Subgroups — #34178752

Order 20224.

618 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₂²⋊(C₇₉⋊C₈)	C₂	1	2	yes	maximal
2	C₂×#19098285	C₂	1	2	yes	maximal, complemented
3	C₇₉⋊#1273	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇₉⋊#1273	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₄×M₁₆×C₇₉	C₂	1	1	yes	maximal, characteristic
6	C₂×C₂²⋊(C₇₉⋊C₈)	C₄	1	4	yes	—
7	C₂×C₄×C₇₉⋊C₈	C₂²	1	2	yes	—
8	C₂×C₄×C₇₉⋊C₈	C₂²	1	2	yes	—
9	#19098318	C₄	1	4	yes	—
10	#19098285	—	2	4	no	—
11	C₂²×M₁₆×C₇₉	C₄	1	2	yes	—
12	C₂²×C₄²×C₇₉	C₂²	1	1	yes	characteristic
13	C₄×M₁₆×C₇₉	—	2	4	no	—
14	C₂×#97×C₇₉	C₄	1	2	yes	—
15	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
17	C₂²×C₇₉⋊C₈	C₂×C₄	1	4	yes	—
18	C₂²×C₇₉⋊C₈	C₂×C₄	1	4	yes	—
19	C₂²⋊(C₇₉⋊C₈)	—	2	8	no	—
20	C₄×C₇₉⋊C₈	—	2	2	no	—
21	C₄×C₇₉⋊C₈	—	2	2	no	—
22	C₄×C₇₉⋊C₈	—	2	2	no	—
23	C₄×C₇₉⋊C₈	—	2	2	no	—
24	C₂³×C₄×C₇₉	C₂×C₄	1	2	yes	—
25	C₂³×C₄×C₇₉	C₂×C₄	1	1	yes	characteristic
26	C₂×M₁₆×C₇₉	—	2	4	no	—
27	C₂×M₁₆×C₇₉	—	2	4	no	—
28	C₂×M₁₆×C₇₉	—	2	4	no	—
29	C₂²×C₈×C₇₉	C₂×C₄	1	2	yes	—
30	C₂²×C₈×C₇₉	C₂×C₄	1	2	yes	—
31	C₂×C₄²×C₇₉	—	2	2	no	—
32	C₂×C₄²×C₇₉	—	2	1	no	—
33	C₂×C₄²×C₇₉	—	2	1	no	—
34	C₂×C₄²×C₇₉	C₂×C₄	1	2	yes	—
35	C₂×C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
37	#97×C₇₉	—	2	4	no	—
38	C₄×C₈×C₇₉	—	2	2	no	—
39	C₄×C₈×C₇₉	D₄	1	4	yes	—
40	C₂×C₇₉⋊C₈	—	2	4	no	—
41	C₂×C₇₉⋊C₈	—	2	4	no	—
42	C₂×C₇₉⋊C₈	—	2	4	no	—
43	C₂×C₇₉⋊C₈	—	2	4	no	—
44	C₂×C₇₉⋊C₈	—	2	4	no	—
45	C₂×C₇₉⋊C₈	M₁₆	1	8	yes	complemented
46	C₂⁴×C₇₉	C₄²	1	1	yes	characteristic
47	C₂²×C₄×C₇₉	—	2	1	no	—
48	C₂²×C₄×C₇₉	—	2	1	no	—
49	C₂²×C₄×C₇₉	—	2	2	no	—
50	C₂²×C₄×C₇₉	—	2	2	no	—
51	C₂²×C₄×C₇₉	—	2	2	no	—
52	C₂²×C₄×C₇₉	—	2	4	no	—
53	C₂²×C₄×C₇₉	—	2	4	no	—
54	C₂²×C₄×C₇₉	C₄²	1	2	yes	—
55	C₂²×C₄×C₇₉	C₄²	1	1	yes	characteristic
56	C₂²×C₄×C₇₉	M₁₆	1	4	yes	—
57	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
58	C₂²×C₄×C₇₉	C₂²×C₄	1	2	yes	—
59	M₁₆×C₇₉	—	2	16	no	—
60	C₂×C₈×C₇₉	—	2	2	no	—
61	C₂×C₈×C₇₉	—	2	2	no	—
