Subgroups — #34440963

Order 20224.

1128 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23776503	C₂	1	2	yes	maximal, complemented
2	#23765021	C₂	1	6	yes	maximal, complemented
3	C₂×#19098345	C₂	1	6	yes	maximal, complemented
4	C₂×#568×C₇₉	C₂	1	1	yes	maximal, characteristic
5	#19098980	—	2	8	no	—
6	C₂×#14579101	C₂²	1	12	yes	complemented
7	#19098345	C₂²	1	4	yes	—
8	#19098345	C₂²	1	12	yes	complemented
9	C₂²×#41×C₇₉	C₄	1	1	yes	characteristic
10	#568×C₇₉	—	2	8	no	—
11	C₂×#131×C₇₉	C₄	1	3	yes	—
12	C₂×#131×C₇₉	C₄	1	1	yes	characteristic
13	C₂×#131×C₇₉	C₂²	1	3	yes	—
14	C₂×#131×C₇₉	C₂²	1	1	yes	characteristic
15	C₂²×M₁₆×C₇₉	C₄	1	3	yes	—
16	C₂²×M₁₆×C₇₉	C₂²	1	3	yes	—
17	#14579101	—	2	12	no	—
18	#14579101	C₂³	1	8	yes	—
19	C₂×C₇₉⋊C₁₆	—	2	12	no	—
20	C₂×#41×C₇₉	—	2	4	no	—
21	C₂×#41×C₇₉	—	2	4	no	—
22	C₂×#41×C₇₉	C₈	1	1	yes	characteristic
23	C₂×#41×C₇₉	C₈	1	3	yes	—
24	C₂×#41×C₇₉	C₂×C₄	1	3	yes	—
25	C₂×#41×C₇₉	C₂×C₄	1	1	yes	characteristic
26	C₂²×Q₈×C₇₉	C₈	1	1	yes	characteristic
27	C₂²×D₄×C₇₉	C₈	1	3	yes	—
28	C₂³×C₄×C₇₉	C₂×C₄	1	3	yes	—
29	#131×C₇₉	—	2	4	no	—
30	#131×C₇₉	—	2	12	no	—
31	#131×C₇₉	—	2	12	no	—
32	#131×C₇₉	D₄	1	8	yes	—
33	C₂×M₁₆×C₇₉	—	2	12	no	—
34	C₂×M₁₆×C₇₉	—	2	12	no	—
35	C₂×M₁₆×C₇₉	C₂×C₄	1	3	yes	—
36	C₂×M₁₆×C₇₉	C₂×C₄	1	3	yes	—
37	C₂×M₁₆×C₇₉	C₂×C₄	1	3	yes	—
38	C₂×M₁₆×C₇₉	C₂×C₄	1	3	yes	—
39	C₂×M₁₆×C₇₉	C₂³	1	1	yes	characteristic
40	C₂×M₁₆×C₇₉	C₂³	1	3	yes	—
41	C₂²×C₈×C₇₉	C₂×C₄	1	6	yes	—
42	C₂²×C₈×C₇₉	C₂×C₄	1	3	yes	—
43	C₂²×C₈×C₇₉	C₂³	1	3	yes	—
44	C₇₉⋊C₁₆	—	2	8	no	—
45	C₂⁴×C₇₉	C₂×C₈	1	3	yes	—
46	#41×C₇₉	—	2	4	no	—
47	#41×C₇₉	—	2	12	no	—
48	#41×C₇₉	—	2	6	no	—
49	#41×C₇₉	—	2	6	no	—
50	#41×C₇₉	C₂²⋊C₄	1	4	yes	—
51	#41×C₇₉	M₁₆	1	4	yes	—
52	C₂×Q₈×C₇₉	—	2	4	no	—
53	C₂×Q₈×C₇₉	C₂×C₈	1	3	yes	—
54	C₂×Q₈×C₇₉	C₂×C₈	1	1	yes	characteristic
55	C₂×D₄×C₇₉	—	2	12	no	—
56	C₂×D₄×C₇₉	C₂×C₈	1	6	yes	—
57	C₂×D₄×C₇₉	C₂×C₈	1	3	yes	—
58	C₂×D₄×C₇₉	C₂×C₈	1	3	yes	—
59	C₂²×C₄×C₇₉	—	2	6	no	—
60	C₂²×C₄×C₇₉	—	2	6	no	—
61	C₂²×C₄×C₇₉	C₂×C₈	1	3	yes	—
62	C₂²×C₄×C₇₉	C₂×C₈	1	3	yes	—
63	C₂²×C₄×C₇₉	C₂×C₈	1	3	yes	—
