Subgroups — C₂×#23763564 #34288676

Order 20224.

1182 conjugacy classes of subgroups.

The subgroup lattice has 236 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	2	yes	maximal
2	C₂²×#14579093	C₂	1	2	yes	maximal
3	C₂×#19098429	C₂	1	2	yes	maximal
4	#23763564	C₂	1	8	yes	maximal, direct product factor
5	C₂²×C₄×C₈×C₇₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₄⋊Dic₇₉	C₂²	1	4	yes	—
7	C₂×#14579095	—	2	8	no	—
8	C₂×#14579095	C₂²	1	8	yes	—
9	C₂×#14579093	—	2	8	no	—
10	C₂×#14579093	C₂²	1	8	yes	—
11	#19098429	—	2	4	no	—
12	#19098429	C₂²	1	8	yes	—
13	C₂×C₇₉⋊#95	—	2	4	no	—
14	C₂³×C₈×C₇₉	C₂²	1	1	yes	characteristic
15	C₂³×C₈×C₇₉	C₂²	1	1	yes	characteristic
16	C₂²×C₄²×C₇₉	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₈×C₇₉	C₄	1	8	yes	complemented
18	C₂×C₄×C₈×C₇₉	C₂²	1	4	yes	—
19	C₂³×Dic₇₉	—	2	4	no	—
20	C₂×C₄⋊Dic₇₉	—	2	8	no	—
21	C₂×C₄⋊Dic₇₉	—	2	8	no	—
22	C₂×C₄⋊Dic₇₉	—	2	4	no	—
23	C₂×C₄⋊Dic₇₉	C₂³	1	8	yes	—
24	#14579095	—	2	16	no	—
25	#14579093	—	2	16	no	—
26	C₇₉⋊#95	—	2	16	no	—
27	C₂³×C₄×C₇₉	D₄	1	2	yes	—
28	C₂³×C₄×C₇₉	C₂³	1	1	yes	characteristic
29	C₂²×C₈×C₇₉	C₂×C₄	1	4	yes	—
30	C₂²×C₈×C₇₉	C₂×C₄	1	4	yes	—
31	C₂²×C₈×C₇₉	D₄	1	4	yes	—
32	C₂²×C₈×C₇₉	D₄	1	4	yes	—
33	C₂²×C₈×C₇₉	Q₈	1	4	yes	—
34	C₂²×C₈×C₇₉	Q₈	1	4	yes	—
35	C₂²×C₈×C₇₉	C₂³	1	2	yes	—
36	C₂²×C₈×C₇₉	C₂³	1	2	yes	—
37	C₂×C₄²×C₇₉	C₂×C₄	1	4	yes	—
38	C₂×C₄²×C₇₉	D₄	1	2	yes	—
39	C₂×C₄²×C₇₉	Q₈	1	4	yes	—
40	C₂×C₄²×C₇₉	C₂³	1	2	yes	—
41	C₄×C₈×C₇₉	C₂×C₄	1	16	yes	complemented
42	C₂²×Dic₇₉	—	2	8	no	—
43	C₂²×Dic₇₉	—	4	8	no	—
44	C₂²×Dic₇₉	—	4	16	no	—
45	C₄⋊Dic₇₉	—	2	8	no	—
46	C₄⋊Dic₇₉	—	2	8	no	—
47	C₄⋊Dic₇₉	—	4	8	no	—
48	C₂⁴×C₇₉	C₂×D₄	1	1	yes	characteristic
49	C₂²×C₄×C₇₉	—	2	8	no	—
50	C₂²×C₄×C₇₉	C₄⋊C₄	1	8	yes	—
51	C₂²×C₄×C₇₉	C₂²×C₄	1	2	yes	—
52	C₂²×C₄×C₇₉	C₂×D₄	1	4	yes	—
53	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄×C₇₉	C₂×D₄	1	2	yes	—
55	C₂²×C₄×C₇₉	C₂×D₄	1	2	yes	—
56	C₂²×C₄×C₇₉	C₂×Q₈	1	2	yes	—
57	C₂²×C₄×C₇₉	C₂×Q₈	1	2	yes	—
58	C₂²×C₄×C₇₉	C₂×Q₈	1	2	yes	—
59	C₂²×C₄×C₇₉	C₂⁴	1	1	yes	characteristic
60	C₂×C₈×C₇₉	C₄⋊C₄	1	16	yes	—
61	C₂×C₈×C₇₉	C₄⋊C₄	1	16	yes	—
62	C₂×C₈×C₇₉	C₂²×C₄	1	4	yes	—
63	C₂×C₈×C₇₉	C₂²×C₄	1	4	yes	—
64	C₂×C₈×C₇₉	C₂×D₄	1	4	yes	—
