Subgroups — #34187167

Order 20224.

586 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23764195	C₂	1	2	yes	maximal, complemented
2	C₂×#19098316	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×D₄⋊(C₇₉⋊C₈)	C₂	1	2	yes	maximal, complemented
4	C₂×#19098285	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₄×D₈×C₇₉	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊(C₇₉⋊C₈)	C₂²	1	2	yes	—
7	C₂×C₂²⋊(C₇₉⋊C₈)	—	2	4	no	—
8	C₂×C₄×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
9	C₂×C₄×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
10	#19098316	—	2	2	no	—
11	D₄⋊(C₇₉⋊C₈)	—	2	4	no	—
12	#19098285	—	2	2	no	—
13	C₂²×D₈×C₇₉	C₄	1	1	yes	characteristic
14	C₂×C₄×D₄×C₇₉	C₂²	1	2	yes	—
15	C₄×D₈×C₇₉	—	2	2	no	—
16	C₄×D₈×C₇₉	—	2	2	no	—
17	C₂×#107×C₇₉	C₄	1	1	yes	characteristic
18	C₂×#102×C₇₉	C₄	1	2	yes	—
19	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
20	C₂²×C₇₉⋊C₈	—	2	1	no	—
21	C₂²×C₇₉⋊C₈	—	2	2	no	—
22	C₂²×C₇₉⋊C₈	D₄	1	2	yes	—
23	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
24	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
25	C₂²⋊(C₇₉⋊C₈)	—	4	8	no	—
26	C₄×C₇₉⋊C₈	—	2	1	no	—
27	C₄×C₇₉⋊C₈	—	2	1	no	—
28	C₄×C₇₉⋊C₈	—	2	1	no	—
29	C₄×C₇₉⋊C₈	D₄	1	2	yes	—
30	C₂²×D₄×C₇₉	C₂×C₄	1	2	yes	—
31	C₂³×C₄×C₇₉	—	2	2	no	—
32	C₂×D₈×C₇₉	—	2	4	no	—
33	C₂×D₈×C₇₉	—	2	2	no	—
34	C₂²×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
36	C₄×D₄×C₇₉	—	2	2	no	—
37	C₄×D₄×C₇₉	—	2	2	no	—
38	C₄×D₄×C₇₉	—	2	2	no	—
39	C₄×D₄×C₇₉	—	2	2	no	—
40	C₂×C₄⋊C₄×C₇₉	C₂×C₄	1	2	yes	—
41	C₂×C₂²⋊C₄×C₇₉	—	2	2	no	—
42	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
43	#107×C₇₉	—	2	2	no	—
44	#102×C₇₉	—	2	4	no	—
45	C₄×C₈×C₇₉	D₄	1	2	yes	—
46	C₄×C₈×C₇₉	D₄	1	2	yes	—
47	C₂×C₇₉⋊C₈	—	2	2	no	—
48	C₂×C₇₉⋊C₈	—	2	2	no	—
49	C₂×C₇₉⋊C₈	—	2	2	no	—
50	C₂×C₇₉⋊C₈	—	4	2	no	—
51	C₂×C₇₉⋊C₈	—	4	2	no	—
52	C₂×C₇₉⋊C₈	—	4	4	no	—
53	C₂×C₇₉⋊C₈	D₈	1	4	yes	complemented
54	C₂⁴×C₇₉	—	2	2	no	—
55	C₂×D₄×C₇₉	—	2	2	no	—
56	C₂×D₄×C₇₉	—	2	2	no	—
57	C₂×D₄×C₇₉	—	2	2	no	—
58	C₂×D₄×C₇₉	—	2	2	no	—
59	C₂×D₄×C₇₉	—	2	2	no	—
60	C₂×D₄×C₇₉	M₁₆	1	4	yes	—
61	C₂²×C₄×C₇₉	—	2	2	no	—
62	C₂²×C₄×C₇₉	—	4	2	no	—
63	C₂²×C₄×C₇₉	—	4	2	no	—
64	C₂²×C₄×C₇₉	—	4	4	no	—
65	C₂²×C₄×C₇₉	—	4	4	no	—
66	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₇₉	#41	1	1	yes	characteristic
