Subgroups — #2273×D₇₉ #34446472

Order 20224.

5790 conjugacy classes of subgroups.

The subgroup lattice has 242 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₂²⋊C₄×D₇₉	C₂	1	3	yes	maximal
2	#23765753	C₂	1	3	yes	maximal, complemented
3	#23765752	C₂	1	12	yes	maximal, complemented
4	C₄×C₂²⋊C₄×D₇₉	C₂	1	12	yes	maximal, complemented
5	C₇₉×#2273	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂⁵×D₇₉	C₂²	1	1	yes	characteristic
7	C₂³×C₄×D₇₉	C₂²	1	6	yes	—
8	C₂²×C₂²⋊Dic₇₉	C₂²	1	3	yes	—
9	C₂²×(C₂×D₇₉)⋊C₄	C₂²	1	6	yes	—
10	C₂×C₂²⋊C₄×D₇₉	C₄	1	48	yes	complemented
11	C₂×C₂²⋊C₄×D₇₉	C₂²	1	12	yes	—
12	C₂×C₂²⋊C₄×D₇₉	C₂²	1	12	yes	—
13	C₂×C₄²×D₇₉	C₂²	1	8	yes	complemented
14	C₂²⋊C₄×Dic₇₉	C₂²	1	24	yes	complemented
15	C₄×(C₂×D₇₉)⋊C₄	C₂²	1	24	yes	complemented
16	C₄×C₂²⋊Dic₇₉	C₂²	1	12	yes	complemented
17	#19098410	C₂²	1	24	yes	complemented
18	#95×D₇₉	C₂²	1	8	yes	complemented
19	C₂²×C₂²⋊C₄×C₇₉	C₂²	1	3	yes	—
20	C₄×C₂²⋊C₄×C₇₉	C₂²	1	12	yes	complemented
21	C₂⁴×D₇₉	—	4	8	no	—
22	C₂⁴×D₇₉	C₂×C₄	1	24	yes	—
23	C₂⁴×D₇₉	C₂³	1	3	yes	—
24	C₂⁴×D₇₉	C₂³	1	3	yes	—
25	C₂³×Dic₇₉	C₂³	1	6	yes	—
26	C₂²×C₄×D₇₉	—	4	24	no	—
27	C₂²×C₄×D₇₉	C₂×C₄	1	48	yes	complemented
28	C₂²×C₄×D₇₉	C₂³	1	12	yes	—
29	C₂²×C₄×D₇₉	C₂³	1	12	yes	—
30	C₂×C₂²⋊Dic₇₉	C₂×C₄	1	24	yes	complemented
31	C₂×C₂²⋊Dic₇₉	C₂³	1	12	yes	—
32	C₂×(C₂×D₇₉)⋊C₄	C₂×C₄	1	48	yes	complemented
33	C₂×(C₂×D₇₉)⋊C₄	C₂³	1	24	yes	—
34	C₂²⋊C₄×D₇₉	C₂×C₄	1	96	yes	complemented
35	C₂²⋊C₄×D₇₉	C₂×C₄	1	96	yes	complemented
36	C₂×C₄×Dic₇₉	C₂³	1	24	yes	complemented
37	C₄²×D₇₉	—	4	16	no	—
38	C₇₉⋊#95	C₂³	1	24	yes	complemented
39	C₂⁵×C₇₉	C₂³	1	1	yes	characteristic
40	C₂³×C₄×C₇₉	C₂³	1	6	yes	—
41	C₂×C₂²⋊C₄×C₇₉	C₂×C₄	1	24	yes	complemented
42	C₂×C₂²⋊C₄×C₇₉	C₂³	1	12	yes	—
43	C₂×C₄²×C₇₉	C₂³	1	8	yes	complemented
44	#95×C₇₉	C₂³	1	8	yes	complemented
45	C₂³×D₇₉	—	2	24	no	—