65	C₂×C₈×C₇₉	C₂×D₄	1	4	yes	—
66	C₂×C₈×C₇₉	C₂×Q₈	1	4	yes	—
67	C₂×C₈×C₇₉	C₂×Q₈	1	4	yes	—
68	C₄²×C₇₉	C₄⋊C₄	1	8	yes	—
69	C₄²×C₇₉	C₂²×C₄	1	4	yes	—
70	C₄²×C₇₉	C₂×Q₈	1	4	yes	—
71	C₂³×C₇₉	C₂×C₄⋊C₄	1	2	yes	—
72	C₂³×C₇₉	#127	1	2	yes	—
73	C₂³×C₇₉	C₄⋊Q₈	1	2	yes	—
74	C₂³×C₇₉	#135	1	8	yes	—
75	C₂³×C₇₉	C₂²×D₄	1	1	yes	characteristic
76	C₂×C₄×C₇₉	—	2	8	no	—
77	C₂×C₄×C₇₉	—	2	16	no	—
78	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	8	yes	—
79	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	4	yes	—
80	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	4	yes	—
81	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	2	yes	—
82	C₂×C₄×C₇₉	#127	1	2	yes	—
83	C₂×C₄×C₇₉	C₄⋊Q₈	1	4	yes	—
84	C₂×C₄×C₇₉	C₄⋊Q₈	1	4	yes	—
85	C₂×C₄×C₇₉	C₄⋊Q₈	1	2	yes	—
86	C₂×C₄×C₇₉	C₂³×C₄	1	1	yes	characteristic
87	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
88	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
89	C₂×C₄×C₇₉	C₂²×Q₈	1	1	yes	characteristic
90	C₂×C₄×C₇₉	C₂²×Q₈	1	1	yes	characteristic
91	C₂×C₄×C₇₉	C₂²×Q₈	1	1	yes	characteristic
92	C₂×Dic₇₉	—	4	8	no	—
93	C₂×Dic₇₉	—	4	16	no	—
94	C₂×Dic₇₉	—	8	16	no	—
95	C₈×C₇₉	C₂×C₄⋊C₄	1	8	yes	—
96	C₈×C₇₉	C₂×C₄⋊C₄	1	8	yes	—
97	C₂²×C₇₉	#384	1	4	yes	—
98	C₂²×C₇₉	#427	1	8	yes	—
99	C₂²×C₇₉	#495	1	8	yes	—
100	C₂²×C₇₉	#499	1	8	yes	—
101	C₂²×C₇₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
102	C₂²×C₇₉	C₂×#127	1	1	yes	characteristic
103	C₂²×C₇₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
104	C₂²×C₇₉	C₂×#135	1	4	yes	—
105	C₄×C₇₉	—	2	8	no	—
106	C₄×C₇₉	#384	1	4	yes	—
107	C₄×C₇₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
108	C₄×C₇₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
109	C₄×C₇₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
110	C₄×C₇₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
111	Dic₇₉	—	8	8	no	—
112	C₂×#1846	—	79	1	no	maximal, Sylow
113	C₂×C₇₉	#1846	1	8	yes	complemented
114	C₂×C₇₉	C₂×#384	1	1	yes	characteristic
115	C₂×C₇₉	C₂×#427	1	2	yes	—
116	C₂×C₇₉	C₂×#495	1	2	yes	—
117	C₂×C₇₉	C₂×#499	1	2	yes	—
118	C₂²×#107	—	79	2	no	—
119	C₂²×#106	—	79	2	no	—
120	C₂²×C₄×C₈	D₇₉	1	1	yes	characteristic
121	C₂×#382	—	79	2	no	—