69	D₈×C₇₉	—	2	8	no	—
70	C₂×C₈×C₇₉	—	2	2	no	—
71	C₂×C₈×C₇₉	—	2	2	no	—
72	C₂×C₈×C₇₉	C₂²⋊C₄	1	2	yes	—
73	C₂×C₈×C₇₉	C₂²⋊C₄	1	2	yes	—
74	C₄⋊C₄×C₇₉	—	2	2	no	—
75	C₄⋊C₄×C₇₉	—	2	2	no	—
76	C₂²⋊C₄×C₇₉	—	4	2	no	—
77	C₂²⋊C₄×C₇₉	—	4	2	no	—
78	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
79	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
80	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
81	C₄²×C₇₉	#41	1	1	yes	characteristic
82	C₂³×C₇₉	—	2	4	no	—
83	C₂³×C₇₉	—	4	2	no	—
84	C₂³×C₇₉	—	4	2	no	—
85	C₂³×C₇₉	—	4	4	no	—
86	C₂³×C₇₉	—	4	4	no	—
87	C₂³×C₇₉	C₄×D₄	1	1	yes	characteristic
88	D₄×C₇₉	—	2	4	no	—
89	D₄×C₇₉	—	2	4	no	—
90	D₄×C₇₉	—	4	2	no	—
91	D₄×C₇₉	—	4	2	no	—
92	C₂×C₄×C₇₉	—	2	2	no	—
93	C₂×C₄×C₇₉	—	4	2	no	—
94	C₂×C₄×C₇₉	—	4	4	no	—
95	C₂×C₄×C₇₉	—	8	4	no	—
96	C₂×C₄×C₇₉	—	8	2	no	—
97	C₂×C₄×C₇₉	—	8	2	no	—
98	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
99	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
100	C₂×C₄×C₇₉	#121	1	1	yes	characteristic
101	C₂×C₄×C₇₉	#121	1	1	yes	characteristic
102	C₂×C₄×C₇₉	#124	1	1	yes	characteristic
103	C₂×C₄×C₇₉	#127	1	1	yes	characteristic
104	C₂×C₄×C₇₉	C₂×M₁₆	1	2	yes	—
105	C₂×C₄×C₇₉	#131	1	2	yes	—
106	C₂×C₄×C₇₉	C₂×D₈	1	2	yes	—
107	C₂×C₄×C₇₉	#135	1	2	yes	—
108	C₈×C₇₉	—	2	2	no	—
109	C₈×C₇₉	—	2	2	no	—
110	C₇₉⋊C₈	—	2	4	no	—
111	C₇₉⋊C₈	—	4	2	no	—
112	C₇₉⋊C₈	—	8	2	no	—
113	C₂²×C₇₉	—	4	8	no	—
114	C₂²×C₇₉	—	4	4	no	—
115	C₂²×C₇₉	—	8	4	no	—
116	C₂²×C₇₉	—	8	2	no	—
117	C₂²×C₇₉	—	8	2	no	—
118	C₂²×C₇₉	#390	1	1	yes	characteristic
119	C₂²×C₇₉	#435	1	2	yes	—
120	C₂²×C₇₉	C₄×D₈	1	2	yes	—
121	C₂²×C₇₉	#444	1	2	yes	—
122	C₄×C₇₉	—	2	1	no	—
123	C₄×C₇₉	—	2	1	no	—
124	C₄×C₇₉	—	8	2	no	—
125	C₄×C₇₉	#407	1	1	yes	characteristic
126	C₄×C₇₉	#408	1	1	yes	characteristic
127	C₄×C₇₉	#435	1	2	yes	—
128	C₄×C₇₉	#466	1	2	yes	—
129	C₄×C₇₉	#493	1	1	yes	characteristic
130	C₄×C₇₉	#495	1	1	yes	characteristic
131	#9663	—	79	1	no	maximal, Sylow
132	C₂×C₇₉	—	8	4	no	—
133	C₂×C₇₉	#1581	1	4	yes	—
134	C₂×C₇₉	#1955	1	1	yes	characteristic
135	C₂×C₇₉	#1970	1	1	yes	characteristic
136	C₂×C₇₉	#1975	1	1	yes	characteristic
137	C₂×C₄×D₈	D₇₉	1	1	yes	characteristic
138	#1954	—	79	2	no	—
139	C₂×#335	—	79	1	no	—
140	C₂×D₄⋊C₈	—	79	2	no	—
