Subgroups — C₂×#23766018 #34446883

Order 20224.

3502 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14579120	C₂	1	2	yes	maximal
2	#23766018	C₂	1	16	yes	maximal, direct product factor
3	C₂×#19098462	C₂	1	8	yes	maximal, complemented
4	C₂×#19098423	C₂	1	4	yes	maximal
5	C₂³×C₄⋊C₄×C₇₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#14579120	—	2	16	no	—
7	C₂×#14579120	C₂²	1	16	yes	—
8	C₂²×C₄⋊Dic₇₉	C₂²	1	4	yes	—
9	C₂²×C₄⋊Dic₇₉	C₂²	1	8	yes	—
10	C₂²×C₂²⋊Dic₇₉	C₂²	1	4	yes	—
11	C₂²×C₄×Dic₇₉	C₂²	1	4	yes	—
12	#19098462	C₂²	1	64	yes	complemented
13	#19098423	C₂²	1	32	yes	—
14	C₂×C₇₉⋊#95	C₂²	1	8	yes	—
15	C₂⁴×C₄×C₇₉	C₂²	1	2	yes	—
16	C₂⁴×C₄×C₇₉	C₂²	1	1	yes	characteristic
17	C₂²×C₄⋊C₄×C₇₉	C₄	1	16	yes	complemented
18	C₂²×C₄⋊C₄×C₇₉	C₂²	1	4	yes	—
19	C₂²×C₄⋊C₄×C₇₉	C₂²	1	8	yes	—
20	C₂³×Dic₇₉	C₂³	1	8	yes	—
21	#14579120	—	2	64	no	—
22	C₂×C₄⋊Dic₇₉	—	2	16	no	—
23	C₂×C₄⋊Dic₇₉	—	4	16	no	—
24	C₂×C₄⋊Dic₇₉	C₂³	1	16	yes	—
25	C₂×C₄⋊Dic₇₉	C₂³	1	32	yes	—
26	C₂×C₂²⋊Dic₇₉	—	2	16	no	—
27	C₂×C₂²⋊Dic₇₉	C₂³	1	16	yes	—
28	C₂×C₄×Dic₇₉	—	2	16	no	—
29	C₂×C₄×Dic₇₉	C₂³	1	16	yes	—
30	C₇₉⋊#95	C₂³	1	32	yes	—
31	C₂⁵×C₇₉	C₂³	1	1	yes	characteristic
32	C₂³×C₄×C₇₉	—	4	8	no	—
33	C₂³×C₄×C₇₉	C₂×C₄	1	16	yes	—
34	C₂³×C₄×C₇₉	C₂×C₄	1	8	yes	—
35	C₂³×C₄×C₇₉	D₄	1	8	yes	—
36	C₂³×C₄×C₇₉	Q₈	1	8	yes	—
37	C₂³×C₄×C₇₉	C₂³	1	4	yes	—
38	C₂³×C₄×C₇₉	C₂³	1	8	yes	—
39	C₂³×C₄×C₇₉	C₂³	1	2	yes	—
40	C₂³×C₄×C₇₉	C₂³	1	4	yes	—
41	C₂×C₄⋊C₄×C₇₉	C₂×C₄	1	32	yes	—
42	C₂×C₄⋊C₄×C₇₉	C₂×C₄	1	64	yes	complemented
43	C₂×C₄⋊C₄×C₇₉	C₂³	1	16	yes	—
44	C₂²×Dic₇₉	—	2	16	no	—
45	C₂²×Dic₇₉	—	2	16	no	—
46	C₂²×Dic₇₉	—	2	16	no	—
47	C₂²×Dic₇₉	C₂⁴	1	16	yes	—
48	C₄⋊Dic₇₉	—	2	32	no	—
49	C₄⋊Dic₇₉	—	4	32	no	—
50	C₂²⋊Dic₇₉	—	2	32	no	—
51	C₄×Dic₇₉	—	2	32	no	—
52	C₂⁴×C₇₉	—	2	4	no	—
53	C₂⁴×C₇₉	—	2	4	no	—
