Subgroups — #34187040

Order 20224.

606 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#14579161	C₂	1	1	yes	maximal, characteristic
2	C₄²×C₇₉⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₂×#19098293	C₂	1	4	yes	maximal
4	C₇₉×#1842	C₂	1	1	yes	maximal, characteristic
5	C₂×#14579161	C₄	1	1	yes	characteristic
6	C₂×C₄×Dic₁₅₈	C₂²	1	2	yes	—
7	C₂×C₄×C₇₉⋊C₈	—	2	2	no	—
8	C₂×C₄×C₇₉⋊C₈	C₂²	1	2	yes	—
9	#19098469	C₄	1	2	yes	—
10	#19098461	—	2	2	no	—
11	#19098433	C₄	1	1	yes	characteristic
12	C₄×C₄⋊Dic₇₉	—	2	2	no	—
13	#19098293	—	2	8	no	—
14	C₂×C₄⋊C₈×C₇₉	—	2	2	no	—
15	C₂×C₄⋊C₈×C₇₉	C₂²	1	2	yes	—
16	C₄³×C₇₉	C₂²	1	1	yes	characteristic
17	C₂²×Dic₁₅₈	C₂×C₄	1	2	yes	—
18	C₂²×C₇₉⋊C₈	—	2	2	no	—
19	#14579161	—	2	2	no	—
20	#14579161	C₈	1	4	yes	complemented
21	C₄×Dic₁₅₈	—	2	4	no	—
22	C₄×Dic₁₅₈	—	2	4	no	—
23	C₂×C₄⋊Dic₇₉	—	2	2	no	—
24	C₂×C₄⋊Dic₇₉	—	2	2	no	—
25	C₂×C₄⋊Dic₇₉	C₂×C₄	1	2	yes	—
26	C₂×Dic₇₉⋊C₄	—	2	2	no	—
27	C₂×C₄×Dic₇₉	—	2	2	no	—
28	C₄×C₇₉⋊C₈	—	2	8	no	—
29	C₄×C₇₉⋊C₈	—	2	2	no	—
30	C₄×C₇₉⋊C₈	D₄	1	4	yes	—
31	C₇₉⋊#95	—	2	4	no	—
32	C₂²×C₈×C₇₉	—	2	2	no	—
33	C₂×C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄²×C₇₉	D₄	1	2	yes	—
36	C₂×C₄²×C₇₉	D₄	1	2	yes	—
37	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
38	C₄⋊C₈×C₇₉	—	2	4	no	—
39	C₄⋊C₈×C₇₉	—	4	4	no	—
40	C₂×Dic₁₅₈	—	2	4	no	—
41	C₂×Dic₁₅₈	—	2	2	no	—
42	C₂×Dic₁₅₈	—	2	4	no	—
43	C₂×Dic₁₅₈	C₂×C₈	1	4	yes	—
44	C₂²×Dic₇₉	—	2	2	no	—
45	C₂²×Dic₇₉	—	2	2	no	—
46	C₂×C₇₉⋊C₈	—	2	8	no	—
47	C₂×C₇₉⋊C₈	—	2	4	no	—
48	C₄⋊Dic₇₉	—	2	8	no	—
49	C₄⋊Dic₇₉	—	2	4	no	—
50	C₄⋊Dic₇₉	—	4	4	no	—
51	Dic₇₉⋊C₄	—	4	4	no	—
52	C₄×Dic₇₉	—	4	2	no	—
53	C₄×Dic₇₉	—	4	2	no	—
54	C₂²×C₄×C₇₉	C₂²⋊C₄	1	2	yes	—
55	C₂²×C₄×C₇₉	C₂²⋊C₄	1	2	yes	—
56	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
57	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
59	C₂×C₈×C₇₉	—	4	2	no	—
60	C₂×C₈×C₇₉	—	4	4	no	—
61	C₄²×C₇₉	—	2	4	no	—
62	C₄²×C₇₉	—	2	2	no	—
63	C₄²×C₇₉	C₂×C₈	1	2	yes	—
64	C₄²×C₇₉	M₁₆	1	2	yes	—
65	C₄²×C₇₉	SD₁₆	1	4	yes	—
66	C₄²×C₇₉	Q₁₆	1	4	yes	—
67	C₄²×C₇₉	C₂×D₄	1	2	yes	—
68	C₄²×C₇₉	#41	1	2	yes	—
69	C₂³×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
70	C₂×C₄×C₇₉	—	2	2	no	—
71	C₂×C₄×C₇₉	—	2	2	no	—
72	C₂×C₄×C₇₉	—	2	2	no	—
73	C₂×C₄×C₇₉	C₂²⋊C₈	1	4	yes	—
74	C₂×C₄×C₇₉	#103	1	4	yes	—
75	C₂×C₄×C₇₉	#103	1	4	yes	—
76	C₂×C₄×C₇₉	#104	1	4	yes	—
77	C₂×C₄×C₇₉	C₄×D₄	1	2	yes	—
78	C₂×C₄×C₇₉	#121	1	2	yes	—
79	C₂×C₄×C₇₉	#124	1	1	yes	characteristic
80	C₂×C₄×C₇₉	#127	1	1	yes	characteristic
81	C₂×C₄×C₇₉	C₂²×C₈	1	1	yes	characteristic
