Subgroups — #34187309

Order 20224.

478 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23763517	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊(C₇₉⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	C₂×#19098312	C₂	1	2	yes	maximal
4	C₂×#19098309	C₂	1	2	yes	maximal
5	C₇₉×#2862	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊(C₇₉⋊C₈)	C₂²	1	2	yes	—
7	C₂×C₄⋊(C₇₉⋊C₈)	C₂²	1	2	yes	—
8	C₂×C₄×C₇₉⋊C₈	—	2	1	no	—
9	#19098318	—	2	1	no	—
10	#19098318	—	2	2	no	—
11	#19098312	—	2	4	no	—
12	#19098309	—	2	4	no	—
13	C₂×C₄⋊Q₈×C₇₉	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊Q₈×C₇₉	C₂²	1	1	yes	characteristic
15	#391×C₇₉	—	2	2	no	—
16	#391×C₇₉	C₄	1	4	yes	—
17	#382×C₇₉	—	2	1	no	—
18	#382×C₇₉	—	2	1	no	—
19	C₄³×C₇₉	C₂²	1	1	yes	characteristic
20	C₂²×C₇₉⋊C₈	—	2	2	no	—
21	C₂²×C₇₉⋊C₈	—	2	2	no	—
22	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
23	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
24	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
25	C₄⋊(C₇₉⋊C₈)	D₄	1	4	yes	—
26	C₄×C₇₉⋊C₈	—	4	2	no	—
27	C₂²×Q₈×C₇₉	—	2	1	no	—
28	C₂²×Q₈×C₇₉	C₂×C₄	1	2	yes	—
29	C₄⋊Q₈×C₇₉	—	2	2	no	—
30	C₄⋊Q₈×C₇₉	—	2	2	no	—
31	C₄⋊Q₈×C₇₉	—	2	2	no	—
32	C₄⋊Q₈×C₇₉	—	2	2	no	—
33	C₂×C₄⋊C₄×C₇₉	—	2	1	no	—
34	C₂×C₄⋊C₄×C₇₉	—	2	1	no	—
35	C₂×C₄⋊C₄×C₇₉	—	2	1	no	—
36	C₂×C₄⋊C₄×C₇₉	C₂×C₄	1	2	yes	—
37	C₂×C₄²×C₇₉	C₂×C₄	1	2	yes	—
38	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₇₉	D₄	1	2	yes	—
40	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
41	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
42	#95×C₇₉	—	2	4	no	—
43	#95×C₇₉	—	2	4	no	—
44	C₂×C₇₉⋊C₈	—	2	4	no	—
45	C₂×C₇₉⋊C₈	—	4	2	no	—
46	C₂×C₇₉⋊C₈	—	4	2	no	—
47	C₂×C₇₉⋊C₈	—	4	4	no	—
48	C₂×C₇₉⋊C₈	—	4	2	no	—
49	C₂×Q₈×C₇₉	—	2	4	no	—
50	C₂×Q₈×C₇₉	—	4	4	no	—
51	C₂×Q₈×C₇₉	—	4	1	no	—
52	C₂×Q₈×C₇₉	—	4	1	no	—
53	C₂×Q₈×C₇₉	—	4	4	no	—
54	C₂²×C₄×C₇₉	—	2	2	no	—
55	C₂²×C₄×C₇₉	—	2	2	no	—
56	C₂²×C₄×C₇₉	C₂²⋊C₄	1	2	yes	—
57	C₂²×C₄×C₇₉	C₂²⋊C₄	1	2	yes	—
58	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
59	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
61	C₄⋊C₄×C₇₉	—	2	4	no	—
62	C₄⋊C₄×C₇₉	—	4	2	no	—
63	C₄⋊C₄×C₇₉	—	4	2	no	—
64	C₄⋊C₄×C₇₉	—	4	1	no	—
65	C₄⋊C₄×C₇₉	—	4	1	no	—
66	C₄²×C₇₉	—	2	4	no	—
67	C₄²×C₇₉	—	2	2	no	—
68	C₄²×C₇₉	—	4	2	no	—
69	C₄²×C₇₉	SD₁₆	1	2	yes	—
70	C₄²×C₇₉	Q₁₆	1	2	yes	—
71	C₄²×C₇₉	C₂×D₄	1	2	yes	—
72	C₄²×C₇₉	#41	1	2	yes	—
73	C₂³×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
74	Q₈×C₇₉	—	8	4	no	—
75	Q₈×C₇₉	—	8	2	no	—
76	Q₈×C₇₉	—	8	2	no	—