54	C₂⁴×C₇₉	C₂²×C₄	1	4	yes	—
55	C₂⁴×C₇₉	C₂×D₄	1	4	yes	—
56	C₂⁴×C₇₉	C₂×Q₈	1	4	yes	—
57	C₂⁴×C₇₉	C₂⁴	1	2	yes	—
58	C₂⁴×C₇₉	C₂⁴	1	1	yes	characteristic
59	C₂²×C₄×C₇₉	—	2	16	no	—
60	C₂²×C₄×C₇₉	—	4	32	no	—
61	C₂²×C₄×C₇₉	—	4	16	no	—
62	C₂²×C₄×C₇₉	C₄⋊C₄	1	64	yes	complemented
63	C₂²×C₄×C₇₉	C₂²×C₄	1	16	yes	—
64	C₂²×C₄×C₇₉	C₂²×C₄	1	32	yes	—
65	C₂²×C₄×C₇₉	C₂²×C₄	1	8	yes	—
66	C₂²×C₄×C₇₉	C₂²×C₄	1	16	yes	—
67	C₂²×C₄×C₇₉	C₂×D₄	1	8	yes	—
68	C₂²×C₄×C₇₉	C₂×D₄	1	16	yes	—
69	C₂²×C₄×C₇₉	C₂×Q₈	1	8	yes	—
70	C₂²×C₄×C₇₉	C₂×Q₈	1	16	yes	—
71	C₂²×C₄×C₇₉	C₂⁴	1	8	yes	—
72	C₂²×C₄×C₇₉	C₂⁴	1	4	yes	—
73	C₄⋊C₄×C₇₉	C₂²×C₄	1	64	yes	—
74	C₂³×C₇₉	—	2	8	no	—
75	C₂³×C₇₉	—	2	8	no	—
76	C₂³×C₇₉	—	2	16	no	—
77	C₂³×C₇₉	—	2	16	no	—
78	C₂³×C₇₉	—	2	4	no	—
79	C₂³×C₇₉	—	2	4	no	—
80	C₂³×C₇₉	C₂×C₄⋊C₄	1	16	yes	—
81	C₂³×C₇₉	C₂³×C₄	1	4	yes	—
82	C₂³×C₇₉	C₂³×C₄	1	2	yes	—
83	C₂³×C₇₉	C₂²×D₄	1	4	yes	—
84	C₂³×C₇₉	C₂²×D₄	1	2	yes	—
85	C₂³×C₇₉	C₂²×Q₈	1	4	yes	—
86	C₂³×C₇₉	C₂²×Q₈	1	2	yes	—
87	C₂³×C₇₉	H(5,𝔽₂)	1	4	yes	—
88	C₂³×C₇₉	2₋¹⁺⁴	1	4	yes	—
89	C₂³×C₇₉	C₂⁵	1	1	yes	characteristic
90	C₂×C₄×C₇₉	—	2	32	no	—
91	C₂×C₄×C₇₉	—	2	16	no	—
92	C₂×C₄×C₇₉	—	4	32	no	—
93	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	32	yes	—
94	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	64	yes	complemented
95	C₂×C₄×C₇₉	C₂³×C₄	1	16	yes	—
96	C₂×C₄×C₇₉	C₂³×C₄	1	8	yes	—
97	C₂×C₄×C₇₉	C₂²×D₄	1	8	yes	—
98	C₂×C₄×C₇₉	C₂²×Q₈	1	8	yes	—
99	C₂×Dic₇₉	—	2	16	no	—
100	C₂×Dic₇₉	—	2	16	no	—
101	C₂×Dic₇₉	—	2	16	no	—
102	C₂×Dic₇₉	—	4	32	no	—
103	C₂²×C₇₉	—	2	32	no	—
104	C₂²×C₇₉	—	2	4	no	—
105	C₂²×C₇₉	—	2	4	no	—
106	C₂²×C₇₉	—	2	8	no	—
107	C₂²×C₇₉	—	2	8	no	—
108	C₂²×C₇₉	C₂²×C₄⋊C₄	1	8	yes	—
109	C₂²×C₇₉	C₂²×C₄⋊C₄	1	4	yes	—
110	C₂²×C₇₉	#518	1	4	yes	—
111	C₂²×C₇₉	#519	1	4	yes	—
112	C₂²×C₇₉	#537	1	8	yes	—
