Subgroups — C₂×#23765578 #34443024

Order 20224.

682 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14579095	C₂	1	1	yes	maximal, characteristic, complemented
2	#23765578	C₂	1	8	yes	maximal, direct product factor
3	C₂×#19098388	C₂	1	2	yes	maximal, complemented
4	C₂×#19098364	C₂	1	2	yes	maximal, complemented
5	C₂×C₂²⋊(C₇₉⋊C₁₆)	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#466×C₇₉	C₂	1	1	yes	maximal, characteristic
7	C₂²×C₄⋊Dic₇₉	—	2	1	no	—
8	C₂²×C₇₉⋊C₁₆	C₂²	1	2	yes	—
9	C₂×#14579095	—	2	2	no	—
10	C₂×#14579095	—	2	1	no	—
11	C₂×#14579095	C₂²	1	2	yes	—
12	C₂×#14579095	C₂²	1	4	yes	complemented
13	#19098388	C₂²	1	8	yes	complemented
14	#19098364	C₂²	1	8	yes	complemented
15	C₂²⋊(C₇₉⋊C₁₆)	C₂²	1	4	yes	complemented
16	C₂²×D₈×C₇₉	C₂²	1	1	yes	characteristic
17	C₂³×C₈×C₇₉	C₂²	1	1	yes	characteristic
18	C₂×#121×C₇₉	—	2	1	no	—
19	#466×C₇₉	—	2	2	no	—
20	#466×C₇₉	C₂²	1	4	yes	—
21	C₂×#107×C₇₉	C₂²	1	1	yes	characteristic
22	C₂×#102×C₇₉	—	2	1	no	—
23	C₂³×Dic₇₉	—	4	1	no	—
24	C₂×C₄⋊Dic₇₉	—	2	2	no	—
25	C₂×C₄⋊Dic₇₉	—	2	4	no	—
26	C₂×C₄⋊Dic₇₉	—	4	1	no	—
27	C₂×C₄⋊Dic₇₉	—	4	2	no	—
28	C₂×C₇₉⋊C₁₆	—	2	4	no	—
29	C₂×C₇₉⋊C₁₆	C₂³	1	4	yes	—
30	#14579095	—	2	2	no	—
31	#14579095	—	2	4	no	—
32	#14579095	C₂³	1	4	yes	—
33	C₂²×D₄×C₇₉	—	2	1	no	—
34	C₂²×D₄×C₇₉	—	4	1	no	—
35	C₂³×C₄×C₇₉	D₄	1	1	yes	characteristic
36	C₂×D₈×C₇₉	—	2	1	no	—
37	C₂×D₈×C₇₉	—	2	4	no	—
38	C₂×D₈×C₇₉	C₂³	1	2	yes	—
39	C₂²×C₈×C₇₉	—	2	4	no	—
40	C₂²×C₈×C₇₉	D₄	1	2	yes	—
41	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
42	C₂²×C₈×C₇₉	C₂³	1	2	yes	—
43	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
44	#121×C₇₉	—	2	4	no	—
45	#121×C₇₉	—	4	2	no	—
46	C₂×C₄⋊C₄×C₇₉	—	2	1	no	—
47	C₂×C₂²⋊C₄×C₇₉	—	4	1	no	—
48	#107×C₇₉	—	2	1	no	—
49	#107×C₇₉	C₂³	1	2	yes	—
50	#102×C₇₉	—	2	4	no	—
51	C₂²×Dic₇₉	—	4	4	no	—
52	C₂²×Dic₇₉	—	4	2	no	—
53	C₂²×Dic₇₉	—	8	4	no	—
54	C₄⋊Dic₇₉	—	2	4	no	—
55	C₄⋊Dic₇₉	—	4	4	no	—
56	C₄⋊Dic₇₉	—	4	2	no	—
57	C₇₉⋊C₁₆	—	2	4	no	—
58	C₂⁴×C₇₉	—	4	1	no	—
59	C₂⁴×C₇₉	D₈	1	1	yes	characteristic
60	C₂×D₄×C₇₉	—	2	2	no	—
61	C₂×D₄×C₇₉	—	4	4	no	—
62	C₂×D₄×C₇₉	—	4	2	no	—
63	C₂×D₄×C₇₉	—	4	1	no	—
64	C₂×D₄×C₇₉	—	4	2	no	—
65	C₂×D₄×C₇₉	—	8	4	no	—
66	C₂²×C₄×C₇₉	—	2	2	no	—
67	C₂²×C₄×C₇₉	—	2	2	no	—
68	C₂²×C₄×C₇₉	—	4	1	no	—
69	C₂²×C₄×C₇₉	D₈	1	1	yes	characteristic
