Subgroups — #34444969

Order 20224.

676 conjugacy classes of subgroups.

The subgroup lattice has 264 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	2	yes	maximal, complemented
2	#23775122	C₂	1	2	yes	maximal, complemented
3	#23765653	C₂	1	1	yes	maximal, characteristic, complemented
4	#23765491	C₂	1	2	yes	maximal, complemented
5	#23765446	C₂	1	2	yes	maximal, complemented
6	#23765342	C₂	1	4	yes	maximal, complemented
7	C₄×D₁₂₆₄	C₂	1	1	yes	maximal, characteristic, complemented
8	D₄×C₁₆×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#14579183	C₂²	1	4	yes	—
10	#19099218	—	2	2	no	—
11	#19098995	—	2	8	no	—
12	C₂×Dic₆₃₂	C₂²	1	2	yes	—
13	C₂×Dic₆₃₂	C₂²	1	2	yes	—
14	C₂×#14579106	C₂²	1	4	yes	—
15	C₂×D₁₂₆₄	C₂²	1	1	yes	characteristic
16	#19098943	C₂²	1	2	yes	—
17	#19098804	C₂²	1	4	yes	—
18	#19098769	—	2	2	no	—
19	#19098727	—	2	4	no	—
20	C₄×D₆₃₂	C₂²	1	2	yes	—
21	#19098389	C₂²	1	1	yes	characteristic
22	#19098368	C₂²	1	4	yes	—
23	#19098365	C₂²	1	2	yes	—
24	C₂²×C₁₆×C₇₉	C₂²	1	2	yes	—
25	C₈×D₄×C₇₉	C₂²	1	1	yes	characteristic
26	C₄⋊C₁₆×C₇₉	C₂²	1	1	yes	characteristic
27	C₂²⋊C₁₆×C₇₉	C₂²	1	2	yes	—
28	C₄×C₁₆×C₇₉	C₂²	1	1	yes	characteristic
29	C₂×#11929624	—	2	4	no	—
30	#14579183	—	2	8	no	—
31	#14579183	—	4	4	no	—
32	C₂×Dic₃₁₆	—	2	2	no	—
33	C₂×Dic₃₁₆	C₂³	1	4	yes	—
34	C₂×#11929613	—	2	4	no	—
35	C₂×D₆₃₂	C₂³	1	2	yes	—
36	#14579161	—	2	2	no	—
37	#14579143	—	4	4	no	—
38	#14579140	—	2	4	no	—
39	#14579132	—	4	2	no	—
40	C₄×D₃₁₆	—	2	2	no	—
41	C₄⋊C₄×D₇₉	—	4	2	no	—
42	Dic₆₃₂	—	2	4	no	—
43	Dic₆₃₂	—	4	2	no	—
44	#14579106	—	2	8	no	—
45	D₁₂₆₄	D₄	1	4	yes	complemented
46	#14579095	C₂³	1	2	yes	—
47	#14579086	—	2	4	no	—
48	#14579083	—	2	2	no	—
49	C₂²×C₈×C₇₉	C₂³	1	2	yes	—
50	C₄×D₄×C₇₉	D₄	1	1	yes	characteristic
51	C₂×C₁₆×C₇₉	—	2	4	no	—
52	C₂×C₁₆×C₇₉	C₂³	1	2	yes	—
53	C₂×C₁₆×C₇₉	C₂³	1	1	yes	characteristic
54	C₂×C₁₆×C₇₉	C₂³	1	1	yes	characteristic
55	C₄⋊C₈×C₇₉	D₄	1	1	yes	characteristic
56	C₂²⋊C₈×C₇₉	D₄	1	2	yes	—
57	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
58	#11929624	—	4	8	no	—
59	#11929624	—	8	4	no	—
60	C₂×Dic₁₅₈	—	2	4	no	—
61	C₂×D₃₁₆	—	2	2	no	—
62	C₂×C₂²⋊D₇₉	—	4	4	no	—
