Subgroups — C₂×#23763556 #34288666

Order 20224.

1358 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14579092	C₂	1	4	yes	maximal
2	C₂²×C₄×C₇₉⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₂×#19098433	C₂	1	1	yes	maximal, characteristic
4	#23763556	C₂	1	8	yes	maximal, direct product factor
5	C₂×#384×C₇₉	C₂	1	1	yes	maximal, characteristic
6	C₂³×C₇₉⋊C₈	C₂²	1	2	yes	—
7	C₂²×C₄⋊Dic₇₉	—	2	2	no	—
8	C₂²×C₄⋊Dic₇₉	C₂²	1	2	yes	—
9	C₂×#14579092	—	2	16	no	—
10	C₂×#14579092	C₂²	1	16	yes	—
11	C₂×C₄×C₇₉⋊C₈	C₄	1	8	yes	complemented
12	C₂×C₄×C₇₉⋊C₈	C₂²	1	4	yes	—
13	#19098433	—	2	2	no	—
14	#19098433	C₂²	1	4	yes	—
15	C₂²×C₄⋊C₄×C₇₉	—	2	2	no	—
16	C₂²×C₄⋊C₄×C₇₉	C₂²	1	2	yes	—
17	C₂²×C₄²×C₇₉	C₂²	1	1	yes	characteristic
18	#384×C₇₉	—	2	2	no	—
19	#384×C₇₉	C₂²	1	4	yes	—
20	C₂³×Dic₇₉	—	2	2	no	—
21	C₂²×C₇₉⋊C₈	C₂×C₄	1	8	yes	—
22	C₂²×C₇₉⋊C₈	D₄	1	8	yes	—
23	C₂²×C₇₉⋊C₈	Q₈	1	8	yes	—
24	C₂²×C₇₉⋊C₈	C₂³	1	4	yes	—
25	C₂×C₄⋊Dic₇₉	—	2	8	no	—
26	C₂×C₄⋊Dic₇₉	—	2	2	no	—
27	C₂×C₄⋊Dic₇₉	—	2	8	no	—
28	C₂×C₄⋊Dic₇₉	—	4	8	no	—
29	C₂×C₄⋊Dic₇₉	C₂³	1	4	yes	—
30	#14579092	—	2	32	no	—
31	C₄×C₇₉⋊C₈	C₂×C₄	1	16	yes	complemented
32	C₂³×C₄×C₇₉	—	2	2	no	—
33	C₂³×C₄×C₇₉	D₄	1	2	yes	—
34	C₂³×C₄×C₇₉	C₂³	1	1	yes	characteristic
35	C₂×C₄⋊C₄×C₇₉	—	2	8	no	—
36	C₂×C₄⋊C₄×C₇₉	—	2	8	no	—
37	C₂×C₄⋊C₄×C₇₉	—	2	2	no	—
38	C₂×C₄⋊C₄×C₇₉	—	4	8	no	—
39	C₂×C₄⋊C₄×C₇₉	C₂³	1	4	yes	—
40	C₂×C₄²×C₇₉	C₂×C₄	1	4	yes	—
41	C₂×C₄²×C₇₉	D₄	1	2	yes	—
42	C₂×C₄²×C₇₉	Q₈	1	4	yes	—
43	C₂×C₄²×C₇₉	C₂³	1	2	yes	—
44	C₂²×Dic₇₉	—	2	4	no	—
45	C₂²×Dic₇₉	—	4	4	no	—
46	C₂²×Dic₇₉	—	4	8	no	—
47	C₂×C₇₉⋊C₈	C₄⋊C₄	1	32	yes	complemented
48	C₂×C₇₉⋊C₈	C₂²×C₄	1	8	yes	—
49	C₂×C₇₉⋊C₈	C₂×D₄	1	8	yes	—
50	C₂×C₇₉⋊C₈	C₂×Q₈	1	8	yes	—
51	C₄⋊Dic₇₉	—	2	8	no	—
52	C₄⋊Dic₇₉	—	4	4	no	—
53	C₄⋊Dic₇₉	—	4	16	no	—
54	C₂⁴×C₇₉	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₇₉	—	2	4	no	—
56	C₂²×C₄×C₇₉	—	4	4	no	—
57	C₂²×C₄×C₇₉	—	4	8	no	—
58	C₂²×C₄×C₇₉	C₄⋊C₄	1	8	yes	—
59	C₂²×C₄×C₇₉	SD₁₆	1	8	yes	—
60	C₂²×C₄×C₇₉	SD₁₆	1	8	yes	—
61	C₂²×C₄×C₇₉	C₂²×C₄	1	2	yes	—
62	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₇₉	C₂×D₄	1	4	yes	—
64	C₂²×C₄×C₇₉	C₂×D₄	1	4	yes	—
65	C₂²×C₄×C₇₉	C₂×Q₈	1	4	yes	—
66	C₂²×C₄×C₇₉	C₂×Q₈	1	2	yes	—
67	C₂²×C₄×C₇₉	C₂⁴	1	1	yes	characteristic
68	C₄⋊C₄×C₇₉	—	2	8	no	—
69	C₄⋊C₄×C₇₉	—	4	4	no	—
70	C₄⋊C₄×C₇₉	—	4	16	no	—
71	C₄²×C₇₉	C₄⋊C₄	1	8	yes	—
72	C₄²×C₇₉	C₂²×C₄	1	4	yes	—
73	C₄²×C₇₉	C₂×Q₈	1	4	yes	—
74	C₂³×C₇₉	C₂×C₄⋊C₄	1	2	yes	—
75	C₂³×C₇₉	#127	1	2	yes	—
76	C₂³×C₇₉	C₄⋊Q₈	1	2	yes	—
77	C₂³×C₇₉	C₂×SD₁₆	1	4	yes	—
78	C₂³×C₇₉	C₂×SD₁₆	1	4	yes	—
79	C₂³×C₇₉	C₂²×D₄	1	1	yes	characteristic
80	C₂×C₄×C₇₉	—	4	4	no	—
