Subgroups — C₂×#23765962 #34446780

Order 20224.

5694 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14579158	C₂	1	1	yes	maximal, characteristic
2	C₂²×#14579123	C₂	1	6	yes	maximal, complemented
3	#23765962	C₂	1	16	yes	maximal, direct product factor
4	C₂×#19098464	C₂	1	6	yes	maximal, complemented
5	C₂×C₄²×Dic₇₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#384×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×D₃₁₆	C₂²	1	6	yes	—
8	C₂³×C₄×D₇₉	C₂²	1	4	yes	—
9	C₂×#14579158	C₄	1	16	yes	complemented
10	C₂×#14579158	C₂²	1	8	yes	—
11	C₂×#14579123	—	2	48	no	—
12	C₂×#14579123	C₂²	1	48	yes	complemented
13	C₂²×(C₂×D₇₉)⋊C₄	C₂²	1	12	yes	—
14	C₂²×C₄×Dic₇₉	C₂²	1	3	yes	—
15	C₂²×C₄×Dic₇₉	C₂²	1	3	yes	—
16	#19098464	C₂²	1	48	yes	complemented
17	C₄²×Dic₇₉	C₂²	1	8	yes	complemented
18	C₂²×C₄⋊C₄×C₇₉	C₂²	1	6	yes	—
19	C₂²×C₄²×C₇₉	C₂²	1	1	yes	characteristic
20	#384×C₇₉	C₂²	1	8	yes	complemented
21	C₂⁴×D₇₉	C₂³	1	4	yes	—
22	C₂²×D₃₁₆	—	2	24	no	—
23	C₂²×D₃₁₆	—	2	24	no	—
24	C₂²×D₃₁₆	C₂×C₄	1	48	yes	complemented
25	C₂²×D₃₁₆	C₂³	1	24	yes	—
26	C₂³×Dic₇₉	C₂³	1	3	yes	—
27	C₂³×Dic₇₉	C₂³	1	1	yes	characteristic
28	C₂²×C₄×D₇₉	—	2	48	no	—
29	C₂²×C₄×D₇₉	C₂³	1	16	yes	—
30	#14579158	C₂×C₄	1	64	yes	complemented
31	#14579123	—	2	192	no	—
32	C₂×(C₂×D₇₉)⋊C₄	—	2	48	no	—
33	C₂×(C₂×D₇₉)⋊C₄	C₂³	1	48	yes	—
34	C₂×C₄×Dic₇₉	—	2	12	no	—
35	C₂×C₄×Dic₇₉	D₄	1	24	yes	complemented
36	C₂×C₄×Dic₇₉	C₂³	1	12	yes	—
37	C₂×C₄×Dic₇₉	C₂³	1	12	yes	—
38	C₂³×C₄×C₇₉	C₂³	1	4	yes	—
39	C₂³×C₄×C₇₉	C₂³	1	3	yes	—
40	C₂×C₄⋊C₄×C₇₉	—	2	24	no	—
41	C₂×C₄⋊C₄×C₇₉	C₂³	1	24	yes	—
42	C₂×C₄²×C₇₉	C₂×C₄	1	8	yes	complemented
43	C₂×C₄²×C₇₉	C₂³	1	4	yes	—
44	C₂³×D₇₉	—	2	24	no	—
45	C₂³×D₇₉	—	2	24	no	—
46	C₂³×D₇₉	C₂²×C₄	1	16	yes	—
47	C₂³×D₇₉	C₂⁴	1	8	yes	—
48	C₂×D₃₁₆	—	2	192	no	—
49	C₂×D₃₁₆	—	2	48	no	—
50	C₂×D₃₁₆	—	2	48	no	—
51	C₂×D₃₁₆	C₂²×C₄	1	96	yes	complemented
52	C₂²×Dic₇₉	—	2	12	no	—
53	C₂²×Dic₇₉	C₂×D₄	1	12	yes	—
54	C₂²×Dic₇₉	C₂×D₄	1	12	yes	—
55	C₂²×Dic₇₉	C₂⁴	1	6	yes	—
56	C₂²×Dic₇₉	C₂⁴	1	2	yes	—
57	C₂×C₄×D₇₉	—	2	96	no	—
58	C₂×C₄×D₇₉	—	4	64	no	—
59	(C₂×D₇₉)⋊C₄	—	2	96	no	—
60	C₄×Dic₇₉	—	2	24	no	—
61	C₄×Dic₇₉	C₂×D₄	1	48	yes	complemented
62	C₂⁴×C₇₉	C₂⁴	1	1	yes	characteristic
63	C₂²×C₄×C₇₉	—	2	24	no	—
64	C₂²×C₄×C₇₉	C₂²×C₄	1	12	yes	—
65	C₂²×C₄×C₇₉	C₂×D₄	1	12	yes	—
