Subgroups — #35153244

Order 20224.

1974 conjugacy classes of subgroups.

The subgroup lattice has 314 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23772489	C₂	1	8	yes	maximal, complemented
2	#23772485	C₂	1	6	yes	maximal, complemented
3	#23772481	C₂	1	2	yes	maximal, complemented
4	Q₈×#11929608	C₂	1	8	yes	maximal, complemented
5	#23772033	C₂	1	6	yes	maximal
6	C₂²×C₄×Q₈×C₇₉	C₂	1	1	yes	maximal, characteristic
7	#19098694	C₂²	1	48	yes	complemented
8	Q₈×C₇₉⋊C₈	C₂²	1	16	yes	complemented
9	#19098620	C₂²	1	12	yes	—
10	#19098620	C₂²	1	12	yes	—
11	#19098616	C₂²	1	12	yes	—
12	#19098616	C₂²	1	12	yes	—
13	#19098587	C₂²	1	12	yes	—
14	#19098527	C₂²	1	4	yes	—
15	C₄×#11929608	C₂²	1	12	yes	—
16	C₂³×Q₈×C₇₉	C₄	1	1	yes	characteristic
17	C₂×C₄×Q₈×C₇₉	C₄	1	4	yes	—
18	C₂×C₄×Q₈×C₇₉	C₄	1	4	yes	—
19	C₂×C₄×Q₈×C₇₉	C₄	1	3	yes	—
20	C₂×C₄×Q₈×C₇₉	C₄	1	1	yes	characteristic
21	C₂×C₄×Q₈×C₇₉	C₂²	1	1	yes	characteristic
22	C₂×C₄×Q₈×C₇₉	C₂²	1	3	yes	—
23	C₂×C₄×Q₈×C₇₉	C₂²	1	4	yes	—
24	C₂×C₄×Q₈×C₇₉	C₂²	1	4	yes	—
25	C₂²×C₄⋊C₄×C₇₉	C₄	1	3	yes	—
26	C₂²×C₄²×C₇₉	C₂²	1	3	yes	—
27	C₂×#11929608	C₂³	1	8	yes	—
28	C₄⋊(C₇₉⋊C₈)	C₂³	1	24	yes	—
29	C₄⋊(C₇₉⋊C₈)	C₂³	1	24	yes	—
30	C₂²⋊(C₇₉⋊C₈)	C₂³	1	8	yes	—
31	C₂²⋊(C₇₉⋊C₈)	C₂³	1	8	yes	—
32	#14579072	C₂³	1	24	yes	—
33	C₄×C₇₉⋊C₈	C₂³	1	24	yes	—
34	C₂²×Q₈×C₇₉	—	2	8	no	—
35	C₂²×Q₈×C₇₉	C₂×C₄	1	4	yes	—
36	C₂²×Q₈×C₇₉	C₂×C₄	1	4	yes	—
37	C₂²×Q₈×C₇₉	C₂×C₄	1	3	yes	—
38	C₂²×Q₈×C₇₉	C₂×C₄	1	1	yes	characteristic
39	C₂³×C₄×C₇₉	C₂×C₄	1	3	yes	—
40	C₂³×C₄×C₇₉	C₂×C₄	1	3	yes	—
41	C₂³×C₄×C₇₉	C₂³	1	1	yes	characteristic
42	C₄×Q₈×C₇₉	C₂×C₄	1	12	yes	—
43	C₄×Q₈×C₇₉	C₂×C₄	1	12	yes	—
44	C₄×Q₈×C₇₉	C₂×C₄	1	12	yes	—
45	C₄×Q₈×C₇₉	C₂×C₄	1	4	yes	—
46	C₄×Q₈×C₇₉	C₂×C₄	1	4	yes	—
47	C₄×Q₈×C₇₉	C₂×C₄	1	4	yes	—
48	C₄×Q₈×C₇₉	C₂³	1	4	yes	—
49	C₄×Q₈×C₇₉	C₂³	1	12	yes	—
50	C₂×C₄⋊C₄×C₇₉	C₂×C₄	1	12	yes	—
51	C₂×C₄⋊C₄×C₇₉	C₂×C₄	1	6	yes	—
52	C₂×C₄⋊C₄×C₇₉	C₂×C₄	1	12	yes	—
53	C₂×C₄⋊C₄×C₇₉	C₂×C₄	1	3	yes	—
54	C₂×C₄⋊C₄×C₇₉	C₂×C₄	1	3	yes	—
55	C₂×C₄²×C₇₉	C₂×C₄	1	6	yes	—
56	C₂×C₄²×C₇₉	C₂×C₄	1	3	yes	—
57	C₂×C₄²×C₇₉	C₂×C₄	1	6	yes	—
58	C₂×C₄²×C₇₉	C₂×C₄	1	3	yes	—