62	C₂×C₈×C₇₉	—	2	2	no	—
63	C₂×C₈×C₇₉	—	2	2	no	—
64	C₂×C₈×C₇₉	C₂²⋊C₄	1	4	yes	—
65	C₂×C₈×C₇₉	M₁₆	1	4	yes	—
66	C₄²×C₇₉	—	2	1	no	—
67	C₄²×C₇₉	—	2	1	no	—
68	C₄²×C₇₉	—	2	2	no	—
69	C₄²×C₇₉	—	2	2	no	—
70	C₄²×C₇₉	C₂×D₄	1	2	yes	—
71	C₄²×C₇₉	#41	1	1	yes	characteristic
72	C₄²×C₇₉	#41	1	1	yes	characteristic
73	C₂³×C₇₉	—	2	1	no	—
74	C₂³×C₇₉	—	2	2	no	—
75	C₂³×C₇₉	—	2	1	no	—
76	C₂³×C₇₉	—	2	2	no	—
77	C₂³×C₇₉	#97	1	2	yes	—
78	C₂³×C₇₉	C₂×C₄²	1	1	yes	characteristic
79	C₂×C₄×C₇₉	—	2	1	no	—
80	C₂×C₄×C₇₉	—	2	1	no	—
81	C₂×C₄×C₇₉	—	2	4	no	—
82	C₂×C₄×C₇₉	—	2	4	no	—
83	C₂×C₄×C₇₉	—	2	2	no	—
84	C₂×C₄×C₇₉	—	2	2	no	—
85	C₂×C₄×C₇₉	—	2	2	no	—
86	C₂×C₄×C₇₉	—	4	4	no	—
87	C₂×C₄×C₇₉	—	4	4	no	—
88	C₂×C₄×C₇₉	#97	1	2	yes	—
89	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
90	C₂×C₄×C₇₉	#117	1	1	yes	characteristic
91	C₂×C₄×C₇₉	C₄×D₄	1	2	yes	—
92	C₂×C₄×C₇₉	C₄×D₄	1	2	yes	—
93	C₂×C₄×C₇₉	C₂×M₁₆	1	4	yes	—
94	C₂×C₄×C₇₉	C₂×M₁₆	1	2	yes	—
95	C₂×C₄×C₇₉	C₂×M₁₆	1	2	yes	—
96	C₂×C₄×C₇₉	#131	1	4	yes	—
97	C₈×C₇₉	—	2	4	no	—
98	C₈×C₇₉	—	2	4	no	—
99	C₇₉⋊C₈	—	2	8	no	—
100	C₇₉⋊C₈	—	4	4	no	—
101	C₂²×C₇₉	—	2	4	no	—
102	C₂²×C₇₉	—	2	2	no	—
103	C₂²×C₇₉	—	4	1	no	—
104	C₂²×C₇₉	—	4	1	no	—
105	C₂²×C₇₉	—	4	2	no	—
106	C₂²×C₇₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
107	C₂²×C₇₉	C₂×#97	1	1	yes	characteristic
108	C₂²×C₇₉	C₄×M₁₆	1	2	yes	—
109	C₂²×C₇₉	C₄×M₁₆	1	1	yes	characteristic
110	C₂²×C₇₉	#405	1	2	yes	—
111	C₄×C₇₉	—	2	2	no	—
112	C₄×C₇₉	—	4	2	no	—
113	C₄×C₇₉	#407	1	2	yes	—
114	C₄×C₇₉	#432	1	2	yes	—
115	C₄×C₇₉	#435	1	4	yes	—
116	C₄×C₇₉	#436	1	4	yes	—
117	#8147	—	79	1	no	maximal, Sylow
118	C₂×C₇₉	—	4	2	no	—
119	C₂×C₇₉	#1451	1	2	yes	—
120	C₂×C₇₉	#1453	1	1	yes	characteristic
121	C₂×C₇₉	#1455	1	1	yes	characteristic
122	C₂×C₇₉	#1749	1	1	yes	characteristic
123	C₂×C₇₉	#1753	1	2	yes	—
124	C₂×C₄×M₁₆	D₇₉	1	1	yes	characteristic
125	C₄×C₂²⋊C₈	—	79	2	no	—
126	C₂×#322	—	79	2	no	—
127	#1273	—	79	1	no	—
128	#1273	—	79	1	no	—
