Subgroups — #34204188

Order 20224.

566 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₈×Dic₇₉	C₂	1	1	yes	maximal, characteristic
2	C₂×Dic₇₉⋊C₁₆	C₂	1	2	yes	maximal, complemented
3	C₂×C₂²⋊(C₇₉⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
4	#23761477	C₂	1	2	yes	maximal, complemented
5	C₂×C₂²⋊C₁₆×C₇₉	C₂	1	1	yes	maximal, characteristic
6	C₂³×C₇₉⋊C₈	C₄	1	1	yes	characteristic
7	C₂²×C₄×Dic₇₉	C₄	1	1	yes	characteristic
8	C₂²×C₇₉⋊C₁₆	C₂²	1	2	yes	—
9	C₂×C₈×Dic₇₉	—	2	4	no	—
10	C₂×C₈×Dic₇₉	C₄	1	1	yes	characteristic
11	C₂×C₈×Dic₇₉	C₄	1	1	yes	characteristic
12	C₂×C₈×Dic₇₉	C₂²	1	1	yes	characteristic
13	C₂×C₈×Dic₇₉	C₂²	1	1	yes	characteristic
14	Dic₇₉⋊C₁₆	—	2	4	no	—
15	C₂²⋊(C₇₉⋊C₁₆)	—	2	2	no	—
16	C₂³×C₈×C₇₉	C₂²	1	1	yes	characteristic
17	C₂²×C₁₆×C₇₉	C₂²	1	2	yes	—
18	C₂²⋊C₁₆×C₇₉	—	2	2	no	—
19	C₂³×Dic₇₉	C₈	1	1	yes	characteristic
20	C₂³×Dic₇₉	C₈	1	1	yes	characteristic
21	C₂²×C₇₉⋊C₈	—	2	2	no	—
22	C₂²×C₇₉⋊C₈	—	2	2	no	—
23	C₂²×C₇₉⋊C₈	—	2	2	no	—
24	C₂²×C₇₉⋊C₈	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₇₉⋊C₈	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄×Dic₇₉	—	2	2	no	—
27	C₂×C₄×Dic₇₉	—	2	2	no	—
28	C₂×C₄×Dic₇₉	C₈	1	1	yes	characteristic
29	C₂×C₄×Dic₇₉	C₈	1	1	yes	characteristic
30	C₂×C₄×Dic₇₉	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄×Dic₇₉	C₂×C₄	1	1	yes	characteristic
32	C₂×C₇₉⋊C₁₆	—	2	2	no	—
33	C₂×C₇₉⋊C₁₆	—	2	2	no	—
34	C₂×C₇₉⋊C₁₆	—	2	2	no	—
35	C₈×Dic₇₉	—	2	2	no	—
36	C₈×Dic₇₉	—	2	2	no	—
37	C₈×Dic₇₉	—	2	2	no	—
38	C₈×Dic₇₉	D₄	1	4	yes	—
39	C₂³×C₄×C₇₉	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₈×C₇₉	—	2	2	no	—
41	C₂²×C₈×C₇₉	—	2	2	no	—
42	C₂²×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
43	C₂²×C₈×C₇₉	D₄	1	2	yes	—
44	C₂²×C₈×C₇₉	Q₈	1	2	yes	—
45	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
46	C₂×C₁₆×C₇₉	—	2	2	no	—
47	C₂×C₁₆×C₇₉	—	2	2	no	—
48	C₂×C₁₆×C₇₉	—	2	2	no	—
49	C₂²×Dic₇₉	—	2	2	no	—
50	C₂²×Dic₇₉	—	2	2	no	—
51	C₂²×Dic₇₉	—	2	2	no	—
52	C₂²×Dic₇₉	—	2	2	no	—
53	C₂²×Dic₇₉	—	2	2	no	—
54	C₂²×Dic₇₉	C₁₆	1	4	yes	complemented
55	C₂²×Dic₇₉	C₂×C₈	1	1	yes	characteristic
56	C₂²×Dic₇₉	C₂×C₈	1	1	yes	characteristic
