Subgroups — #35559763

Order 21248.

578 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	#24917977	C₂	1	1	yes	maximal, characteristic
2	C₂×#19128081	C₂	1	1	yes	maximal, characteristic
3	#24909709	C₂	1	4	yes	maximal, complemented
4	#24909691	C₂	1	4	yes	maximal, complemented
5	#24909184	C₂	1	4	yes	maximal
6	C₄×M₃₂×C₈₃	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₃₃₂	—	2	1	no	—
8	C₂²×Dic₃₃₂	C₂²	1	2	yes	—
9	#19128258	—	2	1	no	—
10	C₂×#14581792	—	2	1	no	—
11	#19128106	C₂²	1	2	yes	—
12	#19128103	—	2	1	no	—
13	#19128103	C₂²	1	2	yes	—
14	#19128099	—	2	1	no	—
15	#19128081	—	2	1	no	—
16	#19128081	—	2	1	no	—
17	#19128081	C₂²	1	2	yes	—
18	#19128081	C₂²	1	2	yes	—
19	#19127739	—	2	1	no	—
20	#19127739	C₂²	1	2	yes	—
21	#19127506	C₂²	1	16	yes	—
22	C₂×M₃₂×C₈₃	C₂²	1	2	yes	—
23	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
24	#346×C₈₃	C₂²	1	2	yes	—
25	C₄×C₁₆×C₈₃	C₂²	1	2	yes	—
26	C₂²×Dic₁₆₆	—	2	2	no	—
27	C₂×Dic₃₃₂	—	2	2	no	—
28	C₂×Dic₃₃₂	—	2	2	no	—
29	C₂×Dic₃₃₂	—	2	2	no	—
30	C₂×Dic₃₃₂	—	2	8	no	—
31	C₂×Dic₃₃₂	—	4	4	no	—
32	C₂×Dic₃₃₂	C₂³	1	4	yes	—
33	#14581792	—	2	2	no	—
34	#14581792	—	2	2	no	—
35	#14581792	—	2	2	no	—
36	#14581792	—	4	2	no	—
37	#14581784	—	2	2	no	—
38	#14581771	—	4	2	no	—
39	C₄×Dic₁₆₆	—	4	2	no	—
40	#14581751	—	2	2	no	—
41	C₂×C₄⋊Dic₈₃	—	4	2	no	—
42	#14581726	—	2	2	no	—
43	#14581726	C₂³	1	4	yes	—
44	#14581724	—	2	2	no	—
45	#14581724	—	2	2	no	—
46	C₂²×C₈×C₈₃	D₄	1	1	yes	characteristic
47	C₂²×C₈×C₈₃	C₂³	1	1	yes	characteristic
48	C₂×C₄²×C₈₃	D₄	1	1	yes	characteristic
49	M₃₂×C₈₃	—	2	4	no	—
50	C₂×C₁₆×C₈₃	C₂³	1	4	yes	—
51	C₄×C₈×C₈₃	D₄	1	1	yes	characteristic
52	C₄×C₈×C₈₃	D₄	1	1	yes	characteristic
53	C₄×C₈×C₈₃	C₂³	1	1	yes	characteristic
54	C₄×C₈×C₈₃	C₂³	1	1	yes	characteristic
55	C₂×Dic₁₆₆	—	2	4	no	—
56	C₂×Dic₁₆₆	—	4	4	no	—
57	C₂×Dic₁₆₆	—	4	4	no	—
58	C₂×Dic₁₆₆	—	4	2	no	—
59	C₂×Dic₁₆₆	—	4	2	no	—
60	C₂²×Dic₈₃	—	4	2	no	—
61	Dic₃₃₂	—	2	8	no	—
62	Dic₃₃₂	—	4	4	no	—
63	Dic₃₃₂	—	4	4	no	—
64	Dic₃₃₂	—	4	4	no	—
65	C₄⋊Dic₈₃	—	2	4	no	—
66	C₄⋊Dic₈₃	—	4	2	no	—
67	C₄⋊Dic₈₃	—	4	4	no	—
68	C₄⋊Dic₈₃	—	4	4	no	—
69	Dic₈₃⋊C₄	—	4	4	no	—
70	C₂²⋊Dic₈₃	—	4	2	no	—
71	C₄×Dic₈₃	—	4	2	no	—
72	C₂²×C₄×C₈₃	D₈	1	2	yes	—
73	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
74	C₂×C₈×C₈₃	SD₁₆	1	4	yes	—
75	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
76	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
77	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
78	C₂×C₈×C₈₃	#41	1	4	yes	—
79	C₂×C₈×C₈₃	C₂⁴	1	1	yes	characteristic
80	C₄²×C₈₃	D₈	1	2	yes	—
81	C₄²×C₈₃	C₂×D₄	1	1	yes	characteristic
82	C₄²×C₈₃	C₂×D₄	1	1	yes	characteristic
83	C₁₆×C₈₃	—	2	4	no	—
84	C₂³×C₈₃	C₂×D₈	1	1	yes	characteristic
85	C₂×C₄×C₈₃	—	2	4	no	—
86	C₂×C₄×C₈₃	#124	1	2	yes	—
