Subgroups — #34293078

Order 20224.

854 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19098616	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×#11929608	C₂	1	1	yes	maximal, characteristic
3	#23764119	C₂	1	4	yes	maximal, complemented
4	#23763585	C₂	1	4	yes	maximal
5	#23763257	C₂	1	4	yes	maximal, complemented
6	C₇₉×#2283	C₂	1	1	yes	maximal, characteristic
7	C₂²×#11929608	C₂²	1	2	yes	—
8	#19098616	—	2	4	no	—
9	C₂×C₄⋊(C₇₉⋊C₈)	C₂²	1	8	yes	—
10	C₂×C₄⋊(C₇₉⋊C₈)	C₂²	1	2	yes	—
11	C₂×C₂²⋊(C₇₉⋊C₈)	C₂²	1	2	yes	—
12	C₄×#11929608	—	2	4	no	—
13	C₂×#14579072	C₂²	1	2	yes	—
14	C₂×C₄×C₇₉⋊C₈	C₂²	1	2	yes	—
15	C₂×C₄×C₇₉⋊C₈	C₂²	1	2	yes	—
16	#19098318	C₂²	1	8	yes	—
17	C₂²×C₄²×C₇₉	C₂²	1	1	yes	characteristic
18	#383×C₇₉	C₄	1	2	yes	—
19	#382×C₇₉	C₄	1	2	yes	—
20	#381×C₇₉	C₄	1	2	yes	—
21	#380×C₇₉	C₄	1	2	yes	—
22	C₄×C₄⋊C₄×C₇₉	C₂²	1	2	yes	—
23	C₄×C₂²⋊C₄×C₇₉	C₂²	1	2	yes	—
24	#376×C₇₉	C₂²	1	2	yes	—
25	C₂×#11929608	—	2	4	no	—
26	C₂×#11929608	—	2	8	no	—
27	C₂²×C₇₉⋊C₈	C₂³	1	4	yes	—
28	C₂²×C₇₉⋊C₈	C₂³	1	4	yes	—
29	C₄⋊(C₇₉⋊C₈)	—	2	4	no	—
30	C₄⋊(C₇₉⋊C₈)	—	2	16	no	—
31	C₂²⋊(C₇₉⋊C₈)	—	2	4	no	—
32	#14579072	—	2	4	no	—
33	C₄×C₇₉⋊C₈	—	2	4	no	—
34	C₄×C₇₉⋊C₈	—	2	4	no	—
35	C₂³×C₄×C₇₉	C₂×C₄	1	1	yes	characteristic
36	C₂³×C₄×C₇₉	C₂×C₄	1	1	yes	characteristic
37	C₂³×C₄×C₇₉	C₂³	1	1	yes	characteristic
38	C₂×C₄⋊C₄×C₇₉	C₂×C₄	1	4	yes	—
39	C₂×C₂²⋊C₄×C₇₉	C₂×C₄	1	4	yes	—
40	C₂×C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄²×C₇₉	D₄	1	4	yes	—
43	C₂×C₄²×C₇₉	Q₈	1	4	yes	—
44	C₂×C₄²×C₇₉	C₂³	1	4	yes	—
45	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
46	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
47	#95×C₇₉	C₂×C₄	1	4	yes	—
48	#95×C₇₉	C₂×C₄	1	4	yes	—
49	#95×C₇₉	C₂×C₄	1	4	yes	—
50	#95×C₇₉	C₂×C₄	1	4	yes	—
51	#11929608	—	2	16	no	—
52	C₂×C₇₉⋊C₈	—	2	4	no	—
53	C₂×C₇₉⋊C₈	—	2	4	no	—
54	C₂×C₇₉⋊C₈	—	2	4	no	—
55	C₂×C₇₉⋊C₈	—	2	8	no	—
56	C₂×C₇₉⋊C₈	#41	1	8	yes	—
57	C₂⁴×C₇₉	C₂²×C₄	1	1	yes	characteristic
58	C₂²×C₄×C₇₉	—	2	4	no	—
59	C₂²×C₄×C₇₉	—	2	4	no	—
60	C₂²×C₄×C₇₉	—	4	2	no	—
61	C₂²×C₄×C₇₉	C₄⋊C₄	1	4	yes	—
62	C₂²×C₄×C₇₉	C₄⋊C₄	1	4	yes	—
63	C₂²×C₄×C₇₉	C₂²×C₄	1	8	yes	—