122	#1846	—	79	8	no	—
123	C₇₉	C₂×#1846	1	1	yes	characteristic, complemented, Sylow
124	C₂³×C₈	C₂×D₇₉	1	1	yes	characteristic
125	C₂³×C₈	C₂×D₇₉	1	1	yes	characteristic
126	C₂²×C₄⋊C₄	—	79	4	no	—
127	C₂²×C₄²	C₂×D₇₉	1	1	yes	characteristic
128	C₂×#107	—	79	8	no	—
129	C₂×#107	—	158	8	no	—
130	C₂×#106	—	79	8	no	—
131	C₂×#106	—	158	8	no	—
132	C₂×C₄×C₈	Dic₇₉	1	8	yes	complemented
133	C₂×C₄×C₈	C₂×D₇₉	1	4	yes	—
134	#382	—	79	8	no	—
135	#382	—	158	4	no	—
136	C₂×#95	—	158	4	no	—
137	C₂³×C₄	—	158	4	no	—
138	C₂³×C₄	D₃₁₆	1	2	yes	—
139	C₂³×C₄	C₂²×D₇₉	1	1	yes	characteristic
140	C₂²×C₈	Dic₁₅₈	1	4	yes	—
141	C₂²×C₈	Dic₁₅₈	1	4	yes	—
142	C₂²×C₈	D₃₁₆	1	4	yes	—
143	C₂²×C₈	D₃₁₆	1	4	yes	—
144	C₂²×C₈	C₂×Dic₇₉	1	4	yes	—
145	C₂²×C₈	C₂×Dic₇₉	1	4	yes	—
146	C₂²×C₈	C₂²×D₇₉	1	2	yes	—
147	C₂²×C₈	C₂²×D₇₉	1	2	yes	—
148	C₂×C₄⋊C₄	—	79	8	no	—
149	C₂×C₄⋊C₄	—	158	4	no	—
150	C₂×C₄⋊C₄	—	158	8	no	—
151	C₂×C₄⋊C₄	—	158	8	no	—
152	C₂×C₄²	Dic₁₅₈	1	4	yes	—
153	C₂×C₄²	D₃₁₆	1	2	yes	—
154	C₂×C₄²	C₂×Dic₇₉	1	4	yes	—
155	C₂×C₄²	C₂²×D₇₉	1	2	yes	—
156	#107	—	158	16	no	—
157	#106	—	158	16	no	—
158	C₄×C₈	C₂×Dic₇₉	1	16	yes	complemented
159	#95	—	158	16	no	—
160	C₂⁴	C₂×D₃₁₆	1	1	yes	characteristic
161	C₂²×C₄	—	2	8	no	—
162	C₂²×C₄	—	316	8	no	—
163	C₂²×C₄	—	316	16	no	—
164	C₂²×C₄	—	158	8	no	—
165	C₂²×C₄	C₄⋊Dic₇₉	1	8	yes	—
166	C₂²×C₄	C₂²×Dic₇₉	1	2	yes	—
167	C₂²×C₄	C₂×D₃₁₆	1	4	yes	—
168	C₂²×C₄	C₂×D₃₁₆	1	1	yes	characteristic
169	C₂²×C₄	C₂×D₃₁₆	1	2	yes	—
170	C₂²×C₄	C₂×D₃₁₆	1	2	yes	—
171	C₂²×C₄	C₂×Dic₁₅₈	1	2	yes	—
172	C₂²×C₄	C₂×Dic₁₅₈	1	2	yes	—
173	C₂²×C₄	C₂×Dic₁₅₈	1	2	yes	—
174	C₂²×C₄	C₂³×D₇₉	1	1	yes	characteristic
175	C₂×C₈	C₄⋊Dic₇₉	1	16	yes	—
176	C₂×C₈	C₄⋊Dic₇₉	1	16	yes	—
177	C₂×C₈	C₂²×Dic₇₉	1	4	yes	—
178	C₂×C₈	C₂²×Dic₇₉	1	4	yes	—
179	C₂×C₈	C₂×D₃₁₆	1	4	yes	—
180	C₂×C₈	C₂×D₃₁₆	1	4	yes	—
181	C₂×C₈	C₂×Dic₁₅₈	1	4	yes	—
182	C₂×C₈	C₂×Dic₁₅₈	1	4	yes	—
183	C₄⋊C₄	—	316	8	no	—
184	C₄⋊C₄	—	158	8	no	—
185	C₄⋊C₄	—	158	8	no	—
186	C₄²	C₄⋊Dic₇₉	1	8	yes	—
187	C₄²	C₂²×Dic₇₉	1	4	yes	—
188	C₄²	C₂×Dic₁₅₈	1	4	yes	—
189	C₂³	C₂×C₄⋊Dic₇₉	1	2	yes	—
190	C₂³	#14579158	1	2	yes	—
191	C₂³	#14579161	1	2	yes	—
192	C₂³	#14579183	1	8	yes	—
193	C₂³	C₂²×D₃₁₆	1	1	yes	characteristic
194	C₂×C₄	—	2	8	no	—
