Subgroups — #34204733

Order 20224.

438 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14579101	C₂	1	1	yes	maximal, characteristic
2	C₂×Dic₇₉⋊C₁₆	C₂	1	2	yes	maximal, complemented
3	C₈×C₂²⋊Dic₇₉	C₂	1	1	yes	maximal, characteristic
4	#23761477	C₂	1	2	yes	maximal, complemented
5	C₂×C₂²⋊C₁₆×C₇₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#14579101	—	2	4	no	—
7	C₂×#14579101	—	2	2	no	—
8	C₂²×C₇₉⋊C₁₆	C₂²	1	2	yes	—
9	C₂×C₂²⋊(C₇₉⋊C₈)	C₄	1	1	yes	characteristic
10	C₂×C₈×Dic₇₉	C₂²	1	2	yes	—
11	C₄×C₂²⋊Dic₇₉	C₄	1	1	yes	characteristic
12	Dic₇₉⋊C₁₆	—	2	4	no	—
13	#19098317	C₄	1	2	yes	—
14	C₂³×C₈×C₇₉	C₂²	1	1	yes	characteristic
15	C₂²×C₁₆×C₇₉	C₂²	1	1	yes	characteristic
16	C₂²×C₁₆×C₇₉	C₂²	1	1	yes	characteristic
17	C₂²⋊C₁₆×C₇₉	—	2	2	no	—
18	C₂²×C₇₉⋊C₈	C₂×C₄	1	2	yes	—
19	C₂×C₂²⋊Dic₇₉	C₈	1	2	yes	—
20	C₂×C₄×Dic₇₉	C₂×C₄	1	2	yes	—
21	#14579101	—	2	8	no	—
22	C₂×C₇₉⋊C₁₆	—	2	2	no	—
23	C₂×C₇₉⋊C₁₆	—	2	2	no	—
24	C₂×C₇₉⋊C₁₆	D₄	1	4	yes	—
25	C₂²⋊(C₇₉⋊C₈)	—	2	2	no	—
26	C₈×Dic₇₉	—	2	2	no	—
27	C₈×Dic₇₉	—	2	2	no	—
28	C₇₉⋊#95	C₈	1	1	yes	characteristic
29	C₇₉⋊#95	C₈	1	1	yes	characteristic
30	C₂³×C₄×C₇₉	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₈×C₇₉	—	2	2	no	—
32	C₂²×C₈×C₇₉	—	2	2	no	—
33	C₂²×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₈×C₇₉	D₄	1	2	yes	—
35	C₂²×C₈×C₇₉	Q₈	1	2	yes	—
36	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
37	C₂×C₁₆×C₇₉	—	2	2	no	—
38	C₂×C₁₆×C₇₉	—	2	1	no	—
39	C₂×C₁₆×C₇₉	—	2	1	no	—
40	C₂×C₁₆×C₇₉	—	2	2	no	—
41	C₂²×Dic₇₉	C₂×C₈	1	2	yes	—
42	C₂²×Dic₇₉	C₂×C₈	1	2	yes	—
43	C₂×C₇₉⋊C₈	—	2	2	no	—
44	C₂×C₇₉⋊C₈	—	2	2	no	—
45	C₂×C₇₉⋊C₈	—	2	2	no	—
46	C₂²⋊Dic₇₉	—	2	4	no	—
47	C₄×Dic₇₉	—	2	2	no	—
48	C₄×Dic₇₉	—	2	2	no	—
49	C₇₉⋊C₁₆	—	2	4	no	—
50	C₂⁴×C₇₉	C₂×C₈	1	1	yes	characteristic
51	C₂²×C₄×C₇₉	—	2	2	no	—
52	C₂²×C₄×C₇₉	—	2	2	no	—
53	C₂²×C₄×C₇₉	C₄⋊C₄	1	2	yes	—
54	C₂²×C₄×C₇₉	C₂×C₈	1	1	yes	characteristic
55	C₂²×C₄×C₇₉	M₁₆	1	2	yes	—
56	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₈×C₇₉	—	2	2	no	—
58	C₂×C₈×C₇₉	—	2	4	no	—
59	C₂×C₈×C₇₉	—	2	2	no	—
60	C₂×C₈×C₇₉	—	2	2	no	—
61	C₂×C₈×C₇₉	C₄⋊C₄	1	2	yes	—
62	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
63	C₂×C₈×C₇₉	C₂×D₄	1	2	yes	—
64	C₂×C₈×C₇₉	C₂×Q₈	1	1	yes	characteristic
65	C₂×C₈×C₇₉	#41	1	2	yes	—
66	C₁₆×C₇₉	—	4	1	no	—
67	C₁₆×C₇₉	—	4	1	no	—
68	C₂³×C₇₉	—	2	2	no	—
69	C₂³×C₇₉	—	2	2	no	—
70	C₂³×C₇₉	—	2	1	no	—
71	C₂³×C₇₉	—	2	1	no	—
72	C₂³×C₇₉	C₄⋊C₈	1	2	yes	—
73	C₂³×C₇₉	C₂²×C₈	1	1	yes	characteristic
74	C₂×C₄×C₇₉	—	2	2	no	—
75	C₂×C₄×C₇₉	—	2	2	no	—
76	C₂×C₄×C₇₉	—	2	2	no	—
77	C₂×C₄×C₇₉	—	2	4	no	—