64	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
68	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₇₉	C₂×D₄	1	2	yes	—
70	C₂²×C₄×C₇₉	C₂×Q₈	1	2	yes	—
71	C₂²×C₄×C₇₉	C₂⁴	1	1	yes	characteristic
72	C₄⋊C₄×C₇₉	—	2	8	no	—
73	C₂²⋊C₄×C₇₉	—	4	4	no	—
74	C₄²×C₇₉	—	2	8	no	—
75	C₄²×C₇₉	C₄⋊C₄	1	4	yes	—
76	C₄²×C₇₉	C₄⋊C₄	1	4	yes	—
77	C₄²×C₇₉	C₂×D₄	1	2	yes	—
78	C₄²×C₇₉	C₂×D₄	1	2	yes	—
79	C₄²×C₇₉	C₂×Q₈	1	2	yes	—
80	C₄²×C₇₉	C₂×Q₈	1	2	yes	—
81	C₂³×C₇₉	—	2	2	no	—
82	C₂³×C₇₉	—	2	2	no	—
83	C₂³×C₇₉	—	2	2	no	—
84	C₂³×C₇₉	C₂×C₄⋊C₄	1	2	yes	—
85	C₂³×C₇₉	C₂³×C₄	1	1	yes	characteristic
86	C₂×C₄×C₇₉	—	2	2	no	—
87	C₂×C₄×C₇₉	—	2	2	no	—
88	C₂×C₄×C₇₉	—	2	2	no	—
89	C₂×C₄×C₇₉	—	2	2	no	—
90	C₂×C₄×C₇₉	—	2	4	no	—
91	C₂×C₄×C₇₉	—	2	4	no	—
92	C₂×C₄×C₇₉	—	2	8	no	—
93	C₂×C₄×C₇₉	—	2	8	no	—
94	C₂×C₄×C₇₉	—	2	8	no	—
95	C₂×C₄×C₇₉	—	2	8	no	—
96	C₂×C₄×C₇₉	—	4	4	no	—
97	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	2	yes	—
98	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	2	yes	—
99	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	2	yes	—
100	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	2	yes	—
101	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	2	yes	—
102	C₂×C₄×C₇₉	#131	1	4	yes	—
103	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
104	C₂×C₄×C₇₉	C₂²×Q₈	1	1	yes	characteristic
105	C₂×C₄×C₇₉	C₂×#41	1	4	yes	—
106	C₇₉⋊C₈	—	2	8	no	—
107	C₇₉⋊C₈	—	4	4	no	—
108	C₂²×C₇₉	—	2	2	no	—
109	C₂²×C₇₉	—	2	2	no	—
110	C₂²×C₇₉	—	2	2	no	—
111	C₂²×C₇₉	—	4	4	no	—
112	C₂²×C₇₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
113	C₂²×C₇₉	C₄×#41	1	2	yes	—
114	C₂²×C₇₉	C₂×#131	1	2	yes	—
115	C₂²×C₇₉	#568	1	2	yes	—
116	C₄×C₇₉	—	2	2	no	—
117	C₄×C₇₉	—	2	2	no	—
118	C₄×C₇₉	—	2	2	no	—
119	C₄×C₇₉	—	2	2	no	—
120	C₄×C₇₉	—	4	8	no	—
121	C₄×C₇₉	#424	1	4	yes	—
122	C₄×C₇₉	#525	1	2	yes	—
123	C₄×C₇₉	#533	1	2	yes	—
124	#21354	—	79	1	no	maximal, Sylow
125	C₂×C₇₉	—	4	2	no	—
126	C₂×C₇₉	#2296	1	1	yes	characteristic
127	C₂×C₇₉	C₂×#424	1	1	yes	characteristic
128	C₂×C₇₉	#2905	1	1	yes	characteristic
129	C₂×C₇₉	#2969	1	4	yes	—
130	C₂×#431	—	79	1	no	—
131	C₂×C₄×M₁₆	—	79	1	no	—
