C₂²×C₄	1	2	yes	—
81	C₂²×Dic₇₉	C₂²×C₄	1	2	yes	—
82	C₂×C₇₉⋊C₈	—	2	2	no	—
83	C₂×C₇₉⋊C₈	—	2	2	no	—
84	C₂×C₇₉⋊C₈	—	2	1	no	—
85	C₂×C₇₉⋊C₈	—	2	1	no	—
86	C₂×C₇₉⋊C₈	C₂²×C₄	1	4	yes	—
87	C₂×C₇₉⋊C₈	C₂²×C₄	1	2	yes	—
88	C₂×C₇₉⋊C₈	C₂²×C₄	1	2	yes	—
89	C₂×C₇₉⋊C₈	C₂×Q₈	1	2	yes	—
90	C₂×C₇₉⋊C₈	#41	1	2	yes	—
91	C₄⋊Dic₇₉	—	2	4	no	—
92	C₄⋊Dic₇₉	—	4	4	no	—
93	C₄×Dic₇₉	—	2	4	no	—
94	C₄×Dic₇₉	—	2	2	no	—
95	C₄×Dic₇₉	—	2	1	no	—
96	C₄×Dic₇₉	—	2	2	no	—
97	C₄×Dic₇₉	—	2	1	no	—
98	C₄×Dic₇₉	—	4	2	no	—
99	C₄×Dic₇₉	C₂×Q₈	1	2	yes	—
100	C₄×Dic₇₉	#41	1	2	yes	—
101	C₂×Q₈×C₇₉	—	2	4	no	—
102	C₂×Q₈×C₇₉	C₄²	1	4	yes	—
103	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
104	C₂²×C₄×C₇₉	C₂²×C₄	1	2	yes	—
105	C₂²×C₄×C₇₉	C₂²×C₄	1	2	yes	—
106	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
107	C₂²×C₄×C₇₉	C₂⁴	1	1	yes	characteristic
108	C₂×C₈×C₇₉	—	2	2	no	—
109	C₂×C₈×C₇₉	—	2	2	no	—
110	C₂×C₈×C₇₉	—	2	1	no	—
111	C₂×C₈×C₇₉	—	2	1	no	—
112	C₂×C₈×C₇₉	C₂²×C₄	1	2	yes	—
113	C₂×C₈×C₇₉	C₂²×C₄	1	4	yes	—
114	C₂×C₈×C₇₉	C₂²×C₄	1	2	yes	—
115	C₂×C₈×C₇₉	C₂×Q₈	1	2	yes	—
116	C₂×C₈×C₇₉	#41	1	2	yes	—
117	C₄⋊C₄×C₇₉	C₄²	1	8	yes	—
118	C₄⋊C₄×C₇₉	C₄²	1	4	yes	—
119	C₄²×C₇₉	C₂²×C₄	1	4	yes	—
120	C₄²×C₇₉	C₂²×C₄	1	2	yes	—
121	C₄²×C₇₉	C₂²×C₄	1	2	yes	—
122	C₄²×C₇₉	C₂²×C₄	1	4	yes	—
123	C₂³×C₇₉	C₂³×C₄	1	1	yes	characteristic
124	Q₈×C₇₉	—	2	8	no	—
125	C₂×C₄×C₇₉	—	2	4	no	—
126	C₂×C₄×C₇₉	—	2	2	no	—
127	C₂×C₄×C₇₉	—	2	4	no	—
128	C₂×C₄×C₇₉	C₂×C₄²	1	2	yes	—
129	C₂×C₄×C₇₉	C₂×C₄²	1	4	yes	—
130	C₂×C₄×C₇₉	C₂×C₄²	1	4	yes	—
131	C₂×C₄×C₇₉	C₂×C₄²	1	2	yes	—
132	C₂×C₄×C₇₉	#131	1	4	yes	—
133	C₂×C₄×C₇₉	C₂³×C₄	1	1	yes	characteristic
134	C₂×C₄×C₇₉	C₂³×C₄	1	1	yes	characteristic
135	C₂×C₄×C₇₉	C₂²×Q₈	1	1	yes	characteristic
136	C₂×C₄×C₇₉	C₂×#41	1	1	yes	characteristic
137	C₂×C₄×C₇₉	C₂×#41	1	1	yes	characteristic
138	C₂×C₄×C₇₉	C₂×#41	1	1	yes	characteristic
139	C₂×Dic₇₉	—	2	2	no	—
140	C₂×Dic₇₉	—	2	2	no	—
141	C₂×Dic₇₉	—	2	4	no	—
142	C₂×Dic₇₉	—	2	2	no	—
143	C₂×Dic₇₉	—	2	2	no	—
144	C₂×Dic₇₉	—	2	4	no	—
145	C₂×Dic₇₉	—	2	2	no	—
146	C₂×Dic₇₉	—	2	4	no	—
147	C₂×Dic₇₉	C₄×Q₈	1	4	yes	—
148	C₈×C₇₉	—	2	4	no	—
149	C₈×C₇₉	—	2	2	no	—
150	C₈×C₇₉	C₄×Q₈	1	4	yes	—
151	C₇₉⋊C₈	—	2	4	no	—
