Subgroups — C₂×#23773914 #35157874

Order 20224.

1942 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19099120	C₂	1	2	yes	maximal
2	#23773914	C₂	1	16	yes	maximal, direct product factor
3	C₂×#19098968	C₂	1	2	yes	maximal, complemented
4	C₂×#19098965	C₂	1	4	yes	maximal, complemented
5	C₂×#19098961	C₂	1	2	yes	maximal, complemented
6	C₂×#19098601	C₂	1	4	yes	maximal
7	C₂²×C₄×C₈×C₇₉	C₂	1	1	yes	maximal, characteristic
8	#19099120	—	2	8	no	—
9	#19099120	C₂²	1	16	yes	—
10	C₂×#14579161	C₂²	1	4	yes	—
11	C₂×#14579153	C₂²	1	4	yes	—
12	C₂×#14579140	—	2	4	no	—
13	C₂×C₄×Dic₁₅₈	—	2	4	no	—
14	C₂×#14579120	C₂²	1	4	yes	—
15	#19098968	C₂²	1	16	yes	complemented
16	#19098965	C₂²	1	32	yes	complemented
17	#19098961	C₂²	1	16	yes	complemented
18	#19098601	C₂²	1	32	yes	—
19	C₂×#14579095	C₂²	1	8	yes	—
20	C₂×#14579093	C₂²	1	8	yes	—
21	C₂³×C₈×C₇₉	C₂²	1	2	yes	—
22	C₂²×C₄²×C₇₉	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
24	C₂×C₄×C₈×C₇₉	C₂²	1	2	yes	—
25	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
26	C₂×C₄×C₈×C₇₉	C₂²	1	8	yes	—
27	C₂²×Dic₁₅₈	—	2	4	no	—
28	#14579161	—	2	8	no	—
29	#14579161	C₂³	1	16	yes	—
30	#14579153	—	2	8	no	—
31	#14579153	C₂³	1	16	yes	—
32	#14579140	—	2	32	no	—
33	C₄×Dic₁₅₈	—	2	32	no	—
34	#14579120	—	2	8	no	—
35	#14579120	C₂³	1	16	yes	—
36	C₂×C₄⋊Dic₇₉	—	2	4	no	—
37	C₂×C₄⋊Dic₇₉	C₂³	1	8	yes	—
38	C₂×Dic₇₉⋊C₄	—	2	8	no	—
39	C₂×C₂²⋊Dic₇₉	—	2	4	no	—
40	C₂×C₄×Dic₇₉	—	2	4	no	—
41	#14579095	C₂³	1	32	yes	—
42	#14579093	C₂³	1	32	yes	—
43	C₂³×C₄×C₇₉	D₄	1	2	yes	—
44	C₂³×C₄×C₇₉	C₂³	1	1	yes	characteristic
45	C₂²×C₈×C₇₉	Q₈	1	16	yes	complemented
46	C₂²×C₈×C₇₉	C₂³	1	2	yes	—
47	C₂²×C₈×C₇₉	C₂³	1	2	yes	—
48	C₂²×C₈×C₇₉	C₂³	1	8	yes	—
49	C₂×C₄²×C₇₉	D₄	1	2	yes	—
50	C₂×C₄²×C₇₉	D₄	1	4	yes	—
51	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
52	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
53	C₂×C₄²×C₇₉	C₂³	1	4	yes	—
54	C₄×C₈×C₇₉	C₂³	1	4	yes	—
55	C₄×C₈×C₇₉	C₂³	1	8	yes	—
56	C₄×C₈×C₇₉	C₂³	1	4	yes	—
57	C₂×Dic₁₅₈	—	2	16	no	—
58	C₂×Dic₁₅₈	—	4	16	no	—
59	C₂²×Dic₇₉	—	2	8	no	—
60	C₄⋊Dic₇₉	—	2	16	no	—
61	C₄⋊Dic₇₉	—	2	8	no	—
62	C₄⋊Dic₇₉	C₂⁴	1	16	yes	—
63	Dic₇₉⋊C₄	—	2	32	no	—
64	C₂²⋊Dic₇₉	—	2	16	no	—
