Subgroups — #34186841

Order 20224.

378 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23764118	C₂	1	2	yes	maximal
2	C₄×#14579093	C₂	1	1	yes	maximal, characteristic
3	C₂×#19098316	C₂	1	1	yes	maximal, characteristic
4	#23761240	C₂	1	2	yes	maximal
5	C₂×#322×C₇₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#14579093	C₄	1	2	yes	—
7	C₂×C₄⋊(C₇₉⋊C₈)	C₂²	1	2	yes	—
8	C₂×Dic₇₉⋊C₈	—	2	2	no	—
9	C₄×C₄⋊Dic₇₉	C₂²	1	2	yes	—
10	#19098326	C₄	1	1	yes	characteristic
11	#19098326	C₄	1	1	yes	characteristic
12	#19098317	—	2	2	no	—
13	#19098316	—	2	2	no	—
14	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
17	#322×C₇₉	—	2	2	no	—
18	C₂²×C₇₉⋊C₈	—	2	2	no	—
19	C₂×C₄⋊Dic₇₉	C₂×C₄	1	2	yes	—
20	C₂×C₄⋊Dic₇₉	C₂×C₄	1	2	yes	—
21	C₂×C₄×Dic₇₉	—	2	2	no	—
22	#14579093	—	2	4	no	—
23	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
24	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
25	Dic₇₉⋊C₈	—	4	4	no	—
26	C₇₉⋊#95	—	2	2	no	—
27	C₂²×C₈×C₇₉	—	2	1	no	—
28	C₂²×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
31	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
32	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
33	C₄×C₈×C₇₉	—	2	1	no	—
34	C₄×C₈×C₇₉	—	2	1	no	—
35	C₄×C₈×C₇₉	—	2	1	no	—
36	C₄×C₈×C₇₉	D₄	1	2	yes	—
37	C₄×C₈×C₇₉	Q₈	1	2	yes	—
38	C₄×C₈×C₇₉	Q₈	1	2	yes	—
39	C₂²×Dic₇₉	—	2	2	no	—
40	C₂²×Dic₇₉	—	2	2	no	—
41	C₂×C₇₉⋊C₈	—	4	2	no	—
42	C₂×C₇₉⋊C₈	—	4	2	no	—
43	C₂×C₇₉⋊C₈	—	4	2	no	—
44	C₄⋊Dic₇₉	—	2	2	no	—
45	C₄⋊Dic₇₉	—	2	2	no	—
46	C₄⋊Dic₇₉	—	2	2	no	—
47	C₄⋊Dic₇₉	M₁₆	1	4	yes	—
48	C₄×Dic₇₉	—	4	2	no	—
49	C₄×Dic₇₉	—	4	2	no	—
50	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
51	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
52	C₂²×C₄×C₇₉	#41	1	1	yes	characteristic
53	C₂×C₈×C₇₉	—	2	2	no	—
54	C₂×C₈×C₇₉	—	2	2	no	—
55	C₂×C₈×C₇₉	—	2	1	no	—
56	C₂×C₈×C₇₉	—	2	1	no	—
57	C₂×C₈×C₇₉	—	2	2	no	—
58	C₂×C₈×C₇₉	—	2	2	no	—
59	C₂×C₈×C₇₉	—	2	2	no	—
60	C₂×C₈×C₇₉	—	2	2	no	—
61	C₂×C₈×C₇₉	—	2	2	no	—
62	C₂×C₈×C₇₉	C₄⋊C₄	1	2	yes	—
63	C₂×C₈×C₇₉	C₄⋊C₄	1	2	yes	—
64	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
65	C₄²×C₇₉	C₂×Q₈	1	1	yes	characteristic
66	C₄²×C₇₉	C₂×Q₈	1	1	yes	characteristic
67	C₄²×C₇₉	#41	1	1	yes	characteristic
68	C₂³×C₇₉	C₄×D₄	1	1	yes	characteristic
69	C₂×C₄×C₇₉	—	2	1	no	—
70	C₂×C₄×C₇₉	—	2	1	no	—
71	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
72	C₂×C₄×C₇₉	C₄×Q₈	1	1	yes	characteristic
73	C₂×C₄×C₇₉	#121	1	1	yes	characteristic
74	C₂×C₄×C₇₉	C₂²⋊Q₈	1	1	yes	characteristic
75	C₂×C₄×C₇₉	#125	1	1	yes	characteristic
76	C₂×C₄×C₇₉	C₄⋊Q₈	1	1	yes	characteristic