132	#2283	D₇₉	1	1	yes	characteristic
133	#1939	—	79	4	no	—
134	#1851	—	79	4	no	—
135	#1768	—	79	4	no	—
136	C₇₉	#21354	1	1	yes	characteristic, complemented, Sylow
137	C₂²×M₁₆	—	79	2	no	—
138	C₂²×C₄²	C₂×D₇₉	1	1	yes	characteristic
139	#431	—	158	4	no	—
140	C₂×C₄⋊C₈	—	79	8	no	—
141	C₂×C₄⋊C₈	—	79	2	no	—
142	C₂×C₂²⋊C₈	—	79	2	no	—
143	C₄×M₁₆	—	158	4	no	—
144	C₂×#97	—	79	2	no	—
145	C₂×C₄×C₈	—	79	2	no	—
146	C₂×C₄×C₈	—	79	2	no	—
147	#383	Dic₇₉	1	2	yes	—
148	#382	Dic₇₉	1	2	yes	—
149	#381	Dic₇₉	1	2	yes	—
150	#380	Dic₇₉	1	2	yes	—
151	C₄×C₄⋊C₄	C₂×D₇₉	1	2	yes	—
152	C₄×C₂²⋊C₄	C₂×D₇₉	1	2	yes	—
153	#376	C₂×D₇₉	1	2	yes	—
154	#336	—	79	8	no	—
155	C₂³×C₄	C₂×Dic₇₉	1	1	yes	characteristic
156	C₂³×C₄	C₂×Dic₇₉	1	1	yes	characteristic
157	C₂³×C₄	C₂²×D₇₉	1	1	yes	characteristic
158	C₂×M₁₆	—	158	8	no	—
159	C₂×M₁₆	—	158	4	no	—
160	C₂²×C₈	—	79	4	no	—
161	C₂²×C₈	—	79	4	no	—
162	C₂×C₄⋊C₄	C₂×Dic₇₉	1	4	yes	—
163	C₂×C₂²⋊C₄	C₂×Dic₇₉	1	4	yes	—
164	C₂×C₄²	Dic₁₅₈	1	4	yes	—
165	C₂×C₄²	D₃₁₆	1	4	yes	—
166	C₂×C₄²	C₂×Dic₇₉	1	1	yes	characteristic
167	C₂×C₄²	C₂×Dic₇₉	1	1	yes	characteristic
168	C₂×C₄²	C₂²×D₇₉	1	4	yes	—
169	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
170	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
171	C₄⋊C₈	—	158	16	no	—
172	C₄⋊C₈	—	158	4	no	—
173	C₂²⋊C₈	—	158	4	no	—
174	#97	—	158	4	no	—
175	C₄×C₈	—	158	4	no	—
176	C₄×C₈	—	158	4	no	—
177	#95	C₂×Dic₇₉	1	4	yes	—
178	#95	C₂×Dic₇₉	1	4	yes	—
179	#95	C₂×Dic₇₉	1	4	yes	—
180	#95	C₂×Dic₇₉	1	4	yes	—
181	C₂⁴	C₂²×Dic₇₉	1	1	yes	characteristic
182	C₂²×C₄	—	2	4	no	—
183	C₂²×C₄	—	2	4	no	—
184	C₂²×C₄	—	4	2	no	—
185	C₂²×C₄	C₄⋊Dic₇₉	1	4	yes	—
186	C₂²×C₄	C₄⋊Dic₇₉	1	4	yes	—
187	C₂²×C₄	C₂²×Dic₇₉	1	8	yes	—
188	C₂²×C₄	C₂²×Dic₇₉	1	1	yes	characteristic
189	C₂²×C₄	C₂²×Dic₇₉	1	1	yes	characteristic
190	C₂²×C₄	C₂²×Dic₇₉	1	1	yes	characteristic
191	C₂²×C₄	C₂²×Dic₇₉	1	1	yes	characteristic
192	C₂²×C₄	C₂²×Dic₇₉	1	1	yes	characteristic
193	C₂²×C₄	C₂×D₃₁₆	1	2	yes	—
194	C₂²×C₄	C₂×Dic₁₅₈	1	2	yes	—
195	C₂²×C₄	C₂³×D₇₉	1	1	yes	characteristic
196	M₁₆	—	158	16	no	—
197	C₂×C₈	—	79	8	no	—
198	C₂×C₈	—	158	8	no	—
199	C₂×C₈	—	158	4	no	—