70	C₂²×C₄×C₇₉	D₈	1	2	yes	—
71	C₂²×C₄×C₇₉	C₂×D₄	1	2	yes	—
72	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
73	D₈×C₇₉	—	2	4	no	—
74	D₈×C₇₉	—	4	2	no	—
75	C₂×C₈×C₇₉	—	2	4	no	—
76	C₂×C₈×C₇₉	—	2	4	no	—
77	C₂×C₈×C₇₉	—	2	2	no	—
78	C₂×C₈×C₇₉	C₂×D₄	1	2	yes	—
79	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
80	C₂×C₈×C₇₉	#41	1	4	yes	—
81	C₂×C₈×C₇₉	C₂⁴	1	1	yes	characteristic
82	C₄⋊C₄×C₇₉	—	2	2	no	—
83	C₄⋊C₄×C₇₉	—	4	1	no	—
84	C₂²⋊C₄×C₇₉	—	4	4	no	—
85	C₂³×C₇₉	—	2	2	no	—
86	C₂³×C₇₉	—	2	2	no	—
87	C₂³×C₇₉	—	4	2	no	—
88	C₂³×C₇₉	—	8	2	no	—
89	C₂³×C₇₉	—	8	2	no	—
90	C₂³×C₇₉	—	8	2	no	—
91	C₂³×C₇₉	D₁₆	1	4	yes	—
92	C₂³×C₇₉	C₂×D₈	1	2	yes	—
93	C₂³×C₇₉	C₂×D₈	1	1	yes	characteristic
94	D₄×C₇₉	—	4	2	no	—
95	D₄×C₇₉	—	4	2	no	—
96	D₄×C₇₉	—	8	8	no	—
97	D₄×C₇₉	—	8	2	no	—
98	C₂×C₄×C₇₉	—	2	2	no	—
99	C₂×C₄×C₇₉	—	2	2	no	—
100	C₂×C₄×C₇₉	—	2	2	no	—
101	C₂×C₄×C₇₉	—	2	4	no	—
102	C₂×C₄×C₇₉	—	4	2	no	—
103	C₂×C₄×C₇₉	—	8	2	no	—
104	C₂×C₄×C₇₉	#123	1	2	yes	—
105	C₂×C₄×C₇₉	C₂×D₈	1	2	yes	—
106	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
107	C₂×C₄×C₇₉	#137	1	2	yes	—
108	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
109	C₂×Dic₇₉	—	4	4	no	—
110	C₂×Dic₇₉	—	8	8	no	—
111	C₂×Dic₇₉	—	8	4	no	—
112	C₈×C₇₉	—	2	2	no	—
113	C₈×C₇₉	#123	1	2	yes	—
114	C₈×C₇₉	C₂×#41	1	2	yes	—
115	C₂²×C₇₉	—	2	2	no	—
116	C₂²×C₇₉	—	2	2	no	—
117	C₂²×C₇₉	—	2	8	no	—
118	C₂²×C₇₉	—	8	2	no	—
119	C₂²×C₇₉	—	8	2	no	—
120	C₂²×C₇₉	—	8	2	no	—
121	C₂²×C₇₉	—	16	4	no	—
122	C₂²×C₇₉	#480	1	2	yes	—
123	C₂²×C₇₉	C₂×D₁₆	1	4	yes	—
124	C₂²×C₇₉	C₂×D₁₆	1	2	yes	—
125	C₂²×C₇₉	#510	1	2	yes	—
126	C₂²×C₇₉	C₂²×D₈	1	1	yes	characteristic
127	C₄×C₇₉	—	2	2	no	—
128	C₄×C₇₉	—	8	2	no	—
129	C₄×C₇₉	#480	1	2	yes	—
130	C₄×C₇₉	C₂×#123	1	1	yes	characteristic
131	C₄×C₇₉	C₂×#137	1	1	yes	characteristic
132	Dic₇₉	—	8	4	no	—
133	C₂×#2232	—	79	1	no	maximal, Sylow
134	C₂×C₇₉	—	2	4	no	—
135	C₂×C₇₉	—	16	2	no	—
136	C₂×C₇₉	#2232	1	4	yes	complemented
137	C₂×C₇₉	C₂×#480	1	1	yes	characteristic
138	C₂×C₇₉	C₂²×D₁₆	1	1	yes	characteristic
139	C₂×C₇₉	C₂×#510	1	1	yes	characteristic
140	C₂×#466	D₇₉	1	1	yes	characteristic
141	C₂²×#107	—	79	1	no	—
142	#2232	—	79	8	no	—
143	C₂×#366	—	79	2	no	—
144	C₂×#357	—	79	2	no	—
145	C₂×C₂²⋊C₁₆	—	79	1	no	—
146	C₇₉	C₂×#2232	1	1	yes	characteristic, complemented, Sylow
