Subgroups — #34286409

Order 20224.

1222 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19098616	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄²×Dic₇₉	C₂	1	1	yes	maximal, characteristic
3	#23763255	C₂	1	4	yes	maximal, complemented
4	#23763164	C₂	1	4	yes	maximal
5	#23763145	C₂	1	2	yes	maximal, complemented
6	C₄×#14579071	C₂	1	2	yes	maximal, complemented
7	C₂×#431×C₇₉	C₂	1	1	yes	maximal, characteristic
8	C₂²×C₄×Dic₇₉	C₄	1	2	yes	—
9	C₂²×C₄×Dic₇₉	C₄	1	2	yes	—
10	C₂²×C₄×Dic₇₉	C₂²	1	2	yes	—
11	#19098616	C₄	1	8	yes	complemented
12	C₂×C₄⋊(C₇₉⋊C₈)	C₂²	1	2	yes	—
13	C₂×C₂²⋊(C₇₉⋊C₈)	C₂²	1	2	yes	—
14	C₂×#14579071	C₂²	1	8	yes	—
15	C₂×C₄×C₇₉⋊C₈	C₂²	1	2	yes	—
16	C₄²×Dic₇₉	C₄	1	2	yes	—
17	C₄²×Dic₇₉	C₄	1	2	yes	—
18	C₄²×Dic₇₉	C₂²	1	2	yes	—
19	C₄²×Dic₇₉	C₂²	1	1	yes	characteristic
20	C₄²×Dic₇₉	C₂²	1	1	yes	characteristic
21	#19098317	C₂²	1	8	yes	—
22	C₂²×C₄²×C₇₉	C₂²	1	1	yes	characteristic
23	#431×C₇₉	C₄	1	8	yes	complemented
24	C₂×C₄⋊C₈×C₇₉	C₂²	1	2	yes	—
25	C₂×C₂²⋊C₈×C₇₉	C₂²	1	2	yes	—
26	C₂×C₄×C₈×C₇₉	C₂²	1	2	yes	—
27	C₂³×Dic₇₉	C₂×C₄	1	4	yes	—
28	C₂²×C₇₉⋊C₈	C₂³	1	4	yes	—
29	C₂×C₄×Dic₇₉	—	2	8	no	—
30	C₂×C₄×Dic₇₉	—	2	8	no	—
31	C₂×C₄×Dic₇₉	—	2	8	no	—
32	C₂×C₄×Dic₇₉	C₂×C₄	1	4	yes	—
33	C₂×C₄×Dic₇₉	C₂×C₄	1	4	yes	—
34	C₂×C₄×Dic₇₉	C₂×C₄	1	2	yes	—
35	C₂×C₄×Dic₇₉	C₂×C₄	1	2	yes	—
36	C₂×C₄×Dic₇₉	C₂×C₄	1	4	yes	—
37	C₂×C₄×Dic₇₉	C₂×C₄	1	2	yes	—
38	C₂×C₄×Dic₇₉	C₂×C₄	1	2	yes	—
39	C₂×C₄×Dic₇₉	C₂³	1	2	yes	—
40	C₂×C₄×Dic₇₉	C₂³	1	2	yes	—
41	C₄⋊(C₇₉⋊C₈)	C₂×C₄	1	8	yes	complemented
42	C₂²⋊(C₇₉⋊C₈)	C₂×C₄	1	8	yes	—
43	#14579071	—	2	16	no	—
44	C₄×C₇₉⋊C₈	C₂×C₄	1	8	yes	complemented
45	C₂³×C₄×C₇₉	C₂×C₄	1	1	yes	characteristic
46	C₂³×C₄×C₇₉	C₂×C₄	1	1	yes	characteristic
47	C₂³×C₄×C₇₉	C₂³	1	1	yes	characteristic
48	C₂²×C₈×C₇₉	C₂³	1	4	yes	—
49	C₂×C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄²×C₇₉	C₂×C₄	1	4	yes	—
52	C₂×C₄²×C₇₉	C₂×C₄	1	4	yes	—
53	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
54	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
55	C₄⋊C₈×C₇₉	C₂×C₄	1	8	yes	complemented
56	C₂²⋊C₈×C₇₉	C₂×C₄	1	8	yes	—