64	C₂²×C₄×C₇₉	C₂²×C₄	1	3	yes	—
65	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₇₉	C₂²×C₄	1	3	yes	—
67	M₁₆×C₇₉	—	2	6	no	—
68	M₁₆×C₇₉	—	2	6	no	—
69	M₁₆×C₇₉	—	2	2	no	—
70	M₁₆×C₇₉	—	2	6	no	—
71	M₁₆×C₇₉	—	2	6	no	—
72	M₁₆×C₇₉	C₂²⋊C₄	1	12	yes	—
73	C₂×C₈×C₇₉	—	2	12	no	—
74	C₂×C₈×C₇₉	—	2	6	no	—
75	C₂×C₈×C₇₉	C₂²×C₄	1	3	yes	—
76	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₈×C₇₉	C₂²×C₄	1	3	yes	—
78	C₂×C₈×C₇₉	C₂×D₄	1	12	yes	—
79	C₂×C₈×C₇₉	C₂⁴	1	1	yes	characteristic
80	C₂³×C₇₉	—	2	6	no	—
81	C₂³×C₇₉	—	8	3	no	—
82	C₂³×C₇₉	C₂²×C₈	1	1	yes	characteristic
83	C₂³×C₇₉	C₂²×C₈	1	3	yes	—
84	C₂³×C₇₉	C₂²×C₈	1	3	yes	—
85	Q₈×C₇₉	—	2	6	no	—
86	Q₈×C₇₉	C₂²⋊C₈	1	4	yes	—
87	D₄×C₇₉	—	2	12	no	—
88	D₄×C₇₉	—	2	6	no	—
89	D₄×C₇₉	C₂²⋊C₈	1	12	yes	—
90	C₂×C₄×C₇₉	—	2	6	no	—
91	C₂×C₄×C₇₉	—	2	3	no	—
92	C₂×C₄×C₇₉	—	2	1	no	—
93	C₂×C₄×C₇₉	—	2	1	no	—
94	C₂×C₄×C₇₉	—	2	3	no	—
95	C₂×C₄×C₇₉	—	2	6	no	—
96	C₂×C₄×C₇₉	—	2	6	no	—
97	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	6	yes	—
98	C₂×C₄×C₇₉	C₂²×C₈	1	1	yes	characteristic
99	C₂×C₄×C₇₉	C₂²×C₈	1	3	yes	—
100	C₂×C₄×C₇₉	C₂²×C₈	1	3	yes	—
101	C₂×C₄×C₇₉	C₂×M₁₆	1	6	yes	—
102	C₂×C₄×C₇₉	C₂³×C₄	1	1	yes	characteristic
103	C₈×C₇₉	—	2	3	no	—
104	C₈×C₇₉	—	2	1	no	—
105	C₈×C₇₉	C₂×C₂²⋊C₄	1	6	yes	—
106	C₈×C₇₉	C₂²×D₄	1	2	yes	—
107	C₂²×C₇₉	—	2	1	no	—
108	C₂²×C₇₉	—	2	3	no	—
109	C₂²×C₇₉	—	4	3	no	—
110	C₂²×C₇₉	—	4	1	no	—
111	C₂²×C₇₉	—	4	3	no	—
112	C₂²×C₇₉	—	8	6	no	—
113	C₂²×C₇₉	C₂×C₂²⋊C₈	1	6	yes	—
114	C₂²×C₇₉	C₂³×C₈	1	1	yes	characteristic
115	C₄×C₇₉	—	2	1	no	—
116	C₄×C₇₉	—	2	3	no	—
117	C₄×C₇₉	C₂×C₂²⋊C₈	1	6	yes	—
118	C₄×C₇₉	C₂²×C₂²⋊C₄	1	1	yes	characteristic
119	C₄×C₇₉	C₂²×M₁₆	1	1	yes	characteristic
120	#33337	—	79	1	no	maximal, Sylow
121	C₂×C₇₉	—	2	1	no	—
122	C₂×C₇₉	—	2	6	no	—
123	C₂×C₇₉	—	4	3	no	—
124	C₂×C₇₉	—	4	1	no	—
125	C₂×C₇₉	C₂²×C₂²⋊C₈	1	1	yes	characteristic
126	C₂×#568	D₇₉	1	1	yes	characteristic
127	#3407	—	79	2	no	—
128	#2115	—	79	6	no	—
129	C₂×#349	—	79	6	no	—
130	C₇₉	#33337	1	1	yes	characteristic, complemented, Sylow
131	C₂²×#41	Dic₇₉	1	1	yes	characteristic
