Subgroups — #34298616

Order 20224.

798 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19098620	C₂	1	1	yes	maximal, characteristic
2	C₂×#19098616	C₂	1	1	yes	maximal, characteristic
3	#23764124	C₂	1	4	yes	maximal, complemented
4	#23764119	C₂	1	4	yes	maximal, complemented
5	#23763384	C₂	1	4	yes	maximal
6	C₇₉×#2285	C₂	1	1	yes	maximal, characteristic
7	#19098620	—	2	4	no	—
8	#19098616	—	2	4	no	—
9	C₂×C₄⋊(C₇₉⋊C₈)	C₂²	1	2	yes	—
10	C₂×C₄⋊(C₇₉⋊C₈)	C₂²	1	2	yes	—
11	C₂×C₄⋊(C₇₉⋊C₈)	C₂²	1	8	yes	—
12	C₂×C₂²⋊(C₇₉⋊C₈)	C₂²	1	2	yes	—
13	C₂×C₂²⋊(C₇₉⋊C₈)	C₂²	1	2	yes	—
14	C₂×#14579072	C₂²	1	2	yes	—
15	C₂×C₄×C₇₉⋊C₈	C₂²	1	2	yes	—
16	#19098318	C₂²	1	8	yes	—
17	C₂×#117×C₇₉	C₂²	1	2	yes	—
18	C₂²×C₄²×C₇₉	C₂²	1	1	yes	characteristic
19	#384×C₇₉	C₄	1	2	yes	—
20	#382×C₇₉	C₄	1	2	yes	—
21	#380×C₇₉	C₄	1	2	yes	—
22	#380×C₇₉	C₄	1	2	yes	—
23	C₄×C₄⋊C₄×C₇₉	C₂²	1	2	yes	—
24	C₄×C₄⋊C₄×C₇₉	C₂²	1	2	yes	—
25	C₂²×C₇₉⋊C₈	C₂³	1	4	yes	—
26	C₂²×C₇₉⋊C₈	C₂³	1	4	yes	—
27	C₄⋊(C₇₉⋊C₈)	—	2	4	no	—
28	C₄⋊(C₇₉⋊C₈)	—	2	4	no	—
29	C₄⋊(C₇₉⋊C₈)	—	4	8	no	—
30	C₂²⋊(C₇₉⋊C₈)	—	2	4	no	—
31	C₂²⋊(C₇₉⋊C₈)	—	2	4	no	—
32	#14579072	—	2	4	no	—
33	C₄×C₇₉⋊C₈	—	2	4	no	—
34	C₂³×C₄×C₇₉	C₂×C₄	1	1	yes	characteristic
35	C₂³×C₄×C₇₉	C₂×C₄	1	1	yes	characteristic
36	C₂³×C₄×C₇₉	C₂³	1	1	yes	characteristic
37	#117×C₇₉	—	4	4	no	—
38	C₂×C₄⋊C₄×C₇₉	C₂×C₄	1	4	yes	—
39	C₂×C₄⋊C₄×C₇₉	C₂×C₄	1	4	yes	—
40	C₂×C₄⋊C₄×C₇₉	C₂×C₄	1	4	yes	—
41	C₂×C₂²⋊C₄×C₇₉	C₂×C₄	1	4	yes	—
42	C₂×C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄²×C₇₉	D₄	1	4	yes	—
45	C₂×C₄²×C₇₉	Q₈	1	4	yes	—
46	C₂×C₄²×C₇₉	C₂³	1	4	yes	—
47	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
48	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
49	#95×C₇₉	C₂×C₄	1	4	yes	—
50	#95×C₇₉	C₂×C₄	1	4	yes	—
51	C₂×C₇₉⋊C₈	—	2	4	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	4	no	—
56	C₂×C₇₉⋊C₈	—	2	4	no	—
57	C₂⁴×C₇₉	C₂²×C₄	1	1	yes	characteristic
58	C₂²×C₄×C₇₉	—	2	4	no	—
59	C₂²×C₄×C₇₉	—	2	2	no	—
60	C₂²×C₄×C₇₉	—	2	2	no	—
61	C₂²×C₄×C₇₉	—	4	2	no	—
62	C₂²×C₄×C₇₉	C₄⋊C₄	1	4	yes	—
63	C₂²×C₄×C₇₉	C₄⋊C₄	1	4	yes	—
64	C₂²×C₄×C₇₉	C₂²×C₄	1	8	yes	—
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	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₇₉	C₂×D₄	1	2	yes	—