152	C₇₉⋊C₈	—	2	2	no	—
153	C₇₉⋊C₈	C₄×Q₈	1	4	yes	—
154	C₂²×C₇₉	C₂²×C₄²	1	1	yes	characteristic
155	C₂²×C₇₉	C₂×C₄×Q₈	1	1	yes	characteristic
156	C₂²×C₇₉	C₄×#41	1	1	yes	characteristic
157	C₂²×C₇₉	C₂×#131	1	2	yes	—
158	C₂²×C₇₉	#568	1	2	yes	—
159	C₄×C₇₉	—	2	4	no	—
160	C₄×C₇₉	—	2	2	no	—
161	C₄×C₇₉	—	2	4	no	—
162	C₄×C₇₉	#405	1	4	yes	—
163	C₄×C₇₉	C₂×C₄×Q₈	1	1	yes	characteristic
164	C₄×C₇₉	C₂×C₄×Q₈	1	1	yes	characteristic
165	C₄×C₇₉	C₄×#41	1	1	yes	characteristic
166	C₄×C₇₉	C₄×#41	1	1	yes	characteristic
167	Dic₇₉	—	2	4	no	—
168	Dic₇₉	—	4	2	no	—
169	Dic₇₉	—	4	4	no	—
170	#20521	—	79	1	no	maximal, Sylow
171	C₂×C₇₉	C₄²×Q₈	1	1	yes	characteristic
172	C₂×C₇₉	C₂×#405	1	1	yes	characteristic
173	C₂×C₇₉	#2875	1	1	yes	characteristic
174	C₂×C₇₉	#2962	1	2	yes	—
175	C₂×C₇₉	#2969	1	2	yes	—
176	C₂×#446	D₇₉	1	1	yes	characteristic
177	C₂×C₈×Q₈	—	79	1	no	—
178	#2295	—	79	1	no	—
179	#1771	—	79	2	no	—
180	#1770	—	79	2	no	—
181	#1770	—	79	1	no	—
182	C₈×C₄⋊C₄	—	79	1	no	—
183	#1768	—	79	2	no	—
184	C₄×C₄⋊C₈	—	79	1	no	—
185	#1726	—	79	2	no	—
186	C₄×#97	—	79	1	no	—
187	C₇₉	#20521	1	1	yes	characteristic, complemented, Sylow
188	C₂×C₄×Q₈	C₂×D₇₉	1	1	yes	characteristic
189	#446	Dic₇₉	1	8	yes	complemented
190	C₈×Q₈	—	79	8	no	—
191	C₂×C₄⋊C₈	—	79	1	no	—
192	C₂×C₄⋊C₈	—	79	2	no	—
193	C₂×C₄⋊C₈	C₂×D₇₉	1	1	yes	characteristic
194	C₂×C₄⋊C₈	C₂×D₇₉	1	2	yes	—
195	C₂×#97	—	79	1	no	—
196	C₂×#97	—	79	1	no	—
197	C₂×#97	—	79	1	no	—
198	C₂×#97	—	79	1	no	—
199	C₂×#97	—	79	2	no	—
200	C₂×#97	C₂×D₇₉	1	2	yes	—
201	C₂×C₄×C₈	—	79	1	no	—
202	C₂×C₄×C₈	—	79	2	no	—
203	C₂×C₄×C₈	—	79	2	no	—
204	C₂×C₄×C₈	—	79	2	no	—
205	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
206	#391	—	79	2	no	—
207	#389	—	79	2	no	—
208	#387	—	79	4	no	—
209	C₄×C₄⋊C₄	—	79	1	no	—
210	C₄×C₄⋊C₄	—	79	2	no	—
211	#376	—	79	2	no	—
212	C₄³	—	79	1	no	—
213	#336	—	79	2	no	—
214	#336	—	79	2	no	—
215	#336	—	79	2	no	—
216	C₂²×Q₈	C₄×D₇₉	1	1	yes	characteristic
217	C₂²×C₈	—	79	1	no	—
218	C₂²×C₈	—	79	2	no	—
219	C₂²×C₈	—	79	1	no	—
220	C₂²×C₈	C₂²×D₇₉	1	2	yes	—
221	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
222	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