132	#568	—	2	8	no	—
133	C₂×#131	Dic₇₉	1	3	yes	—
134	C₂×#131	Dic₇₉	1	1	yes	characteristic
135	C₂×#131	C₂×D₇₉	1	3	yes	—
136	C₂×#131	C₂×D₇₉	1	1	yes	characteristic
137	C₂²×M₁₆	Dic₇₉	1	3	yes	—
138	C₂²×M₁₆	C₂×D₇₉	1	3	yes	—
139	#504	—	158	8	no	—
140	C₂×M₃₂	—	79	12	no	—
141	#349	—	79	4	no	—
142	#349	—	79	12	no	—
143	C₂×#41	—	2	4	no	—
144	C₂×#41	—	2	4	no	—
145	C₂×#41	C₇₉⋊C₈	1	1	yes	characteristic
146	C₂×#41	C₇₉⋊C₈	1	3	yes	—
147	C₂×#41	C₂×Dic₇₉	1	3	yes	—
148	C₂×#41	C₂×Dic₇₉	1	1	yes	characteristic
149	C₂²×Q₈	C₇₉⋊C₈	1	1	yes	characteristic
150	C₂²×D₄	C₇₉⋊C₈	1	3	yes	—
151	C₂³×C₄	C₂×Dic₇₉	1	3	yes	—
152	#131	—	2	4	no	—
153	#131	—	2	12	no	—
154	#131	—	2	12	no	—
155	#131	C₂²⋊D₇₉	1	8	yes	—
156	C₂×M₁₆	—	2	12	no	—
157	C₂×M₁₆	—	2	12	no	—
158	C₂×M₁₆	C₂×Dic₇₉	1	3	yes	—
159	C₂×M₁₆	C₂×Dic₇₉	1	3	yes	—
160	C₂×M₁₆	C₂×Dic₇₉	1	3	yes	—
161	C₂×M₁₆	C₂×Dic₇₉	1	3	yes	—
162	C₂×M₁₆	C₂²×D₇₉	1	1	yes	characteristic
163	C₂×M₁₆	C₂²×D₇₉	1	3	yes	—
164	C₂²×C₈	C₂×Dic₇₉	1	6	yes	—
165	C₂²×C₈	C₂×Dic₇₉	1	3	yes	—
166	C₂²×C₈	C₂²×D₇₉	1	3	yes	—
167	M₃₂	—	79	8	no	—
168	M₃₂	—	158	12	no	—
169	C₂×C₁₆	—	158	12	no	—
170	C₂⁴	C₂×C₇₉⋊C₈	1	3	yes	—
171	#41	—	2	4	no	—
172	#41	—	2	12	no	—
173	#41	—	2	6	no	—
174	#41	—	2	6	no	—
175	#41	C₂²⋊Dic₇₉	1	4	yes	—
176	#41	#11929608	1	4	yes	—
177	C₂×Q₈	—	2	4	no	—
178	C₂×Q₈	C₂×C₇₉⋊C₈	1	3	yes	—
179	C₂×Q₈	C₂×C₇₉⋊C₈	1	1	yes	characteristic
180	C₂×D₄	—	2	12	no	—
181	C₂×D₄	C₂×C₇₉⋊C₈	1	6	yes	—
182	C₂×D₄	C₂×C₇₉⋊C₈	1	3	yes	—
183	C₂×D₄	C₂×C₇₉⋊C₈	1	3	yes	—
184	C₂²×C₄	—	2	6	no	—
185	C₂²×C₄	—	2	6	no	—
186	C₂²×C₄	C₂×C₇₉⋊C₈	1	3	yes	—
187	C₂²×C₄	C₂×C₇₉⋊C₈	1	3	yes	—
188	C₂²×C₄	C₂×C₇₉⋊C₈	1	3	yes	—
189	C₂²×C₄	C₂²×Dic₇₉	1	3	yes	—
190	C₂²×C₄	C₂²×Dic₇₉	1	1	yes	characteristic
191	C₂²×C₄	C₂²×Dic₇₉	1	3	yes	—
192	M₁₆	—	2	6	no	—
193	M₁₆	—	2	6	no	—
194	M₁₆	—	2	6	no	—
195	M₁₆	—	2	2	no	—
196	M₁₆	—	2	6	no	—
197	M₁₆	C₂²⋊Dic₇₉	1	12	yes	—
198	C₂×C₈	—	2	6	no	—
199	C₂×C₈	—	2	12	no	—
200	C₂×C₈	C₂²×Dic₇₉	1	3	yes	—
201	C₂×C₈	C₂²×Dic₇₉	1	1	yes	characteristic
202	C₂×C₈	C₂²×Dic₇₉	1	3	yes	—
203	C₂×C₈	C₂×C₂²⋊D₇₉	1	12	yes	—