113	C₂²×C₇₉	#544	1	8	yes	—
114	C₂²×C₇₉	C₂⁴×C₄	1	1	yes	characteristic
115	C₂²×C₇₉	C₂³×D₄	1	1	yes	characteristic
116	C₂²×C₇₉	C₂³×Q₈	1	1	yes	characteristic
117	C₂²×C₇₉	C₂×H(5,𝔽₂)	1	2	yes	—
118	C₂²×C₇₉	C₂×2₋¹⁺⁴	1	2	yes	—
119	C₄×C₇₉	—	2	16	no	—
120	C₄×C₇₉	C₂²×C₄⋊C₄	1	16	yes	—
121	Dic₇₉	—	4	16	no	—
122	C₂×#2316	—	79	1	no	maximal, Sylow
123	C₂×C₇₉	—	2	8	no	—
124	C₂×C₇₉	#2316	1	8	yes	complemented
125	C₂×C₇₉	C₂³×C₄⋊C₄	1	1	yes	characteristic
126	C₂×C₇₉	C₂×#518	1	1	yes	characteristic
127	C₂×C₇₉	C₂×#519	1	1	yes	characteristic
128	C₂×C₇₉	C₂×#537	1	2	yes	—
129	C₂×C₇₉	C₂×#544	1	2	yes	—
130	C₂²×#117	—	79	2	no	—
131	C₂³×C₄⋊C₄	D₇₉	1	1	yes	characteristic
132	#2316	—	79	16	no	—
133	C₂×#389	—	79	8	no	—
134	C₂×#380	—	79	4	no	—
135	C₇₉	C₂×#2316	1	1	yes	characteristic, complemented, Sylow
136	C₂⁴×C₄	C₂×D₇₉	1	2	yes	—
137	C₂⁴×C₄	C₂×D₇₉	1	1	yes	characteristic
138	C₂×#117	—	79	16	no	—
139	C₂×#117	—	158	16	no	—
140	C₂²×C₄⋊C₄	—	79	8	no	—
141	C₂²×C₄⋊C₄	—	79	4	no	—
142	C₂²×C₄⋊C₄	Dic₇₉	1	16	yes	complemented
143	C₂²×C₄⋊C₄	C₂×D₇₉	1	4	yes	—
144	C₂²×C₄⋊C₄	C₂×D₇₉	1	8	yes	—
145	C₂²×C₂²⋊C₄	—	79	4	no	—
146	C₂²×C₄²	—	79	4	no	—
147	#389	—	79	64	no	—
148	#380	—	79	32	no	—
149	C₂×#95	—	79	8	no	—
150	C₂⁵	C₂²×D₇₉	1	1	yes	characteristic
151	C₂³×C₄	—	4	8	no	—
152	C₂³×C₄	—	79	8	no	—
153	C₂³×C₄	Dic₁₅₈	1	8	yes	—
154	C₂³×C₄	D₃₁₆	1	8	yes	—
155	C₂³×C₄	C₂×Dic₇₉	1	16	yes	—
156	C₂³×C₄	C₂×Dic₇₉	1	8	yes	—
157	C₂³×C₄	C₂²×D₇₉	1	4	yes	—
158	C₂³×C₄	C₂²×D₇₉	1	8	yes	—
159	C₂³×C₄	C₂²×D₇₉	1	4	yes	—
160	C₂³×C₄	C₂²×D₇₉	1	2	yes	—
161	#117	—	158	64	no	—
162	C₂×C₄⋊C₄	—	316	16	no	—
163	C₂×C₄⋊C₄	—	79	32	no	—
164	C₂×C₄⋊C₄	—	79	16	no	—
165	C₂×C₄⋊C₄	—	158	16	no	—
166	C₂×C₄⋊C₄	C₂×Dic₇₉	1	32	yes	—
167	C₂×C₄⋊C₄	C₂×Dic₇₉	1	64	yes	complemented
168	C₂×C₄⋊C₄	C₂²×D₇₉	1	16	yes	—