223	C₄×Q₈	C₄×D₇₉	1	4	yes	—
224	C₄×Q₈	C₂×Dic₇₉	1	4	yes	—
225	C₂×C₄⋊C₄	—	79	4	no	—
226	C₂×C₄⋊C₄	—	79	2	no	—
227	C₂×C₄⋊C₄	C₄×D₇₉	1	2	yes	—
228	C₂×C₄⋊C₄	C₄×D₇₉	1	1	yes	characteristic
229	C₂×C₄²	—	79	2	no	—
230	C₂×C₄²	—	79	1	no	—
231	C₂×C₄²	—	79	2	no	—
232	C₂×C₄²	—	79	1	no	—
233	C₂×C₄²	—	79	2	no	—
234	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
235	C₂×C₄²	C₂²×D₇₉	1	2	yes	—
236	C₄⋊C₈	—	79	4	no	—
237	C₄⋊C₈	—	79	8	no	—
238	C₄⋊C₈	C₂×Dic₇₉	1	4	yes	—
239	C₄⋊C₈	C₂×Dic₇₉	1	8	yes	—
240	#97	—	79	4	no	—
241	#97	—	158	2	no	—
242	#97	—	158	2	no	—
243	#97	—	158	4	no	—
244	#97	—	158	2	no	—
245	#97	C₂×Dic₇₉	1	8	yes	—
246	C₄×C₈	—	79	4	no	—
247	C₄×C₈	—	79	8	no	—
248	C₄×C₈	—	158	4	no	—
249	C₄×C₈	—	158	4	no	—
250	C₄×C₈	C₂×Dic₇₉	1	4	yes	—
251	#95	—	79	4	no	—
252	#95	—	79	4	no	—
253	#95	—	79	2	no	—
254	#95	—	79	4	no	—
255	C₂×Q₈	—	2	4	no	—
256	C₂×Q₈	C₄×Dic₇₉	1	4	yes	—
257	C₂²×C₄	—	79	2	no	—
258	C₂²×C₄	—	79	4	no	—
259	C₂²×C₄	—	79	2	no	—
260	C₂²×C₄	C₂×C₄×D₇₉	1	2	yes	—
261	C₂²×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
262	C₂²×C₄	C₂×C₄×D₇₉	1	2	yes	—
263	C₂²×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
264	C₂²×C₄	C₂³×D₇₉	1	1	yes	characteristic
265	C₂×C₈	—	2	2	no	—
266	C₂×C₈	—	2	2	no	—
267	C₂×C₈	—	2	1	no	—
268	C₂×C₈	—	2	1	no	—
269	C₂×C₈	—	79	2	no	—
270	C₂×C₈	—	79	2	no	—
271	C₂×C₈	—	79	2	no	—
272	C₂×C₈	—	79	4	no	—
273	C₂×C₈	—	79	2	no	—
274	C₂×C₈	—	158	2	no	—
275	C₂×C₈	—	158	1	no	—
276	C₂×C₈	—	158	2	no	—
277	C₂×C₈	—	158	1	no	—
278	C₂×C₈	C₂²×Dic₇₉	1	2	yes	—
279	C₂×C₈	C₂²×Dic₇₉	1	4	yes	—
280	C₂×C₈	C₂²×Dic₇₉	1	2	yes	—
281	C₂×C₈	Q₈×D₇₉	1	2	yes	—
282	C₂×C₈	#11929625	1	2	yes	—
283	C₄⋊C₄	—	316	4	no	—
284	C₄⋊C₄	—	158	4	no	—
285	C₄⋊C₄	C₄×Dic₇₉	1	8	yes	—
286	C₄⋊C₄	C₄×Dic₇₉	1	4	yes	—
287	C₄²	—	316	2	no	—
288	C₄²	—	79	2	no	—
289	C₄²	—	79	2	no	—
290	C₄²	—	158	1	no	—
291	C₄²	—	158	2	no	—
292	C₄²	—	158	2	no	—
293	C₄²	—	158	1	no	—
294	C₄²	—	158	4	no	—
295	C₄²	C₂×C₄×D₇₉	1	4	yes	—
296	C₄²	C₂×C₄×D₇₉	1	2	yes	—
297	C₄²	C₂²×Dic₇₉	1	2	yes	—
298	C₄²	C₂²×Dic₇₉	1	4	yes	—
299	C₂³	C₂²×C₄×D₇₉	1	1	yes	characteristic
300	Q₈	—	2	8	no	—
301	C₂×C₄	—	2	4	no	—
302	C₂×C₄	—	2	2	no	—