65	C₄×Dic₇₉	—	2	16	no	—
66	C₂⁴×C₇₉	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₇₉	—	2	8	no	—
68	C₂²×C₄×C₇₉	C₂×D₄	1	2	yes	—
69	C₂²×C₄×C₇₉	C₂×D₄	1	8	yes	—
70	C₂²×C₄×C₇₉	C₂×D₄	1	2	yes	—
71	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₇₉	C₂×D₄	1	2	yes	—
73	C₂²×C₄×C₇₉	C₂×Q₈	1	8	yes	—
74	C₂²×C₄×C₇₉	C₂⁴	1	1	yes	characteristic
75	C₂²×C₄×C₇₉	C₂⁴	1	2	yes	—
76	C₂×C₈×C₇₉	C₂×Q₈	1	8	yes	—
77	C₂×C₈×C₇₉	C₂×Q₈	1	8	yes	—
78	C₂×C₈×C₇₉	C₂×Q₈	1	32	yes	complemented
79	C₂×C₈×C₇₉	C₂⁴	1	4	yes	—
80	C₂×C₈×C₇₉	C₂⁴	1	4	yes	—
81	C₄²×C₇₉	C₂×D₄	1	8	yes	—
82	C₄²×C₇₉	C₂×D₄	1	2	yes	—
83	C₄²×C₇₉	C₂×D₄	1	2	yes	—
84	C₄²×C₇₉	C₂⁴	1	2	yes	—
85	C₄²×C₇₉	C₂⁴	1	2	yes	—
86	C₂³×C₇₉	—	2	4	no	—
87	C₂³×C₇₉	C₄⋊Q₈	1	4	yes	—
88	C₂³×C₇₉	C₂²×D₄	1	2	yes	—
89	C₂³×C₇₉	C₂²×D₄	1	1	yes	characteristic
90	C₂×C₄×C₇₉	—	2	4	no	—
91	C₂×C₄×C₇₉	—	2	16	no	—
92	C₂×C₄×C₇₉	C₄⋊Q₈	1	8	yes	—
93	C₂×C₄×C₇₉	C₄⋊Q₈	1	4	yes	—
94	C₂×C₄×C₇₉	#135	1	8	yes	—
95	C₂×C₄×C₇₉	C₂²×D₄	1	4	yes	—
96	C₂×C₄×C₇₉	C₂²×D₄	1	2	yes	—
97	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
98	C₂×C₄×C₇₉	C₂²×D₄	1	4	yes	—
99	C₂×C₄×C₇₉	C₂²×Q₈	1	4	yes	—
100	C₂×C₄×C₇₉	C₂²×Q₈	1	8	yes	—
101	C₂×C₄×C₇₉	C₂⁵	1	1	yes	characteristic
102	C₂×Dic₇₉	—	2	16	no	—
103	C₂×Dic₇₉	—	4	16	no	—
104	Dic₁₅₈	—	4	32	no	—
105	C₈×C₇₉	C₂²×Q₈	1	8	yes	—
106	C₈×C₇₉	C₂²×Q₈	1	8	yes	—
107	C₂²×C₇₉	—	2	8	no	—
108	C₂²×C₇₉	—	2	4	no	—
109	C₂²×C₇₉	C₂×C₄⋊Q₈	1	4	yes	—
110	C₂²×C₇₉	C₂×C₄⋊Q₈	1	2	yes	—
111	C₂²×C₇₉	C₂×#135	1	4	yes	—
112	C₂²×C₇₉	C₂³×D₄	1	1	yes	characteristic
113	C₄×C₇₉	—	2	4	no	—
114	C₄×C₇₉	C₂×C₄⋊Q₈	1	4	yes	—
115	C₄×C₇₉	C₂×C₄⋊Q₈	1	2	yes	—
116	C₄×C₇₉	C₂×#135	1	8	yes	—
117	C₄×C₇₉	C₂³×Q₈	1	2	yes	—
118	Dic₇₉	—	4	16	no	—
119	C₂×#3160	—	79	1	no	maximal, Sylow
120	C₂×C₇₉	—	2	4	no	—
121	C₂×C₇₉	#3160	1	4	yes	complemented
122	C₂×C₇₉	C₂²×C₄⋊Q₈	1	1	yes	characteristic
123	C₂×C₇₉	C₂²×#135	1	2	yes	—
124	C₂×#533	—	79	2	no	—
125	#3160	—	79	16	no	—
126	C₂×#500	—	79	2	no	—
127	C₂×#499	—	79	4	no	—
128	C₂×#498	—	79	2	no	—
129	C₂×#427	—	79	4	no	—
130	C₂²×C₄×C₈	D₇₉	1	1	yes	characteristic
131	C₇₉	C₂×#3160	1	1	yes	characteristic, complemented, Sylow
132	C₂³×C₈	C₂×D₇₉	1	2	yes	—
133	#533	—	79	16	no	—
134	#533	—	158	8	no	—