77	C₂×C₄×C₇₉	—	2	2	no	—
78	C₂×C₄×C₇₉	—	2	2	no	—
79	C₂×C₄×C₇₉	—	2	2	no	—
80	C₂×C₄×C₇₉	—	2	2	no	—
81	C₂×C₄×C₇₉	—	2	2	no	—
82	C₂×C₄×C₇₉	—	2	2	no	—
83	C₂×C₄×C₇₉	—	4	2	no	—
84	C₂×C₄×C₇₉	—	4	2	no	—
85	C₂×C₄×C₇₉	—	4	4	no	—
86	C₂×C₄×C₇₉	—	4	4	no	—
87	C₂×C₄×C₇₉	#103	1	4	yes	—
88	C₂×C₄×C₇₉	C₄×D₄	1	2	yes	—
89	C₂×C₄×C₇₉	#121	1	2	yes	—
90	C₂×C₄×C₇₉	#124	1	1	yes	characteristic
91	C₂×C₄×C₇₉	#127	1	1	yes	characteristic
92	C₂×C₄×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
93	C₂×C₄×C₇₉	C₂×Q₁₆	1	1	yes	characteristic
94	C₂×C₄×C₇₉	#135	1	4	yes	—
95	C₂×C₄×C₇₉	#137	1	2	yes	—
96	C₇₉⋊C₈	—	4	4	no	—
97	C₇₉⋊C₈	—	8	2	no	—
98	C₂²×C₇₉	#390	1	1	yes	characteristic
99	C₂²×C₇₉	#413	1	2	yes	—
100	C₂²×C₇₉	C₂×#103	1	1	yes	characteristic
101	C₂²×C₇₉	#416	1	2	yes	—
102	C₂²×C₇₉	#419	1	1	yes	characteristic
103	C₄×C₇₉	—	2	4	no	—
104	C₄×C₇₉	—	2	2	no	—
105	C₄×C₇₉	—	4	2	no	—
106	C₄×C₇₉	—	8	2	no	—
107	C₄×C₇₉	—	8	2	no	—
108	C₄×C₇₉	#461	1	2	yes	—
109	C₄×C₇₉	#462	1	2	yes	—
110	C₄×C₇₉	#488	1	2	yes	—
111	C₄×C₇₉	#495	1	2	yes	—
112	#9693	—	79	1	no	maximal, Sylow
113	C₂×C₇₉	#1542	1	2	yes	—
114	C₂×C₇₉	#1554	1	2	yes	—
115	C₂×C₇₉	#1958	1	1	yes	characteristic
116	C₂×C₇₉	#1967	1	1	yes	characteristic
117	C₂×C₇₉	#1981	1	1	yes	characteristic
118	#2862	D₇₉	1	1	yes	characteristic
119	#1831	—	79	1	no	—
120	C₄×C₄⋊C₈	—	79	1	no	—
121	C₂×#333	—	79	2	no	—
122	C₂×#332	—	79	2	no	—
123	C₇₉	#9693	1	1	yes	characteristic, complemented, Sylow
124	C₂×C₄⋊Q₈	C₂×D₇₉	1	1	yes	characteristic
125	C₂×C₄⋊Q₈	C₂×D₇₉	1	1	yes	characteristic
126	C₂×C₄⋊C₈	—	79	2	no	—
127	C₂×C₄⋊C₈	—	79	2	no	—
128	C₂×C₄×C₈	—	158	1	no	—
129	#391	—	2	2	no	—
130	#391	Dic₇₉	1	4	yes	—
131	#382	—	2	1	no	—
132	#382	—	2	1	no	—
133	C₄³	C₂×D₇₉	1	1	yes	characteristic
134	#336	—	158	2	no	—
135	#336	—	158	1	no	—
136	#333	—	158	4	no	—
137	#332	—	158	4	no	—
138	C₂²×Q₈	—	2	1	no	—
139	C₂²×Q₈	C₂×Dic₇₉	1	2	yes	—
140	C₂²×C₈	—	158	2	no	—
141	C₂²×C₈	—	158	2	no	—
142	C₄⋊Q₈	—	2	2	no	—
143	C₄⋊Q₈	—	2	2	no	—
144	C₄⋊Q₈	—	2	2	no	—
145	C₄⋊Q₈	—	2	2	no	—
146	C₂×C₄⋊C₄	—	2	1	no	—
147	C₂×C₄⋊C₄	—	2	1	no	—
148	C₂×C₄⋊C₄	—	2	1	no	—
149	C₂×C₄⋊C₄	C₂×Dic₇₉	1	2	yes	—
150	C₂×C₄²	C₂×Dic₇₉	1	2	yes	—
151	C₂×C₄²	C₂²⋊D₇₉	1	1	yes	characteristic
152	C₂×C₄²	C₂²⋊D₇₉	1	2	yes	—
153	C₂×C₄²	C₂²⋊D₇₉	1	1	yes	characteristic
154	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
155	C₄⋊C₈	—	79	4	no	—
156	C₄⋊C₈	—	158	2	no	—
157	C₄⋊C₈	—	158	2	no	—
158	C₄⋊C₈	—	158	2	no	—
159	C₄×C₈	—	316	2	no	—
160	#95	—	2	4	no	—
161	#95	—	2	4	no	—
162	C₂×Q₈	—	2	4	no	—
163	C₂×Q₈	—	4	4	no	—