57	C₄×C₈×C₇₉	C₂×C₄	1	8	yes	complemented
58	C₂²×Dic₇₉	—	2	8	no	—
59	C₂²×Dic₇₉	—	2	8	no	—
60	C₂²×Dic₇₉	—	2	8	no	—
61	C₂²×Dic₇₉	C₂²×C₄	1	4	yes	—
62	C₂²×Dic₇₉	C₂²×C₄	1	4	yes	—
63	C₂×C₇₉⋊C₈	—	2	8	no	—
64	C₂×C₇₉⋊C₈	C₂²×C₄	1	8	yes	—
65	C₄×Dic₇₉	—	2	8	no	—
66	C₄×Dic₇₉	—	2	8	no	—
67	C₄×Dic₇₉	—	2	4	no	—
68	C₄×Dic₇₉	—	2	4	no	—
69	C₄×Dic₇₉	—	2	8	no	—
70	C₄×Dic₇₉	—	2	4	no	—
71	C₄×Dic₇₉	—	2	4	no	—
72	C₄×Dic₇₉	M₁₆	1	8	yes	—
73	C₄×Dic₇₉	#41	1	8	yes	—
74	C₂⁴×C₇₉	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₄×C₇₉	—	2	4	no	—
76	C₂²×C₄×C₇₉	—	2	4	no	—
77	C₂²×C₄×C₇₉	C₄²	1	4	yes	—
78	C₂²×C₄×C₇₉	C₄²	1	4	yes	—
79	C₂²×C₄×C₇₉	M₁₆	1	8	yes	—
80	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
81	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
82	C₂²×C₄×C₇₉	C₂²×C₄	1	2	yes	—
83	C₂²×C₄×C₇₉	C₂²×C₄	1	2	yes	—
84	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
85	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
86	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
87	C₂²×C₄×C₇₉	C₂⁴	1	1	yes	characteristic
88	C₂×C₈×C₇₉	—	2	8	no	—
89	C₂×C₈×C₇₉	C₂²×C₄	1	8	yes	—
90	C₄²×C₇₉	C₄²	1	4	yes	—
91	C₄²×C₇₉	C₄²	1	4	yes	—
92	C₄²×C₇₉	C₂²×C₄	1	2	yes	—
93	C₄²×C₇₉	C₂²×C₄	1	2	yes	—
94	C₄²×C₇₉	C₂²×C₄	1	2	yes	—
95	C₄²×C₇₉	C₂²×C₄	1	2	yes	—
96	C₂³×C₇₉	—	2	2	no	—
97	C₂³×C₇₉	—	2	2	no	—
98	C₂³×C₇₉	C₂×C₄²	1	2	yes	—
99	C₂³×C₇₉	C₂×M₁₆	1	4	yes	—
100	C₂³×C₇₉	C₂³×C₄	1	1	yes	characteristic
101	C₂×C₄×C₇₉	—	2	2	no	—
102	C₂×C₄×C₇₉	—	2	2	no	—
103	C₂×C₄×C₇₉	—	2	4	no	—
104	C₂×C₄×C₇₉	—	2	4	no	—
105	C₂×C₄×C₇₉	—	2	8	no	—
106	C₂×C₄×C₇₉	#97	1	16	yes	complemented
107	C₂×C₄×C₇₉	C₂×C₄²	1	2	yes	—
108	C₂×C₄×C₇₉	C₂×C₄²	1	2	yes	—
109	C₂×C₄×C₇₉	C₂×C₄²	1	2	yes	—
110	C₂×C₄×C₇₉	C₂×C₄²	1	2	yes	—
111	C₂×C₄×C₇₉	C₂×C₄²	1	2	yes	—
112	C₂×C₄×C₇₉	C₂×M₁₆	1	4	yes	—
113	C₂×C₄×C₇₉	C₂×M₁₆	1	4	yes	—
114	C₂×C₄×C₇₉	C₂×M₁₆	1	4	yes	—
115	C₂×C₄×C₇₉	C₂³×C₄	1	1	yes	characteristic
116	C₂×C₄×C₇₉	C₂³×C₄	1	1	yes	characteristic
117	C₂×C₄×C₇₉	C₂×#41	1	4	yes	—
118	C₂×Dic₇₉	—	2	32	no	—
119	C₂×Dic₇₉	—	2	4	no	—
120	C₂×Dic₇₉	—	2	4	no	—
121	C₂×Dic₇₉	—	2	4	no	—