129	C₇₉	#8147	1	1	yes	characteristic, complemented, Sylow
130	C₂²×M₁₆	Dic₇₉	1	2	yes	—
131	C₂²×C₄²	C₂×D₇₉	1	1	yes	characteristic
132	C₂×C₂²⋊C₈	—	79	4	no	—
133	C₄×M₁₆	—	2	4	no	—
134	C₂×#97	Dic₇₉	1	2	yes	—
135	C₂×C₄×C₈	—	79	2	no	—
136	C₂×C₄×C₈	—	79	2	no	—
137	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
138	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
139	#336	—	79	4	no	—
140	#322	—	158	4	no	—
141	C₂³×C₄	C₄×D₇₉	1	2	yes	—
142	C₂³×C₄	C₂×Dic₇₉	1	1	yes	characteristic
143	C₂×M₁₆	—	2	4	no	—
144	C₂×M₁₆	—	2	4	no	—
145	C₂×M₁₆	—	2	4	no	—
146	C₂²×C₈	—	79	4	no	—
147	C₂²×C₈	—	79	4	no	—
148	C₂²×C₈	C₂×Dic₇₉	1	2	yes	—
149	C₂²×C₈	C₂×Dic₇₉	1	2	yes	—
150	C₂×C₄²	—	2	2	no	—
151	C₂×C₄²	—	2	1	no	—
152	C₂×C₄²	—	2	1	no	—
153	C₂×C₄²	C₄×D₇₉	1	2	yes	—
154	C₂×C₄²	C₂×Dic₇₉	1	1	yes	characteristic
155	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
156	C₂²⋊C₈	—	158	8	no	—
157	#97	—	2	4	no	—
158	C₄×C₈	—	2	2	no	—
159	C₄×C₈	—	158	2	no	—
160	C₄×C₈	—	158	2	no	—
161	C₄×C₈	—	158	2	no	—
162	C₄×C₈	—	158	2	no	—
163	C₄×C₈	C₂²⋊D₇₉	1	4	yes	—
164	C₂⁴	C₄×Dic₇₉	1	1	yes	characteristic
165	C₂²×C₄	—	2	1	no	—
166	C₂²×C₄	—	2	1	no	—
167	C₂²×C₄	—	2	2	no	—
168	C₂²×C₄	—	2	2	no	—
169	C₂²×C₄	—	2	2	no	—
170	C₂²×C₄	—	2	4	no	—
171	C₂²×C₄	—	2	4	no	—
172	C₂²×C₄	C₄×Dic₇₉	1	2	yes	—
173	C₂²×C₄	C₄×Dic₇₉	1	1	yes	characteristic
174	C₂²×C₄	#11929608	1	4	yes	—
175	C₂²×C₄	C₂×C₄×D₇₉	1	2	yes	—
176	C₂²×C₄	C₂²×Dic₇₉	1	1	yes	characteristic
177	M₁₆	—	2	16	no	—
178	C₂×C₈	—	2	2	no	—
179	C₂×C₈	—	2	2	no	—
180	C₂×C₈	—	2	2	no	—
181	C₂×C₈	—	2	2	no	—
182	C₂×C₈	—	79	8	no	—
183	C₂×C₈	—	158	4	no	—
184	C₂×C₈	—	158	4	no	—
185	C₂×C₈	—	158	4	no	—
186	C₂×C₈	—	158	4	no	—
187	C₂×C₈	—	158	4	no	—
188	C₂×C₈	C₂²⋊Dic₇₉	1	4	yes	—
189	C₂×C₈	#11929608	1	4	yes	—
190	C₄²	—	2	1	no	—
191	C₄²	—	2	1	no	—
192	C₄²	—	2	2	no	—
193	C₄²	—	2	2	no	—
194	C₄²	C₂×C₂²⋊D₇₉	1	2	yes	—
195	C₄²	#11929624	1	1	yes	characteristic
196	C₄²	#11929624	1	1	yes	characteristic
197	C₂³	—	2	1	no	—
198	C₂³	—	2	2	no	—