71	C₂²×C₄×C₇₉	C₂×Q₈	1	2	yes	—
72	C₂²×C₄×C₇₉	C₂⁴	1	1	yes	characteristic
73	C₄⋊C₄×C₇₉	—	2	8	no	—
74	C₄⋊C₄×C₇₉	—	4	4	no	—
75	C₄⋊C₄×C₇₉	—	4	4	no	—
76	C₂²⋊C₄×C₇₉	—	4	4	no	—
77	C₄²×C₇₉	—	2	8	no	—
78	C₄²×C₇₉	C₄⋊C₄	1	4	yes	—
79	C₄²×C₇₉	C₄⋊C₄	1	4	yes	—
80	C₄²×C₇₉	C₂×D₄	1	2	yes	—
81	C₄²×C₇₉	C₂×D₄	1	2	yes	—
82	C₄²×C₇₉	C₂×Q₈	1	2	yes	—
83	C₄²×C₇₉	C₂×Q₈	1	2	yes	—
84	C₂³×C₇₉	—	2	1	no	—
85	C₂³×C₇₉	—	2	1	no	—
86	C₂³×C₇₉	—	2	2	no	—
87	C₂³×C₇₉	—	2	2	no	—
88	C₂³×C₇₉	C₂×C₄⋊C₄	1	2	yes	—
89	C₂³×C₇₉	C₂³×C₄	1	1	yes	characteristic
90	C₂×C₄×C₇₉	—	2	1	no	—
91	C₂×C₄×C₇₉	—	2	1	no	—
92	C₂×C₄×C₇₉	—	2	2	no	—
93	C₂×C₄×C₇₉	—	2	2	no	—
94	C₂×C₄×C₇₉	—	2	2	no	—
95	C₂×C₄×C₇₉	—	2	4	no	—
96	C₂×C₄×C₇₉	—	2	4	no	—
97	C₂×C₄×C₇₉	—	2	8	no	—
98	C₂×C₄×C₇₉	—	2	8	no	—
99	C₂×C₄×C₇₉	—	2	8	no	—
100	C₂×C₄×C₇₉	—	2	8	no	—
101	C₂×C₄×C₇₉	—	4	4	no	—
102	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	2	yes	—
103	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	2	yes	—
104	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	2	yes	—
105	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	2	yes	—
106	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	2	yes	—
107	C₂×C₄×C₇₉	#131	1	4	yes	—
108	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
109	C₂×C₄×C₇₉	C₂²×Q₈	1	1	yes	characteristic
110	C₂×C₄×C₇₉	H(5,𝔽₂)	1	2	yes	—
111	C₂×C₄×C₇₉	2₋¹⁺⁴	1	2	yes	—
112	C₇₉⋊C₈	—	4	4	no	—
113	C₇₉⋊C₈	—	4	4	no	—
114	C₂²×C₇₉	—	2	2	no	—
115	C₂²×C₇₉	—	2	2	no	—
116	C₂²×C₇₉	—	2	2	no	—
117	C₂²×C₇₉	—	4	2	no	—
118	C₂²×C₇₉	—	4	2	no	—
119	C₂²×C₇₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
120	C₂²×C₇₉	#518	1	1	yes	characteristic
121	C₂²×C₇₉	#519	1	1	yes	characteristic
122	C₂²×C₇₉	C₂×#131	1	2	yes	—
123	C₂²×C₇₉	#568	1	2	yes	—
124	C₄×C₇₉	—	2	2	no	—
125	C₄×C₇₉	—	2	2	no	—
126	C₄×C₇₉	—	2	2	no	—
127	C₄×C₇₉	—	2	2	no	—
128	C₄×C₇₉	—	4	8	no	—
129	C₄×C₇₉	#424	1	4	yes	—
130	C₄×C₇₉	#537	1	2	yes	—
131	C₄×C₇₉	#544	1	2	yes	—
132	#21954	—	79	1	no	maximal, Sylow
133	C₂×C₇₉	—	4	2	no	—
134	C₂×C₇₉	#2316	1	1	yes	characteristic
135	C₂×C₇₉	C₂×#424	1	1	yes	characteristic
136	C₂×C₇₉	#2905	1	1	yes	characteristic
137	C₂×C₇₉	#2981	1	2	yes	—
138	C₂×C₇₉	#2985	1	2	yes	—
139	C₂×#432	—	79	1	no	—
140	C₂×#431	—	79	1	no	—