122	C₂×Dic₇₉	—	2	4	no	—
123	C₂×Dic₇₉	#117	1	8	yes	—
124	C₂×Dic₇₉	C₂×M₁₆	1	8	yes	—
125	C₈×C₇₉	—	2	8	no	—
126	C₇₉⋊C₈	—	2	8	no	—
127	C₂²×C₇₉	—	2	2	no	—
128	C₂²×C₇₉	—	2	8	no	—
129	C₂²×C₇₉	—	2	2	no	—
130	C₂²×C₇₉	C₂×#97	1	4	yes	—
131	C₂²×C₇₉	C₂²×C₄²	1	1	yes	characteristic
132	C₂²×C₇₉	C₂×#117	1	2	yes	—
133	C₂²×C₇₉	C₂²×M₁₆	1	2	yes	—
134	C₂²×C₇₉	C₂²×M₁₆	1	2	yes	—
135	C₄×C₇₉	—	2	2	no	—
136	C₄×C₇₉	—	2	2	no	—
137	C₄×C₇₉	C₂×#97	1	4	yes	—
138	C₄×C₇₉	C₂×#97	1	4	yes	—
139	C₄×C₇₉	C₄×M₁₆	1	4	yes	—
140	C₄×C₇₉	C₄×#41	1	2	yes	—
141	C₄×C₇₉	C₄×#41	1	2	yes	—
142	Dic₇₉	—	2	8	no	—
143	Dic₇₉	—	2	8	no	—
144	#20463	—	79	1	no	maximal, Sylow
145	C₂×C₇₉	—	2	4	no	—
146	C₂×C₇₉	C₄×#117	1	1	yes	characteristic
147	C₂×C₇₉	C₂²×#97	1	1	yes	characteristic
148	C₂×C₇₉	C₂×C₄×M₁₆	1	1	yes	characteristic
149	C₂×C₇₉	#2920	1	4	yes	—
150	C₂×#431	—	79	1	no	—
151	C₂×#431	D₇₉	1	1	yes	characteristic
152	C₂×C₄³	—	79	1	no	—
153	#1767	—	79	4	no	—
154	#1749	—	79	4	no	—
155	#1744	—	79	2	no	—
156	C₄×#97	—	79	2	no	—
157	C₇₉	#20463	1	1	yes	characteristic, complemented, Sylow
158	C₂²×C₄²	—	79	2	no	—
159	C₂²×C₄²	—	79	2	no	—
160	C₂²×C₄²	—	79	2	no	—
161	C₂²×C₄²	C₂×D₇₉	1	1	yes	characteristic
162	#431	—	79	8	no	—
163	#431	Dic₇₉	1	8	yes	complemented
164	C₂×C₄⋊C₈	—	79	2	no	—
165	C₂×C₄⋊C₈	C₂×D₇₉	1	2	yes	—
166	C₂×C₂²⋊C₈	—	79	2	no	—
167	C₂×C₂²⋊C₈	C₂×D₇₉	1	2	yes	—
168	C₂×#97	—	79	8	no	—
169	C₂×C₄×C₈	—	79	2	no	—
170	C₂×C₄×C₈	C₂×D₇₉	1	2	yes	—
171	C₄³	—	79	2	no	—
172	C₄³	—	79	1	no	—
173	C₄³	—	79	1	no	—
174	C₄³	—	79	2	no	—
175	C₄³	—	79	2	no	—
176	#336	—	79	8	no	—
177	C₂³×C₄	—	79	4	no	—
178	C₂³×C₄	C₄×D₇₉	1	1	yes	characteristic
179	C₂³×C₄	C₄×D₇₉	1	1	yes	characteristic
180	C₂³×C₄	C₂²×D₇₉	1	1	yes	characteristic
181	C₂²×C₈	—	79	4	no	—
182	C₂²×C₈	C₂²×D₇₉	1	4	yes	—
183	C₂×C₄²	—	79	2	no	—
184	C₂×C₄²	—	79	2	no	—
185	C₂×C₄²	—	79	4	no	—
186	C₂×C₄²	—	79	4	no	—
187	C₂×C₄²	—	79	2	no	—
188	C₂×C₄²	—	79	4	no	—
189	C₂×C₄²	—	79	2	no	—
190	C₂×C₄²	—	79	2	no	—
191	C₂×C₄²	—	79	2	no	—
192	C₂×C₄²	—	158	8	no	—
193	C₂×C₄²	—	158	8	no	—
194	C₂×C₄²	—	158	8	no	—
195	C₂×C₄²	C₄×D₇₉	1	1	yes	characteristic