78	C₂×C₄×C₇₉	C₄⋊C₈	1	2	yes	—
79	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
80	C₂×C₄×C₇₉	C₄×D₄	1	2	yes	—
81	C₂×C₄×C₇₉	C₂×M₁₆	1	1	yes	characteristic
82	C₂×C₄×C₇₉	#131	1	2	yes	—
83	C₂×Dic₇₉	—	2	2	no	—
84	C₂×Dic₇₉	—	2	2	no	—
85	C₂×Dic₇₉	—	2	2	no	—
86	C₂×Dic₇₉	—	2	2	no	—
87	C₂×Dic₇₉	—	2	2	no	—
88	C₂×Dic₇₉	M₃₂	1	4	yes	complemented
89	C₈×C₇₉	—	2	2	no	—
90	C₈×C₇₉	#120	1	1	yes	characteristic
91	C₈×C₇₉	C₂²⋊Q₈	1	2	yes	—
92	C₈×C₇₉	#123	1	1	yes	characteristic
93	C₇₉⋊C₈	—	4	2	no	—
94	C₂²×C₇₉	—	2	2	no	—
95	C₂²×C₇₉	—	2	2	no	—
96	C₂²×C₇₉	—	2	2	no	—
97	C₂²×C₇₉	—	4	2	no	—
98	C₂²×C₇₉	—	4	2	no	—
99	C₂²×C₇₉	C₂×C₄⋊C₈	1	1	yes	characteristic
100	C₂²×C₇₉	C₈×D₄	1	2	yes	—
101	C₂²×C₇₉	C₂×M₃₂	1	2	yes	—
102	C₂²×C₇₉	#504	1	1	yes	characteristic
103	C₂²×C₇₉	#504	1	1	yes	characteristic
104	C₄×C₇₉	—	2	2	no	—
105	C₄×C₇₉	#385	1	1	yes	characteristic
106	C₄×C₇₉	#424	1	1	yes	characteristic
107	C₄×C₇₉	#435	1	2	yes	—
108	Dic₇₉	—	2	4	no	—
109	Dic₇₉	—	4	2	no	—
110	#12913	—	79	1	no	maximal, Sylow
111	C₂×C₇₉	—	4	2	no	—
112	C₂×C₇₉	#1850	1	1	yes	characteristic
113	C₂×C₇₉	#2150	1	1	yes	characteristic
114	C₂×C₇₉	#2151	1	1	yes	characteristic
115	C₂×C₇₉	#2168	1	2	yes	—
116	C₂×C₇₉	#2169	1	2	yes	—
117	C₂²×M₃₂	—	79	1	no	—
118	C₂×C₄⋊C₁₆	—	79	2	no	—
119	C₂×C₂²⋊C₁₆	D₇₉	1	1	yes	characteristic
120	C₈×C₂²⋊C₄	—	79	1	no	—
121	#1374	—	79	2	no	—
122	C₇₉	#12913	1	1	yes	characteristic, complemented, Sylow
123	C₂³×C₈	C₂×D₇₉	1	1	yes	characteristic
124	C₂×M₃₂	—	158	4	no	—
125	C₂×M₃₂	—	158	2	no	—
126	C₂²×C₁₆	—	79	2	no	—
127	C₂²×C₁₆	C₂×D₇₉	1	1	yes	characteristic
128	C₂²×C₁₆	C₂×D₇₉	1	1	yes	characteristic
129	C₂×C₂²⋊C₈	—	79	1	no	—
130	C₂×C₄×C₈	—	79	2	no	—
131	C₄×C₂²⋊C₄	—	79	1	no	—
132	C₄⋊C₁₆	—	158	4	no	—
133	C₂²⋊C₁₆	—	2	2	no	—
134	#336	—	79	2	no	—
135	C₂³×C₄	C₄×D₇₉	1	1	yes	characteristic
136	C₂²×C₈	—	2	2	no	—
137	C₂²×C₈	—	2	2	no	—
138	C₂²×C₈	—	79	2	no	—
139	C₂²×C₈	Dic₁₅₈	1	2	yes	—
140	C₂²×C₈	C₄×D₇₉	1	1	yes	characteristic
141	C₂²×C₈	C₂²⋊D₇₉	1	2	yes	—
142	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
143	C₂×C₂²⋊C₄	—	79	2	no	—
144	C₂×C₄²	—	79	2	no	—
145	M₃₂	—	158	8	no	—
146	C₂×C₁₆	—	2	2	no	—
147	C₂×C₁₆	—	2	1	no	—
148	C₂×C₁₆	—	2	1	no	—
149	C₂×C₁₆	—	2	2	no	—
150	C₂×C₁₆	—	79	4	no	—
151	C₂×C₁₆	—	158	2	no	—
152	C₂×C₁₆	—	158	2	no	—
153	C₂²⋊C₈	—	158	2	no	—
154	C₄×C₈	—	158	2	no	—
155	C₄×C₈	—	158	2	no	—
156	#95	—	79	1	no	—
157	#95	—	79	1	no	—
158	C₂⁴	C₈×D₇₉	1	1	yes	characteristic
159	C₂²×C₄	—	2	2	no	—
160	C₂²×C₄	—	2	2	no	—
161	C₂²×C₄	—	79	2	no	—
162	C₂²×C₄	—	79	2	no	—
163	C₂²×C₄	Dic₇₉⋊C₄	1	2	yes	—
164	C₂²×C₄	C₈×D₇₉	1	1	yes	characteristic