81	C₂×C₄×C₇₉	—	4	8	no	—
82	C₂×C₄×C₇₉	—	8	8	no	—
83	C₂×C₄×C₇₉	#106	1	32	yes	complemented
84	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	8	yes	—
85	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	2	yes	—
86	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	8	yes	—
87	C₂×C₄×C₇₉	#127	1	2	yes	—
88	C₂×C₄×C₇₉	C₄⋊Q₈	1	2	yes	—
89	C₂×C₄×C₇₉	C₄⋊Q₈	1	8	yes	—
90	C₂×C₄×C₇₉	C₂×SD₁₆	1	8	yes	—
91	C₂×C₄×C₇₉	C₂×SD₁₆	1	8	yes	—
92	C₂×C₄×C₇₉	C₂³×C₄	1	1	yes	characteristic
93	C₂×C₄×C₇₉	C₂²×D₄	1	2	yes	—
94	C₂×C₄×C₇₉	C₂²×Q₈	1	2	yes	—
95	C₂×C₄×C₇₉	C₂²×Q₈	1	1	yes	characteristic
96	C₂×Dic₇₉	—	4	4	no	—
97	C₂×Dic₇₉	—	4	8	no	—
98	C₂×Dic₇₉	—	8	8	no	—
99	C₇₉⋊C₈	C₂×C₄⋊C₄	1	16	yes	complemented
100	C₂²×C₇₉	#384	1	4	yes	—
101	C₂²×C₇₉	C₂×#106	1	8	yes	—
102	C₂²×C₇₉	#492	1	4	yes	—
103	C₂²×C₇₉	#492	1	4	yes	—
104	C₂²×C₇₉	#498	1	8	yes	—
105	C₂²×C₇₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
106	C₂²×C₇₉	C₂×#127	1	1	yes	characteristic
107	C₂²×C₇₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
108	C₂²×C₇₉	C₂²×SD₁₆	1	2	yes	—
109	C₂²×C₇₉	C₂²×SD₁₆	1	2	yes	—
110	C₄×C₇₉	—	8	4	no	—
111	C₄×C₇₉	#384	1	4	yes	—
112	C₄×C₇₉	C₂×#106	1	16	yes	complemented
113	C₄×C₇₉	C₂²×C₄⋊C₄	1	2	yes	—
114	C₄×C₇₉	C₂×C₄⋊Q₈	1	2	yes	—
115	Dic₇₉	—	8	4	no	—
116	C₂×#1844	—	79	1	no	maximal, Sylow
117	C₂×C₇₉	#1844	1	8	yes	complemented
118	C₂×C₇₉	C₂×#384	1	1	yes	characteristic
119	C₂×C₇₉	C₂²×#106	1	2	yes	—
120	C₂×C₇₉	C₂×#492	1	1	yes	characteristic
121	C₂×C₇₉	C₂×#492	1	1	yes	characteristic
122	C₂×C₇₉	C₂×#498	1	2	yes	—
123	C₂²×#106	—	79	4	no	—
124	C₂²×C₄×C₈	—	79	1	no	—
125	C₂×#384	—	79	1	no	—
126	C₂×#384	D₇₉	1	1	yes	characteristic
127	#1844	—	79	8	no	—
128	C₇₉	C₂×#1844	1	1	yes	characteristic, complemented, Sylow
129	C₂³×C₈	—	79	2	no	—
130	C₂²×C₄⋊C₄	—	2	2	no	—
131	C₂²×C₄⋊C₄	—	79	2	no	—
132	C₂²×C₄⋊C₄	—	158	2	no	—
133	C₂²×C₄⋊C₄	C₂×D₇₉	1	2	yes	—
134	C₂²×C₄²	C₂×D₇₉	1	1	yes	characteristic
135	C₂×#106	—	79	16	no	—
136	C₂×#106	—	158	16	no	—
137	C₂×C₄×C₈	—	79	4	no	—
138	C₂×C₄×C₈	—	79	8	no	—
139	#384	—	2	2	no	—
140	#384	—	79	4	no	—
141	#384	—	158	2	no	—
142	#384	C₂×D₇₉	1	4	yes	—
143	C₂³×C₄	—	2	2	no	—
144	C₂³×C₄	—	158	2	no	—
145	C₂³×C₄	C₂²⋊D₇₉	1	2	yes	—
146	C₂³×C₄	C₂²×D₇₉	1	1	yes	characteristic
147	C₂²×C₈	—	79	8	no	—
148	C₂²×C₈	—	79	8	no	—
149	C₂²×C₈	—	79	8	no	—
150	C₂²×C₈	—	79	4	no	—
151	C₂×C₄⋊C₄	—	2	8	no	—
152	C₂×C₄⋊C₄	—	2	8	no	—
153	C₂×C₄⋊C₄	—	2	2	no	—
154	C₂×C₄⋊C₄	—	4	8	no	—
155	C₂×C₄⋊C₄	—	316	8	no	—
156	C₂×C₄⋊C₄	—	79	4	no	—
157	C₂×C₄⋊C₄	—	158	2	no	—
158	C₂×C₄⋊C₄	—	158	8	no	—
159	C₂×C₄⋊C₄	—	158	8	no	—
160	C₂×C₄⋊C₄	C₂²×D₇₉	1	4	yes	—