57	C₂²×Dic₇₉	C₂×C₈	1	1	yes	characteristic
58	C₂²×Dic₇₉	C₂×C₈	1	1	yes	characteristic
59	C₂×C₇₉⋊C₈	—	2	8	no	—
60	C₂×C₇₉⋊C₈	—	2	1	no	—
61	C₂×C₇₉⋊C₈	—	2	1	no	—
62	C₂×C₇₉⋊C₈	—	2	1	no	—
63	C₂×C₇₉⋊C₈	—	2	1	no	—
64	C₂×C₇₉⋊C₈	C₂²⋊C₄	1	2	yes	—
65	C₂×C₇₉⋊C₈	M₁₆	1	2	yes	—
66	C₄×Dic₇₉	—	2	2	no	—
67	C₄×Dic₇₉	—	2	2	no	—
68	C₄×Dic₇₉	—	2	1	no	—
69	C₄×Dic₇₉	—	2	1	no	—
70	C₄×Dic₇₉	C₂²⋊C₄	1	2	yes	—
71	C₄×Dic₇₉	M₁₆	1	2	yes	—
72	C₇₉⋊C₁₆	—	4	2	no	—
73	C₂⁴×C₇₉	C₂×C₈	1	1	yes	characteristic
74	C₂²×C₄×C₇₉	—	2	1	no	—
75	C₂²×C₄×C₇₉	—	2	1	no	—
76	C₂²×C₄×C₇₉	—	2	1	no	—
77	C₂²×C₄×C₇₉	—	2	1	no	—
78	C₂²×C₄×C₇₉	C₄⋊C₄	1	2	yes	—
79	C₂²×C₄×C₇₉	C₂×C₈	1	1	yes	characteristic
80	C₂²×C₄×C₇₉	M₁₆	1	2	yes	—
81	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₈×C₇₉	—	2	4	no	—
83	C₂×C₈×C₇₉	—	2	4	no	—
84	C₂×C₈×C₇₉	—	2	1	no	—
85	C₂×C₈×C₇₉	—	2	1	no	—
86	C₂×C₈×C₇₉	C₄⋊C₄	1	2	yes	—
87	C₂×C₈×C₇₉	C₂×D₄	1	2	yes	—
88	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
89	C₂×C₈×C₇₉	C₂×Q₈	1	1	yes	characteristic
90	C₂×C₈×C₇₉	#41	1	2	yes	—
91	C₁₆×C₇₉	—	4	2	no	—
92	C₂³×C₇₉	—	2	1	no	—
93	C₂³×C₇₉	—	2	1	no	—
94	C₂³×C₇₉	—	2	1	no	—
95	C₂³×C₇₉	—	2	1	no	—
96	C₂³×C₇₉	C₄⋊C₈	1	2	yes	—
97	C₂³×C₇₉	C₂×C₁₆	1	2	yes	—
98	C₂³×C₇₉	M₃₂	1	2	yes	—
99	C₂³×C₇₉	C₂²×C₈	1	1	yes	characteristic
100	C₂×C₄×C₇₉	—	2	1	no	—
101	C₂×C₄×C₇₉	—	2	1	no	—
102	C₂×C₄×C₇₉	—	2	2	no	—
103	C₂×C₄×C₇₉	—	2	2	no	—
104	C₂×C₄×C₇₉	—	2	2	no	—
105	C₂×C₄×C₇₉	—	2	2	no	—
106	C₂×C₄×C₇₉	C₄⋊C₈	1	2	yes	—
107	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
108	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
109	C₂×C₄×C₇₉	#117	1	1	yes	characteristic
110	C₂×C₄×C₇₉	C₂×M₁₆	1	1	yes	characteristic
111	C₂×C₄×C₇₉	C₂×M₁₆	1	1	yes	characteristic
112	C₂×C₄×C₇₉	C₂×M₁₆	1	1	yes	characteristic
113	C₂×Dic₇₉	—	2	8	no	—
114	C₂×Dic₇₉	—	2	1	no	—
115	C₂×Dic₇₉	—	2	1	no	—
116	C₂×Dic₇₉	—	2	1	no	—
117	C₂×Dic₇₉	—	2	1	no	—
118	C₂×Dic₇₉	—	2	1	no	—
119	C₂×Dic₇₉	—	2	1	no	—
120	C₂×Dic₇₉	—	2	8	no	—
121	C₂×Dic₇₉	C₂²⋊C₈	1	2	yes	—
122	C₂×Dic₇₉	C₂²⋊C₈	1	2	yes	—
123	C₂×Dic₇₉	C₂×C₁₆	1	2	yes	—
124	C₂×Dic₇₉	C₂×C₁₆	1	2	yes	—
125	C₂×Dic₇₉	M₃₂	1	2	yes	—
126	C₂×Dic₇₉	M₃₂	1	2	yes	—
127	C₈×C₇₉	—	2	2	no	—
128	C₈×C₇₉	#121	1	2	yes	—
