Subgroups — #34443675

Order 20224.

834 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19098995	C₂	1	4	yes	maximal, complemented
2	#23773828	C₂	1	2	yes	maximal, complemented
3	#23765661	C₂	1	4	yes	maximal, complemented
4	#23765656	C₂	1	2	yes	maximal, complemented
5	#23765653	C₂	1	1	yes	maximal, characteristic, complemented
6	#23765650	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₄×C₁₆×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#14579183	—	2	2	no	—
9	C₂×#14579183	C₂²	1	4	yes	—
10	#19099113	—	2	2	no	—
11	#19098995	—	2	16	no	—
12	C₂×Dic₆₃₂	C₂²	1	4	yes	—
13	C₂×#14579106	C₂²	1	8	yes	—
14	C₂×D₁₂₆₄	C₂²	1	4	yes	—
15	#19098947	—	2	2	no	—
16	#19098947	C₂²	1	4	yes	—
17	#19098943	C₂²	1	2	yes	—
18	#19098940	C₂²	1	2	yes	—
19	#19098936	—	2	2	no	—
20	C₂²×C₁₆×C₇₉	C₂²	1	2	yes	—
21	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
22	C₄×C₁₆×C₇₉	C₂²	1	1	yes	characteristic
23	C₄×C₁₆×C₇₉	C₂²	1	1	yes	characteristic
24	C₄×C₁₆×C₇₉	C₂²	1	2	yes	—
25	C₂×#11929624	—	2	4	no	—
26	#14579183	—	2	16	no	—
27	#14579183	—	4	8	no	—
28	C₂×Dic₃₁₆	—	2	2	no	—
29	C₂×Dic₃₁₆	C₂³	1	4	yes	—
30	C₂×#11929613	—	2	4	no	—
31	C₂×#11929613	—	2	4	no	—
32	C₂×D₆₃₂	—	2	2	no	—
33	C₂×D₆₃₂	C₂³	1	4	yes	—
34	#14579161	—	2	2	no	—
35	#14579158	—	2	2	no	—
36	#14579149	—	2	4	no	—
37	#14579135	—	4	4	no	—
38	C₄×D₃₁₆	—	4	4	no	—
39	Dic₆₃₂	—	2	8	no	—
40	#14579106	—	2	16	no	—
41	D₁₂₆₄	—	2	8	no	—
42	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
43	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
44	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
45	C₂×C₁₆×C₇₉	D₄	1	8	yes	complemented
46	C₂×C₁₆×C₇₉	C₂³	1	2	yes	—
47	C₂×C₁₆×C₇₉	C₂³	1	2	yes	—
48	C₄×C₈×C₇₉	D₄	1	1	yes	characteristic
49	C₄×C₈×C₇₉	D₄	1	1	yes	characteristic
50	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
51	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
52	#11929624	—	4	8	no	—
53	#11929624	—	4	8	no	—
54	C₂×Dic₁₅₈	—	2	4	no	—
55	C₂×D₃₁₆	—	2	4	no	—
56	C₂×D₃₁₆	—	4	4	no	—
57	C₂×C₂²⋊D₇₉	—	4	4	no	—
58	C₂×C₄×D₇₉	—	4	4	no	—
59	Dic₃₁₆	—	2	8	no	—
60	Dic₃₁₆	—	4	4	no	—
61	#11929613	—	4	8	no	—
62	#11929613	—	4	8	no	—
63	D₆₃₂	—	2	8	no	—
64	D₆₃₂	—	4	4	no	—
65	C₄⋊Dic₇₉	—	4	4	no	—
66	(C₂×D₇₉)⋊C₄	—	4	8	no	—
67	C₂²×C₄×C₇₉	D₈	1	2	yes	—
68	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
69	C₂×C₈×C₇₉	D₈	1	4	yes	—
70	C₂×C₈×C₇₉	C₂×D₄	1	4	yes	—
71	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
72	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
73	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
74	C₂×C₈×C₇₉	C₂⁴	1	1	yes	characteristic
75	C₄²×C₇₉	D₈	1	2	yes	—
76	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