59	C₂×C₄²×C₇₉	C₂³	1	6	yes	—
60	C₂×C₄²×C₇₉	C₂³	1	3	yes	—
61	C₂×C₄²×C₇₉	C₂³	1	6	yes	—
62	C₂×C₄²×C₇₉	C₂³	1	3	yes	—
63	#11929608	—	2	16	no	—
64	C₂×C₇₉⋊C₈	C₂⁴	1	16	yes	—
65	C₂⁴×C₇₉	C₂²×C₄	1	1	yes	characteristic
66	C₂×Q₈×C₇₉	—	2	16	no	—
67	C₂×Q₈×C₇₉	—	2	16	no	—
68	C₂×Q₈×C₇₉	—	2	12	no	—
69	C₂×Q₈×C₇₉	M₁₆	1	8	yes	complemented
70	C₂×Q₈×C₇₉	C₂²×C₄	1	12	yes	—
71	C₂×Q₈×C₇₉	C₂²×C₄	1	4	yes	—
72	C₂²×C₄×C₇₉	—	2	12	no	—
73	C₂²×C₄×C₇₉	—	2	4	no	—
74	C₂²×C₄×C₇₉	—	4	6	no	—
75	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
76	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
77	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
78	C₂²×C₄×C₇₉	C₂²×C₄	1	6	yes	—
79	C₂²×C₄×C₇₉	C₂²×C₄	1	3	yes	—
80	C₂²×C₄×C₇₉	C₂²×C₄	1	6	yes	—
81	C₂²×C₄×C₇₉	C₂²×C₄	1	6	yes	—
82	C₂²×C₄×C₇₉	C₂²×C₄	1	6	yes	—
83	C₂²×C₄×C₇₉	C₂²×C₄	1	3	yes	—
84	C₂²×C₄×C₇₉	C₂²×C₄	1	3	yes	—
85	C₂²×C₄×C₇₉	C₂²×C₄	1	3	yes	—
86	C₂²×C₄×C₇₉	C₂⁴	1	1	yes	characteristic
87	C₂²×C₄×C₇₉	C₂⁴	1	1	yes	characteristic
88	C₂²×C₄×C₇₉	C₂⁴	1	1	yes	characteristic
89	C₄⋊C₄×C₇₉	C₂²×C₄	1	24	yes	—
90	C₄⋊C₄×C₇₉	C₂²×C₄	1	12	yes	—
91	C₄⋊C₄×C₇₉	C₂²×C₄	1	12	yes	—
92	C₄²×C₇₉	C₂²×C₄	1	6	yes	—
93	C₄²×C₇₉	C₂²×C₄	1	6	yes	—
94	C₄²×C₇₉	C₂²×C₄	1	6	yes	—
95	C₄²×C₇₉	C₂²×C₄	1	6	yes	—
96	C₄²×C₇₉	C₂²×C₄	1	6	yes	—
97	C₄²×C₇₉	C₂²×C₄	1	6	yes	—
98	C₄²×C₇₉	C₂⁴	1	6	yes	—
99	C₄²×C₇₉	C₂⁴	1	6	yes	—
100	C₂³×C₇₉	—	2	2	no	—
101	C₂³×C₇₉	—	2	2	no	—
102	C₂³×C₇₉	—	2	2	no	—
103	C₂³×C₇₉	C₂³×C₄	1	1	yes	characteristic
104	C₂³×C₇₉	C₂³×C₄	1	1	yes	characteristic
105	C₂³×C₇₉	C₂³×C₄	1	1	yes	characteristic
106	Q₈×C₇₉	—	2	24	no	—
107	Q₈×C₇₉	C₂×M₁₆	1	16	yes	—
108	C₂×C₄×C₇₉	—	2	2	no	—
109	C₂×C₄×C₇₉	—	2	6	no	—
110	C₂×C₄×C₇₉	—	2	12	no	—
111	C₂×C₄×C₇₉	—	2	2	no	—
112	C₂×C₄×C₇₉	—	2	2	no	—
113	C₂×C₄×C₇₉	—	2	12	no	—
114	C₂×C₄×C₇₉	—	2	12	no	—
115	C₂×C₄×C₇₉	—	2	2	no	—
116	C₂×C₄×C₇₉	—	2	2	no	—
117	C₂×C₄×C₇₉	—	4	6	no	—
118	C₂×C₄×C₇₉	—	4	3	no	—
119	C₂×C₄×C₇₉	—	4	3	no	—
120	C₂×C₄×C₇₉	C₂×M₁₆	1	12	yes	—
121	C₂×C₄×C₇₉	C₂³×C₄	1	1	yes	characteristic
122	C₂×C₄×C₇₉	C₂³×C₄	1	1	yes	characteristic
123	C₂×C₄×C₇₉	C₂³×C₄	1	6	yes	—
124	C₂×C₄×C₇₉	C₂³×C₄	1	6	yes	—
