Subgroups — #35423310

Order 21248.

876 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×Q₈×C₈₃⋊C₈	C₂	1	2	yes	maximal
2	#24908182	C₂	1	6	yes	maximal
3	#24907580	C₂	1	3	yes	maximal
4	C₄×C₄⋊(C₈₃⋊C₈)	C₂	1	3	yes	maximal
5	C₄²×Q₈×C₈₃	C₂	1	1	yes	maximal, characteristic
6	Q₈×C₈₃⋊C₈	—	2	8	no	—
7	C₂×C₄⋊(C₈₃⋊C₈)	C₂²	1	6	yes	—
8	C₂×C₄⋊(C₈₃⋊C₈)	C₂²	1	6	yes	—
9	C₂×C₄×C₈₃⋊C₈	C₂²	1	6	yes	—
10	#19127456	C₂²	1	1	yes	characteristic
11	#19127456	C₂²	1	3	yes	—
12	#19127456	C₂²	1	6	yes	—
13	C₂×C₄×Q₈×C₈₃	C₄	1	1	yes	characteristic
14	C₂×C₄×Q₈×C₈₃	C₄	1	1	yes	characteristic
15	C₂×C₄×Q₈×C₈₃	C₂²	1	1	yes	characteristic
16	C₄×C₄⋊C₄×C₈₃	C₄	1	3	yes	—
17	C₄×C₄⋊C₄×C₈₃	C₄	1	3	yes	—
18	C₄×C₄⋊C₄×C₈₃	C₂²	1	3	yes	—
19	C₄³×C₈₃	C₂²	1	3	yes	—
20	C₂²×C₈₃⋊C₈	C₂³	1	6	yes	—
21	C₂²×C₈₃⋊C₈	C₂³	1	2	yes	—
22	C₄⋊(C₈₃⋊C₈)	—	2	12	no	—
23	C₄⋊(C₈₃⋊C₈)	—	2	12	no	—
24	C₄×C₈₃⋊C₈	—	2	12	no	—
25	C₂²×Q₈×C₈₃	C₂×C₄	1	1	yes	characteristic
26	C₄×Q₈×C₈₃	—	2	4	no	—
27	C₄×Q₈×C₈₃	C₈	1	8	yes	complemented
28	C₄×Q₈×C₈₃	D₄	1	4	yes	—
29	C₄×Q₈×C₈₃	Q₈	1	4	yes	—
30	C₂×C₄⋊C₄×C₈₃	C₂×C₄	1	3	yes	—
31	C₂×C₄⋊C₄×C₈₃	C₂×C₄	1	3	yes	—
32	C₂×C₄⋊C₄×C₈₃	C₂×C₄	1	3	yes	—
33	C₂×C₄²×C₈₃	C₂×C₄	1	3	yes	—
34	C₂×C₄²×C₈₃	C₂×C₄	1	3	yes	—
35	C₂×C₄²×C₈₃	C₂×C₄	1	3	yes	—
36	C₂×C₄²×C₈₃	C₂×C₄	1	3	yes	—
37	C₂×C₄²×C₈₃	C₂³	1	1	yes	characteristic
38	C₂×C₄²×C₈₃	C₂³	1	3	yes	—
39	C₂×C₄²×C₈₃	C₂³	1	3	yes	—
40	#95×C₈₃	C₂×C₄	1	6	yes	—
41	C₂×C₈₃⋊C₈	—	2	2	no	—
42	C₂×C₈₃⋊C₈	—	2	2	no	—
43	C₂×C₈₃⋊C₈	—	2	6	no	—
44	C₂×C₈₃⋊C₈	—	2	6	no	—
45	C₂×C₈₃⋊C₈	—	2	6	no	—
46	C₂×C₈₃⋊C₈	#41	1	4	yes	—
47	C₂×Q₈×C₈₃	C₄⋊C₄	1	4	yes	—
48	C₂×Q₈×C₈₃	C₂×C₈	1	4	yes	—
49	C₂×Q₈×C₈₃	M₁₆	1	4	yes	—
50	C₂²×C₄×C₈₃	C₂²×C₄	1	3	yes	—
51	C₂²×C₄×C₈₃	C₂²×C₄	1	3	yes	—
52	C₂²×C₄×C₈₃	C₂²×C₄	1	3	yes	—
53	C₂²×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
54	C₂²×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₈₃	C₂²×C₄	1	3	yes	—
56	C₂²×C₄×C₈₃	C₂⁴	1	1	yes	characteristic
57	C₄⋊C₄×C₈₃	—	2	6	no	—
58	C₄⋊C₄×C₈₃	C₄⋊C₄	1	12	yes	—
59	C₄⋊C₄×C₈₃	C₂×C₈	1	12	yes	—