141	C₂×#322	—	79	1	no	—
142	C₇₉	#9663	1	1	yes	characteristic, complemented, Sylow
143	C₂²×D₈	Dic₇₉	1	1	yes	characteristic
144	C₂×C₄×D₄	C₂×D₇₉	1	2	yes	—
145	C₄×D₈	—	2	2	no	—
146	C₄×D₈	—	2	2	no	—
147	C₂×#107	Dic₇₉	1	1	yes	characteristic
148	C₂×C₄⋊C₈	—	79	2	no	—
149	C₂×#102	Dic₇₉	1	2	yes	—
150	C₂×C₂²⋊C₈	—	158	4	no	—
151	C₂×C₄×C₈	—	79	1	no	—
152	C₂×C₄×C₈	—	79	1	no	—
153	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
154	#335	—	158	2	no	—
155	D₄⋊C₈	—	158	4	no	—
156	#322	—	158	2	no	—
157	C₂²×D₄	C₂×Dic₇₉	1	2	yes	—
158	C₂³×C₄	—	2	2	no	—
159	C₂×D₈	—	2	4	no	—
160	C₂×D₈	—	2	2	no	—
161	C₂²×C₈	—	79	2	no	—
162	C₂²×C₈	—	158	2	no	—
163	C₂²×C₈	—	158	1	no	—
164	C₂²×C₈	C₂×Dic₇₉	1	1	yes	characteristic
165	C₂²×C₈	C₂×Dic₇₉	1	1	yes	characteristic
166	C₄×D₄	—	2	2	no	—
167	C₄×D₄	—	2	2	no	—
168	C₄×D₄	—	2	2	no	—
169	C₄×D₄	—	2	2	no	—
170	C₂×C₄⋊C₄	C₂×Dic₇₉	1	2	yes	—
171	C₂×C₂²⋊C₄	—	2	2	no	—
172	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
173	#107	—	2	2	no	—
174	C₄⋊C₈	—	158	2	no	—
175	C₄⋊C₈	—	158	2	no	—
176	#102	—	2	4	no	—
177	C₂²⋊C₈	—	316	8	no	—
178	C₄×C₈	—	79	2	no	—
179	C₄×C₈	—	158	1	no	—
180	C₄×C₈	—	158	1	no	—
181	C₄×C₈	—	158	1	no	—
182	C₄×C₈	C₂²⋊D₇₉	1	2	yes	—
183	C₄×C₈	C₂²⋊D₇₉	1	2	yes	—
184	C₂⁴	—	2	2	no	—
185	C₂×D₄	—	2	2	no	—
186	C₂×D₄	—	2	2	no	—
187	C₂×D₄	—	2	2	no	—
188	C₂×D₄	—	2	2	no	—
189	C₂×D₄	—	2	2	no	—
190	C₂×D₄	#11929608	1	4	yes	—
191	C₂²×C₄	—	2	2	no	—
192	C₂²×C₄	—	4	2	no	—
193	C₂²×C₄	—	4	2	no	—
194	C₂²×C₄	—	4	4	no	—
195	C₂²×C₄	—	4	4	no	—
196	C₂²×C₄	C₂²×Dic₇₉	1	1	yes	characteristic
197	C₂²×C₄	D₄×D₇₉	1	1	yes	characteristic
198	C₂²×C₄	#11929623	1	1	yes	characteristic
199	D₈	—	2	8	no	—
200	C₂×C₈	—	2	2	no	—
201	C₂×C₈	—	2	2	no	—
202	C₂×C₈	—	316	2	no	—
203	C₂×C₈	—	316	2	no	—
204	C₂×C₈	—	316	4	no	—
205	C₂×C₈	—	79	4	no	—
206	C₂×C₈	—	158	2	no	—
207	C₂×C₈	—	158	2	no	—
208	C₂×C₈	—	158	2	no	—
209	C₂×C₈	C₂²⋊Dic₇₉	1	2	yes	—
210	C₂×C₈	C₂²⋊Dic₇₉	1	2	yes	—
211	C₄⋊C₄	—	2	2	no	—
212	C₄⋊C₄	—	2	2	no	—
213	C₂²⋊C₄	—	4	2	no	—
214	C₂²⋊C₄	—	4	2	no	—
215	C₄²	D₄×D₇₉	1	1	yes	characteristic
216	C₄²	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
217	C₄²	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
218	C₄²	#11929623	1	1	yes	characteristic
219	C₂³	—	2	4	no	—
220	C₂³	—	4	2	no	—