129	C₈×C₇₉	C₂²⋊Q₈	1	2	yes	—
130	C₇₉⋊C₈	—	2	2	no	—
131	C₇₉⋊C₈	—	2	2	no	—
132	C₂²×C₇₉	—	2	2	no	—
133	C₂²×C₇₉	—	2	2	no	—
134	C₂²×C₇₉	—	2	2	no	—
135	C₂²×C₇₉	—	2	2	no	—
136	C₂²×C₇₉	—	2	2	no	—
137	C₂²×C₇₉	—	2	2	no	—
138	C₂²×C₇₉	C₄⋊C₁₆	1	4	yes	complemented
139	C₂²×C₇₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
140	C₂²×C₇₉	C₂×C₄⋊C₈	1	1	yes	characteristic
141	C₂²×C₇₉	#431	1	1	yes	characteristic
142	C₂²×C₇₉	C₂²×C₁₆	1	1	yes	characteristic
143	C₂²×C₇₉	C₂×M₃₂	1	1	yes	characteristic
144	C₂²×C₇₉	C₂×M₃₂	1	1	yes	characteristic
145	C₂²×C₇₉	C₂×M₃₂	1	1	yes	characteristic
146	C₄×C₇₉	—	2	2	no	—
147	C₄×C₇₉	#380	1	1	yes	characteristic
148	C₄×C₇₉	#407	1	1	yes	characteristic
149	C₄×C₇₉	#423	1	1	yes	characteristic
150	C₄×C₇₉	#432	1	1	yes	characteristic
151	Dic₇₉	—	2	2	no	—
152	Dic₇₉	—	2	2	no	—
153	Dic₇₉	—	2	2	no	—
154	Dic₇₉	C₂²⋊C₁₆	1	4	yes	complemented
155	#12783	—	79	1	no	maximal, Sylow
156	C₂×C₇₉	—	2	4	no	—
157	C₂×C₇₉	#1797	1	1	yes	characteristic
158	C₂×C₇₉	C₂×C₂²⋊C₁₆	1	1	yes	characteristic
159	C₂×C₇₉	#2112	1	1	yes	characteristic
160	C₂×C₇₉	C₂×C₄⋊C₁₆	1	1	yes	characteristic
161	C₂×C₇₉	#2150	1	1	yes	characteristic
162	C₂×C₇₉	#2162	1	1	yes	characteristic
163	C₂×C₇₉	#2163	1	1	yes	characteristic
164	C₂²×C₄×C₈	—	79	1	no	—
165	C₂×C₄⋊C₁₆	—	79	2	no	—
166	C₂×C₂²⋊C₁₆	—	79	1	no	—
167	C₂×C₂²⋊C₁₆	D₇₉	1	1	yes	characteristic
168	#1374	—	79	2	no	—
169	C₇₉	#12783	1	1	yes	characteristic, complemented, Sylow
170	C₂³×C₈	—	79	1	no	—
171	C₂³×C₈	C₂×D₇₉	1	1	yes	characteristic
172	C₂²×C₄²	—	79	1	no	—
173	C₂²×C₁₆	—	79	2	no	—
174	C₂²×C₁₆	C₂×D₇₉	1	2	yes	—
175	C₂×C₄×C₈	—	79	1	no	—
176	C₂×C₄×C₈	—	79	1	no	—
177	C₂×C₄×C₈	—	79	1	no	—
178	C₂×C₄×C₈	—	79	1	no	—
179	C₂×C₄×C₈	—	158	4	no	—
180	C₄⋊C₁₆	—	158	4	no	—
181	C₂²⋊C₁₆	—	2	2	no	—
182	C₂²⋊C₁₆	—	158	2	no	—
183	C₂³×C₄	—	79	1	no	—
184	C₂³×C₄	—	79	1	no	—
185	C₂³×C₄	C₄×D₇₉	1	1	yes	characteristic
186	C₂²×C₈	—	2	2	no	—
187	C₂²×C₈	—	2	2	no	—
188	C₂²×C₈	—	79	1	no	—
189	C₂²×C₈	—	79	1	no	—
190	C₂²×C₈	—	158	2	no	—
191	C₂²×C₈	—	158	2	no	—
192	C₂²×C₈	—	158	2	no	—
193	C₂²×C₈	Dic₁₅₈	1	2	yes	—
194	C₂²×C₈	C₄×D₇₉	1	1	yes	characteristic
195	C₂²×C₈	C₂²⋊D₇₉	1	2	yes	—