60	C₄²×C₈₃	—	2	6	no	—
61	C₄²×C₈₃	—	2	6	no	—
62	C₄²×C₈₃	—	2	6	no	—
63	C₄²×C₈₃	—	2	2	no	—
64	C₄²×C₈₃	—	4	3	no	—
65	C₄²×C₈₃	C₂×C₈	1	12	yes	—
66	C₄²×C₈₃	C₂×D₄	1	6	yes	—
67	C₄²×C₈₃	C₂×Q₈	1	6	yes	—
68	C₂³×C₈₃	C₂³×C₄	1	1	yes	characteristic
69	Q₈×C₈₃	C₄⋊C₈	1	16	yes	complemented
70	C₂×C₄×C₈₃	—	2	3	no	—
71	C₂×C₄×C₈₃	—	2	1	no	—
72	C₂×C₄×C₈₃	—	2	1	no	—
73	C₂×C₄×C₈₃	—	2	1	no	—
74	C₂×C₄×C₈₃	—	2	3	no	—
75	C₂×C₄×C₈₃	—	2	3	no	—
76	C₂×C₄×C₈₃	—	2	1	no	—
77	C₂×C₄×C₈₃	—	2	3	no	—
78	C₂×C₄×C₈₃	—	2	3	no	—
79	C₂×C₄×C₈₃	—	2	3	no	—
80	C₂×C₄×C₈₃	—	2	3	no	—
81	C₂×C₄×C₈₃	C₂×C₄⋊C₄	1	6	yes	—
82	C₂×C₄×C₈₃	C₂×C₄⋊C₄	1	6	yes	—
83	C₂×C₄×C₈₃	C₂²×C₈	1	2	yes	—
84	C₂×C₄×C₈₃	C₂²×C₈	1	6	yes	—
85	C₂×C₄×C₈₃	C₂²×C₈	1	6	yes	—
86	C₂×C₄×C₈₃	C₂×M₁₆	1	6	yes	—
87	C₂×C₄×C₈₃	#131	1	2	yes	—
88	C₂×C₄×C₈₃	C₂²×D₄	1	1	yes	characteristic
89	C₂×C₄×C₈₃	C₂²×Q₈	1	1	yes	characteristic
90	C₂×C₄×C₈₃	C₂×#41	1	2	yes	—
91	C₂×C₄×C₈₃	2₋¹⁺⁴	1	2	yes	—
92	C₈₃⋊C₈	—	2	4	no	—
93	C₈₃⋊C₈	—	4	6	no	—
94	C₂²×C₈₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
95	C₂²×C₈₃	C₄×#41	1	1	yes	characteristic
96	C₂²×C₈₃	#519	1	1	yes	characteristic
97	C₂²×C₈₃	C₂³×C₈	1	1	yes	characteristic
98	C₂²×C₈₃	C₂²×M₁₆	1	1	yes	characteristic
99	C₂²×C₈₃	C₂×#131	1	1	yes	characteristic
100	C₂²×C₈₃	#568	1	1	yes	characteristic
101	C₄×C₈₃	—	2	6	no	—
102	C₄×C₈₃	—	2	6	no	—
103	C₄×C₈₃	—	2	2	no	—
104	C₄×C₈₃	—	2	2	no	—
105	C₄×C₈₃	—	4	3	no	—
106	C₄×C₈₃	C₂×C₄⋊C₈	1	12	yes	—
107	C₄×C₈₃	#548	1	2	yes	—
108	C₄×C₈₃	#557	1	2	yes	—
109	#22734	—	83	1	no	maximal, Sylow
110	C₂×C₈₃	#2351	1	1	yes	characteristic
111	C₂×C₈₃	C₂²×C₄⋊C₈	1	1	yes	characteristic
112	C₂×C₈₃	#2906	1	1	yes	characteristic
113	C₂×C₈₃	C₈×#41	1	1	yes	characteristic
114	C₂×C₈₃	#2971	1	1	yes	characteristic
115	C₂×C₈₃	#2982	1	1	yes	characteristic
116	C₂×C₈₃	#2984	1	1	yes	characteristic
117	C₂×C₈×Q₈	—	83	2	no	—
118	C₄²×Q₈	D₈₃	1	1	yes	characteristic
119	#1939	—	83	6	no	—
120	#1831	—	83	3	no	—
121	C₄×C₄⋊C₈	—	83	3	no	—
122	C₈₃	#22734	1	1	yes	characteristic, complemented, Sylow