46	C₂³×D₇₉	—	4	96	no	—
47	C₂³×D₇₉	—	4	12	no	—
48	C₂³×D₇₉	C₄²	1	64	yes	complemented
49	C₂³×D₇₉	C₂²×C₄	1	24	yes	—
50	C₂³×D₇₉	C₂²×C₄	1	12	yes	—
51	C₂³×D₇₉	C₂²×C₄	1	24	yes	—
52	C₂³×D₇₉	C₂²×C₄	1	12	yes	—
53	C₂³×D₇₉	C₂⁴	1	3	yes	—
54	C₂³×D₇₉	C₂⁴	1	1	yes	characteristic
55	C₂²×Dic₇₉	—	4	12	no	—
56	C₂²×Dic₇₉	C₂²×C₄	1	24	yes	complemented
57	C₂²×Dic₇₉	C₂⁴	1	12	yes	—
58	C₂×C₄×D₇₉	—	2	48	no	—
59	C₂×C₄×D₇₉	—	4	24	no	—
60	C₂×C₄×D₇₉	—	4	96	no	—
61	C₂×C₄×D₇₉	C₂²×C₄	1	48	yes	—
62	C₂×C₄×D₇₉	C₂²×C₄	1	48	yes	complemented
63	C₂²⋊Dic₇₉	C₂²×C₄	1	48	yes	complemented
64	(C₂×D₇₉)⋊C₄	C₂²×C₄	1	96	yes	complemented
65	C₄×Dic₇₉	—	4	24	no	—
66	C₂⁴×C₇₉	—	4	4	no	—
67	C₂⁴×C₇₉	C₂²×C₄	1	12	yes	—
68	C₂⁴×C₇₉	C₂⁴	1	3	yes	—
69	C₂²×C₄×C₇₉	—	4	12	no	—
70	C₂²×C₄×C₇₉	C₂²×C₄	1	24	yes	complemented
71	C₂²×C₄×C₇₉	C₂⁴	1	12	yes	—
72	C₂²⋊C₄×C₇₉	C₂²×C₄	1	48	yes	complemented
73	C₄²×C₇₉	—	4	8	no	—
74	C₂³×C₇₉	—	2	12	no	—
75	C₂³×C₇₉	—	4	24	no	—
76	C₂³×C₇₉	C₂×C₄²	1	16	yes	complemented
77	C₂³×C₇₉	C₂³×C₄	1	12	yes	—
78	C₂³×C₇₉	C₂³×C₄	1	6	yes	—
79	C₂³×C₇₉	C₂⁵	1	1	yes	characteristic
80	C₂²×D₇₉	—	2	12	no	—
81	C₂²×D₇₉	—	2	48	no	—
82	C₂²×D₇₉	—	2	96	no	—
83	C₂²×D₇₉	—	4	128	no	—
84	C₂²×D₇₉	—	4	24	no	—
85	C₂²×D₇₉	—	4	48	no	—
86	C₂²×D₇₉	C₂×C₄²	1	48	yes	—
87	C₂²×D₇₉	C₂×C₄²	1	48	yes	complemented
88	C₂²×D₇₉	C₂³×C₄	1	12	yes	—
89	C₂²×D₇₉	C₂³×C₄	1	6	yes	—
90	C₂²×D₇₉	C₂³×C₄	1	6	yes	—
91	C₂²×D₇₉	H(5,𝔽₂)	1	8	yes	—
92	C₂×C₄×C₇₉	—	2	24	no	—
93	C₂×C₄×C₇₉	—	4	24	no	—
94	C₂×C₄×C₇₉	C₂³×C₄	1	24	yes	—
95	C₂×Dic₇₉	—	2	24	no	—
96	C₂×Dic₇₉	—	4	24	no	—
97	C₂×Dic₇₉	C₂³×C₄	1	24	yes	—
98	C₄×D₇₉	—	2	96	no	—
99	C₄×D₇₉	—	4	48	no	—
100	C₂²×C₇₉	—	2	12	no	—
101	C₂²×C₇₉	—	2	24	no	—
102	C₂²×C₇₉	—	4	16	no	—
103	C₂²×C₇₉	C₂²×C₄²	1	12	yes	—
104	C₂²×C₇₉	C₂⁴×C₄	1	3	yes	—
105	C₂²×C₇₉	C₂×H(5,𝔽₂)	1	4	yes	—
106	C₂×D₇₉	—	2	24	no	—
107	C₂×D₇₉	—	2	12	no	—
