Subgroups — #34206526

Order 20224.

470 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19098379	C₂	1	4	yes	maximal
2	C₄×#14579101	C₂	1	1	yes	maximal, characteristic
3	#23763572	C₂	1	1	yes	maximal, characteristic
4	C₂×#494×C₇₉	C₂	1	1	yes	maximal, characteristic
5	C₂×#14579101	C₂²	1	2	yes	—
6	C₂×#14579097	—	2	1	no	—
7	C₂×#14579097	C₂²	1	2	yes	—
8	#19098587	—	2	1	no	—
9	#19098379	—	2	8	no	—
10	#19098339	—	2	1	no	—
11	C₄×C₇₉⋊C₁₆	—	2	1	no	—
12	C₂²×Q₁₆×C₇₉	—	2	1	no	—
13	C₂²×Q₁₆×C₇₉	C₂²	1	2	yes	—
14	C₂×C₄⋊Q₈×C₇₉	—	2	1	no	—
15	#494×C₇₉	—	2	1	no	—
16	#494×C₇₉	—	2	1	no	—
17	#494×C₇₉	C₄	1	2	yes	—
18	#494×C₇₉	C₄	1	2	yes	—
19	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
20	C₂×#11929608	—	2	2	no	—
21	#14579101	—	2	2	no	—
22	#14579101	D₄	1	4	yes	—
23	C₂×C₇₉⋊C₁₆	—	2	2	no	—
24	#14579097	—	2	4	no	—
25	#14579097	—	4	2	no	—
26	C₄⋊(C₇₉⋊C₈)	—	4	2	no	—
27	C₂²×Q₈×C₇₉	—	2	2	no	—
28	C₂×Q₁₆×C₇₉	—	2	2	no	—
29	C₂×Q₁₆×C₇₉	—	2	2	no	—
30	C₂×Q₁₆×C₇₉	—	2	4	no	—
31	C₂×Q₁₆×C₇₉	—	2	2	no	—
32	C₂×Q₁₆×C₇₉	—	2	4	no	—
33	C₂×Q₁₆×C₇₉	—	4	2	no	—
34	C₂×Q₁₆×C₇₉	—	4	2	no	—
35	C₂×Q₁₆×C₇₉	C₂×C₄	1	4	yes	—
36	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
37	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
38	C₄⋊Q₈×C₇₉	—	2	2	no	—
39	C₄⋊Q₈×C₇₉	—	2	2	no	—
40	C₄⋊Q₈×C₇₉	—	4	2	no	—
41	C₂×C₄⋊C₄×C₇₉	—	4	2	no	—
42	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
43	C₄×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
44	C₄×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
45	C₄×C₈×C₇₉	D₄	1	1	yes	characteristic
46	C₄×C₈×C₇₉	D₄	1	1	yes	characteristic
47	#11929608	—	4	2	no	—
48	#11929608	—	4	2	no	—
49	C₂×C₇₉⋊C₈	—	4	2	no	—
50	C₇₉⋊C₁₆	—	2	4	no	—
51	C₂×Q₈×C₇₉	—	2	4	no	—
52	C₂×Q₈×C₇₉	—	4	2	no	—
53	C₂×Q₈×C₇₉	—	4	2	no	—
54	C₂×Q₈×C₇₉	—	4	2	no	—
55	C₂×Q₈×C₇₉	—	4	2	no	—
56	C₂×Q₈×C₇₉	—	4	2	no	—
57	C₂²×C₄×C₇₉	—	4	2	no	—
58	C₂²×C₄×C₇₉	SD₁₆	1	2	yes	—
59	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
60	Q₁₆×C₇₉	—	2	8	no	—
61	Q₁₆×C₇₉	—	4	4	no	—
62	Q₁₆×C₇₉	—	4	4	no	—
63	Q₁₆×C₇₉	—	4	4	no	—
64	C₂×C₈×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
65	C₂×C₈×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
66	C₂×C₈×C₇₉	D₈	1	2	yes	—
67	C₂×C₈×C₇₉	SD₁₆	1	2	yes	—
68	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
70	C₂×C₈×C₇₉	C₂×D₄	1	2	yes	—
71	C₂×C₈×C₇₉	#41	1	2	yes	—
72	C₄⋊C₄×C₇₉	—	4	4	no	—
73	C₄⋊C₄×C₇₉	—	4	4	no	—
74	C₄²×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
75	C₄²×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
76	C₄²×C₇₉	D₈	1	2	yes	—
77	C₂³×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
78	Q₈×C₇₉	—	4	4	no	—
79	Q₈×C₇₉	—	4	4	no	—