141	#2285	D₇₉	1	1	yes	characteristic
142	#1940	—	79	4	no	—
143	#1939	—	79	4	no	—
144	#1798	—	79	4	no	—
145	C₇₉	#21954	1	1	yes	characteristic, complemented, Sylow
146	C₂×#117	C₂×D₇₉	1	2	yes	—
147	C₂²×C₄²	C₂×D₇₉	1	1	yes	characteristic
148	#432	—	158	4	no	—
149	#431	—	158	4	no	—
150	C₂×C₄⋊C₈	—	79	8	no	—
151	C₂×C₄⋊C₈	—	79	2	no	—
152	C₂×C₄⋊C₈	—	79	2	no	—
153	C₂×C₂²⋊C₈	—	79	2	no	—
154	C₂×C₂²⋊C₈	—	79	2	no	—
155	C₂×#97	—	79	2	no	—
156	C₂×C₄×C₈	—	79	2	no	—
157	#384	Dic₇₉	1	2	yes	—
158	#382	Dic₇₉	1	2	yes	—
159	#380	Dic₇₉	1	2	yes	—
160	#380	Dic₇₉	1	2	yes	—
161	C₄×C₄⋊C₄	C₂×D₇₉	1	2	yes	—
162	C₄×C₄⋊C₄	C₂×D₇₉	1	2	yes	—
163	#336	—	79	8	no	—
164	C₂³×C₄	C₂×Dic₇₉	1	1	yes	characteristic
165	C₂³×C₄	C₂×Dic₇₉	1	1	yes	characteristic
166	C₂³×C₄	C₂²×D₇₉	1	1	yes	characteristic
167	C₂²×C₈	—	79	4	no	—
168	C₂²×C₈	—	79	4	no	—
169	#117	—	4	4	no	—
170	C₂×C₄⋊C₄	C₂×Dic₇₉	1	4	yes	—
171	C₂×C₄⋊C₄	C₂×Dic₇₉	1	4	yes	—
172	C₂×C₄⋊C₄	C₂×Dic₇₉	1	4	yes	—
173	C₂×C₂²⋊C₄	C₂×Dic₇₉	1	4	yes	—
174	C₂×C₄²	Dic₁₅₈	1	4	yes	—
175	C₂×C₄²	D₃₁₆	1	4	yes	—
176	C₂×C₄²	C₂×Dic₇₉	1	1	yes	characteristic
177	C₂×C₄²	C₂×Dic₇₉	1	1	yes	characteristic
178	C₂×C₄²	C₂²×D₇₉	1	4	yes	—
179	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
180	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
181	C₄⋊C₈	—	316	8	no	—
182	C₄⋊C₈	—	158	4	no	—
183	C₄⋊C₈	—	158	4	no	—
184	C₂²⋊C₈	—	158	4	no	—
185	C₂²⋊C₈	—	158	4	no	—
186	#97	—	158	4	no	—
187	C₄×C₈	—	158	4	no	—
188	#95	C₂×Dic₇₉	1	4	yes	—
189	#95	C₂×Dic₇₉	1	4	yes	—
190	C₂⁴	C₂²×Dic₇₉	1	1	yes	characteristic
191	C₂²×C₄	—	2	2	no	—
192	C₂²×C₄	—	2	2	no	—
193	C₂²×C₄	—	2	4	no	—
194	C₂²×C₄	—	4	2	no	—
195	C₂²×C₄	C₄⋊Dic₇₉	1	4	yes	—
196	C₂²×C₄	C₄⋊Dic₇₉	1	4	yes	—
197	C₂²×C₄	C₂²×Dic₇₉	1	8	yes	—
198	C₂²×C₄	C₂²×Dic₇₉	1	1	yes	characteristic
199	C₂²×C₄	C₂²×Dic₇₉	1	1	yes	characteristic
200	C₂²×C₄	C₂²×Dic₇₉	1	1	yes	characteristic
201	C₂²×C₄	C₂²×Dic₇₉	1	1	yes	characteristic
202	C₂²×C₄	C₂²×Dic₇₉	1	1	yes	characteristic
203	C₂²×C₄	C₂×D₃₁₆	1	2	yes	—
204	C₂²×C₄	C₂×Dic₁₅₈	1	2	yes	—
205	C₂²×C₄	C₂³×D₇₉	1	1	yes	characteristic
206	C₂×C₈	—	158	4	no	—
207	C₂×C₈	—	158	4	no	—
208	C₂×C₈	—	158	4	no	—
209	C₂×C₈	—	158	4	no	—
210	C₂×C₈	—	158	4	no	—
211	C₂×C₈	—	158	4	no	—
212	C₄⋊C₄	—	2	8	no	—