125	C₂×C₄×C₇₉	C₂³×C₄	1	6	yes	—
126	C₂×C₄×C₇₉	C₂³×C₄	1	1	yes	characteristic
127	C₂×C₄×C₇₉	C₂³×C₄	1	6	yes	—
128	C₂×C₄×C₇₉	2₋¹⁺⁴	1	4	yes	—
129	C₂×C₄×C₇₉	C₂⁵	1	1	yes	characteristic
130	C₇₉⋊C₈	—	2	16	no	—
131	C₂²×C₇₉	—	2	8	no	—
132	C₂²×C₇₉	—	2	1	no	—
133	C₂²×C₇₉	—	2	1	no	—
134	C₂²×C₇₉	—	2	1	no	—
135	C₂²×C₇₉	—	2	1	no	—
136	C₂²×C₇₉	—	2	1	no	—
137	C₂²×C₇₉	—	2	1	no	—
138	C₂²×C₇₉	#519	1	2	yes	—
139	C₂²×C₇₉	C₂²×M₁₆	1	2	yes	—
140	C₂²×C₇₉	#568	1	2	yes	—
141	C₂²×C₇₉	C₂⁴×C₄	1	1	yes	characteristic
142	C₄×C₇₉	—	2	1	no	—
143	C₄×C₇₉	—	2	1	no	—
144	C₄×C₇₉	—	2	6	no	—
145	C₄×C₇₉	—	2	6	no	—
146	C₄×C₇₉	—	2	6	no	—
147	C₄×C₇₉	#519	1	2	yes	—
148	C₄×C₇₉	C₂²×M₁₆	1	12	yes	—
149	C₄×C₇₉	C₂×2₋¹⁺⁴	1	2	yes	—
150	#60248	—	79	1	no	maximal, Sylow
151	C₂×C₇₉	—	2	2	no	—
152	C₂×C₇₉	—	2	2	no	—
153	C₂×C₇₉	—	2	2	no	—
154	C₂×C₇₉	C₂×#519	1	1	yes	characteristic
155	C₂×C₇₉	C₂³×M₁₆	1	1	yes	characteristic
156	C₂×C₇₉	C₂×#568	1	1	yes	characteristic
157	C₂²×C₄×Q₈	D₇₉	1	1	yes	characteristic
158	#2984	—	79	8	no	—
159	#2983	—	79	6	no	—
160	#2982	—	79	2	no	—
161	Q₈×M₁₆	—	79	8	no	—
162	#2920	—	79	6	no	—
163	C₇₉	#60248	1	1	yes	characteristic, complemented, Sylow
164	C₂³×Q₈	Dic₇₉	1	1	yes	characteristic
165	C₂×C₄×Q₈	Dic₇₉	1	4	yes	—
166	C₂×C₄×Q₈	Dic₇₉	1	4	yes	—
167	C₂×C₄×Q₈	Dic₇₉	1	3	yes	—
168	C₂×C₄×Q₈	Dic₇₉	1	1	yes	characteristic
169	C₂×C₄×Q₈	C₂×D₇₉	1	4	yes	—
170	C₂×C₄×Q₈	C₂×D₇₉	1	1	yes	characteristic
171	C₂×C₄×Q₈	C₂×D₇₉	1	3	yes	—
172	C₂×C₄×Q₈	C₂×D₇₉	1	4	yes	—
173	C₂²×C₄⋊C₄	Dic₇₉	1	3	yes	—
174	C₂²×C₄²	C₂×D₇₉	1	3	yes	—
175	#446	—	79	48	no	—
176	C₈×Q₈	—	79	16	no	—
177	#432	—	79	12	no	—
178	#432	—	79	12	no	—
179	#431	—	79	12	no	—
180	#431	—	79	12	no	—
181	#423	—	79	12	no	—
182	#407	—	79	4	no	—
183	C₄×M₁₆	—	79	12	no	—
184	C₂²×Q₈	—	2	8	no	—
185	C₂²×Q₈	C₂×Dic₇₉	1	4	yes	—
186	C₂²×Q₈	C₂×Dic₇₉	1	4	yes	—
187	C₂²×Q₈	C₂×Dic₇₉	1	3	yes	—
188	C₂²×Q₈	C₂×Dic₇₉	1	1	yes	characteristic
189	C₂³×C₄	C₂×Dic₇₉	1	3	yes	—
190	C₂³×C₄	C₂×Dic₇₉	1	3	yes	—
191	C₂³×C₄	C₂²×D₇₉	1	1	yes	characteristic
192	C₂×M₁₆	—	79	8	no	—
193	C₄×Q₈	C₂×Dic₇₉	1	12	yes	—
194	C₄×Q₈	C₂×Dic₇₉	1	12	yes	—