221	C₂³	—	4	4	no	—
222	C₂³	—	4	4	no	—
223	C₂³	—	4	2	no	—
224	C₂³	D₄×Dic₇₉	1	1	yes	characteristic
225	D₄	—	2	4	no	—
226	D₄	—	2	4	no	—
227	D₄	—	4	2	no	—
228	D₄	—	4	2	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	4	4	no	—
231	C₂×C₄	—	4	2	no	—
232	C₂×C₄	—	8	2	no	—
233	C₂×C₄	—	8	2	no	—
234	C₂×C₄	—	8	4	no	—
235	C₂×C₄	C₂×C₂²⋊Dic₇₉	1	1	yes	characteristic
236	C₂×C₄	D₄×Dic₇₉	1	1	yes	characteristic
237	C₂×C₄	#14579132	1	1	yes	characteristic
238	C₂×C₄	#14579132	1	1	yes	characteristic
239	C₂×C₄	#14579148	1	1	yes	characteristic
240	C₂×C₄	#14579157	1	1	yes	characteristic
241	C₂×C₄	C₂×#11929608	1	2	yes	—
242	C₂×C₄	#14579168	1	2	yes	—
243	C₂×C₄	D₈×D₇₉	1	2	yes	—
244	C₂×C₄	#14579180	1	2	yes	—
245	C₈	—	2	2	no	—
246	C₈	—	2	2	no	—
247	C₈	—	316	2	no	—
248	C₈	—	632	2	no	—
249	C₈	—	158	4	no	—
250	C₂²	—	4	8	no	—
251	C₂²	—	4	4	no	—
252	C₂²	—	8	2	no	—
253	C₂²	—	8	4	no	—
254	C₂²	—	8	2	no	—
255	C₂²	#19098466	1	1	yes	characteristic
256	C₂²	#19098635	1	2	yes	—
257	C₂²	D₈×Dic₇₉	1	2	yes	—
258	C₂²	#19098685	1	2	yes	—
259	C₄	—	2	1	no	—
260	C₄	—	2	1	no	—
261	C₄	—	8	2	no	—
262	C₄	#19098527	1	1	yes	characteristic
263	C₄	#19098530	1	1	yes	characteristic
264	C₄	#19098635	1	2	yes	—
265	C₄	#19098796	1	2	yes	—
266	C₄	#19098938	1	1	yes	characteristic
267	C₄	#19098946	1	1	yes	characteristic
268	C₂	—	8	4	no	—
269	C₂	#23762230	1	4	yes	—
270	C₂	#23764199	1	1	yes	characteristic
271	C₂	#23764267	1	1	yes	characteristic
272	C₂	#23764294	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₇₉
commutator	C₂²×C₄×C₇₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₂×C₄×D₈×C₇₉	C₂
socle	C₂³×C₇₉	C₄×D₄
prime radical	C₂²×D₈×C₇₉	C₄
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	C₂×C₄×D₈	D₇₉
hypocenter	C₇₉	#9663
LCS	G ▷ C₂²×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄×D₈

Complements

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

N	H ≅ G/N	type
1. #23764195	C₂	semidirect
2. C₂×#19098316	C₂	semidirect
3. C₂×D₄⋊(C₇₉⋊C₈)	C₂	semidirect
4. C₂×#19098285	C₂	semidirect
53. C₂×C₇₉⋊C₈	D₈	semidirect
142. C₇₉	#9663	semidirect

Maximal subgroups

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

subgroup	index
#23764195	2
C₂×#19098316	2
C₂×D₄⋊(C₇₉⋊C₈)	2
C₂×#19098285	2
C₂×C₄×D₈×C₇₉	2
#9663	79