108	C₂×D₇₉	—	2	96	no	—
109	C₂×D₇₉	—	4	48	no	—
110	C₂×D₇₉	C₂²×C₄²	1	12	yes	—
111	C₂×D₇₉	C₂²×C₄²	1	4	yes	—
112	C₂×D₇₉	#518	1	24	yes	—
113	C₄×C₇₉	—	2	24	no	—
114	Dic₇₉	—	2	24	no	—
115	C₂×#2273	—	79	1	no	maximal, Sylow
116	C₂×C₇₉	—	2	12	no	—
117	C₂×C₇₉	C₂³×C₄²	1	1	yes	characteristic
118	C₂×C₇₉	C₂×#518	1	6	yes	—
119	D₇₉	—	2	12	no	—
120	D₇₉	#2273	1	8	yes	direct product factor
121	C₂³×C₂²⋊C₄	—	79	3	no	—
122	#2273	—	79	12	no	—
123	#2273	—	79	3	no	—
124	#2273	D₇₉	1	1	yes	characteristic, direct product factor
125	C₂×C₄×C₂²⋊C₄	—	79	12	no	—
126	C₇₉	C₂×#2273	1	1	yes	characteristic, complemented, Sylow
127	C₂⁶	—	79	1	no	—
128	C₂⁴×C₄	—	79	6	no	—
129	C₂²×C₂²⋊C₄	—	79	3	no	—
130	C₂²×C₂²⋊C₄	—	79	6	no	—
131	C₂²×C₂²⋊C₄	—	79	12	no	—
132	C₂²×C₂²⋊C₄	—	79	12	no	—
133	C₂²×C₂²⋊C₄	—	79	48	no	—
134	C₂²×C₂²⋊C₄	C₂×D₇₉	1	3	yes	—
135	C₂²×C₄²	—	79	8	no	—
136	C₄×C₂²⋊C₄	—	79	12	no	—
137	C₄×C₂²⋊C₄	—	79	24	no	—
138	C₄×C₂²⋊C₄	—	79	24	no	—
139	C₄×C₂²⋊C₄	—	79	24	no	—
140	C₄×C₂²⋊C₄	C₂×D₇₉	1	12	yes	complemented
141	C₂×#95	—	79	8	no	—
142	C₂⁵	—	316	8	no	—
143	C₂⁵	—	79	24	no	—
144	C₂⁵	—	79	3	no	—
145	C₂⁵	—	79	3	no	—
146	C₂⁵	C₂²×D₇₉	1	1	yes	characteristic
147	C₂³×C₄	—	316	24	no	—
148	C₂³×C₄	—	79	6	no	—
149	C₂³×C₄	—	79	48	no	—
150	C₂³×C₄	—	79	12	no	—
151	C₂³×C₄	—	79	12	no	—
152	C₂³×C₄	C₂²×D₇₉	1	6	yes	—
153	C₂×C₂²⋊C₄	—	79	24	no	—
154	C₂×C₂²⋊C₄	—	79	12	no	—
155	C₂×C₂²⋊C₄	—	79	48	no	—
156	C₂×C₂²⋊C₄	—	79	24	no	—
157	C₂×C₂²⋊C₄	—	79	96	no	—
158	C₂×C₂²⋊C₄	—	79	96	no	—
159	C₂×C₂²⋊C₄	C₄×D₇₉	1	24	yes	complemented
160	C₂×C₂²⋊C₄	C₂²×D₇₉	1	12	yes	—
161	C₂×C₄²	—	316	16	no	—
162	C₂×C₄²	—	79	24	no	—
163	C₂×C₄²	C₂²×D₇₉	1	8	yes	complemented
164	#95	—	79	24	no	—
165	#95	C₂²×D₇₉	1	8	yes	complemented
166	C₂⁴	—	4	4	no	—
167	C₂⁴	—	316	12	no	—
168	C₂⁴	—	316	96	no	—
169	C₂⁴	—	79	64	no	—
170	C₂⁴	—	79	12	no	—