147	C₂²×D₈	C₂×D₇₉	1	1	yes	characteristic
148	C₂³×C₈	C₂×D₇₉	1	1	yes	characteristic
149	C₂×#121	—	2	1	no	—
150	C₂²×C₄⋊C₄	—	158	1	no	—
151	C₂²×C₁₆	—	79	2	no	—
152	#466	—	2	2	no	—
153	#466	C₂×D₇₉	1	4	yes	—
154	C₂×#107	—	79	2	no	—
155	C₂×#107	—	79	4	no	—
156	C₂×#107	—	158	1	no	—
157	C₂×#107	—	158	2	no	—
158	C₂×#107	C₂×D₇₉	1	1	yes	characteristic
159	C₂×#102	—	2	1	no	—
160	#366	—	79	8	no	—
161	#357	—	79	8	no	—
162	C₂²⋊C₁₆	—	79	4	no	—
163	C₂²×D₄	—	2	1	no	—
164	C₂²×D₄	—	4	1	no	—
165	C₂³×C₄	—	316	1	no	—
166	C₂³×C₄	C₂²⋊D₇₉	1	1	yes	characteristic
167	C₂×D₈	—	2	1	no	—
168	C₂×D₈	—	2	4	no	—
169	C₂×D₈	C₂²×D₇₉	1	2	yes	—
170	C₂²×C₈	—	2	4	no	—
171	C₂²×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
172	C₂²×C₈	C₂²⋊D₇₉	1	2	yes	—
173	C₂²×C₈	C₂²×D₇₉	1	2	yes	—
174	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
175	#121	—	2	4	no	—
176	#121	—	4	2	no	—
177	C₂×C₄⋊C₄	—	2	1	no	—
178	C₂×C₄⋊C₄	—	316	2	no	—
179	C₂×C₄⋊C₄	—	316	1	no	—
180	C₂×C₄⋊C₄	—	158	4	no	—
181	C₂×C₄⋊C₄	—	158	2	no	—
182	C₂×C₂²⋊C₄	—	4	1	no	—
183	C₂×C₁₆	—	79	4	no	—
184	C₂×C₁₆	—	158	4	no	—
185	#107	—	2	1	no	—
186	#107	—	79	4	no	—
187	#107	—	158	4	no	—
188	#107	—	158	2	no	—
189	#107	C₂²×D₇₉	1	2	yes	—
190	#102	—	2	4	no	—
191	C₂⁴	—	4	1	no	—
192	C₂⁴	#11929609	1	1	yes	characteristic
193	C₂×D₄	—	2	2	no	—
194	C₂×D₄	—	4	4	no	—
195	C₂×D₄	—	4	2	no	—
196	C₂×D₄	—	4	1	no	—
197	C₂×D₄	—	4	2	no	—
198	C₂×D₄	—	8	4	no	—
199	C₂²×C₄	—	2	2	no	—
200	C₂²×C₄	—	2	2	no	—
201	C₂²×C₄	—	4	1	no	—
202	C₂²×C₄	—	316	4	no	—
203	C₂²×C₄	—	316	2	no	—
204	C₂²×C₄	—	632	4	no	—
205	C₂²×C₄	#11929609	1	2	yes	—
206	C₂²×C₄	#11929609	1	1	yes	characteristic
207	C₂²×C₄	C₂×C₂²⋊D₇₉	1	2	yes	—
208	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
209	D₈	—	2	4	no	—
210	D₈	—	4	2	no	—
211	C₂×C₈	—	2	2	no	—
212	C₂×C₈	—	2	4	no	—
213	C₂×C₈	—	2	4	no	—
214	C₂×C₈	C₂×C₂²⋊D₇₉	1	2	yes	—
215	C₂×C₈	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
216	C₂×C₈	#11929623	1	4	yes	—
217	C₂×C₈	C₂³×D₇₉	1	1	yes	characteristic
218	C₄⋊C₄	—	2	2	no	—
219	C₄⋊C₄	—	4	1	no	—
220	C₄⋊C₄	—	316	4	no	—
221	C₄⋊C₄	—	316	2	no	—
222	C₄⋊C₄	—	158	4	no	—
223	C₂²⋊C₄	—	4	4	no	—
224	C₁₆	—	158	4	no	—
225	C₂³	—	2	2	no	—
226	C₂³	—	2	2	no	—
227	C₂³	—	4	2	no	—
228	C₂³	—	8	2	no	—
229	C₂³	—	8	2	no	—
230	C₂³	—	8	2	no	—
231	C₂³	#14579102	1	4	yes	—
232	C₂³	C₂×#11929609	1	2	yes	—