213	C₄⋊C₄	—	4	4	no	—
214	C₄⋊C₄	—	4	4	no	—
215	C₂²⋊C₄	—	4	4	no	—
216	C₄²	—	2	8	no	—
217	C₄²	C₄⋊Dic₇₉	1	4	yes	—
218	C₄²	C₄⋊Dic₇₉	1	4	yes	—
219	C₄²	C₂×D₃₁₆	1	2	yes	—
220	C₄²	C₂×D₃₁₆	1	2	yes	—
221	C₄²	C₂×Dic₁₅₈	1	2	yes	—
222	C₄²	C₂×Dic₁₅₈	1	2	yes	—
223	C₂³	—	2	1	no	—
224	C₂³	—	2	1	no	—
225	C₂³	—	2	2	no	—
226	C₂³	—	2	2	no	—
227	C₂³	C₂×C₄⋊Dic₇₉	1	2	yes	—
228	C₂³	C₂³×Dic₇₉	1	1	yes	characteristic
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	4	no	—
235	C₂×C₄	—	2	4	no	—
236	C₂×C₄	—	2	8	no	—
237	C₂×C₄	—	2	8	no	—
238	C₂×C₄	—	2	8	no	—
239	C₂×C₄	—	2	8	no	—
240	C₂×C₄	—	4	4	no	—
241	C₂×C₄	C₂×C₄⋊Dic₇₉	1	2	yes	—
242	C₂×C₄	C₂×C₄⋊Dic₇₉	1	2	yes	—
243	C₂×C₄	C₂×C₄⋊Dic₇₉	1	2	yes	—
244	C₂×C₄	C₂×C₄⋊Dic₇₉	1	2	yes	—
245	C₂×C₄	C₂×C₄⋊Dic₇₉	1	2	yes	—
246	C₂×C₄	#14579168	1	4	yes	—
247	C₂×C₄	C₂²×D₃₁₆	1	1	yes	characteristic
248	C₂×C₄	C₂²×Dic₁₅₈	1	1	yes	characteristic
249	C₂×C₄	#14579206	1	2	yes	—
250	C₂×C₄	#14579209	1	2	yes	—
251	C₈	—	316	4	no	—
252	C₈	—	316	4	no	—
253	C₂²	—	2	2	no	—
254	C₂²	—	2	2	no	—
255	C₂²	—	2	2	no	—
256	C₂²	—	4	2	no	—
257	C₂²	—	4	2	no	—
258	C₂²	C₂²×C₄⋊Dic₇₉	1	1	yes	characteristic
259	C₂²	#19099038	1	1	yes	characteristic
260	C₂²	#19099041	1	1	yes	characteristic
261	C₂²	C₂×#14579168	1	2	yes	—
262	C₂²	#19099330	1	2	yes	—
263	C₄	—	2	2	no	—
264	C₄	—	2	2	no	—
265	C₄	—	2	2	no	—
266	C₄	—	2	2	no	—
267	C₄	—	4	8	no	—
268	C₄	#19098590	1	4	yes	—
269	C₄	#19099144	1	2	yes	—
270	C₄	#19099194	1	2	yes	—
271	C₂	—	4	2	no	—
272	C₂	#23766018	1	1	yes	characteristic
273	C₂	C₂×#19098590	1	1	yes	characteristic
274	C₂	#23771964	1	1	yes	characteristic
275	C₂	#23772476	1	2	yes	—
276	C₂	#23772495	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₇₉
commutator, socle	C₂³×C₇₉	C₂³×C₄
Fitting	C₇₉×#2285	C₂
prime radical	C₂⁴×C₇₉	C₂²×C₄
cosocle	C₂²×C₄×C₇₉	C₂⁴
hypercenter	#2285	D₇₉
hypocenter	C₇₉	#21954
LCS	G ▷ C₂³×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂²×C₄ ◁ #2285

Complements

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

N	H ≅ G/N	type
3. #23764124	C₂	semidirect
4. #23764119	C₂	semidirect
145. C₇₉	#21954	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19098620	2
C₂×#19098616	2
#23764124	2
#23764119	2
#23763384	2
C₇₉×#2285	2
#21954	79