171	C₂⁴	—	79	24	no	—
172	C₂⁴	—	79	1	no	—
173	C₂⁴	—	79	3	no	—
174	C₂⁴	—	79	24	no	—
175	C₂⁴	—	79	12	no	—
176	C₂⁴	—	158	24	no	—
177	C₂⁴	C₂×C₄×D₇₉	1	12	yes	—
178	C₂⁴	C₂³×D₇₉	1	3	yes	—
179	C₂²×C₄	—	4	12	no	—
180	C₂²×C₄	—	316	24	no	—
181	C₂²×C₄	—	316	96	no	—
182	C₂²×C₄	—	316	12	no	—
183	C₂²×C₄	—	79	24	no	—
184	C₂²×C₄	—	79	48	no	—
185	C₂²×C₄	—	79	12	no	—
186	C₂²×C₄	—	79	48	no	—
187	C₂²×C₄	—	158	48	no	—
188	C₂²×C₄	C₂×C₄×D₇₉	1	24	yes	complemented
189	C₂²×C₄	C₂³×D₇₉	1	12	yes	—
190	C₂²⋊C₄	—	79	48	no	—
191	C₂²⋊C₄	—	79	96	no	—
192	C₂²⋊C₄	C₂×C₄×D₇₉	1	48	yes	complemented
193	C₄²	—	4	8	no	—
194	C₄²	—	316	24	no	—
195	C₂³	—	2	12	no	—
196	C₂³	—	4	24	no	—
197	C₂³	—	316	24	no	—
198	C₂³	—	316	48	no	—
199	C₂³	—	316	128	no	—
200	C₂³	—	79	48	no	—
201	C₂³	—	79	8	no	—
202	C₂³	—	79	6	no	—
203	C₂³	—	79	48	no	—
204	C₂³	—	79	12	no	—
205	C₂³	—	79	6	no	—
206	C₂³	—	158	48	no	—
207	C₂³	—	158	96	no	—
208	C₂³	—	158	12	no	—
209	C₂³	C₄²×D₇₉	1	16	yes	complemented
210	C₂³	C₂²×C₄×D₇₉	1	12	yes	—
211	C₂³	C₂²×C₄×D₇₉	1	6	yes	—
212	C₂³	C₂⁴×D₇₉	1	1	yes	characteristic
213	C₂×C₄	—	2	24	no	—
214	C₂×C₄	—	4	24	no	—
215	C₂×C₄	—	316	48	no	—
216	C₂×C₄	—	316	24	no	—
217	C₂×C₄	—	79	24	no	—
218	C₂×C₄	—	158	96	no	—
219	C₂×C₄	—	158	24	no	—
220	C₂×C₄	C₂²×C₄×D₇₉	1	24	yes	—
221	C₂²	—	2	12	no	—
222	C₂²	—	2	24	no	—
223	C₂²	—	4	16	no	—
224	C₂²	—	316	48	no	—
225	C₂²	—	79	4	no	—
226	C₂²	—	79	24	no	—
227	C₂²	—	79	12	no	—
228	C₂²	—	158	96	no	—
229	C₂²	—	158	12	no	—
230	C₂²	—	158	24	no	—
231	C₂²	C₂×C₄²×D₇₉	1	12	yes	—
232	C₂²	C₂³×C₄×D₇₉	1	3	yes	—
233	C₂²	H(5,𝔽₂)×D₇₉	1	4	yes	—
234	C₄	—	2	24	no	—
235	C₄	—	158	24	no	—
236	C₂	—	2	12	no	—
237	C₂	—	79	8	no	—
238	C₂	—	158	12	no	—
239	C₂	C₂²×C₄²×D₇₉	1	1	yes	characteristic
240	C₂	#518×D₇₉	1	6	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂⁴×D₇₉
commutator	C₂×C₇₉	C₂³×C₄²