233	C₂³	C₂×#11929609	1	1	yes	characteristic
234	D₄	—	4	2	no	—
235	D₄	—	4	2	no	—
236	D₄	—	8	8	no	—
237	D₄	—	8	2	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	4	no	—
242	C₂×C₄	—	4	2	no	—
243	C₂×C₄	—	8	2	no	—
244	C₂×C₄	—	316	4	no	—
245	C₂×C₄	—	632	8	no	—
246	C₂×C₄	—	632	4	no	—
247	C₂×C₄	#14579142	1	2	yes	—
248	C₂×C₄	C₂×#11929609	1	2	yes	—
249	C₂×C₄	C₂×#11929609	1	1	yes	characteristic
250	C₂×C₄	#14579191	1	2	yes	—
251	C₂×C₄	C₂²×C₂²⋊D₇₉	1	1	yes	characteristic
252	C₈	—	2	2	no	—
253	C₈	#14579142	1	2	yes	—
254	C₈	C₂×#11929623	1	2	yes	—
255	C₂²	—	2	2	no	—
256	C₂²	—	2	8	no	—
257	C₂²	—	2	2	no	—
258	C₂²	—	8	2	no	—
259	C₂²	—	8	2	no	—
260	C₂²	—	8	2	no	—
261	C₂²	—	16	4	no	—
262	C₂²	#19098874	1	2	yes	—
263	C₂²	C₂×#14579102	1	4	yes	—
264	C₂²	C₂×#14579102	1	2	yes	—
265	C₂²	#19099003	1	2	yes	—
266	C₂²	C₂²×#11929609	1	1	yes	characteristic
267	C₄	—	2	2	no	—
268	C₄	—	8	2	no	—
269	C₄	—	632	4	no	—
270	C₄	#19098874	1	2	yes	—
271	C₄	C₂×#14579142	1	1	yes	characteristic
272	C₄	C₂×#14579191	1	1	yes	characteristic
273	C₂	—	2	4	no	—
274	C₂	—	16	2	no	—
275	C₂	#23765578	1	4	yes	direct product factor
276	C₂	C₂×#19098874	1	1	yes	characteristic
277	C₂	C₂²×#14579102	1	1	yes	characteristic
278	C₂	C₂×#19099003	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#11929609
commutator, cosocle	C₂×C₈×C₇₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₇₉
Fitting, prime radical	C₂×#466×C₇₉	C₂
socle	C₂³×C₇₉	C₂×D₈
hypercenter	C₂×#466	D₇₉
hypocenter	C₇₉	C₂×#2232
LCS	G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂×#466

Complements

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

N	H ≅ G/N	type
1. C₂²×#14579095	C₂	semidirect
2. #23765578	C₂	direct (N × H)
3. C₂×#19098388	C₂	semidirect
4. C₂×#19098364	C₂	semidirect
5. C₂×C₂²⋊(C₇₉⋊C₁₆)	C₂	semidirect
12. C₂×#14579095	C₂²	semidirect
13. #19098388	C₂²	semidirect
14. #19098364	C₂²	semidirect
15. C₂²⋊(C₇₉⋊C₁₆)	C₂²	semidirect
136. C₂×C₇₉	#2232	semidirect
146. C₇₉	C₂×#2232	semidirect
275. C₂	#23765578	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×#14579095	2
#23765578	2
C₂×#19098388	2
C₂×#19098364	2
C₂×C₂²⋊(C₇₉⋊C₁₆)	2
C₂×#466×C₇₉	2
C₂×#2232	79

Subgroups — C2×#23765578 #34443024

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#23765578 #34443024