123	C₂×C₄×Q₈	Dic₈₃	1	1	yes	characteristic
124	C₂×C₄×Q₈	Dic₈₃	1	1	yes	characteristic
125	C₂×C₄×Q₈	C₂×D₈₃	1	1	yes	characteristic
126	C₈×Q₈	—	166	8	no	—
127	C₂×C₄⋊C₈	—	83	6	no	—
128	C₂×C₄⋊C₈	—	83	6	no	—
129	C₂×C₄×C₈	—	83	6	no	—
130	C₄×C₄⋊C₄	Dic₈₃	1	3	yes	—
131	C₄×C₄⋊C₄	Dic₈₃	1	3	yes	—
132	C₄×C₄⋊C₄	C₂×D₈₃	1	3	yes	—
133	C₄³	C₂×D₈₃	1	3	yes	—
134	#336	—	83	3	no	—
135	#336	—	83	6	no	—
136	#336	—	83	1	no	—
137	C₂²×Q₈	C₂×Dic₈₃	1	1	yes	characteristic
138	C₂²×C₈	—	83	2	no	—
139	C₂²×C₈	—	83	6	no	—
140	C₄×Q₈	—	2	4	no	—
141	C₄×Q₈	C₈₃⋊C₈	1	8	yes	complemented
142	C₄×Q₈	Dic₁₆₆	1	4	yes	—
143	C₄×Q₈	D₃₃₂	1	4	yes	—
144	C₂×C₄⋊C₄	C₂×Dic₈₃	1	3	yes	—
145	C₂×C₄⋊C₄	C₂×Dic₈₃	1	3	yes	—
146	C₂×C₄⋊C₄	C₂×Dic₈₃	1	3	yes	—
147	C₂×C₄²	C₂×Dic₈₃	1	3	yes	—
148	C₂×C₄²	C₂×Dic₈₃	1	3	yes	—
149	C₂×C₄²	C₂×Dic₈₃	1	3	yes	—
150	C₂×C₄²	C₂×Dic₈₃	1	3	yes	—
151	C₂×C₄²	C₂²×D₈₃	1	1	yes	characteristic
152	C₂×C₄²	C₂²×D₈₃	1	3	yes	—
153	C₂×C₄²	C₂²×D₈₃	1	3	yes	—
154	C₄⋊C₈	—	166	12	no	—
155	C₄⋊C₈	—	166	12	no	—
156	C₄×C₈	—	166	12	no	—
157	#95	C₂×Dic₈₃	1	6	yes	—
158	C₂×Q₈	C₄⋊Dic₈₃	1	4	yes	—
159	C₂×Q₈	C₂×C₈₃⋊C₈	1	4	yes	—
160	C₂×Q₈	#13050574	1	4	yes	—
161	C₂²×C₄	C₂²×Dic₈₃	1	3	yes	—
162	C₂²×C₄	C₂²×Dic₈₃	1	3	yes	—
163	C₂²×C₄	C₂²×Dic₈₃	1	1	yes	characteristic
164	C₂²×C₄	C₂²×Dic₈₃	1	1	yes	characteristic
165	C₂²×C₄	C₂²×Dic₈₃	1	3	yes	—
166	C₂²×C₄	C₂²×Dic₈₃	1	3	yes	—
167	C₂²×C₄	C₂³×D₈₃	1	1	yes	characteristic
168	C₂×C₈	—	83	4	no	—
169	C₂×C₈	—	166	6	no	—
170	C₂×C₈	—	166	6	no	—
171	C₂×C₈	—	166	6	no	—
172	C₂×C₈	—	166	2	no	—
173	C₂×C₈	—	166	2	no	—
174	C₄⋊C₄	—	2	6	no	—
175	C₄⋊C₄	C₄⋊Dic₈₃	1	12	yes	—
176	C₄⋊C₄	C₂×C₈₃⋊C₈	1	12	yes	—
177	C₄²	—	2	6	no	—
178	C₄²	—	2	6	no	—
179	C₄²	—	2	6	no	—
180	C₄²	—	2	2	no	—
181	C₄²	—	4	3	no	—
182	C₄²	C₂×C₈₃⋊C₈	1	12	yes	—
183	C₄²	C₂×D₃₃₂	1	6	yes	—
184	C₄²	C₂×Dic₁₆₆	1	6	yes	—
185	C₂³	C₂³×Dic₈₃	1	1	yes	characteristic
186	Q₈	C₄⋊(C₈₃⋊C₈)	1	16	yes	complemented
187	C₂×C₄	—	2	3	no	—
188	C₂×C₄	—	2	1	no	—
189	C₂×C₄	—	2	1	no	—
190	C₂×C₄	—	2	1	no	—
191	C₂×C₄	—	2	3	no	—