169	C₂×C₂²⋊C₄	—	79	16	no	—
170	C₂×C₂²⋊C₄	—	158	16	no	—
171	C₂×C₄²	—	79	16	no	—
172	C₂×C₄²	—	158	16	no	—
173	#95	—	79	32	no	—
174	C₂⁴	—	2	4	no	—
175	C₂⁴	—	2	4	no	—
176	C₂⁴	C₂²×Dic₇₉	1	4	yes	—
177	C₂⁴	C₂×D₃₁₆	1	4	yes	—
178	C₂⁴	C₂×Dic₁₅₈	1	4	yes	—
179	C₂⁴	C₂³×D₇₉	1	2	yes	—
180	C₂⁴	C₂³×D₇₉	1	1	yes	characteristic
181	C₂²×C₄	—	2	16	no	—
182	C₂²×C₄	—	4	16	no	—
183	C₂²×C₄	—	4	32	no	—
184	C₂²×C₄	—	79	16	no	—
185	C₂²×C₄	—	158	16	no	—
186	C₂²×C₄	—	158	16	no	—
187	C₂²×C₄	—	158	16	no	—
188	C₂²×C₄	C₄⋊Dic₇₉	1	64	yes	complemented
189	C₂²×C₄	C₂²×Dic₇₉	1	16	yes	—
190	C₂²×C₄	C₂²×Dic₇₉	1	32	yes	—
191	C₂²×C₄	C₂²×Dic₇₉	1	8	yes	—
192	C₂²×C₄	C₂²×Dic₇₉	1	16	yes	—
193	C₂²×C₄	C₂×D₃₁₆	1	8	yes	—
194	C₂²×C₄	C₂×D₃₁₆	1	16	yes	—
195	C₂²×C₄	C₂×Dic₁₅₈	1	8	yes	—
196	C₂²×C₄	C₂×Dic₁₅₈	1	16	yes	—
197	C₂²×C₄	C₂³×D₇₉	1	8	yes	—
198	C₂²×C₄	C₂³×D₇₉	1	4	yes	—
199	C₄⋊C₄	—	316	32	no	—
200	C₄⋊C₄	—	158	32	no	—
201	C₄⋊C₄	C₂²×Dic₇₉	1	64	yes	—
202	C₂²⋊C₄	—	158	32	no	—
203	C₄²	—	158	32	no	—
204	C₂³	—	2	4	no	—
205	C₂³	—	2	4	no	—
206	C₂³	—	2	16	no	—
207	C₂³	—	2	16	no	—
208	C₂³	—	2	8	no	—
209	C₂³	—	2	8	no	—
210	C₂³	C₂×C₄⋊Dic₇₉	1	16	yes	—
211	C₂³	C₂³×Dic₇₉	1	4	yes	—
212	C₂³	C₂³×Dic₇₉	1	2	yes	—
213	C₂³	C₂²×D₃₁₆	1	4	yes	—
214	C₂³	C₂²×D₃₁₆	1	2	yes	—
215	C₂³	C₂²×Dic₁₅₈	1	4	yes	—
216	C₂³	C₂²×Dic₁₅₈	1	2	yes	—
217	C₂³	#14579206	1	4	yes	—
218	C₂³	#14579209	1	4	yes	—
219	C₂³	C₂⁴×D₇₉	1	1	yes	characteristic
220	C₂×C₄	—	2	16	no	—
221	C₂×C₄	—	2	32	no	—
222	C₂×C₄	—	4	32	no	—
223	C₂×C₄	—	316	32	no	—
224	C₂×C₄	—	158	16	no	—
225	C₂×C₄	—	158	16	no	—
226	C₂×C₄	—	158	16	no	—
227	C₂×C₄	C₂×C₄⋊Dic₇₉	1	32	yes	—
228	C₂×C₄	C₂×C₄⋊Dic₇₉	1	64	yes	complemented
229	C₂×C₄	C₂³×Dic₇₉	1	16	yes	—
230	C₂×C₄	C₂³×Dic₇₉	1	8	yes	—
231	C₂×C₄	C₂²×D₃₁₆	1	8	yes	—
232	C₂×C₄	C₂²×Dic₁₅₈	1	8	yes	—