195	C₂×C₄	—	2	16	no	—
196	C₂×C₄	—	316	16	no	—
197	C₂×C₄	—	316	8	no	—
198	C₂×C₄	—	632	16	no	—
199	C₂×C₄	C₂×C₄⋊Dic₇₉	1	8	yes	—
200	C₂×C₄	C₂×C₄⋊Dic₇₉	1	2	yes	—
201	C₂×C₄	C₂×C₄⋊Dic₇₉	1	4	yes	—
202	C₂×C₄	C₂×C₄⋊Dic₇₉	1	4	yes	—
203	C₂×C₄	#14579158	1	2	yes	—
204	C₂×C₄	#14579161	1	2	yes	—
205	C₂×C₄	#14579161	1	4	yes	—
206	C₂×C₄	#14579161	1	4	yes	—
207	C₂×C₄	C₂³×Dic₇₉	1	1	yes	characteristic
208	C₂×C₄	C₂²×D₃₁₆	1	1	yes	characteristic
209	C₂×C₄	C₂²×D₃₁₆	1	1	yes	characteristic
210	C₂×C₄	C₂²×Dic₁₅₈	1	1	yes	characteristic
211	C₂×C₄	C₂²×Dic₁₅₈	1	1	yes	characteristic
212	C₂×C₄	C₂²×Dic₁₅₈	1	1	yes	characteristic
213	C₈	C₂×C₄⋊Dic₇₉	1	8	yes	—
214	C₈	C₂×C₄⋊Dic₇₉	1	8	yes	—
215	C₂²	#19098433	1	4	yes	—
216	C₂²	#19098601	1	8	yes	—
217	C₂²	#19098947	1	8	yes	—
218	C₂²	#19098965	1	8	yes	—
219	C₂²	C₂²×C₄⋊Dic₇₉	1	1	yes	characteristic
220	C₂²	C₂×#14579158	1	1	yes	characteristic
221	C₂²	C₂×#14579161	1	1	yes	characteristic
222	C₂²	C₂×#14579183	1	4	yes	—
223	C₄	—	2	8	no	—
224	C₄	—	632	8	no	—
225	C₄	#19098433	1	4	yes	—
226	C₄	C₂²×C₄⋊Dic₇₉	1	1	yes	characteristic
227	C₄	C₂²×C₄⋊Dic₇₉	1	1	yes	characteristic
228	C₄	C₂×#14579161	1	1	yes	characteristic
229	C₄	C₂×#14579161	1	1	yes	characteristic
230	C₂	#23763564	1	8	yes	direct product factor
231	C₂	C₂×#19098433	1	1	yes	characteristic
232	C₂	C₂×#19098601	1	2	yes	—
233	C₂	C₂×#19098947	1	2	yes	—
234	C₂	C₂×#19098965	1	2	yes	—

	subgroup	G/H
center	C₂⁴	C₂×D₃₁₆
commutator	C₂×C₄×C₇₉	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₇₉
Fitting	C₂²×C₄×C₈×C₇₉	C₂
socle, prime radical	C₂⁴×C₇₉	C₂×D₄
cosocle	C₂²×C₄×C₇₉	C₂⁴
hypercenter	C₂²×C₄×C₈	D₇₉
hypocenter	C₇₉	C₂×#1846
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
4. #23763564	C₂	direct (N × H)
17. C₂×C₄×C₈×C₇₉	C₄	semidirect
41. C₄×C₈×C₇₉	C₂×C₄	semidirect
113. C₂×C₇₉	#1846	semidirect
123. C₇₉	C₂×#1846	semidirect
132. C₂×C₄×C₈	Dic₇₉	semidirect
158. C₄×C₈	C₂×Dic₇₉	semidirect
230. C₂	#23763564	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
C₂²×#14579093	2
C₂×#19098429	2
#23763564	2
C₂²×C₄×C₈×C₇₉	2
C₂×#1846	79

Subgroups — C2×#23763564 #34288676

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#23763564 #34288676