196	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
197	C₂×C₄²	—	79	1	no	—
198	C₂×C₄²	—	79	1	no	—
199	C₂×C₄²	—	79	1	no	—
200	C₂×C₄²	—	79	1	no	—
201	C₂×C₄²	—	158	2	no	—
202	C₂×C₄²	—	158	2	no	—
203	C₂×C₁₆	—	2	2	no	—
204	C₂×C₁₆	—	2	2	no	—
205	C₂×C₁₆	—	2	2	no	—
206	C₂×C₁₆	—	158	2	no	—
207	C₂×C₁₆	—	158	2	no	—
208	C₂×C₁₆	—	158	2	no	—
209	C₄×C₈	—	79	4	no	—
210	C₄×C₈	—	158	2	no	—
211	C₄×C₈	—	158	2	no	—
212	C₄×C₈	—	158	2	no	—
213	C₂⁴	C₈×D₇₉	1	1	yes	characteristic
214	C₂²×C₄	—	2	1	no	—
215	C₂²×C₄	—	2	1	no	—
216	C₂²×C₄	—	2	1	no	—
217	C₂²×C₄	—	2	1	no	—
218	C₂²×C₄	—	79	1	no	—
219	C₂²×C₄	—	79	4	no	—
220	C₂²×C₄	—	79	1	no	—
221	C₂²×C₄	—	79	1	no	—
222	C₂²×C₄	—	79	1	no	—
223	C₂²×C₄	—	158	2	no	—
224	C₂²×C₄	—	158	2	no	—
225	C₂²×C₄	—	158	2	no	—
226	C₂²×C₄	—	158	2	no	—
227	C₂²×C₄	—	158	2	no	—
228	C₂²×C₄	Dic₇₉⋊C₄	1	2	yes	—
229	C₂²×C₄	C₈×D₇₉	1	1	yes	characteristic
230	C₂²×C₄	#11929607	1	2	yes	—
231	C₂²×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
232	C₂×C₈	—	2	1	no	—
233	C₂×C₈	—	2	1	no	—
234	C₂×C₈	—	2	4	no	—
235	C₂×C₈	—	2	4	no	—
236	C₂×C₈	—	79	2	no	—
237	C₂×C₈	—	79	2	no	—
238	C₂×C₈	—	158	8	no	—
239	C₂×C₈	—	158	1	no	—
240	C₂×C₈	—	158	1	no	—
241	C₂×C₈	—	158	1	no	—
242	C₂×C₈	—	158	1	no	—
243	C₂×C₈	Dic₇₉⋊C₄	1	2	yes	—
244	C₂×C₈	D₄×D₇₉	1	2	yes	—
245	C₂×C₈	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
246	C₂×C₈	C₂×Dic₁₅₈	1	1	yes	characteristic
247	C₂×C₈	#11929623	1	2	yes	—
248	C₄²	—	79	2	no	—
249	C₄²	—	79	2	no	—
250	C₄²	—	158	2	no	—
251	C₄²	—	158	2	no	—
252	C₄²	—	158	1	no	—
253	C₄²	—	158	1	no	—
254	C₁₆	—	4	2	no	—
255	C₁₆	—	316	2	no	—
256	C₂³	—	2	1	no	—
257	C₂³	—	2	1	no	—
258	C₂³	—	2	1	no	—
259	C₂³	—	2	1	no	—
260	C₂³	Dic₇₉⋊C₈	1	2	yes	—
261	C₂³	C₁₆×D₇₉	1	2	yes	—
262	C₂³	#14579100	1	2	yes	—
263	C₂³	C₂×C₈×D₇₉	1	1	yes	characteristic
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	79	2	no	—
271	C₂×C₄	—	79	2	no	—
272	C₂×C₄	—	79	2	no	—
273	C₂×C₄	—	79	2	no	—
274	C₂×C₄	—	79	2	no	—
275	C₂×C₄	—	79	2	no	—
276	C₂×C₄	—	158	8	no	—
277	C₂×C₄	—	158	8	no	—
278	C₂×C₄	—	158	1	no	—
279	C₂×C₄	—	158	1	no	—
280	C₂×C₄	—	158	1	no	—
281	C₂×C₄	—	158	1	no	—
282	C₂×C₄	—	158	1	no	—
283	C₂×C₄	—	158	1	no	—