63	C₂×C₄×D₇₉	—	4	4	no	—
64	C₂×C₄×D₇₉	—	4	2	no	—
65	Dic₃₁₆	—	2	4	no	—
66	Dic₃₁₆	—	2	4	no	—
67	Dic₃₁₆	—	8	2	no	—
68	#11929613	—	4	8	no	—
69	D₆₃₂	—	2	2	no	—
70	D₆₃₂	C₂×D₄	1	4	yes	—
71	C₄⋊Dic₇₉	—	2	2	no	—
72	C₄⋊Dic₇₉	—	4	2	no	—
73	C₄⋊Dic₇₉	—	4	2	no	—
74	Dic₇₉⋊C₄	—	4	4	no	—
75	C₂²⋊Dic₇₉	—	4	4	no	—
76	(C₂×D₇₉)⋊C₄	—	4	2	no	—
77	C₂×D₄×C₇₉	D₈	1	1	yes	characteristic
78	C₂²×C₄×C₇₉	C₂×D₄	1	2	yes	—
79	C₂×C₈×C₇₉	—	2	2	no	—
80	C₂×C₈×C₇₉	—	2	2	no	—
81	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
82	C₂×C₈×C₇₉	C₂×D₄	1	2	yes	—
83	C₂×C₈×C₇₉	C₂⁴	1	1	yes	characteristic
84	C₄⋊C₄×C₇₉	D₈	1	1	yes	characteristic
85	C₂²⋊C₄×C₇₉	D₈	1	2	yes	—
86	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
87	C₁₆×C₇₉	—	2	2	no	—
88	C₁₆×C₇₉	—	2	1	no	—
89	C₁₆×C₇₉	C₂×D₄	1	2	yes	—
90	C₂³×C₇₉	C₂×D₈	1	2	yes	—
91	C₂²×D₇₉	—	4	2	no	—
92	D₄×C₇₉	—	2	2	no	—
93	C₂×C₄×C₇₉	—	2	2	no	—
94	C₂×C₄×C₇₉	—	2	2	no	—
95	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
96	C₂×C₄×C₇₉	C₂×D₈	1	2	yes	—
97	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
98	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
99	C₂²⋊D₇₉	—	8	8	no	—
100	C₂×Dic₇₉	—	4	2	no	—
101	C₂×Dic₇₉	—	4	4	no	—
102	D₃₁₆	—	2	4	no	—
103	D₃₁₆	—	4	2	no	—
104	C₄×D₇₉	—	8	8	no	—
105	C₄×D₇₉	—	8	4	no	—
106	Dic₁₅₈	—	4	4	no	—
107	Dic₁₅₈	—	4	4	no	—
108	C₈×C₇₉	—	2	2	no	—
109	C₈×C₇₉	—	2	1	no	—
110	C₈×C₇₉	C₂²×D₄	1	1	yes	characteristic
111	C₈×C₇₉	H(5,𝔽₂)	1	1	yes	characteristic
112	C₂²×C₇₉	—	2	2	no	—
113	C₂²×C₇₉	—	2	2	no	—
114	C₂²×C₇₉	—	2	2	no	—
115	C₂²×C₇₉	C₂²×D₈	1	1	yes	characteristic
116	C₂×D₇₉	—	4	4	no	—
117	C₂×D₇₉	—	8	2	no	—
118	C₂×D₇₉	—	8	2	no	—
119	C₄×C₇₉	—	2	1	no	—
120	C₄×C₇₉	—	2	2	no	—
121	C₄×C₇₉	#508	1	2	yes	complemented
122	C₄×C₇₉	D₄²	1	1	yes	characteristic
123	C₄×C₇₉	#576	1	1	yes	characteristic
124	Dic₇₉	—	8	2	no	—
125	Dic₇₉	—	8	4	no	—
126	#33849	—	79	1	no	maximal, Sylow
127	C₂×C₇₉	—	4	2	no	—
128	C₂×C₇₉	D₄×D₈	1	1	yes	characteristic
129	C₂×C₇₉	C₂×#508	1	1	yes	characteristic
130	C₂×C₇₉	#3417	1	1	yes	characteristic
131	D₇₉	—	8	4	no	—
132	C₂×#508	—	79	2	no	—
133	#3283	—	79	2	no	—
134	#2247	—	79	1	no	—
135	#2216	—	79	2	no	—
136	#2208	—	79	2	no	—