66	C₂²×C₄×C₇₉	#41	1	12	yes	—
67	C₂²×C₄×C₇₉	C₂⁴	1	8	yes	—
68	C₂²×C₄×C₇₉	C₂⁴	1	6	yes	—
69	C₄⋊C₄×C₇₉	—	2	48	no	—
70	C₄²×C₇₉	C₂²×C₄	1	16	yes	complemented
71	C₂³×C₇₉	C₂³×C₄	1	2	yes	—
72	C₂³×C₇₉	C₂²×D₄	1	6	yes	—
73	C₂³×C₇₉	C₂×#41	1	6	yes	—
74	C₂³×C₇₉	C₂⁵	1	1	yes	characteristic
75	C₂²×D₇₉	—	2	96	no	—
76	C₂²×D₇₉	—	2	24	no	—
77	C₂²×D₇₉	—	2	24	no	—
78	C₂²×D₇₉	—	4	16	no	—
79	C₂²×D₇₉	—	4	48	no	—
80	C₂²×D₇₉	C₂³×C₄	1	16	yes	—
81	C₂×C₄×C₇₉	—	2	24	no	—
82	C₂×C₄×C₇₉	—	4	16	no	—
83	C₂×C₄×C₇₉	C₄×D₄	1	48	yes	complemented
84	C₂×C₄×C₇₉	C₂³×C₄	1	12	yes	—
85	C₂×C₄×C₇₉	C₂²×D₄	1	12	yes	—
86	C₂×C₄×C₇₉	C₂×#41	1	12	yes	—
87	C₂×Dic₇₉	—	2	24	no	—
88	C₂×Dic₇₉	—	2	12	no	—
89	C₂×Dic₇₉	#127	1	16	yes	complemented
90	C₂×Dic₇₉	C₂²×D₄	1	12	yes	—
91	C₂×Dic₇₉	C₂²×D₄	1	12	yes	—
92	D₃₁₆	—	2	192	no	—
93	C₄×D₇₉	—	4	64	no	—
94	C₂²×C₇₉	C₂×C₄×D₄	1	12	yes	—
95	C₂²×C₇₉	C₂×#127	1	4	yes	—
96	C₂²×C₇₉	#532	1	12	yes	—
97	C₂²×C₇₉	C₂⁴×C₄	1	1	yes	characteristic
98	C₂²×C₇₉	C₂³×D₄	1	3	yes	—
99	C₂²×C₇₉	C₂²×#41	1	3	yes	—
100	C₂×D₇₉	—	2	48	no	—
101	C₂×D₇₉	—	4	64	no	—
102	C₂×D₇₉	—	4	8	no	—
103	C₂×D₇₉	—	4	24	no	—
104	C₄×C₇₉	—	4	8	no	—
105	C₄×C₇₉	C₂×C₄×D₄	1	24	yes	complemented
106	Dic₇₉	—	2	12	no	—
107	Dic₇₉	C₂×#127	1	8	yes	complemented
108	C₂×C₄×#127	—	79	1	no	maximal, Sylow
109	C₂×C₇₉	C₄×#127	1	8	yes	complemented
110	C₂×C₇₉	C₂²×C₄×D₄	1	3	yes	—
111	C₂×C₇₉	C₂²×#127	1	1	yes	characteristic
112	C₂×C₇₉	C₂×#532	1	3	yes	—
113	D₇₉	—	4	16	no	—
114	C₂²×#127	—	79	1	no	—
115	C₂²×C₄×D₄	—	79	6	no	—
116	C₄×#127	—	79	16	no	—
117	C₂×#390	—	79	6	no	—
118	C₂×#384	D₇₉	1	1	yes	characteristic, complemented
119	C₂×C₄³	—	79	1	no	—
120	C₇₉	C₂×C₄×#127	1	1	yes	characteristic, complemented, Sylow
121	C₂³×D₄	—	79	6	no	—
122	C₂⁴×C₄	—	79	4	no	—
123	C₂×#127	—	79	8	no	—
124	C₂×#127	—	79	16	no	—
125	C₂×C₄×D₄	—	79	48	no	—
126	C₂×C₄×D₄	—	158	48	no	—
127	C₂²×C₄⋊C₄	C₂×D₇₉	1	6	yes	—
128	C₂²×C₂²⋊C₄	—	79	12	no	—
129	C₂²×C₄²	—	79	3	no	—
130	C₂²×C₄²	—	79	3	no	—
131	C₂²×C₄²	C₂×D₇₉	1	1	yes	characteristic
132	#390	—	79	48	no	—
133	#384	C₂×D₇₉	1	8	yes	complemented
134	C₄³	—	79	8	no	—
135	C₂⁵	—	79	4	no	—
136	C₂²×D₄	—	79	48	no	—
137	C₂²×D₄	—	79	24	no	—
138	C₂²×D₄	—	158	24	no	—
139	C₂²×D₄	—	158	24	no	—
140	C₂³×C₄	—	79	1	no	—
141	C₂³×C₄	—	79	3	no	—
142	C₂³×C₄	—	79	16	no	—
143	C₂³×C₄	—	158	48	no	—