Fitting	C₇₉×#2273	C₂
socle, cosocle	C₂³×C₇₉	C₂⁵
prime radical	C₂⁵×D₇₉	C₂²
hypercenter	#2273	D₇₉
hypocenter	C₇₉	C₂×#2273
LCS	G ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ #2273

Complements

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

N	H ≅ G/N	type
2. #23765753	C₂	semidirect
3. #23765752	C₂	semidirect
4. C₄×C₂²⋊C₄×D₇₉	C₂	semidirect
5. C₇₉×#2273	C₂	semidirect
10. C₂×C₂²⋊C₄×D₇₉	C₄	semidirect
13. C₂×C₄²×D₇₉	C₂²	semidirect
14. C₂²⋊C₄×Dic₇₉	C₂²	semidirect
15. C₄×(C₂×D₇₉)⋊C₄	C₂²	semidirect
16. C₄×C₂²⋊Dic₇₉	C₂²	semidirect
17. #19098410	C₂²	semidirect
18. #95×D₇₉	C₂²	semidirect
20. C₄×C₂²⋊C₄×C₇₉	C₂²	semidirect
27. C₂²×C₄×D₇₉	C₂×C₄	semidirect
30. C₂×C₂²⋊Dic₇₉	C₂×C₄	semidirect
32. C₂×(C₂×D₇₉)⋊C₄	C₂×C₄	semidirect
34. C₂²⋊C₄×D₇₉	C₂×C₄	semidirect
35. C₂²⋊C₄×D₇₉	C₂×C₄	semidirect
36. C₂×C₄×Dic₇₉	C₂³	semidirect
38. C₇₉⋊#95	C₂³	semidirect
41. C₂×C₂²⋊C₄×C₇₉	C₂×C₄	semidirect
43. C₂×C₄²×C₇₉	C₂³	semidirect
44. #95×C₇₉	C₂³	semidirect
48. C₂³×D₇₉	C₄²	semidirect
56. C₂²×Dic₇₉	C₂²×C₄	semidirect
62. C₂×C₄×D₇₉	C₂²×C₄	semidirect
63. C₂²⋊Dic₇₉	C₂²×C₄	semidirect
64. (C₂×D₇₉)⋊C₄	C₂²×C₄	semidirect
70. C₂²×C₄×C₇₉	C₂²×C₄	semidirect
72. C₂²⋊C₄×C₇₉	C₂²×C₄	semidirect
76. C₂³×C₇₉	C₂×C₄²	semidirect
87. C₂²×D₇₉	C₂×C₄²	semidirect
120. D₇₉	#2273	direct (N × H)
124. #2273	D₇₉	direct (N × H)
126. C₇₉	C₂×#2273	semidirect
140. C₄×C₂²⋊C₄	C₂×D₇₉	semidirect
159. C₂×C₂²⋊C₄	C₄×D₇₉	semidirect
163. C₂×C₄²	C₂²×D₇₉	semidirect
165. #95	C₂²×D₇₉	semidirect
188. C₂²×C₄	C₂×C₄×D₇₉	semidirect
192. C₂²⋊C₄	C₂×C₄×D₇₉	semidirect
209. C₂³	C₄²×D₇₉	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₂²⋊C₄×D₇₉	2
#23765753	2
#23765752	2
C₄×C₂²⋊C₄×D₇₉	2
C₇₉×#2273	2
C₂×#2273	79

Subgroups — #2273×D79 #34446472

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #2273×D₇₉ #34446472