165	C₂²×C₄	#11929607	1	2	yes	—
166	C₂²×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
167	C₂×C₈	—	2	2	no	—
168	C₂×C₈	—	2	2	no	—
169	C₂×C₈	—	2	4	no	—
170	C₂×C₈	—	2	2	no	—
171	C₂×C₈	—	158	2	no	—
172	C₂×C₈	—	158	2	no	—
173	C₂×C₈	—	158	2	no	—
174	C₂×C₈	Dic₇₉⋊C₄	1	2	yes	—
175	C₂×C₈	D₄×D₇₉	1	2	yes	—
176	C₂×C₈	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
177	C₂×C₈	C₂×Dic₁₅₈	1	1	yes	characteristic
178	C₂×C₈	#11929623	1	2	yes	—
179	C₂²⋊C₄	—	158	4	no	—
180	C₄²	—	158	2	no	—
181	C₄²	—	158	2	no	—
182	C₁₆	—	4	1	no	—
183	C₁₆	—	4	1	no	—
184	C₁₆	—	158	4	no	—
185	C₂³	—	2	2	no	—
186	C₂³	—	2	2	no	—
187	C₂³	—	2	1	no	—
188	C₂³	—	2	1	no	—
189	C₂³	Dic₇₉⋊C₈	1	2	yes	—
190	C₂³	C₂×C₈×D₇₉	1	1	yes	characteristic
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	2	4	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	79	4	no	—
196	C₂×C₄	—	158	2	no	—
197	C₂×C₄	—	158	2	no	—
198	C₂×C₄	—	158	2	no	—
199	C₂×C₄	—	158	2	no	—
200	C₂×C₄	—	158	2	no	—
201	C₂×C₄	Dic₇₉⋊C₈	1	2	yes	—
202	C₂×C₄	C₂×Dic₇₉⋊C₄	1	1	yes	characteristic
203	C₂×C₄	(C₂²⋊D₇₉)⋊C₄	1	2	yes	—
204	C₂×C₄	C₂×#11929607	1	1	yes	characteristic
205	C₂×C₄	#14579169	1	2	yes	—
206	C₈	—	2	2	no	—
207	C₈	—	316	2	no	—
208	C₈	#14579129	1	1	yes	characteristic
209	C₈	#14579138	1	2	yes	—
210	C₈	#14579142	1	1	yes	characteristic
211	C₂²	—	2	2	no	—
212	C₂²	—	2	2	no	—
213	C₂²	—	2	2	no	—
214	C₂²	—	4	2	no	—
215	C₂²	—	4	2	no	—
216	C₂²	C₂×Dic₇₉⋊C₈	1	1	yes	characteristic
217	C₂²	(C₂²⋊D₇₉)⋊C₈	1	2	yes	—
218	C₂²	M₃₂×D₇₉	1	2	yes	—
219	C₂²	#19098979	1	1	yes	characteristic
220	C₂²	#19098979	1	1	yes	characteristic
221	C₄	—	2	2	no	—
222	C₄	—	316	2	no	—
223	C₄	—	158	4	no	—
224	C₄	#19098435	1	1	yes	characteristic
225	C₄	#19098591	1	1	yes	characteristic
226	C₄	#19098631	1	2	yes	—
227	C₂	—	4	2	no	—
228	C₂	#23763577	1	1	yes	characteristic
229	C₂	#23765155	1	1	yes	characteristic
230	C₂	#23765158	1	1	yes	characteristic
231	C₂	#23765228	1	2	yes	—
232	C₂	#23765229	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₇₉
commutator, socle	C₂³×C₇₉	C₂²×C₈
Fitting	C₂×C₂²⋊C₁₆×C₇₉	C₂
prime radical	C₂⁴×C₇₉	C₂×C₈
cosocle	C₂²×C₈×C₇₉	C₂³
hypercenter	C₂×C₂²⋊C₁₆	D₇₉
hypocenter	C₇₉	#12913
LCS	G ▷ C₂³×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₂²⋊C₁₆

Complements

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

N	H ≅ G/N	type
2. C₂×Dic₇₉⋊C₁₆	C₂	semidirect
4. #23761477	C₂	semidirect
88. C₂×Dic₇₉	M₃₂	semidirect
122. C₇₉	#12913	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#14579101	2
C₂×Dic₇₉⋊C₁₆	2
C₈×C₂²⋊Dic₇₉	2
#23761477	2
C₂×C₂²⋊C₁₆×C₇₉	2
#12913	79