77	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
78	C₁₆×C₇₉	C₂×D₄	1	4	yes	—
79	C₁₆×C₇₉	C₂×D₄	1	4	yes	—
80	C₂³×C₇₉	C₂×D₈	1	1	yes	characteristic
81	C₂²×D₇₉	—	4	4	no	—
82	C₂×C₄×C₇₉	—	2	4	no	—
83	C₂×C₄×C₇₉	#127	1	2	yes	—
84	C₂×C₄×C₇₉	C₂×D₈	1	2	yes	—
85	C₂×C₄×C₇₉	C₂×D₈	1	2	yes	—
86	C₂×C₄×C₇₉	C₂×D₈	1	2	yes	—
87	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
88	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
89	C₂²⋊D₇₉	—	8	8	no	—
90	C₂×Dic₇₉	—	4	4	no	—
91	D₃₁₆	—	4	4	no	—
92	D₃₁₆	—	4	4	no	—
93	D₃₁₆	—	8	8	no	—
94	C₄×D₇₉	—	8	8	no	—
95	Dic₁₅₈	—	4	4	no	—
96	Dic₁₅₈	—	4	4	no	—
97	C₈×C₇₉	#127	1	2	yes	—
98	C₈×C₇₉	C₂×D₈	1	2	yes	—
99	C₈×C₇₉	C₂×D₈	1	2	yes	—
100	C₈×C₇₉	C₂²×D₄	1	2	yes	—
101	C₂²×C₇₉	—	2	2	no	—
102	C₂²×C₇₉	#493	1	2	yes	—
103	C₂²×C₇₉	C₂²×D₈	1	1	yes	characteristic
104	C₂×D₇₉	—	8	4	no	—
105	C₂×D₇₉	—	8	8	no	—
106	C₄×C₇₉	—	2	2	no	—
107	C₄×C₇₉	#493	1	2	yes	—
108	C₄×C₇₉	#508	1	4	yes	complemented
109	C₄×C₇₉	C₂×#127	1	1	yes	characteristic
110	C₄×C₇₉	C₂²×D₈	1	1	yes	characteristic
111	Dic₇₉	—	8	4	no	—
112	#33713	—	79	1	no	maximal, Sylow
113	C₂×C₇₉	—	2	2	no	—
114	C₂×C₇₉	C₂×#493	1	1	yes	characteristic
115	C₂×C₇₉	C₂×#508	1	2	yes	—
116	D₇₉	—	16	4	no	—
117	C₂×#508	—	79	4	no	—
118	#3147	—	79	2	no	—
119	#2249	—	79	4	no	—
120	#2248	—	79	2	no	—
121	#2247	—	79	1	no	—
122	#2246	—	79	1	no	—
123	C₂×C₄×C₁₆	D₇₉	1	1	yes	characteristic, complemented
124	C₇₉	#33713	1	1	yes	characteristic, complemented, Sylow
125	C₂×#135	—	79	4	no	—
126	C₂×#135	—	158	2	no	—
127	#532	—	158	2	no	—
128	#508	—	158	16	no	—
129	C₂×Q₃₂	—	79	4	no	—
130	C₂×SD₃₂	—	79	8	no	—
131	C₂×D₁₆	—	79	4	no	—
132	C₂²×C₁₆	C₂×D₇₉	1	2	yes	—
133	#495	—	79	4	no	—
134	#495	—	158	2	no	—
135	#494	—	79	2	no	—
136	#493	—	79	2	no	—
137	#492	—	158	2	no	—
138	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
139	C₄×C₁₆	C₂×D₇₉	1	1	yes	characteristic
140	C₄×C₁₆	C₂×D₇₉	1	1	yes	characteristic
141	C₄×C₁₆	C₂×D₇₉	1	2	yes	—
142	C₂×#41	—	158	4	no	—
143	#135	—	316	8	no	—
144	#135	—	158	16	no	—
145	C₂×Q₁₆	—	79	4	no	—
146	C₂×Q₁₆	—	158	2	no	—
147	C₂×SD₁₆	—	158	4	no	—
148	C₂×SD₁₆	—	158	4	no	—
149	C₂×D₈	—	79	4	no	—
150	C₂×D₈	—	158	2	no	—
151	C₂²×C₈	D₃₁₆	1	1	yes	characteristic
152	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
153	C₄⋊Q₈	—	158	2	no	—
154	#127	—	158	2	no	—
155	#124	—	158	4	no	—
156	#121	—	316	4	no	—
157	C₄×D₄	—	316	4	no	—
158	C₂×C₄²	D₃₁₆	1	1	yes	characteristic
159	Q₃₂	—	158	8	no	—
160	SD₃₂	—	158	16	no	—
161	D₁₆	—	158	8	no	—
162	C₂×C₁₆	D₃₁₆	1	8	yes	complemented
163	C₂×C₁₆	C₂²×D₇₉	1	2	yes	—
164	C₂×C₁₆	C₂²×D₇₉	1	2	yes	—
165	C₄×C₈	D₃₁₆	1	1	yes	characteristic
166	C₄×C₈	D₃₁₆	1	1	yes	characteristic
167	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