161	C₂×C₄²	Dic₁₅₈	1	4	yes	—
162	C₂×C₄²	C₄×D₇₉	1	4	yes	—
163	C₂×C₄²	C₂²⋊D₇₉	1	2	yes	—
164	C₂×C₄²	C₂²×D₇₉	1	2	yes	—
165	#106	—	158	32	no	—
166	C₄×C₈	—	79	16	no	—
167	C₂⁴	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
168	C₂²×C₄	—	2	4	no	—
169	C₂²×C₄	—	4	4	no	—
170	C₂²×C₄	—	4	8	no	—
171	C₂²×C₄	—	316	4	no	—
172	C₂²×C₄	—	316	8	no	—
173	C₂²×C₄	—	158	4	no	—
174	C₂²×C₄	Dic₇₉⋊C₄	1	8	yes	—
175	C₂²×C₄	#11929611	1	8	yes	—
176	C₂²×C₄	#11929612	1	8	yes	—
177	C₂²×C₄	C₂×C₄×D₇₉	1	2	yes	—
178	C₂²×C₄	D₄×D₇₉	1	4	yes	—
179	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
180	C₂²×C₄	C₂×C₂²⋊D₇₉	1	4	yes	—
181	C₂²×C₄	Q₈×D₇₉	1	4	yes	—
182	C₂²×C₄	C₂×Dic₁₅₈	1	2	yes	—
183	C₂²×C₄	C₂³×D₇₉	1	1	yes	characteristic
184	C₂×C₈	—	79	8	no	—
185	C₂×C₈	—	79	32	no	—
186	C₂×C₈	—	79	8	no	—
187	C₂×C₈	—	79	8	no	—
188	C₄⋊C₄	—	2	8	no	—
189	C₄⋊C₄	—	4	4	no	—
190	C₄⋊C₄	—	4	16	no	—
191	C₄⋊C₄	—	316	4	no	—
192	C₄⋊C₄	—	316	16	no	—
193	C₄⋊C₄	—	158	8	no	—
194	C₄²	Dic₇₉⋊C₄	1	8	yes	—
195	C₄²	C₂×C₄×D₇₉	1	4	yes	—
196	C₄²	C₂×Dic₁₅₈	1	4	yes	—
197	C₂³	C₂×Dic₇₉⋊C₄	1	2	yes	—
198	C₂³	#14579157	1	2	yes	—
199	C₂³	#14579159	1	2	yes	—
200	C₂³	C₂×#11929611	1	4	yes	—
201	C₂³	C₂×#11929612	1	4	yes	—
202	C₂³	C₂²×C₂²⋊D₇₉	1	1	yes	characteristic
203	C₂×C₄	—	4	8	no	—
204	C₂×C₄	—	4	4	no	—
205	C₂×C₄	—	8	8	no	—
206	C₂×C₄	—	316	8	no	—
207	C₂×C₄	—	316	4	no	—
208	C₂×C₄	—	632	8	no	—
209	C₂×C₄	#14579092	1	32	yes	complemented
210	C₂×C₄	C₄⋊C₄×D₇₉	1	8	yes	—
211	C₂×C₄	C₂×Dic₇₉⋊C₄	1	2	yes	—
212	C₂×C₄	C₂×Dic₇₉⋊C₄	1	8	yes	—
213	C₂×C₄	#14579157	1	2	yes	—
214	C₂×C₄	#14579159	1	2	yes	—
215	C₂×C₄	#14579160	1	8	yes	—
216	C₂×C₄	C₂×#11929611	1	8	yes	—
217	C₂×C₄	C₂×#11929612	1	8	yes	—
218	C₂×C₄	C₂²×C₄×D₇₉	1	1	yes	characteristic
219	C₂×C₄	C₂×D₄×D₇₉	1	2	yes	—
220	C₂×C₄	C₂×Q₈×D₇₉	1	2	yes	—
221	C₂×C₄	C₂²×Dic₁₅₈	1	1	yes	characteristic
222	C₈	—	79	16	no	—
223	C₂²	#19098432	1	4	yes	—
224	C₂²	C₂×#14579092	1	8	yes	—
225	C₂²	#19098935	1	4	yes	—
226	C₂²	#19098937	1	4	yes	—
227	C₂²	#19098959	1	8	yes	—
228	C₂²	C₂²×Dic₇₉⋊C₄	1	1	yes	characteristic
229	C₂²	C₂×#14579157	1	1	yes	characteristic
230	C₂²	C₂×#14579159	1	1	yes	characteristic
231	C₂²	C₂²×#11929611	1	2	yes	—
232	C₂²	C₂²×#11929612	1	2	yes	—
233	C₄	—	8	4	no	—
234	C₄	—	632	4	no	—
235	C₄	#19098432	1	4	yes	—
236	C₄	C₂×#14579092	1	16	yes	complemented
237	C₄	C₂×C₄⋊C₄×D₇₉	1	2	yes	—
238	C₄	C₂×#14579160	1	2	yes	—
239	C₂	#23763556	1	8	yes	direct product factor
240	C₂	C₂×#19098432	1	1	yes	characteristic
241	C₂	C₂²×#14579092	1	2	yes	—
242	C₂	C₂×#19098935	1	1	yes	characteristic
243	C₂	C₂×#19098937	1	1	yes	characteristic
244	C₂	C₂×#19098959	1	2	yes	—