204	C₂×C₈	C₂³×D₇₉	1	1	yes	characteristic
205	C₁₆	—	158	8	no	—
206	C₂³	—	2	6	no	—
207	C₂³	—	8	3	no	—
208	C₂³	C₂²×C₇₉⋊C₈	1	1	yes	characteristic
209	C₂³	C₂²×C₇₉⋊C₈	1	3	yes	—
210	C₂³	C₂²×C₇₉⋊C₈	1	3	yes	—
211	Q₈	—	2	6	no	—
212	Q₈	C₂²⋊(C₇₉⋊C₈)	1	4	yes	—
213	D₄	—	2	12	no	—
214	D₄	—	2	6	no	—
215	D₄	C₂²⋊(C₇₉⋊C₈)	1	12	yes	—
216	C₂×C₄	—	2	6	no	—
217	C₂×C₄	—	2	3	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	2	6	no	—
220	C₂×C₄	—	2	6	no	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	2	3	no	—
223	C₂×C₄	C₂×C₂²⋊Dic₇₉	1	6	yes	—
224	C₂×C₄	C₂²×C₇₉⋊C₈	1	1	yes	characteristic
225	C₂×C₄	C₂²×C₇₉⋊C₈	1	3	yes	—
226	C₂×C₄	C₂²×C₇₉⋊C₈	1	3	yes	—
227	C₂×C₄	C₂×#11929608	1	6	yes	—
228	C₂×C₄	C₂³×Dic₇₉	1	1	yes	characteristic
229	C₈	—	2	1	no	—
230	C₈	—	2	3	no	—
231	C₈	C₂×C₂²⋊Dic₇₉	1	6	yes	—
232	C₈	C₂²×C₂²⋊D₇₉	1	2	yes	—
233	C₂²	—	2	1	no	—
234	C₂²	—	2	3	no	—
235	C₂²	—	4	3	no	—
236	C₂²	—	4	1	no	—
237	C₂²	—	4	3	no	—
238	C₂²	—	8	6	no	—
239	C₂²	C₂×C₂²⋊(C₇₉⋊C₈)	1	6	yes	—
240	C₂²	C₂³×C₇₉⋊C₈	1	1	yes	characteristic
241	C₄	—	2	1	no	—
242	C₄	—	2	3	no	—
243	C₄	C₂×C₂²⋊(C₇₉⋊C₈)	1	6	yes	—
244	C₄	C₂²×C₂²⋊Dic₇₉	1	1	yes	characteristic
245	C₄	C₂²×#11929608	1	1	yes	characteristic
246	C₂	—	2	1	no	—
247	C₂	—	2	6	no	—
248	C₂	—	4	1	no	—
249	C₂	—	4	3	no	—
250	C₂	C₂²×C₂²⋊(C₇₉⋊C₈)	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂²×C₂²⋊Dic₇₉
commutator	C₂²×C₇₉	C₂³×C₈
Frattini	C₂×C₈	C₂³×D₇₉
Fitting	C₂×#568×C₇₉	C₂
socle	C₂×C₇₉	C₂²×C₂²⋊C₈
prime radical	C₂²×#41×C₇₉	C₄
cosocle	C₂×C₈×C₇₉	C₂⁴
hypercenter	C₂×#568	D₇₉
hypocenter	C₇₉	#33337
LCS	G ▷ C₂²×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₄ ◁ C₂×#131 ◁ C₂×#568

Complements

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

N	H ≅ G/N	type
1. #23776503	C₂	semidirect
2. #23765021	C₂	semidirect
3. C₂×#19098345	C₂	semidirect
6. C₂×#14579101	C₂²	semidirect
8. #19098345	C₂²	semidirect
130. C₇₉	#33337	semidirect

Maximal subgroups

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

subgroup	index
#23776503	2
#23765021	2
C₂×#19098345	2
C₂×#568×C₇₉	2
#33337	79