77	C₂×C₄×C₇₉	C₂×M₁₆	1	2	yes	—
78	C₂×C₄×C₇₉	#131	1	2	yes	—
79	C₂×C₄×C₇₉	AGL(1,ℤ/8)	1	2	yes	—
80	C₂×C₄×C₇₉	#137	1	2	yes	—
81	C₂×Dic₇₉	—	2	4	no	—
82	C₂×Dic₇₉	—	4	2	no	—
83	C₂×Dic₇₉	—	4	2	no	—
84	C₂×Dic₇₉	—	4	2	no	—
85	C₂×Dic₇₉	—	4	2	no	—
86	C₂×Dic₇₉	—	4	2	no	—
87	C₈×C₇₉	—	2	4	no	—
88	C₈×C₇₉	—	2	2	no	—
89	C₈×C₇₉	—	2	2	no	—
90	C₈×C₇₉	—	4	1	no	—
91	C₈×C₇₉	—	4	1	no	—
92	C₇₉⋊C₈	—	8	2	no	—
93	C₂²×C₇₉	#389	1	1	yes	characteristic
94	C₂²×C₇₉	#435	1	2	yes	—
95	C₂²×C₇₉	#440	1	2	yes	—
96	C₂²×C₇₉	#443	1	1	yes	characteristic
97	C₂²×C₇₉	#443	1	1	yes	characteristic
98	C₄×C₇₉	—	2	1	no	—
99	C₄×C₇₉	—	2	1	no	—
100	C₄×C₇₉	#423	1	1	yes	characteristic
101	C₄×C₇₉	#424	1	1	yes	characteristic
102	C₄×C₇₉	#446	1	2	yes	—
103	C₄×C₇₉	#469	1	1	yes	characteristic
104	C₄×C₇₉	#470	1	1	yes	characteristic
105	C₄×C₇₉	#501	1	2	yes	—
106	Dic₇₉	—	4	4	no	—
107	Dic₇₉	—	8	2	no	—
108	#9600	—	79	1	no	maximal, Sylow
109	C₂×C₇₉	#1585	1	2	yes	—
110	C₂×C₇₉	#1588	1	2	yes	—
111	C₂×C₇₉	#1940	1	1	yes	characteristic
112	C₂×C₇₉	#1944	1	1	yes	characteristic
113	C₂×C₇₉	#1945	1	1	yes	characteristic
114	#1939	—	79	2	no	—
115	C₄×#106	—	79	1	no	—
116	C₂×#335	—	79	1	no	—
117	C₂×#322	D₇₉	1	1	yes	characteristic
118	#1279	—	79	2	no	—
119	C₇₉	#9600	1	1	yes	characteristic, complemented, Sylow
120	C₂×#106	—	79	2	no	—
121	C₂×C₄⋊C₈	—	79	2	no	—
122	C₂×C₄⋊C₈	—	158	2	no	—
123	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
124	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
125	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
126	C₄×C₄⋊C₄	—	79	2	no	—
127	#340	—	79	1	no	—
128	#340	—	79	1	no	—
129	#336	—	158	2	no	—
130	#335	—	158	2	no	—
131	#322	—	2	2	no	—
132	C₂²×C₈	—	2	1	no	—
133	C₂²×C₈	—	158	2	no	—
134	C₂²×C₈	C₄×D₇₉	1	1	yes	characteristic
135	C₂²×C₈	C₄×D₇₉	1	1	yes	characteristic
136	C₂²×C₈	D₃₁₆	1	1	yes	characteristic
137	C₂²×C₈	D₃₁₆	1	1	yes	characteristic
138	C₂×C₄⋊C₄	—	79	2	no	—
139	C₂×C₄⋊C₄	—	79	2	no	—
140	C₂×C₄²	—	158	2	no	—
141	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
142	#106	—	158	4	no	—
143	C₄⋊C₈	—	316	4	no	—
144	C₄⋊C₈	—	158	2	no	—
145	C₄⋊C₈	—	158	2	no	—
146	C₄×C₈	—	2	1	no	—
147	C₄×C₈	—	2	1	no	—
148	C₄×C₈	—	2	1	no	—
149	C₄×C₈	Dic₁₅₈	1	2	yes	—
150	C₄×C₈	Dic₁₅₈	1	2	yes	—
151	C₄×C₈	C₂²⋊D₇₉	1	2	yes	—
152	#95	—	158	2	no	—
153	C₂²×C₄	—	158	2	no	—
154	C₂²×C₄	—	158	2	no	—
155	C₂²×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
156	C₂²×C₄	C₂×D₃₁₆	1	1	yes	characteristic
157	C₂²×C₄	#11929624	1	1	yes	characteristic
158	C₂×C₈	—	2	2	no	—
159	C₂×C₈	—	2	2	no	—
160	C₂×C₈	—	2	2	no	—
161	C₂×C₈	—	2	2	no	—