	subgroup	G/H
center	C₂⁴	C₂×C₂²⋊D₇₉
commutator	C₂×C₄×C₇₉	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₇₉
Fitting	C₂×#384×C₇₉	C₂
socle, prime radical	C₂⁴×C₇₉	C₂×D₄
cosocle	C₂²×C₄×C₇₉	C₂⁴
hypercenter	C₂×#384	D₇₉
hypocenter	C₇₉	C₂×#1844
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ C₂×#384

Complements

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

N	H ≅ G/N	type
4. #23763556	C₂	direct (N × H)
11. C₂×C₄×C₇₉⋊C₈	C₄	semidirect
31. C₄×C₇₉⋊C₈	C₂×C₄	semidirect
47. C₂×C₇₉⋊C₈	C₄⋊C₄	semidirect
83. C₂×C₄×C₇₉	#106	semidirect
99. C₇₉⋊C₈	C₂×C₄⋊C₄	semidirect
112. C₄×C₇₉	C₂×#106	semidirect
117. C₂×C₇₉	#1844	semidirect
128. C₇₉	C₂×#1844	semidirect
209. C₂×C₄	#14579092	semidirect
236. C₄	C₂×#14579092	semidirect
239. C₂	#23763556	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#14579092	2
C₂²×C₄×C₇₉⋊C₈	2
C₂×#19098433	2
#23763556	2
C₂×#384×C₇₉	2
C₂×#1844	79

Subgroups — C2×#23763556 #34288666

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#23763556 #34288666