168	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
169	#41	—	316	8	no	—
170	#41	—	316	8	no	—
171	C₂×Q₈	—	158	4	no	—
172	C₂×D₄	—	316	4	no	—
173	C₂×D₄	—	316	4	no	—
174	C₂×D₄	—	158	4	no	—
175	C₂²×C₄	—	316	4	no	—
176	C₂²×C₄	D₆₃₂	1	2	yes	—
177	C₂²×C₄	C₂×D₃₁₆	1	1	yes	characteristic
178	Q₁₆	—	316	4	no	—
179	Q₁₆	—	158	8	no	—
180	SD₁₆	—	316	8	no	—
181	SD₁₆	—	316	8	no	—
182	D₈	—	316	4	no	—
183	D₈	—	158	8	no	—
184	C₂×C₈	D₆₃₂	1	4	yes	—
185	C₂×C₈	C₂×D₃₁₆	1	1	yes	characteristic
186	C₂×C₈	C₂×D₃₁₆	1	4	yes	—
187	C₂×C₈	C₂×D₃₁₆	1	1	yes	characteristic
188	C₂×C₈	C₂×D₃₁₆	1	1	yes	characteristic
189	C₂×C₈	C₂³×D₇₉	1	1	yes	characteristic
190	C₄⋊C₄	—	316	4	no	—
191	C₂²⋊C₄	—	316	8	no	—
192	C₄²	D₆₃₂	1	2	yes	—
193	C₄²	C₂×D₃₁₆	1	1	yes	characteristic
194	C₄²	C₂×D₃₁₆	1	1	yes	characteristic
195	C₁₆	C₂×D₃₁₆	1	4	yes	—
196	C₁₆	C₂×D₃₁₆	1	4	yes	—
197	C₂³	—	316	4	no	—
198	C₂³	C₂×D₆₃₂	1	1	yes	characteristic
199	Q₈	—	316	4	no	—
200	Q₈	—	316	4	no	—
201	D₄	—	316	4	no	—
202	D₄	—	316	4	no	—
203	D₄	—	632	8	no	—
204	D₄	—	632	8	no	—
205	C₂×C₄	—	2	4	no	—
206	C₂×C₄	—	316	4	no	—
207	C₂×C₄	—	632	8	no	—
208	C₂×C₄	#14579158	1	2	yes	—
209	C₂×C₄	C₂×D₆₃₂	1	2	yes	—
210	C₂×C₄	C₂×D₆₃₂	1	2	yes	—
211	C₂×C₄	C₂×D₆₃₂	1	2	yes	—
212	C₂×C₄	C₂×D₆₃₂	1	1	yes	characteristic
213	C₂×C₄	C₂²×D₃₁₆	1	1	yes	characteristic
214	C₈	#14579158	1	2	yes	—
215	C₈	C₂×D₆₃₂	1	2	yes	—
216	C₈	C₂×D₆₃₂	1	2	yes	—
217	C₈	C₂²×D₃₁₆	1	2	yes	—
218	C₂²	—	2	2	no	—
219	C₂²	—	632	8	no	—
220	C₂²	—	632	4	no	—
221	C₂²	#19098940	1	2	yes	—
222	C₂²	C₂²×D₆₃₂	1	1	yes	characteristic
223	C₄	—	2	2	no	—
224	C₄	—	632	4	no	—
225	C₄	#19098940	1	2	yes	—
226	C₄	#19098995	1	4	yes	complemented
227	C₄	C₂×#14579158	1	1	yes	characteristic
228	C₄	C₂²×D₆₃₂	1	1	yes	characteristic
229	C₂	—	2	2	no	—
230	C₂	—	1264	4	no	—
231	C₂	C₂×#19098940	1	1	yes	characteristic
232	C₂	C₂×#19098995	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₆₃₂
commutator, cosocle	C₂×C₈×C₇₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₇₉
Fitting	C₂×C₄×C₁₆×C₇₉	C₂
socle	C₂²×C₇₉	C₂²×D₈
hypercenter	C₂×C₄×C₁₆	D₇₉
hypocenter	C₇₉	#33713
LCS	G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
1. C₂×#19098995	C₂	semidirect
2. #23773828	C₂	semidirect
3. #23765661	C₂	semidirect
4. #23765656	C₂	semidirect
5. #23765653	C₂	semidirect
6. #23765650	C₂	semidirect
7. C₂×C₄×C₁₆×C₇₉	C₂	semidirect
45. C₂×C₁₆×C₇₉	D₄	semidirect
108. C₄×C₇₉	#508	semidirect
123. C₂×C₄×C₁₆	D₇₉	semidirect
124. C₇₉	#33713	semidirect
162. C₂×C₁₆	D₃₁₆	semidirect
226. C₄	#19098995	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19098995	2
#23773828	2
#23765661	2
#23765656	2
#23765653	2
#23765650	2
C₂×C₄×C₁₆×C₇₉	2
#33713	79