164	C₂×Q₈	—	4	1	no	—
165	C₂×Q₈	—	4	1	no	—
166	C₂×Q₈	—	4	4	no	—
167	C₂²×C₄	—	2	2	no	—
168	C₂²×C₄	—	2	2	no	—
169	C₂²×C₄	C₂²⋊Dic₇₉	1	2	yes	—
170	C₂²×C₄	C₂²⋊Dic₇₉	1	2	yes	—
171	C₂²×C₄	C₂²×Dic₇₉	1	1	yes	characteristic
172	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
173	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
174	C₂×C₈	—	316	2	no	—
175	C₂×C₈	—	316	2	no	—
176	C₂×C₈	—	316	2	no	—
177	C₂×C₈	—	316	4	no	—
178	C₂×C₈	—	158	4	no	—
179	C₄⋊C₄	—	2	4	no	—
180	C₄⋊C₄	—	4	2	no	—
181	C₄⋊C₄	—	4	2	no	—
182	C₄⋊C₄	—	4	1	no	—
183	C₄⋊C₄	—	4	1	no	—
184	C₄²	—	2	4	no	—
185	C₄²	—	2	2	no	—
186	C₄²	—	4	2	no	—
187	C₄²	#11929611	1	2	yes	—
188	C₄²	#11929614	1	2	yes	—
189	C₄²	D₄×D₇₉	1	2	yes	—
190	C₄²	#11929623	1	2	yes	—
191	C₂³	C₂×C₂²⋊Dic₇₉	1	1	yes	characteristic
192	Q₈	—	8	4	no	—
193	Q₈	—	8	2	no	—
194	Q₈	—	8	2	no	—
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	4	2	no	—
202	C₂×C₄	—	4	2	no	—
203	C₂×C₄	—	4	4	no	—
204	C₂×C₄	—	4	4	no	—
205	C₂×C₄	#14579085	1	4	yes	—
206	C₂×C₄	D₄×Dic₇₉	1	2	yes	—
207	C₂×C₄	#14579132	1	2	yes	—
208	C₂×C₄	#14579148	1	1	yes	characteristic
209	C₂×C₄	#14579157	1	1	yes	characteristic
210	C₂×C₄	C₂×#11929611	1	1	yes	characteristic
211	C₂×C₄	C₂×#11929614	1	1	yes	characteristic
212	C₂×C₄	#14579181	1	4	yes	—
213	C₂×C₄	#14579192	1	2	yes	—
214	C₈	—	316	4	no	—
215	C₈	—	632	2	no	—
216	C₂²	#19098466	1	1	yes	characteristic
217	C₂²	#19098547	1	2	yes	—
218	C₂²	C₂×#14579085	1	1	yes	characteristic
219	C₂²	#19098559	1	2	yes	—
220	C₂²	#19098572	1	1	yes	characteristic
221	C₄	—	2	2	no	—
222	C₄	—	2	4	no	—
223	C₄	—	4	2	no	—
224	C₄	—	8	2	no	—
225	C₄	—	8	2	no	—
226	C₄	#19098765	1	2	yes	—
227	C₄	#19098770	1	2	yes	—
228	C₄	#19098917	1	2	yes	—
229	C₄	#19098945	1	2	yes	—
230	C₂	#23762057	1	2	yes	—
231	C₂	#23762110	1	2	yes	—
232	C₂	#23764212	1	1	yes	characteristic
233	C₂	#23764250	1	1	yes	characteristic
234	C₂	#23764324	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₇₉
commutator	C₂²×C₄×C₇₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₇₉×#2862	C₂
socle, prime radical	C₂³×C₇₉	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	#2862	D₇₉
hypocenter	C₇₉	#9693
LCS	G ▷ C₂²×C₄×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #2862

Complements

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

N	H ≅ G/N	type
123. C₇₉	#9693	semidirect

Maximal subgroups

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

subgroup	index
#23763517	2
C₄×C₄⋊(C₇₉⋊C₈)	2
C₂×#19098312	2
C₂×#19098309	2
C₇₉×#2862	2
#9693	79