137	#2189	—	79	4	no	—
138	C₄×D₁₆	—	79	1	no	—
139	D₄×C₁₆	D₇₉	1	1	yes	characteristic, complemented
140	C₇₉	#33849	1	1	yes	characteristic, complemented, Sylow
141	C₂×#135	—	79	4	no	—
142	#547	—	158	2	no	—
143	#508	—	158	8	no	—
144	C₂×Q₃₂	—	79	2	no	—
145	C₂×Q₃₂	—	79	2	no	—
146	C₂×SD₃₂	—	79	4	no	—
147	C₂×D₁₆	—	79	1	no	—
148	C₂²×C₁₆	C₂×D₇₉	1	2	yes	—
149	#494	—	79	2	no	—
150	#467	—	79	4	no	—
151	#461	—	158	2	no	—
152	#452	—	158	4	no	—
153	C₄×D₈	—	79	2	no	—
154	C₈×D₄	C₂×D₇₉	1	1	yes	characteristic
155	#366	—	79	1	no	—
156	C₄⋊C₁₆	C₂×D₇₉	1	1	yes	characteristic
157	#358	—	79	4	no	—
158	#357	—	79	2	no	—
159	C₂²⋊C₁₆	C₂×D₇₉	1	2	yes	—
160	C₄×C₁₆	C₂×D₇₉	1	1	yes	characteristic
161	C₂×#41	—	158	4	no	—
162	#135	—	316	4	no	—
163	#135	—	158	8	no	—
164	C₂×Q₁₆	—	79	4	no	—
165	C₂×Q₁₆	—	158	2	no	—
166	C₂×SD₁₆	—	158	4	no	—
167	C₂×D₈	—	79	2	no	—
168	C₂²×C₈	C₂²×D₇₉	1	2	yes	—
169	C₄⋊Q₈	—	158	2	no	—
170	#123	—	316	4	no	—
171	C₂²⋊Q₈	—	158	4	no	—
172	#121	—	316	2	no	—
173	C₄×D₄	—	158	2	no	—
174	C₄×D₄	D₃₁₆	1	1	yes	characteristic
175	C₂×C₄⋊C₄	—	316	2	no	—
176	Q₃₂	—	316	2	no	—
177	Q₃₂	—	158	4	no	—
178	SD₃₂	—	158	8	no	—
179	D₁₆	—	79	4	no	—
180	C₂×C₁₆	—	2	4	no	—
181	C₂×C₁₆	C₂²×D₇₉	1	1	yes	characteristic
182	C₂×C₁₆	C₂²×D₇₉	1	2	yes	—
183	C₂×C₁₆	C₂²×D₇₉	1	1	yes	characteristic
184	#107	—	79	2	no	—
185	C₄⋊C₈	D₃₁₆	1	1	yes	characteristic
186	#103	—	158	4	no	—
187	#102	—	158	2	no	—
188	C₂²⋊C₈	D₃₁₆	1	2	yes	—
189	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
190	#41	—	316	8	no	—
191	#41	—	632	4	no	—
192	C₂×Q₈	—	158	4	no	—
193	C₂×D₄	—	316	4	no	—
194	C₂×D₄	—	158	2	no	—
195	C₂×D₄	D₆₃₂	1	1	yes	characteristic
196	C₂²×C₄	—	316	2	no	—
197	C₂²×C₄	—	316	4	no	—
198	C₂²×C₄	C₂×D₃₁₆	1	2	yes	—
199	Q₁₆	—	632	2	no	—
200	Q₁₆	—	158	4	no	—
201	Q₁₆	—	158	4	no	—
202	SD₁₆	—	316	8	no	—
203	D₈	—	79	4	no	—
204	D₈	—	158	2	no	—
205	C₂×C₈	—	2	2	no	—
206	C₂×C₈	—	2	2	no	—
207	C₂×C₈	C₂×D₃₁₆	1	1	yes	characteristic
208	C₂×C₈	C₂×D₃₁₆	1	2	yes	—
209	C₂×C₈	C₂³×D₇₉	1	1	yes	characteristic
210	C₄⋊C₄	—	316	4	no	—
211	C₄⋊C₄	—	316	2	no	—
212	C₄⋊C₄	—	316	2	no	—
213	C₄⋊C₄	—	158	2	no	—
214	C₄⋊C₄	D₆₃₂	1	1	yes	characteristic