162	C₂×C₈	—	2	2	no	—
163	C₂×C₈	—	2	1	no	—
164	C₂×C₈	—	2	1	no	—
165	C₂×C₈	—	2	2	no	—
166	C₂×C₈	—	2	2	no	—
167	C₂×C₈	—	316	2	no	—
168	C₂×C₈	—	316	2	no	—
169	C₂×C₈	—	316	2	no	—
170	C₂×C₈	Dic₇₉⋊C₄	1	2	yes	—
171	C₂×C₈	Dic₇₉⋊C₄	1	2	yes	—
172	C₄⋊C₄	—	79	4	no	—
173	C₄⋊C₄	—	158	2	no	—
174	C₄⋊C₄	—	158	2	no	—
175	C₄⋊C₄	—	158	2	no	—
176	C₄²	—	316	2	no	—
177	C₄²	—	316	2	no	—
178	C₄²	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
179	C₄²	C₂×Dic₁₅₈	1	1	yes	characteristic
180	C₄²	C₂×Dic₁₅₈	1	1	yes	characteristic
181	C₄²	#11929624	1	1	yes	characteristic
182	C₂³	C₄×D₃₁₆	1	1	yes	characteristic
183	C₂×C₄	—	2	1	no	—
184	C₂×C₄	—	2	1	no	—
185	C₂×C₄	—	316	2	no	—
186	C₂×C₄	—	316	2	no	—
187	C₂×C₄	—	316	2	no	—
188	C₂×C₄	—	316	2	no	—
189	C₂×C₄	—	316	2	no	—
190	C₂×C₄	—	158	4	no	—
191	C₂×C₄	C₂×Dic₇₉⋊C₄	1	1	yes	characteristic
192	C₂×C₄	C₄×Dic₁₅₈	1	1	yes	characteristic
193	C₂×C₄	#14579135	1	1	yes	characteristic
194	C₂×C₄	#14579140	1	1	yes	characteristic
195	C₂×C₄	#14579153	1	1	yes	characteristic
196	C₂×C₄	#14579161	1	1	yes	characteristic
197	C₂×C₄	M₁₆×D₇₉	1	2	yes	—
198	C₂×C₄	#14579169	1	2	yes	—
199	C₂×C₄	#14579189	1	2	yes	—
200	C₂×C₄	#14579194	1	2	yes	—
201	C₈	—	2	4	no	—
202	C₈	—	2	2	no	—
203	C₈	—	2	2	no	—
204	C₈	—	4	1	no	—
205	C₈	—	4	1	no	—
206	C₈	—	632	2	no	—
207	C₂²	#19098461	1	1	yes	characteristic
208	C₂²	#19098636	1	2	yes	—
209	C₂²	#19098665	1	2	yes	—
210	C₂²	#19098681	1	1	yes	characteristic
211	C₂²	#19098681	1	1	yes	characteristic
212	C₄	—	2	1	no	—
213	C₄	—	2	1	no	—
214	C₄	—	316	4	no	—
215	C₄	—	632	2	no	—
216	C₄	#19098586	1	1	yes	characteristic
217	C₄	#19098591	1	1	yes	characteristic
218	C₄	#19098691	1	2	yes	—
219	C₄	#19098816	1	1	yes	characteristic
220	C₄	#19098821	1	1	yes	characteristic
221	C₄	#19098972	1	2	yes	—
222	C₂	#23762259	1	2	yes	—
223	C₂	#23762280	1	2	yes	—
224	C₂	#23764123	1	1	yes	characteristic
225	C₂	#23764143	1	1	yes	characteristic
226	C₂	#23764148	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₃₁₆
commutator	C₂²×C₄×C₇₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₂×#322×C₇₉	C₂
socle, prime radical	C₂³×C₇₉	C₄×D₄
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	C₂×#322	D₇₉
hypocenter	C₇₉	#9600
LCS	G ▷ C₂²×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#322

Complements

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

N	H ≅ G/N	type
119. C₇₉	#9600	semidirect

Maximal subgroups

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

subgroup	index
#23764118	2
C₄×#14579093	2
C₂×#19098316	2
#23761240	2
C₂×#322×C₇₉	2
#9600	79