199	C₂³	—	2	1	no	—
200	C₂³	—	2	2	no	—
201	C₂³	#14579072	1	2	yes	—
202	C₂³	C₂×C₄×Dic₇₉	1	1	yes	characteristic
203	C₂×C₄	—	2	1	no	—
204	C₂×C₄	—	2	1	no	—
205	C₂×C₄	—	2	4	no	—
206	C₂×C₄	—	2	4	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	4	4	no	—
211	C₂×C₄	—	4	4	no	—
212	C₂×C₄	#14579072	1	2	yes	—
213	C₂×C₄	C₂×C₂²⋊Dic₇₉	1	1	yes	characteristic
214	C₂×C₄	#14579120	1	1	yes	characteristic
215	C₂×C₄	C₄×C₂²⋊D₇₉	1	2	yes	—
216	C₂×C₄	C₄×C₂²⋊D₇₉	1	2	yes	—
217	C₂×C₄	M₁₆×D₇₉	1	4	yes	—
218	C₂×C₄	C₂×#11929608	1	2	yes	—
219	C₂×C₄	C₂×#11929608	1	2	yes	—
220	C₂×C₄	#14579169	1	4	yes	—
221	C₈	—	2	4	no	—
222	C₈	—	2	4	no	—
223	C₈	—	316	4	no	—
224	C₈	—	158	8	no	—
225	C₂²	—	2	4	no	—
226	C₂²	—	2	2	no	—
227	C₂²	—	4	1	no	—
228	C₂²	—	4	1	no	—
229	C₂²	—	4	2	no	—
230	C₂²	C₄×C₂²⋊Dic₇₉	1	1	yes	characteristic
231	C₂²	C₂×#14579072	1	1	yes	characteristic
232	C₂²	M₁₆×Dic₇₉	1	2	yes	—
233	C₂²	C₄×#11929608	1	1	yes	characteristic
234	C₂²	#19098521	1	2	yes	—
235	C₄	—	2	2	no	—
236	C₄	—	4	2	no	—
237	C₄	#19098527	1	2	yes	—
238	C₄	#19098620	1	2	yes	—
239	C₄	#19098632	1	4	yes	—
240	C₄	#19098639	1	4	yes	—
241	C₂	—	4	2	no	—
242	C₂	#23761687	1	2	yes	—
243	C₂	#23761697	1	1	yes	characteristic
244	C₂	#23761703	1	1	yes	characteristic
245	C₂	#23763162	1	1	yes	characteristic
246	C₂	#23763181	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄²	C₂²×D₇₉
commutator, socle	C₂³×C₇₉	C₂×C₄²
Fitting	C₂×C₄×M₁₆×C₇₉	C₂
prime radical	C₂⁴×C₇₉	C₄²
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	C₂×C₄×M₁₆	D₇₉
hypocenter	C₇₉	#8147
LCS	G ▷ C₂³×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄² ◁ C₂×C₄×M₁₆

Complements

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

N	H ≅ G/N	type
2. C₂×#19098285	C₂	semidirect
3. C₇₉⋊#1273	C₂	semidirect
4. C₇₉⋊#1273	C₂	semidirect
45. C₂×C₇₉⋊C₈	M₁₆	semidirect
129. C₇₉	#8147	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₂²⋊(C₇₉⋊C₈)	2
C₂×#19098285	2
C₇₉⋊#1273	2
C₂×C₄×M₁₆×C₇₉	2
#8147	79