284	C₂×C₄	Dic₇₉⋊C₈	1	2	yes	—
285	C₂×C₄	C₂²⋊C₄×D₇₉	1	1	yes	characteristic
286	C₂×C₄	C₂×Dic₇₉⋊C₄	1	1	yes	characteristic
287	C₂×C₄	#14579117	1	1	yes	characteristic
288	C₂×C₄	M₁₆×D₇₉	1	1	yes	characteristic
289	C₂×C₄	M₁₆×D₇₉	1	1	yes	characteristic
290	C₂×C₄	C₂×#11929607	1	1	yes	characteristic
291	C₈	—	2	2	no	—
292	C₈	—	158	2	no	—
293	C₈	—	158	2	no	—
294	C₈	#14579133	1	2	yes	—
295	C₈	#14579138	1	2	yes	—
296	C₂²	—	2	2	no	—
297	C₂²	—	2	2	no	—
298	C₂²	—	2	2	no	—
299	C₂²	—	2	2	no	—
300	C₂²	—	2	2	no	—
301	C₂²	—	2	2	no	—
302	C₂²	Dic₇₉⋊C₁₆	1	4	yes	complemented
303	C₂²	C₂²⋊C₈×D₇₉	1	1	yes	characteristic
304	C₂²	C₂×Dic₇₉⋊C₈	1	1	yes	characteristic
305	C₂²	#19098613	1	1	yes	characteristic
306	C₂²	C₂×C₁₆×D₇₉	1	1	yes	characteristic
307	C₂²	M₃₂×D₇₉	1	1	yes	characteristic
308	C₂²	M₃₂×D₇₉	1	1	yes	characteristic
309	C₂²	C₂×#14579100	1	1	yes	characteristic
310	C₄	—	2	2	no	—
311	C₄	—	79	4	no	—
312	C₄	—	158	2	no	—
313	C₄	—	158	2	no	—
314	C₄	—	158	2	no	—
315	C₄	#19098421	1	1	yes	characteristic
316	C₄	#19098525	1	1	yes	characteristic
317	C₄	#19098586	1	1	yes	characteristic
318	C₄	#19098617	1	1	yes	characteristic
319	C₂	—	2	4	no	—
320	C₂	#23763378	1	1	yes	characteristic
321	C₂	C₂²⋊C₁₆×D₇₉	1	1	yes	characteristic
322	C₂	#23765009	1	1	yes	characteristic
323	C₂	C₂×Dic₇₉⋊C₁₆	1	1	yes	characteristic
324	C₂	#23765155	1	1	yes	characteristic
325	C₂	#23765196	1	1	yes	characteristic
326	C₂	#23765200	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₇₉
commutator	C₂²×C₇₉	C₂²×C₁₆
Fitting	C₂×C₂²⋊C₁₆×C₇₉	C₂
socle	C₂³×C₇₉	C₂²×C₈
prime radical	C₂⁴×C₇₉	C₂×C₈
cosocle	C₂²×C₈×C₇₉	C₂³
hypercenter	C₂×C₂²⋊C₁₆	D₇₉
hypocenter	C₇₉	#12783
LCS	G ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₂²⋊C₁₆

Complements

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

N	H ≅ G/N	type
2. C₂×Dic₇₉⋊C₁₆	C₂	semidirect
4. #23761477	C₂	semidirect
54. C₂²×Dic₇₉	C₁₆	semidirect
138. C₂²×C₇₉	C₄⋊C₁₆	semidirect
154. Dic₇₉	C₂²⋊C₁₆	semidirect
169. C₇₉	#12783	semidirect
302. C₂²	Dic₇₉⋊C₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₈×Dic₇₉	2
C₂×Dic₇₉⋊C₁₆	2
C₂×C₂²⋊(C₇₉⋊C₁₆)	2
#23761477	2
C₂×C₂²⋊C₁₆×C₇₉	2
#12783	79