135	C₂×C₄⋊Q₈	—	79	4	no	—
136	C₂×#125	—	79	4	no	—
137	C₂×C₂²⋊Q₈	—	158	4	no	—
138	C₂×C₄×Q₈	—	158	4	no	—
139	C₂×#117	—	79	4	no	—
140	C₂²×C₄²	C₂×D₇₉	1	1	yes	characteristic
141	#500	—	79	16	no	—
142	#499	—	79	32	no	—
143	#498	—	79	16	no	—
144	#427	—	79	32	no	—
145	C₂×#107	—	79	8	no	—
146	C₂×#106	—	79	8	no	—
147	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
148	C₂×C₄×C₈	C₂×D₇₉	1	2	yes	—
149	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
150	C₂×C₄×C₈	C₂×D₇₉	1	8	yes	—
151	C₂²×Q₈	—	158	4	no	—
152	C₂³×C₄	D₃₁₆	1	2	yes	—
153	C₂³×C₄	C₂²×D₇₉	1	1	yes	characteristic
154	C₂²×C₈	Dic₁₅₈	1	16	yes	complemented
155	C₂²×C₈	C₂²×D₇₉	1	2	yes	—
156	C₂²×C₈	C₂²×D₇₉	1	2	yes	—
157	C₂²×C₈	C₂²×D₇₉	1	8	yes	—
158	C₄⋊Q₈	—	79	16	no	—
159	C₄⋊Q₈	—	158	8	no	—
160	#125	—	79	16	no	—
161	#125	—	158	8	no	—
162	C₂²⋊Q₈	—	158	32	no	—
163	C₄×Q₈	—	158	32	no	—
164	#117	—	79	16	no	—
165	#117	—	158	8	no	—
166	C₂×C₄⋊C₄	—	79	8	no	—
167	C₂×C₄⋊C₄	—	158	8	no	—
168	C₂×C₄⋊C₄	—	158	4	no	—
169	C₂×C₂²⋊C₄	—	158	4	no	—
170	C₂×C₄²	—	158	4	no	—
171	C₂×C₄²	D₃₁₆	1	4	yes	—
172	C₂×C₄²	D₃₁₆	1	2	yes	—
173	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
174	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
175	C₂×C₄²	C₂²×D₇₉	1	4	yes	—
176	#107	—	79	32	no	—
177	#106	—	79	32	no	—
178	C₄×C₈	C₂²×D₇₉	1	4	yes	—
179	C₄×C₈	C₂²×D₇₉	1	8	yes	—
180	C₄×C₈	C₂²×D₇₉	1	4	yes	—
181	C₂⁴	C₂×D₃₁₆	1	1	yes	characteristic
182	C₂×Q₈	—	316	16	no	—
183	C₂×Q₈	—	158	16	no	—
184	C₂²×C₄	—	2	8	no	—
185	C₂²×C₄	—	158	8	no	—
186	C₂²×C₄	C₂×D₃₁₆	1	2	yes	—
187	C₂²×C₄	C₂×D₃₁₆	1	8	yes	—
188	C₂²×C₄	C₂×D₃₁₆	1	2	yes	—
189	C₂²×C₄	C₂×D₃₁₆	1	2	yes	—
190	C₂²×C₄	C₂×D₃₁₆	1	1	yes	characteristic
191	C₂²×C₄	C₂×Dic₁₅₈	1	8	yes	—
192	C₂²×C₄	C₂³×D₇₉	1	1	yes	characteristic
193	C₂²×C₄	C₂³×D₇₉	1	2	yes	—
194	C₂×C₈	C₂×Dic₁₅₈	1	8	yes	—
195	C₂×C₈	C₂×Dic₁₅₈	1	8	yes	—
196	C₂×C₈	C₂×Dic₁₅₈	1	32	yes	complemented
197	C₂×C₈	C₂³×D₇₉	1	4	yes	—
198	C₂×C₈	C₂³×D₇₉	1	4	yes	—
199	C₄⋊C₄	—	79	16	no	—
200	C₄⋊C₄	—	158	32	no	—
201	C₄⋊C₄	—	158	8	no	—
202	C₄⋊C₄	—	158	16	no	—
203	C₂²⋊C₄	—	158	16	no	—
204	C₄²	—	158	16	no	—
205	C₄²	C₂×D₃₁₆	1	8	yes	—
206	C₄²	C₂×D₃₁₆	1	2	yes	—
207	C₄²	C₂×D₃₁₆	1	2	yes	—