233	C₂²	—	2	32	no	—
234	C₂²	—	2	8	no	—
235	C₂²	—	2	4	no	—
236	C₂²	—	2	4	no	—
237	C₂²	—	2	8	no	—
238	C₂²	C₂²×C₄⋊Dic₇₉	1	8	yes	—
239	C₂²	C₂²×C₄⋊Dic₇₉	1	4	yes	—
240	C₂²	#19099038	1	4	yes	—
241	C₂²	#19099041	1	4	yes	—
242	C₂²	#19099144	1	8	yes	—
243	C₂²	#19099194	1	8	yes	—
244	C₂²	C₂⁴×Dic₇₉	1	1	yes	characteristic
245	C₂²	C₂³×D₃₁₆	1	1	yes	characteristic
246	C₂²	C₂³×Dic₁₅₈	1	1	yes	characteristic
247	C₂²	C₂×#14579206	1	2	yes	—
248	C₂²	C₂×#14579209	1	2	yes	—
249	C₄	—	2	16	no	—
250	C₄	—	316	16	no	—
251	C₄	C₂²×C₄⋊Dic₇₉	1	16	yes	—
252	C₂	—	2	8	no	—
253	C₂	#23766018	1	8	yes	direct product factor
254	C₂	C₂³×C₄⋊Dic₇₉	1	1	yes	characteristic
255	C₂	C₂×#19099038	1	1	yes	characteristic
256	C₂	C₂×#19099041	1	1	yes	characteristic
257	C₂	C₂×#19099144	1	2	yes	—
258	C₂	C₂×#19099194	1	2	yes	—

	subgroup	G/H
center	C₂⁴	C₂³×D₇₉
commutator	C₂²×C₇₉	C₂⁴×C₄
Frattini	C₂³	C₂⁴×D₇₉
Fitting	C₂³×C₄⋊C₄×C₇₉	C₂
socle	C₂⁴×C₇₉	C₂⁴
prime radical	C₂⁵×C₇₉	C₂³
cosocle	C₂³×C₇₉	C₂⁵
hypercenter	C₂³×C₄⋊C₄	D₇₉
hypocenter	C₇₉	C₂×#2316
LCS	G ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
2. #23766018	C₂	direct (N × H)
3. C₂×#19098462	C₂	semidirect
12. #19098462	C₂²	semidirect
17. C₂²×C₄⋊C₄×C₇₉	C₄	semidirect
42. C₂×C₄⋊C₄×C₇₉	C₂×C₄	semidirect
62. C₂²×C₄×C₇₉	C₄⋊C₄	semidirect
94. C₂×C₄×C₇₉	C₂×C₄⋊C₄	semidirect
124. C₂×C₇₉	#2316	semidirect
135. C₇₉	C₂×#2316	semidirect
142. C₂²×C₄⋊C₄	Dic₇₉	semidirect
167. C₂×C₄⋊C₄	C₂×Dic₇₉	semidirect
188. C₂²×C₄	C₄⋊Dic₇₉	semidirect
228. C₂×C₄	C₂×C₄⋊Dic₇₉	semidirect
253. C₂	#23766018	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×#14579120	2
#23766018	2
C₂×#19098462	2
C₂×#19098423	2
C₂³×C₄⋊C₄×C₇₉	2
C₂×#2316	79

Subgroups — C2×#23766018 #34446883

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#23766018 #34446883