215	C₂²⋊C₄	—	316	4	no	—
216	C₂²⋊C₄	—	316	2	no	—
217	C₂²⋊C₄	D₆₃₂	1	2	yes	—
218	C₄²	C₂×D₃₁₆	1	1	yes	characteristic
219	C₁₆	—	2	1	no	—
220	C₁₆	—	2	2	no	—
221	C₁₆	D₄×D₇₉	1	2	yes	—
222	C₂³	—	316	2	no	—
223	C₂³	C₂×D₆₃₂	1	2	yes	—
224	Q₈	—	316	4	no	—
225	Q₈	—	316	4	no	—
226	D₄	—	2	2	no	—
227	D₄	—	316	2	no	—
228	D₄	—	632	8	no	—
229	D₄	—	158	4	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	316	2	no	—
233	C₂×C₄	—	316	4	no	—
234	C₂×C₄	—	632	8	no	—
235	C₂×C₄	—	632	4	no	—
236	C₂×C₄	C₂×D₆₃₂	1	1	yes	characteristic
237	C₂×C₄	C₂×D₆₃₂	1	2	yes	—
238	C₂×C₄	C₂×D₆₃₂	1	1	yes	characteristic
239	C₂×C₄	C₂²×D₃₁₆	1	1	yes	characteristic
240	C₈	—	2	2	no	—
241	C₈	—	2	1	no	—
242	C₈	C₂×D₄×D₇₉	1	1	yes	characteristic
243	C₈	#14579207	1	1	yes	characteristic
244	C₂²	—	2	2	no	—
245	C₂²	—	2	2	no	—
246	C₂²	—	2	2	no	—
247	C₂²	—	316	4	no	—
248	C₂²	—	632	2	no	—
249	C₂²	—	632	2	no	—
250	C₂²	C₂²×D₆₃₂	1	1	yes	characteristic
251	C₄	—	2	1	no	—
252	C₄	—	2	2	no	—
253	C₄	—	632	2	no	—
254	C₄	—	632	4	no	—
255	C₄	#19098995	1	2	yes	complemented
256	C₄	D₄×D₃₁₆	1	1	yes	characteristic
257	C₄	#19099368	1	1	yes	characteristic
258	C₂	—	4	2	no	—
259	C₂	—	632	4	no	—
260	C₂	D₄×D₆₃₂	1	1	yes	characteristic
261	C₂	C₂×#19098995	1	1	yes	characteristic
262	C₂	#23776553	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₆₃₂
commutator, cosocle	C₂×C₈×C₇₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₇₉
Fitting	D₄×C₁₆×C₇₉	C₂
socle	C₂²×C₇₉	C₂²×D₈
hypercenter	D₄×C₁₆	D₇₉
hypocenter	C₇₉	#33849
LCS	G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ C₈×D₄ ◁ D₄×C₁₆

Complements

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

N	H ≅ G/N	type
1. C₂×#19098995	C₂	semidirect
2. #23775122	C₂	semidirect
3. #23765653	C₂	semidirect
4. #23765491	C₂	semidirect
5. #23765446	C₂	semidirect
6. #23765342	C₂	semidirect
7. C₄×D₁₂₆₄	C₂	semidirect
8. D₄×C₁₆×C₇₉	C₂	semidirect
45. D₁₂₆₄	D₄	semidirect
121. C₄×C₇₉	#508	semidirect
139. D₄×C₁₆	D₇₉	semidirect
140. C₇₉	#33849	semidirect
255. 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
#23775122	2
#23765653	2
#23765491	2
#23765446	2
#23765342	2
C₄×D₁₂₆₄	2
D₄×C₁₆×C₇₉	2
#33849	79