208	C₄²	C₂³×D₇₉	1	2	yes	—
209	C₄²	C₂³×D₇₉	1	2	yes	—
210	C₂³	—	2	4	no	—
211	C₂³	#14579161	1	4	yes	—
212	C₂³	C₂²×D₃₁₆	1	2	yes	—
213	C₂³	C₂²×D₃₁₆	1	1	yes	characteristic
214	Q₈	—	316	32	no	—
215	C₂×C₄	—	2	4	no	—
216	C₂×C₄	—	2	16	no	—
217	C₂×C₄	—	316	16	no	—
218	C₂×C₄	—	158	16	no	—
219	C₂×C₄	#14579161	1	8	yes	—
220	C₂×C₄	#14579161	1	4	yes	—
221	C₂×C₄	#14579183	1	8	yes	—
222	C₂×C₄	C₂²×D₃₁₆	1	4	yes	—
223	C₂×C₄	C₂²×D₃₁₆	1	4	yes	—
224	C₂×C₄	C₂²×D₃₁₆	1	2	yes	—
225	C₂×C₄	C₂²×D₃₁₆	1	1	yes	characteristic
226	C₂×C₄	C₂²×Dic₁₅₈	1	4	yes	—
227	C₂×C₄	C₂²×Dic₁₅₈	1	8	yes	—
228	C₂×C₄	C₂⁴×D₇₉	1	1	yes	characteristic
229	C₈	C₂²×Dic₁₅₈	1	8	yes	—
230	C₈	C₂²×Dic₁₅₈	1	8	yes	—
231	C₂²	—	2	4	no	—
232	C₂²	—	2	8	no	—
233	C₂²	C₂×#14579161	1	4	yes	—
234	C₂²	C₂×#14579161	1	2	yes	—
235	C₂²	C₂×#14579183	1	4	yes	—
236	C₂²	C₂³×D₃₁₆	1	1	yes	characteristic
237	C₄	—	2	4	no	—
238	C₄	—	316	16	no	—
239	C₄	C₂×#14579161	1	2	yes	—
240	C₄	C₂×#14579161	1	4	yes	—
241	C₄	C₂×#14579183	1	8	yes	—
242	C₄	C₂³×Dic₁₅₈	1	2	yes	—
243	C₂	—	2	4	no	—
244	C₂	#23773914	1	4	yes	direct product factor
245	C₂	C₂²×#14579161	1	1	yes	characteristic
246	C₂	C₂²×#14579183	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×D₃₁₆
commutator, cosocle	C₂×C₄×C₇₉	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₇₉
Fitting	C₂²×C₄×C₈×C₇₉	C₂
socle	C₂³×C₇₉	C₂²×D₄
prime radical	C₂⁴×C₇₉	C₂×D₄
hypercenter	C₂²×C₄×C₈	D₇₉
hypocenter	C₇₉	C₂×#3160
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
2. #23773914	C₂	direct (N × H)
3. C₂×#19098968	C₂	semidirect
4. C₂×#19098965	C₂	semidirect
5. C₂×#19098961	C₂	semidirect
15. #19098968	C₂²	semidirect
16. #19098965	C₂²	semidirect
17. #19098961	C₂²	semidirect
45. C₂²×C₈×C₇₉	Q₈	semidirect
78. C₂×C₈×C₇₉	C₂×Q₈	semidirect
121. C₂×C₇₉	#3160	semidirect
131. C₇₉	C₂×#3160	semidirect
154. C₂²×C₈	Dic₁₅₈	semidirect
196. C₂×C₈	C₂×Dic₁₅₈	semidirect
244. C₂	#23773914	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#19099120	2
#23773914	2
C₂×#19098968	2
C₂×#19098965	2
C₂×#19098961	2
C₂×#19098601	2
C₂²×C₄×C₈×C₇₉	2
C₂×#3160	79

Subgroups — C2×#23773914 #35157874

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#23773914 #35157874