82	C₂×C₄×C₇₉	C₂×M₁₆	1	1	yes	characteristic
83	C₂×C₄×C₇₉	#131	1	2	yes	—
84	C₂×C₄×C₇₉	C₂×SD₁₆	1	2	yes	—
85	C₂×C₄×C₇₉	C₂×Q₁₆	1	2	yes	—
86	C₂×Dic₇₉	—	2	4	no	—
87	C₂×Dic₇₉	—	4	2	no	—
88	C₂×Dic₇₉	—	4	2	no	—
89	C₂×Dic₇₉	—	4	2	no	—
90	C₂×Dic₇₉	—	4	4	no	—
91	Dic₁₅₈	—	2	8	no	—
92	Dic₁₅₈	—	4	4	no	—
93	C₈×C₇₉	—	8	2	no	—
94	C₇₉⋊C₈	—	2	8	no	—
95	C₂²×C₇₉	#390	1	1	yes	characteristic
96	C₂²×C₇₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
97	C₂²×C₇₉	#408	1	1	yes	characteristic
98	C₂²×C₇₉	C₂×#103	1	2	yes	—
99	C₂²×C₇₉	C₂×#104	1	2	yes	—
100	C₄×C₇₉	—	2	2	no	—
101	C₄×C₇₉	—	2	4	no	—
102	C₄×C₇₉	Q₈⋊C₈	1	8	yes	complemented
103	C₄×C₇₉	C₈×D₄	1	2	yes	—
104	C₄×C₇₉	#436	1	2	yes	—
105	C₄×C₇₉	#487	1	2	yes	—
106	C₄×C₇₉	#492	1	1	yes	characteristic
107	C₄×C₇₉	#494	1	1	yes	characteristic
108	Dic₇₉	—	4	4	no	—
109	Dic₇₉	—	8	2	no	—
110	#9637	—	79	1	no	maximal, Sylow
111	C₂×C₇₉	C₂×Q₈⋊C₈	1	2	yes	—
112	C₂×C₇₉	#1493	1	2	yes	—
113	C₂×C₇₉	#1954	1	1	yes	characteristic
114	C₂×C₇₉	#1956	1	1	yes	characteristic
115	C₂×C₇₉	#1962	1	1	yes	characteristic
116	C₄×C₄⋊Q₈	—	79	1	no	—
117	#1842	D₇₉	1	1	yes	characteristic
118	C₄²×C₈	—	79	1	no	—
119	C₂×Q₈⋊C₈	—	79	4	no	—
120	C₇₉	#9637	1	1	yes	characteristic, complemented, Sylow
121	C₂×C₄⋊Q₈	—	79	1	no	—
122	C₂×C₄×Q₈	—	79	2	no	—
123	C₂×C₄⋊C₈	—	2	2	no	—
124	C₂×C₄⋊C₈	C₂×D₇₉	1	2	yes	—
125	C₂×C₄×C₈	—	79	2	no	—
126	C₂×C₄×C₈	—	158	2	no	—
127	#391	—	79	2	no	—
128	#389	—	158	2	no	—
129	#384	—	79	1	no	—
130	C₄×C₄⋊C₄	—	158	2	no	—
131	C₄³	C₂×D₇₉	1	1	yes	characteristic
132	Q₈⋊C₈	—	158	8	no	—
133	C₂²×Q₈	—	79	2	no	—
134	C₂²×C₈	—	2	2	no	—
135	C₂²×C₈	—	158	2	no	—
136	C₄⋊Q₈	—	79	4	no	—
137	C₄⋊Q₈	—	158	2	no	—
138	C₄×Q₈	—	158	4	no	—
139	C₄×Q₈	—	158	4	no	—
140	C₂×C₄⋊C₄	—	79	2	no	—
141	C₂×C₄⋊C₄	—	158	2	no	—
142	C₂×C₄⋊C₄	—	158	2	no	—
143	C₂×C₄⋊C₄	—	158	2	no	—
144	C₂×C₄²	—	158	2	no	—
145	C₂×C₄²	C₄×D₇₉	1	1	yes	characteristic
146	C₂×C₄²	C₄×D₇₉	1	1	yes	characteristic
147	C₂×C₄²	D₃₁₆	1	2	yes	—
148	C₂×C₄²	C₂²⋊D₇₉	1	2	yes	—
149	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
150	C₄⋊C₈	—	2	4	no	—
151	C₄⋊C₈	—	4	4	no	—
152	C₄×C₈	—	79	4	no	—
153	C₄×C₈	—	158	8	no	—
154	C₄×C₈	—	158	2	no	—
155	#95	—	158	4	no	—
156	C₂×Q₈	—	79	4	no	—
157	C₂×Q₈	—	158	4	no	—
158	C₂×Q₈	—	158	4	no	—
159	C₂×Q₈	—	158	2	no	—
160	C₂²×C₄	—	158	2	no	—
161	C₂²×C₄	—	158	2	no	—
162	C₂²×C₄	(C₂×D₇₉)⋊C₄	1	2	yes	—
163	C₂²×C₄	(C₂×D₇₉)⋊C₄	1	2	yes	—
164	C₂²×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
165	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
166	C₂²×C₄	C₂×D₃₁₆	1	1	yes	characteristic