200	C₂×C₈	—	158	4	no	—
201	C₂×C₈	—	158	4	no	—
202	C₄⋊C₄	—	2	8	no	—
203	C₂²⋊C₄	—	4	4	no	—
204	C₄²	—	2	8	no	—
205	C₄²	C₄⋊Dic₇₉	1	4	yes	—
206	C₄²	C₄⋊Dic₇₉	1	4	yes	—
207	C₄²	C₂×D₃₁₆	1	2	yes	—
208	C₄²	C₂×D₃₁₆	1	2	yes	—
209	C₄²	C₂×Dic₁₅₈	1	2	yes	—
210	C₄²	C₂×Dic₁₅₈	1	2	yes	—
211	C₂³	—	2	2	no	—
212	C₂³	—	2	2	no	—
213	C₂³	—	2	2	no	—
214	C₂³	C₂×C₄⋊Dic₇₉	1	2	yes	—
215	C₂³	C₂³×Dic₇₉	1	1	yes	characteristic
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	4	no	—
221	C₂×C₄	—	2	8	no	—
222	C₂×C₄	—	2	4	no	—
223	C₂×C₄	—	2	8	no	—
224	C₂×C₄	—	2	8	no	—
225	C₂×C₄	—	2	8	no	—
226	C₂×C₄	—	4	4	no	—
227	C₂×C₄	C₂×C₄⋊Dic₇₉	1	2	yes	—
228	C₂×C₄	C₂×C₄⋊Dic₇₉	1	2	yes	—
229	C₂×C₄	C₂×C₄⋊Dic₇₉	1	2	yes	—
230	C₂×C₄	C₂×C₄⋊Dic₇₉	1	2	yes	—
231	C₂×C₄	C₂×C₄⋊Dic₇₉	1	2	yes	—
232	C₂×C₄	#14579168	1	4	yes	—
233	C₂×C₄	C₂²×D₃₁₆	1	1	yes	characteristic
234	C₂×C₄	C₂²×Dic₁₅₈	1	1	yes	characteristic
235	C₂×C₄	#41×D₇₉	1	4	yes	—
236	C₈	—	316	4	no	—
237	C₈	—	158	8	no	—
238	C₂²	—	2	2	no	—
239	C₂²	—	2	2	no	—
240	C₂²	—	2	2	no	—
241	C₂²	—	4	4	no	—
242	C₂²	C₂²×C₄⋊Dic₇₉	1	1	yes	characteristic
243	C₂²	#41×Dic₇₉	1	2	yes	—
244	C₂²	C₂×#14579168	1	2	yes	—
245	C₂²	#19099330	1	2	yes	—
246	C₄	—	2	2	no	—
247	C₄	—	2	2	no	—
248	C₄	—	2	2	no	—
249	C₄	—	2	2	no	—
250	C₄	—	4	8	no	—
251	C₄	#19098590	1	4	yes	—
252	C₄	#19099075	1	2	yes	—
253	C₄	#19099116	1	2	yes	—
254	C₂	—	4	2	no	—
255	C₂	#23765884	1	1	yes	characteristic
256	C₂	C₂×#19098590	1	1	yes	characteristic
257	C₂	#23771964	1	1	yes	characteristic
258	C₂	#23772405	1	4	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₇₉
commutator, socle	C₂³×C₇₉	C₂³×C₄
Fitting	C₇₉×#2283	C₂
prime radical	C₂⁴×C₇₉	C₂²×C₄
cosocle	C₂²×C₄×C₇₉	C₂⁴
hypercenter	#2283	D₇₉
hypocenter	C₇₉	#21354
LCS	G ▷ C₂³×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂²×C₄ ◁ #2283

Complements

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

N	H ≅ G/N	type
3. #23764119	C₂	semidirect
5. #23763257	C₂	semidirect
136. C₇₉	#21354	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19098616	2
C₂×C₄×#11929608	2
#23764119	2
#23763585	2
#23763257	2
C₇₉×#2283	2
#21354	79