144	C₂³×C₄	C₂²×D₇₉	1	4	yes	—
145	C₂³×C₄	C₂²×D₇₉	1	3	yes	—
146	#127	—	79	64	no	—
147	C₄×D₄	—	158	192	no	—
148	C₂×C₄⋊C₄	—	2	24	no	—
149	C₂×C₄⋊C₄	C₂²×D₇₉	1	24	yes	—
150	C₂×C₂²⋊C₄	—	79	48	no	—
151	C₂×C₂²⋊C₄	—	158	48	no	—
152	C₂×C₄²	—	79	24	no	—
153	C₂×C₄²	—	79	12	no	—
154	C₂×C₄²	—	79	12	no	—
155	C₂×C₄²	—	158	12	no	—
156	C₂×C₄²	C₄×D₇₉	1	8	yes	complemented
157	C₂×C₄²	C₂²×D₇₉	1	4	yes	—
158	C₂⁴	—	79	8	no	—
159	C₂⁴	—	79	16	no	—
160	C₂⁴	—	158	24	no	—
161	C₂⁴	—	158	24	no	—
162	C₂⁴	C₂³×D₇₉	1	1	yes	characteristic
163	C₂×D₄	—	79	96	no	—
164	C₂×D₄	—	158	48	no	—
165	C₂×D₄	—	158	192	no	—
166	C₂×D₄	—	158	48	no	—
167	C₂²×C₄	—	2	24	no	—
168	C₂²×C₄	—	316	64	no	—
169	C₂²×C₄	—	79	2	no	—
170	C₂²×C₄	—	79	12	no	—
171	C₂²×C₄	—	79	6	no	—
172	C₂²×C₄	—	79	12	no	—
173	C₂²×C₄	—	158	96	no	—
174	C₂²×C₄	—	158	12	no	—
175	C₂²×C₄	C₂×C₄×D₇₉	1	12	yes	—
176	C₂²×C₄	D₄×D₇₉	1	12	yes	—
177	C₂²×C₄	#11929625	1	12	yes	—
178	C₂²×C₄	C₂³×D₇₉	1	8	yes	—
179	C₂²×C₄	C₂³×D₇₉	1	6	yes	—
180	C₄⋊C₄	—	2	48	no	—
181	C₂²⋊C₄	—	158	96	no	—
182	C₄²	—	79	48	no	—
183	C₄²	—	158	24	no	—
184	C₄²	C₂×C₄×D₇₉	1	16	yes	complemented
185	C₂³	—	316	16	no	—
186	C₂³	—	316	48	no	—
187	C₂³	—	79	16	no	—
188	C₂³	—	158	96	no	—
189	C₂³	—	158	24	no	—
190	C₂³	—	158	24	no	—
191	C₂³	C₂²×C₄×D₇₉	1	2	yes	—
192	C₂³	C₂×D₄×D₇₉	1	6	yes	—
193	C₂³	C₂×#11929625	1	6	yes	—
194	C₂³	C₂⁴×D₇₉	1	1	yes	characteristic
195	D₄	—	158	192	no	—
196	C₂×C₄	—	2	24	no	—
197	C₂×C₄	—	4	16	no	—
198	C₂×C₄	—	316	64	no	—
199	C₂×C₄	—	79	16	no	—
200	C₂×C₄	—	79	12	no	—
201	C₂×C₄	—	79	12	no	—
202	C₂×C₄	—	158	24	no	—
203	C₂×C₄	—	158	12	no	—
204	C₂×C₄	#14579123	1	48	yes	complemented
205	C₂×C₄	C₂²×C₄×D₇₉	1	12	yes	—
206	C₂×C₄	C₂×D₄×D₇₉	1	12	yes	—
207	C₂×C₄	C₂×#11929625	1	12	yes	—
208	C₂²	—	316	8	no	—
209	C₂²	—	316	24	no	—
210	C₂²	—	316	64	no	—
211	C₂²	—	158	48	no	—
212	C₂²	C₂×#14579123	1	12	yes	—
213	C₂²	#127×D₇₉	1	4	yes	—
214	C₂²	#19099114	1	12	yes	—
215	C₂²	C₂³×C₄×D₇₉	1	1	yes	characteristic
216	C₂²	C₂²×D₄×D₇₉	1	3	yes	—
217	C₂²	C₂²×#11929625	1	3	yes	—
218	C₄	—	4	8	no	—
219	C₄	—	79	8	no	—
220	C₄	—	158	12	no	—
221	C₄	C₂×#14579123	1	24	yes	complemented
222	C₂	—	316	16	no	—
223	C₂	#23765962	1	8	yes	direct product factor
224	C₂	C₂²×#14579123	1	3	yes	—
225	C₂	C₂×#127×D₇₉	1	1	yes	characteristic
226	C₂	C₂×#19099114	1	3	yes	—