192	C₂×C₄	—	2	3	no	—
193	C₂×C₄	—	2	1	no	—
194	C₂×C₄	—	2	3	no	—
195	C₂×C₄	—	2	3	no	—
196	C₂×C₄	—	2	3	no	—
197	C₂×C₄	—	2	3	no	—
198	C₂×C₄	C₂×C₄⋊Dic₈₃	1	6	yes	—
199	C₂×C₄	C₂×C₄⋊Dic₈₃	1	6	yes	—
200	C₂×C₄	C₂²×C₈₃⋊C₈	1	2	yes	—
201	C₂×C₄	C₂²×C₈₃⋊C₈	1	6	yes	—
202	C₂×C₄	C₂²×C₈₃⋊C₈	1	6	yes	—
203	C₂×C₄	C₂×#13050574	1	6	yes	—
204	C₂×C₄	#14581799	1	2	yes	—
205	C₂×C₄	C₂²×D₃₃₂	1	1	yes	characteristic
206	C₂×C₄	C₂²×Dic₁₆₆	1	1	yes	characteristic
207	C₂×C₄	#41×D₈₃	1	2	yes	—
208	C₂×C₄	#14581840	1	2	yes	—
209	C₈	—	332	6	no	—
210	C₈	—	166	4	no	—
211	C₂²	C₂²×C₄⋊Dic₈₃	1	1	yes	characteristic
212	C₂²	#41×Dic₈₃	1	1	yes	characteristic
213	C₂²	#19128179	1	1	yes	characteristic
214	C₂²	C₂³×C₈₃⋊C₈	1	1	yes	characteristic
215	C₂²	C₂²×#13050574	1	1	yes	characteristic
216	C₂²	C₂×#14581799	1	1	yes	characteristic
217	C₂²	#19128468	1	1	yes	characteristic
218	C₄	—	2	6	no	—
219	C₄	—	2	2	no	—
220	C₄	—	2	2	no	—
221	C₄	—	2	6	no	—
222	C₄	—	4	3	no	—
223	C₄	C₂×C₄⋊(C₈₃⋊C₈)	1	12	yes	—
224	C₄	#19128363	1	2	yes	—
225	C₄	#19128427	1	2	yes	—
226	C₂	#24910384	1	1	yes	characteristic
227	C₂	C₂²×C₄⋊(C₈₃⋊C₈)	1	1	yes	characteristic
228	C₂	#24916031	1	1	yes	characteristic
229	C₂	#41×C₈₃⋊C₈	1	1	yes	characteristic
230	C₂	#24916483	1	1	yes	characteristic
231	C₂	#24916544	1	1	yes	characteristic
232	C₂	#24916552	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₈₃
commutator	C₂²×C₈₃	C₂³×C₈
Fitting	C₄²×Q₈×C₈₃	C₂
socle, prime radical	C₂³×C₈₃	C₂³×C₄
cosocle	C₂²×C₄×C₈₃	C₂⁴
hypercenter	C₄²×Q₈	D₈₃
hypocenter	C₈₃	#22734
LCS	G ▷ C₂²×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄²×Q₈

Complements

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

N	H ≅ G/N	type
27. C₄×Q₈×C₈₃	C₈	semidirect
69. Q₈×C₈₃	C₄⋊C₈	semidirect
122. C₈₃	#22734	semidirect
141. C₄×Q₈	C₈₃⋊C₈	semidirect
186. Q₈	C₄⋊(C₈₃⋊C₈)	semidirect

Maximal subgroups

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

subgroup	index
C₂×Q₈×C₈₃⋊C₈	2
#24908182	2
#24907580	2
C₄×C₄⋊(C₈₃⋊C₈)	2
C₄²×Q₈×C₈₃	2
#22734	83