80	Q₈×C₇₉	—	8	2	no	—
81	Q₈×C₇₉	—	8	2	no	—
82	C₂×C₄×C₇₉	—	2	4	no	—
83	C₂×C₄×C₇₉	—	4	4	no	—
84	C₂×C₄×C₇₉	—	8	2	no	—
85	C₂×C₄×C₇₉	—	8	2	no	—
86	C₂×C₄×C₇₉	#102	1	2	yes	—
87	C₂×C₄×C₇₉	#102	1	2	yes	—
88	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
89	C₂×C₄×C₇₉	#124	1	1	yes	characteristic
90	C₂×C₄×C₇₉	#127	1	1	yes	characteristic
91	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
92	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
93	C₂×C₄×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
94	C₈×C₇₉	#102	1	2	yes	—
95	C₈×C₇₉	#102	1	2	yes	—
96	C₈×C₇₉	C₄×D₄	1	2	yes	—
97	C₈×C₇₉	#121	1	2	yes	—
98	C₇₉⋊C₈	—	8	2	no	—
99	C₂²×C₇₉	—	2	2	no	—
100	C₂²×C₇₉	C₂×#102	1	1	yes	characteristic
101	C₂²×C₇₉	#486	1	1	yes	characteristic
102	C₂²×C₇₉	#492	1	1	yes	characteristic
103	C₄×C₇₉	—	2	2	no	—
104	C₄×C₇₉	—	8	4	no	—
105	C₄×C₇₉	#362	1	4	yes	complemented
106	C₄×C₇₉	#390	1	1	yes	characteristic
107	C₄×C₇₉	C₂×#102	1	1	yes	characteristic
108	C₄×C₇₉	#486	1	1	yes	characteristic
109	C₄×C₇₉	#493	1	1	yes	characteristic
110	#13212	—	79	1	no	maximal, Sylow
111	C₂×C₇₉	—	2	1	no	—
112	C₂×C₇₉	—	2	1	no	—
113	C₂×C₇₉	#1957	1	1	yes	characteristic
114	C₂×C₇₉	C₂×#362	1	2	yes	—
115	C₂×#494	D₇₉	1	1	yes	characteristic
116	C₂×#362	—	79	4	no	—
117	C₄×M₃₂	—	79	1	no	—
118	#1848	—	79	1	no	—
119	C₇₉	#13212	1	1	yes	characteristic, complemented, Sylow
120	C₂²×Q₁₆	—	2	1	no	—
121	C₂²×Q₁₆	C₂×D₇₉	1	2	yes	—
122	C₂×C₄⋊Q₈	—	2	1	no	—
123	C₂×M₃₂	—	79	2	no	—
124	#494	—	2	1	no	—
125	#494	—	2	1	no	—
126	#494	Dic₇₉	1	2	yes	—
127	#494	Dic₇₉	1	2	yes	—
128	C₂×#108	—	79	2	no	—
129	C₂×#108	—	158	1	no	—
130	#423	—	158	1	no	—
131	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
132	#362	—	158	8	no	—
133	#346	—	158	1	no	—
134	C₄×C₁₆	—	158	1	no	—
135	C₂²×Q₈	—	2	2	no	—
136	C₂×Q₁₆	—	2	2	no	—
137	C₂×Q₁₆	—	2	2	no	—
138	C₂×Q₁₆	—	2	4	no	—
139	C₂×Q₁₆	—	2	2	no	—
140	C₂×Q₁₆	—	2	4	no	—
141	C₂×Q₁₆	—	4	2	no	—
142	C₂×Q₁₆	—	4	2	no	—
143	C₂×Q₁₆	C₂×Dic₇₉	1	4	yes	—
144	C₂×M₁₆	—	158	2	no	—
145	C₂²×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
146	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
147	C₄⋊Q₈	—	2	2	no	—
148	C₄⋊Q₈	—	2	2	no	—
149	C₄⋊Q₈	—	4	2	no	—
150	C₂×C₄⋊C₄	—	4	2	no	—
151	C₂×C₄²	C₂²⋊D₇₉	1	1	yes	characteristic
152	M₃₂	—	79	4	no	—
153	M₃₂	—	158	2	no	—
154	C₂×C₁₆	—	158	2	no	—
155	#108	—	316	2	no	—
156	#108	—	158	4	no	—
157	C₄⋊C₈	—	316	2	no	—
158	C₄×C₈	C₂×Dic₇₉	1	1	yes	characteristic
159	C₄×C₈	C₂×Dic₇₉	1	1	yes	characteristic
160	C₄×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
161	C₄×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
162	C₂×Q₈	—	2	4	no	—
163	C₂×Q₈	—	4	2	no	—