196	C₂×C₄²	C₄×D₇₉	1	1	yes	characteristic
197	C₂×C₄²	C₄×D₇₉	1	4	yes	—
198	C₂×C₄²	C₂×Dic₇₉	1	4	yes	—
199	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
200	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
201	C₄⋊C₈	—	79	8	no	—
202	C₄⋊C₈	C₂×Dic₇₉	1	8	yes	complemented
203	C₂²⋊C₈	—	79	8	no	—
204	C₂²⋊C₈	C₂×Dic₇₉	1	8	yes	—
205	#97	—	158	16	no	—
206	C₄×C₈	—	79	8	no	—
207	C₄×C₈	C₂×Dic₇₉	1	8	yes	complemented
208	C₂⁴	C₂×C₄×D₇₉	1	1	yes	characteristic
209	C₂²×C₄	—	2	4	no	—
210	C₂²×C₄	—	2	4	no	—
211	C₂²×C₄	—	79	4	no	—
212	C₂²×C₄	—	79	4	no	—
213	C₂²×C₄	—	158	8	no	—
214	C₂²×C₄	—	158	8	no	—
215	C₂²×C₄	—	158	8	no	—
216	C₂²×C₄	C₄×Dic₇₉	1	4	yes	—
217	C₂²×C₄	C₄×Dic₇₉	1	4	yes	—
218	C₂²×C₄	#11929607	1	8	yes	—
219	C₂²×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
220	C₂²×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
221	C₂²×C₄	C₂×C₄×D₇₉	1	2	yes	—
222	C₂²×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
223	C₂²×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
224	C₂²×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
225	C₂²×C₄	C₂²×Dic₇₉	1	2	yes	—
226	C₂²×C₄	C₂³×D₇₉	1	1	yes	characteristic
227	C₂×C₈	—	2	8	no	—
228	C₂×C₈	—	79	8	no	—
229	C₂×C₈	—	158	8	no	—
230	C₂×C₈	C₂²×Dic₇₉	1	8	yes	—
231	C₄²	—	79	8	no	—
232	C₄²	—	79	8	no	—
233	C₄²	—	158	8	no	—
234	C₄²	—	158	8	no	—
235	C₄²	—	158	4	no	—
236	C₄²	—	158	4	no	—
237	C₄²	—	158	8	no	—
238	C₄²	—	158	4	no	—
239	C₄²	—	158	4	no	—
240	C₄²	C₄×Dic₇₉	1	4	yes	—
241	C₄²	C₄×Dic₇₉	1	4	yes	—
242	C₄²	C₂×C₄×D₇₉	1	2	yes	—
243	C₄²	C₂×C₄×D₇₉	1	2	yes	—
244	C₄²	C₂²×Dic₇₉	1	2	yes	—
245	C₄²	C₂²×Dic₇₉	1	2	yes	—
246	C₂³	—	2	2	no	—
247	C₂³	—	2	2	no	—
248	C₂³	C₂×C₄×Dic₇₉	1	2	yes	—
249	C₂³	C₂×#11929607	1	4	yes	—
250	C₂³	C₂²×C₄×D₇₉	1	1	yes	characteristic
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	2	8	no	—
254	C₂×C₄	—	2	4	no	—
255	C₂×C₄	—	2	4	no	—
256	C₂×C₄	—	79	8	no	—
257	C₂×C₄	—	79	8	no	—
258	C₂×C₄	—	158	32	no	—
259	C₂×C₄	—	158	4	no	—
260	C₂×C₄	—	158	4	no	—
261	C₂×C₄	—	158	4	no	—
262	C₂×C₄	—	158	4	no	—
263	C₂×C₄	#14579071	1	16	yes	complemented