195	C₄×Q₈	C₂×Dic₇₉	1	12	yes	—
196	C₄×Q₈	C₂×Dic₇₉	1	4	yes	—
197	C₄×Q₈	C₂×Dic₇₉	1	4	yes	—
198	C₄×Q₈	C₂×Dic₇₉	1	4	yes	—
199	C₄×Q₈	C₂²×D₇₉	1	4	yes	—
200	C₄×Q₈	C₂²×D₇₉	1	12	yes	—
201	C₂×C₄⋊C₄	C₂×Dic₇₉	1	12	yes	—
202	C₂×C₄⋊C₄	C₂×Dic₇₉	1	12	yes	—
203	C₂×C₄⋊C₄	C₂×Dic₇₉	1	6	yes	—
204	C₂×C₄⋊C₄	C₂×Dic₇₉	1	3	yes	—
205	C₂×C₄⋊C₄	C₂×Dic₇₉	1	3	yes	—
206	C₂×C₄²	C₂×Dic₇₉	1	6	yes	—
207	C₂×C₄²	C₂×Dic₇₉	1	3	yes	—
208	C₂×C₄²	C₂×Dic₇₉	1	6	yes	—
209	C₂×C₄²	C₂×Dic₇₉	1	3	yes	—
210	C₂×C₄²	C₂²×D₇₉	1	6	yes	—
211	C₂×C₄²	C₂²×D₇₉	1	3	yes	—
212	C₂×C₄²	C₂²×D₇₉	1	6	yes	—
213	C₂×C₄²	C₂²×D₇₉	1	3	yes	—
214	C₄⋊C₈	—	79	24	no	—
215	C₄⋊C₈	—	79	24	no	—
216	C₂²⋊C₈	—	79	8	no	—
217	C₂²⋊C₈	—	79	8	no	—
218	#97	—	79	24	no	—
219	C₄×C₈	—	79	24	no	—
220	C₂⁴	C₂²×Dic₇₉	1	1	yes	characteristic
221	C₂×Q₈	—	2	16	no	—
222	C₂×Q₈	—	2	16	no	—
223	C₂×Q₈	—	2	12	no	—
224	C₂×Q₈	#11929608	1	8	yes	complemented
225	C₂×Q₈	C₂²×Dic₇₉	1	12	yes	—
226	C₂×Q₈	C₂²×Dic₇₉	1	4	yes	—
227	C₂²×C₄	—	2	4	no	—
228	C₂²×C₄	—	2	12	no	—
229	C₂²×C₄	—	4	6	no	—
230	C₂²×C₄	C₂²×Dic₇₉	1	6	yes	—
231	C₂²×C₄	C₂²×Dic₇₉	1	3	yes	—
232	C₂²×C₄	C₂²×Dic₇₉	1	6	yes	—
233	C₂²×C₄	C₂²×Dic₇₉	1	6	yes	—
234	C₂²×C₄	C₂²×Dic₇₉	1	1	yes	characteristic
235	C₂²×C₄	C₂²×Dic₇₉	1	1	yes	characteristic
236	C₂²×C₄	C₂²×Dic₇₉	1	1	yes	characteristic
237	C₂²×C₄	C₂²×Dic₇₉	1	6	yes	—
238	C₂²×C₄	C₂²×Dic₇₉	1	3	yes	—
239	C₂²×C₄	C₂²×Dic₇₉	1	3	yes	—
240	C₂²×C₄	C₂²×Dic₇₉	1	3	yes	—
241	C₂²×C₄	C₂³×D₇₉	1	1	yes	characteristic
242	C₂²×C₄	C₂³×D₇₉	1	1	yes	characteristic
243	C₂²×C₄	C₂³×D₇₉	1	1	yes	characteristic
244	M₁₆	—	158	16	no	—
245	C₂×C₈	—	79	16	no	—
246	C₄⋊C₄	C₂²×Dic₇₉	1	24	yes	—
247	C₄⋊C₄	C₂²×Dic₇₉	1	12	yes	—
248	C₄⋊C₄	C₂²×Dic₇₉	1	12	yes	—
249	C₄²	C₂²×Dic₇₉	1	6	yes	—
250	C₄²	C₂²×Dic₇₉	1	6	yes	—
251	C₄²	C₂²×Dic₇₉	1	6	yes	—
252	C₄²	C₂²×Dic₇₉	1	6	yes	—
253	C₄²	C₂²×Dic₇₉	1	6	yes	—
254	C₄²	C₂²×Dic₇₉	1	6	yes	—
255	C₄²	C₂³×D₇₉	1	6	yes	—
256	C₄²	C₂³×D₇₉	1	6	yes	—
257	C₂³	—	2	2	no	—
258	C₂³	—	2	2	no	—
259	C₂³	—	2	2	no	—
260	C₂³	C₂³×Dic₇₉	1	1	yes	characteristic
261	C₂³	C₂³×Dic₇₉	1	1	yes	characteristic
262	C₂³	C₂³×Dic₇₉	1	1	yes	characteristic
263	Q₈	—	2	24	no	—