164	C₂×Q₈	—	4	2	no	—
165	C₂×Q₈	—	4	2	no	—
166	C₂×Q₈	—	4	2	no	—
167	C₂×Q₈	—	4	2	no	—
168	C₂²×C₄	—	4	2	no	—
169	C₂²×C₄	#11929612	1	2	yes	—
170	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
171	Q₁₆	—	2	8	no	—
172	Q₁₆	—	4	4	no	—
173	Q₁₆	—	4	4	no	—
174	Q₁₆	—	4	4	no	—
175	M₁₆	—	316	2	no	—
176	M₁₆	—	316	2	no	—
177	C₂×C₈	—	316	2	no	—
178	C₂×C₈	C₂²⋊Dic₇₉	1	1	yes	characteristic
179	C₂×C₈	C₂²⋊Dic₇₉	1	1	yes	characteristic
180	C₂×C₈	#11929609	1	2	yes	—
181	C₂×C₈	#11929612	1	2	yes	—
182	C₂×C₈	C₂²×Dic₇₉	1	1	yes	characteristic
183	C₂×C₈	D₄×D₇₉	1	2	yes	—
184	C₂×C₈	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
185	C₂×C₈	#11929623	1	2	yes	—
186	C₄⋊C₄	—	4	4	no	—
187	C₄⋊C₄	—	4	4	no	—
188	C₄²	C₂²⋊Dic₇₉	1	1	yes	characteristic
189	C₄²	C₂²⋊Dic₇₉	1	1	yes	characteristic
190	C₄²	#11929609	1	2	yes	—
191	C₁₆	—	158	4	no	—
192	C₂³	C₂×#11929612	1	1	yes	characteristic
193	Q₈	—	4	4	no	—
194	Q₈	—	4	4	no	—
195	Q₈	—	8	2	no	—
196	Q₈	—	8	2	no	—
197	C₂×C₄	—	2	4	no	—
198	C₂×C₄	—	4	4	no	—
199	C₂×C₄	—	8	2	no	—
200	C₂×C₄	—	8	2	no	—
201	C₂×C₄	#14579082	1	2	yes	—
202	C₂×C₄	#14579082	1	2	yes	—
203	C₂×C₄	C₂×C₂²⋊Dic₇₉	1	1	yes	characteristic
204	C₂×C₄	#14579148	1	1	yes	characteristic
205	C₂×C₄	#14579157	1	1	yes	characteristic
206	C₂×C₄	C₂×#11929609	1	1	yes	characteristic
207	C₂×C₄	C₂×#11929609	1	1	yes	characteristic
208	C₂×C₄	C₂×#11929612	1	1	yes	characteristic
209	C₈	—	632	2	no	—
210	C₈	#14579082	1	2	yes	—
211	C₈	#14579082	1	2	yes	—
212	C₈	D₄×Dic₇₉	1	2	yes	—
213	C₈	#14579132	1	2	yes	—
214	C₂²	—	2	2	no	—
215	C₂²	C₂×#14579082	1	1	yes	characteristic
216	C₂²	#19098909	1	1	yes	characteristic
217	C₂²	#19098935	1	1	yes	characteristic
218	C₄	—	2	2	no	—
219	C₄	—	8	4	no	—
220	C₄	#19098379	1	4	yes	complemented
221	C₄	#19098466	1	1	yes	characteristic
222	C₄	C₂×#14579082	1	1	yes	characteristic
223	C₄	#19098909	1	1	yes	characteristic
224	C₄	#19098939	1	1	yes	characteristic
225	C₂	—	2	1	no	—
226	C₂	—	2	1	no	—
227	C₂	#23764207	1	1	yes	characteristic
228	C₂	C₂×#19098379	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×#14579082
commutator	C₂×C₈×C₇₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇₉
Fitting	C₂×#494×C₇₉	C₂
socle	C₂²×C₇₉	C₂×#102
prime radical	C₂³×C₇₉	C₂×SD₁₆
cosocle	C₂²×C₈×C₇₉	C₂³
hypercenter	C₂×#494	D₇₉
hypocenter	C₇₉	#13212
LCS	G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#494

Complements

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

N	H ≅ G/N	type
105. C₄×C₇₉	#362	semidirect
119. C₇₉	#13212	semidirect
220. C₄	#19098379	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19098379	2
C₄×#14579101	2
#23763572	2
C₂×#494×C₇₉	2
#13212	79