303	C₂×C₄	—	2	4	no	—
304	C₂×C₄	—	79	4	no	—
305	C₂×C₄	—	158	4	no	—
306	C₂×C₄	—	158	2	no	—
307	C₂×C₄	—	158	2	no	—
308	C₂×C₄	—	158	4	no	—
309	C₂×C₄	—	158	2	no	—
310	C₂×C₄	—	158	4	no	—
311	C₂×C₄	—	158	2	no	—
312	C₂×C₄	—	158	2	no	—
313	C₂×C₄	C₂×C₄×Dic₇₉	1	2	yes	—
314	C₂×C₄	C₂×C₄×Dic₇₉	1	4	yes	—
315	C₂×C₄	C₂×C₄×Dic₇₉	1	4	yes	—
316	C₂×C₄	C₂×C₄×Dic₇₉	1	2	yes	—
317	C₂×C₄	#14579169	1	4	yes	—
318	C₂×C₄	C₂²×C₄×D₇₉	1	1	yes	characteristic
319	C₂×C₄	C₂³×Dic₇₉	1	1	yes	characteristic
320	C₂×C₄	C₂×Q₈×D₇₉	1	1	yes	characteristic
321	C₂×C₄	#41×D₇₉	1	1	yes	characteristic
322	C₂×C₄	#41×D₇₉	1	1	yes	characteristic
323	C₂×C₄	C₂×#11929625	1	1	yes	characteristic
324	C₈	—	2	4	no	—
325	C₈	—	2	2	no	—
326	C₈	—	79	4	no	—
327	C₈	—	158	4	no	—
328	C₈	—	158	2	no	—
329	C₈	Q₈×Dic₇₉	1	4	yes	—
330	C₂²	C₂²×C₄×Dic₇₉	1	1	yes	characteristic
331	C₂²	C₄×Q₈×D₇₉	1	1	yes	characteristic
332	C₂²	C₄×#11929625	1	1	yes	characteristic
333	C₂²	C₂×#14579169	1	2	yes	—
334	C₂²	#19099328	1	2	yes	—
335	C₄	—	2	4	no	—
336	C₄	—	2	2	no	—
337	C₄	—	2	4	no	—
338	C₄	—	316	2	no	—
339	C₄	—	316	4	no	—
340	C₄	—	158	4	no	—
341	C₄	#19098521	1	4	yes	—
342	C₄	C₄×Q₈×D₇₉	1	1	yes	characteristic
343	C₄	C₂×Q₈×Dic₇₉	1	1	yes	characteristic
344	C₄	C₄×#11929625	1	1	yes	characteristic
345	C₄	#41×Dic₇₉	1	1	yes	characteristic
346	C₂	C₄×Q₈×Dic₇₉	1	1	yes	characteristic
347	C₂	C₂×#19098521	1	1	yes	characteristic
348	C₂	#23771826	1	1	yes	characteristic
349	C₂	#23772348	1	2	yes	—
350	C₂	#23772406	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₇₉
commutator	C₂²×C₇₉	C₂²×C₄²
Fitting	C₂×#446×C₇₉	C₂
socle, prime radical	C₂³×C₇₉	C₂³×C₄
cosocle	C₂²×C₄×C₇₉	C₂⁴
hypercenter	C₂×#446	D₇₉
hypocenter	C₇₉	#20521
LCS	G ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#446

Complements

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

N	H ≅ G/N	type
12. Q₈×C₇₉⋊C₈	C₄	semidirect
35. #446×C₇₉	C₄	semidirect
56. #14579071	Q₈	semidirect
187. C₇₉	#20521	semidirect
189. #446	Dic₇₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×Q₈×C₇₉⋊C₈	2
#23765878	2
#23763272	2
#23763270	2
C₈×C₄⋊Dic₇₉	2
#23763259	2
C₄⋊C₈×Dic₇₉	2
#23763088	2
C₄×#14579071	2
C₂×#446×C₇₉	2
#20521	79