	subgroup	G/H
center	C₂⁴	C₂³×D₇₉
commutator	C₂²×C₇₉	C₂⁴×C₄
Frattini	C₂³	C₂⁴×D₇₉
Fitting	C₂×#384×C₇₉	C₂
socle	C₂⁴×C₇₉	C₂⁴
prime radical	C₂²×#14579158	C₂
cosocle	C₂³×C₇₉	C₂⁵
hypercenter	C₂×#384	D₇₉
hypocenter	C₇₉	C₂×C₄×#127
LCS	G ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂⁴ ◁ C₂×#384

Complements

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

N	H ≅ G/N	type
2. C₂²×#14579123	C₂	semidirect
3. #23765962	C₂	direct (N × H)
4. C₂×#19098464	C₂	semidirect
5. C₂×C₄²×Dic₇₉	C₂	semidirect
6. C₂×#384×C₇₉	C₂	semidirect
9. C₂×#14579158	C₄	semidirect
12. C₂×#14579123	C₂²	semidirect
16. #19098464	C₂²	semidirect
17. C₄²×Dic₇₉	C₂²	semidirect
20. #384×C₇₉	C₂²	semidirect
24. C₂²×D₃₁₆	C₂×C₄	semidirect
30. #14579158	C₂×C₄	semidirect
35. C₂×C₄×Dic₇₉	D₄	semidirect
42. C₂×C₄²×C₇₉	C₂×C₄	semidirect
51. C₂×D₃₁₆	C₂²×C₄	semidirect
61. C₄×Dic₇₉	C₂×D₄	semidirect
70. C₄²×C₇₉	C₂²×C₄	semidirect
83. C₂×C₄×C₇₉	C₄×D₄	semidirect
89. C₂×Dic₇₉	#127	semidirect
105. C₄×C₇₉	C₂×C₄×D₄	semidirect
107. Dic₇₉	C₂×#127	semidirect
109. C₂×C₇₉	C₄×#127	semidirect
118. C₂×#384	D₇₉	semidirect
120. C₇₉	C₂×C₄×#127	semidirect
133. #384	C₂×D₇₉	semidirect
156. C₂×C₄²	C₄×D₇₉	semidirect
184. C₄²	C₂×C₄×D₇₉	semidirect
204. C₂×C₄	#14579123	semidirect
221. C₄	C₂×#14579123	semidirect
223. C₂	#23765962	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×#14579158	2
C₂²×#14579123	2
#23765962	2
C₂×#19098464	2
C₂×C₄²×Dic₇₉	2
C₂×#384×C₇₉	2
C₂×C₄×#127	79

Subgroups — C2×#23765962 #34446780

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#23765962 #34446780