167	C₂×C₈	—	4	2	no	—
168	C₂×C₈	—	4	4	no	—
169	C₂×C₈	—	158	8	no	—
170	C₂×C₈	—	158	4	no	—
171	C₄⋊C₄	—	316	4	no	—
172	C₄⋊C₄	—	316	4	no	—
173	C₄⋊C₄	—	158	4	no	—
174	C₄⋊C₄	—	158	8	no	—
175	C₄²	—	2	4	no	—
176	C₄²	—	2	2	no	—
177	C₄²	—	316	2	no	—
178	C₄²	—	316	2	no	—
179	C₄²	C₈×D₇₉	1	2	yes	—
180	C₄²	#11929607	1	2	yes	—
181	C₄²	#11929612	1	4	yes	—
182	C₄²	#11929614	1	4	yes	—
183	C₄²	D₄×D₇₉	1	2	yes	—
184	C₄²	#11929625	1	2	yes	—
185	C₂³	C₂×(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
186	Q₈	—	316	4	no	—
187	Q₈	—	158	8	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	316	2	no	—
192	C₂×C₄	—	316	2	no	—
193	C₂×C₄	—	316	2	no	—
194	C₂×C₄	—	316	4	no	—
195	C₂×C₄	—	158	4	no	—
196	C₂×C₄	(C₂×D₇₉)⋊C₈	1	4	yes	—
197	C₂×C₄	#14579084	1	4	yes	—
198	C₂×C₄	#14579084	1	4	yes	—
199	C₂×C₄	#14579087	1	4	yes	—
200	C₂×C₄	#14579123	1	2	yes	—
201	C₂×C₄	#14579136	1	2	yes	—
202	C₂×C₄	#14579150	1	1	yes	characteristic
203	C₂×C₄	#14579157	1	1	yes	characteristic
204	C₂×C₄	C₂×C₈×D₇₉	1	1	yes	characteristic
205	C₂×C₄	C₂×#11929607	1	1	yes	characteristic
206	C₂×C₄	#14579169	1	2	yes	—
207	C₂×C₄	C₂×#11929612	1	2	yes	—
208	C₂×C₄	C₂×#11929614	1	2	yes	—
209	C₈	—	8	2	no	—
210	C₈	—	158	8	no	—
211	C₂²	#19098464	1	1	yes	characteristic
212	C₂²	C₂×(C₂×D₇₉)⋊C₈	1	1	yes	characteristic
213	C₂²	#19098531	1	1	yes	characteristic
214	C₂²	C₂×#14579084	1	2	yes	—
215	C₂²	C₂×#14579087	1	2	yes	—
216	C₄	—	2	2	no	—
217	C₄	—	2	4	no	—
218	C₄	—	316	4	no	—
219	C₄	—	632	2	no	—
220	C₄	#19098293	1	8	yes	complemented
221	C₄	#19098628	1	2	yes	—
222	C₄	#19098640	1	2	yes	—
223	C₄	#19098915	1	2	yes	—
224	C₄	#19098935	1	1	yes	characteristic
225	C₄	#19098942	1	1	yes	characteristic
226	C₂	C₂×#19098293	1	2	yes	—
227	C₂	#23761854	1	2	yes	—
228	C₂	#23764193	1	1	yes	characteristic
229	C₂	#23764201	1	1	yes	characteristic
230	C₂	#23764227	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₇₉
commutator	C₂×C₄×C₇₉	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₇₉×#1842	C₂
socle, prime radical	C₂³×C₇₉	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	#1842	D₇₉
hypocenter	C₇₉	#9637
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1842

Complements

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

N	H ≅ G/N	type
20. #14579161	C₈	semidirect
102. C₄×C₇₉	Q₈⋊C₈	semidirect
120. C₇₉	#9637	semidirect
220. C₄	#19098293	semidirect

Maximal subgroups

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

subgroup	index
C₄×#14579161	2
C₄²×C₇₉⋊C₈	2
C₂×#19098293	2
C₇₉×#1842	2
#9637	79