264	C₂×C₄	C₂×C₄×Dic₇₉	1	2	yes	—
265	C₂×C₄	C₂×C₄×Dic₇₉	1	2	yes	—
266	C₂×C₄	C₂×C₄×Dic₇₉	1	2	yes	—
267	C₂×C₄	C₂×C₄×Dic₇₉	1	2	yes	—
268	C₂×C₄	C₂×C₄×Dic₇₉	1	2	yes	—
269	C₂×C₄	M₁₆×D₇₉	1	4	yes	—
270	C₂×C₄	C₂×#11929607	1	4	yes	—
271	C₂×C₄	C₂×#11929607	1	4	yes	—
272	C₂×C₄	C₂²×C₄×D₇₉	1	1	yes	characteristic
273	C₂×C₄	C₂³×Dic₇₉	1	1	yes	characteristic
274	C₂×C₄	#41×D₇₉	1	4	yes	—
275	C₈	—	2	8	no	—
276	C₈	—	158	8	no	—
277	C₂²	—	2	8	no	—
278	C₂²	—	2	2	no	—
279	C₂²	—	2	2	no	—
280	C₂²	C₂×#14579071	1	4	yes	—
281	C₂²	C₂²×C₄×Dic₇₉	1	1	yes	characteristic
282	C₂²	#117×D₇₉	1	2	yes	—
283	C₂²	C₂×M₁₆×D₇₉	1	2	yes	—
284	C₂²	C₂²×#11929607	1	2	yes	—
285	C₄	—	2	2	no	—
286	C₄	—	2	2	no	—
287	C₄	—	158	8	no	—
288	C₄	—	158	8	no	—
289	C₄	C₂×#14579071	1	4	yes	—
290	C₄	C₂×#14579071	1	4	yes	—
291	C₄	M₁₆×Dic₇₉	1	4	yes	—
292	C₄	#19099031	1	2	yes	—
293	C₄	#41×Dic₇₉	1	2	yes	—
294	C₂	—	2	4	no	—
295	C₂	#117×Dic₇₉	1	1	yes	characteristic
296	C₂	C₂²×#14579071	1	1	yes	characteristic
297	C₂	C₂×M₁₆×Dic₇₉	1	1	yes	characteristic
298	C₂	#23772028	1	4	yes	—

	subgroup	G/H
center	C₂×C₄²	C₂²×D₇₉
commutator	C₂²×C₇₉	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₇₉
Fitting	C₂×#431×C₇₉	C₂
socle	C₂³×C₇₉	C₂³×C₄
prime radical	C₂⁴×C₇₉	C₂²×C₄
cosocle	C₂²×C₄×C₇₉	C₂⁴
hypercenter	C₂×#431	D₇₉
hypocenter	C₇₉	#20463
LCS	G ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄² ◁ C₂×#431

Complements

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

N	H ≅ G/N	type
3. #23763255	C₂	semidirect
5. #23763145	C₂	semidirect
6. C₄×#14579071	C₂	semidirect
11. #19098616	C₄	semidirect
23. #431×C₇₉	C₄	semidirect
41. C₄⋊(C₇₉⋊C₈)	C₂×C₄	semidirect
44. C₄×C₇₉⋊C₈	C₂×C₄	semidirect
55. C₄⋊C₈×C₇₉	C₂×C₄	semidirect
57. C₄×C₈×C₇₉	C₂×C₄	semidirect
106. C₂×C₄×C₇₉	#97	semidirect
157. C₇₉	#20463	semidirect
163. #431	Dic₇₉	semidirect
202. C₄⋊C₈	C₂×Dic₇₉	semidirect
207. C₄×C₈	C₂×Dic₇₉	semidirect
263. C₂×C₄	#14579071	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19098616	2
C₂×C₄²×Dic₇₉	2
#23763255	2
#23763164	2
#23763145	2
C₄×#14579071	2
C₂×#431×C₇₉	2
#20463	79