264	Q₈	C₂×#11929608	1	16	yes	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	6	no	—
267	C₂×C₄	—	2	12	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	2	12	no	—
273	C₂×C₄	—	2	12	no	—
274	C₂×C₄	—	4	6	no	—
275	C₂×C₄	—	4	3	no	—
276	C₂×C₄	—	4	3	no	—
277	C₂×C₄	C₂×#11929608	1	12	yes	—
278	C₂×C₄	C₂³×Dic₇₉	1	1	yes	characteristic
279	C₂×C₄	C₂³×Dic₇₉	1	1	yes	characteristic
280	C₂×C₄	C₂³×Dic₇₉	1	6	yes	—
281	C₂×C₄	C₂³×Dic₇₉	1	6	yes	—
282	C₂×C₄	C₂³×Dic₇₉	1	6	yes	—
283	C₂×C₄	C₂³×Dic₇₉	1	1	yes	characteristic
284	C₂×C₄	C₂³×Dic₇₉	1	6	yes	—
285	C₂×C₄	#14579209	1	4	yes	—
286	C₂×C₄	C₂⁴×D₇₉	1	1	yes	characteristic
287	C₈	—	158	16	no	—
288	C₂²	—	2	8	no	—
289	C₂²	—	2	1	no	—
290	C₂²	—	2	1	no	—
291	C₂²	—	2	1	no	—
292	C₂²	—	2	1	no	—
293	C₂²	—	2	1	no	—
294	C₂²	—	2	1	no	—
295	C₂²	#19099041	1	2	yes	—
296	C₂²	C₂²×#11929608	1	2	yes	—
297	C₂²	#19099330	1	2	yes	—
298	C₂²	C₂⁴×Dic₇₉	1	1	yes	characteristic
299	C₄	—	2	6	no	—
300	C₄	—	2	6	no	—
301	C₄	—	2	6	no	—
302	C₄	—	2	1	no	—
303	C₄	—	2	1	no	—
304	C₄	#19099041	1	2	yes	—
305	C₄	C₂²×#11929608	1	12	yes	—
306	C₄	C₂×#14579209	1	2	yes	—
307	C₂	—	2	2	no	—
308	C₂	—	2	2	no	—
309	C₂	—	2	2	no	—
310	C₂	C₂×#19099041	1	1	yes	characteristic
311	C₂	C₂³×#11929608	1	1	yes	characteristic
312	C₂	C₂×#19099330	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂⁴×D₇₉
commutator, socle	C₂²×C₇₉	C₂⁴×C₄
Fitting	C₂²×C₄×Q₈×C₇₉	C₂
prime radical	C₂⁴×C₇₉	C₂²×C₄
cosocle	C₂×C₄×C₇₉	C₂⁵
hypercenter	C₂²×C₄×Q₈	D₇₉
hypocenter	C₇₉	#60248
LCS	G ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄×Q₈

Complements

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

N	H ≅ G/N	type
1. #23772489	C₂	semidirect
2. #23772485	C₂	semidirect
3. #23772481	C₂	semidirect
4. Q₈×#11929608	C₂	semidirect
7. #19098694	C₂²	semidirect
8. Q₈×C₇₉⋊C₈	C₂²	semidirect
69. C₂×Q₈×C₇₉	M₁₆	semidirect
163. C₇₉	#60248	semidirect
224. C₂×Q₈	#11929608	semidirect

Maximal subgroups

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

subgroup	index
#23772489	2
#23772485	2
#23772481	2
Q₈×#11929608	2
#23772033	2
C₂²×C₄×Q₈×C₇₉	2
#60248	79