87	C₂×C₄×C₈₃	C₂×D₈	1	2	yes	—
88	C₂×C₄×C₈₃	C₂×D₈	1	1	yes	characteristic
89	C₂×C₄×C₈₃	C₂×D₈	1	2	yes	—
90	C₂×C₄×C₈₃	C₂×SD₁₆	1	2	yes	—
91	C₂×C₄×C₈₃	C₂²×D₄	1	1	yes	characteristic
92	C₂×Dic₈₃	—	4	4	no	—
93	C₂×Dic₈₃	—	4	4	no	—
94	C₂×Dic₈₃	—	8	4	no	—
95	Dic₁₆₆	—	4	4	no	—
96	Dic₁₆₆	—	4	4	no	—
97	Dic₁₆₆	—	8	4	no	—
98	Dic₁₆₆	—	8	4	no	—
99	C₈×C₈₃	#124	1	2	yes	—
100	C₈×C₈₃	C₂×SD₁₆	1	2	yes	—
101	C₈×C₈₃	C₂×SD₁₆	1	2	yes	—
102	C₈×C₈₃	C₂×#41	1	2	yes	—
103	C₂²×C₈₃	—	2	2	no	—
104	C₂²×C₈₃	#486	1	2	yes	—
105	C₂²×C₈₃	C₂²×D₈	1	1	yes	characteristic
106	C₄×C₈₃	—	2	2	no	—
107	C₄×C₈₃	#486	1	2	yes	—
108	C₄×C₈₃	#510	1	4	yes	—
109	C₄×C₈₃	C₂×#124	1	1	yes	characteristic
110	C₄×C₈₃	C₂²×SD₁₆	1	1	yes	characteristic
111	Dic₈₃	—	8	4	no	—
112	Dic₈₃	—	8	4	no	—
113	#33705	—	83	1	no	maximal, Sylow
114	C₂×C₈₃	—	2	2	no	—
115	C₂×C₈₃	C₂×#486	1	1	yes	characteristic
116	C₂×C₈₃	C₂×#510	1	2	yes	—
117	#3160	—	83	1	no	—
118	C₂×#494	—	83	1	no	—
119	#2245	—	83	4	no	—
120	#2241	—	83	4	no	—
121	#2141	—	83	4	no	—
122	C₄×M₃₂	D₈₃	1	1	yes	characteristic
123	C₈₃	#33705	1	1	yes	characteristic, complemented, Sylow
124	C₂²×Q₁₆	—	83	2	no	—
125	C₂²×Q₁₆	—	166	1	no	—
126	#533	—	166	1	no	—
127	C₂×C₄⋊Q₈	—	166	1	no	—
128	C₂×M₃₂	C₂×D₈₃	1	2	yes	—
129	#500	—	83	2	no	—
130	#499	—	83	2	no	—
131	#499	—	166	1	no	—
132	#498	—	166	1	no	—
133	#494	—	83	2	no	—
134	#494	—	83	2	no	—
135	#494	—	166	1	no	—
136	#494	—	166	1	no	—
137	#427	—	83	2	no	—
138	#427	—	166	1	no	—
139	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
140	#358	—	83	16	no	—
141	#346	C₂×D₈₃	1	2	yes	—
142	C₄×C₁₆	C₂×D₈₃	1	2	yes	—
143	C₂²×Q₈	—	166	2	no	—
144	C₂×Q₁₆	—	332	4	no	—
145	C₂×Q₁₆	—	83	4	no	—
146	C₂×Q₁₆	—	166	2	no	—
147	C₂×Q₁₆	—	166	2	no	—
148	C₂×Q₁₆	—	166	2	no	—
149	C₂×Q₁₆	—	166	8	no	—
150	C₂²×C₈	D₃₃₂	1	1	yes	characteristic
151	C₂²×C₈	C₂²×D₈₃	1	1	yes	characteristic
152	C₄⋊Q₈	—	332	2	no	—
153	C₄⋊Q₈	—	166	2	no	—
154	C₄⋊Q₈	—	166	2	no	—
155	C₄⋊Q₈	—	166	2	no	—
156	#125	—	166	2	no	—
157	C₂²⋊Q₈	—	332	2	no	—
158	C₄×Q₈	—	332	2	no	—
159	#117	—	166	2	no	—
160	C₂×C₄⋊C₄	—	332	2	no	—
161	C₂×C₄²	D₃₃₂	1	1	yes	characteristic
162	M₃₂	—	2	4	no	—
163	C₂×C₁₆	C₂²×D₈₃	1	4	yes	—
164	#107	—	83	4	no	—
165	#107	—	166	2	no	—
166	#106	—	166	2	no	—
167	#106	—	166	2	no	—
168	C₄×C₈	D₃₃₂	1	1	yes	characteristic
169	C₄×C₈	D₃₃₂	1	1	yes	characteristic
170	C₄×C₈	C₂²×D₈₃	1	1	yes	characteristic
171	C₄×C₈	C₂²×D₈₃	1	1	yes	characteristic
172	C₂×Q₈	—	332	2	no	—
173	C₂×Q₈	—	332	2	no	—
174	C₂×Q₈	—	332	4	no	—
175	C₂×Q₈	—	332	4	no	—
176	C₂×Q₈	—	166	4	no	—
177	C₂²×C₄	—	332	2	no	—
178	C₂²×C₄	D₆₆₄	1	2	yes	—
179	C₂²×C₄	C₂×D₃₃₂	1	1	yes	characteristic
180	Q₁₆	—	332	4	no	—
181	Q₁₆	—	332	4	no	—
182	Q₁₆	—	332	4	no	—
183	Q₁₆	—	166	8	no	—
184	C₂×C₈	#13050579	1	4	yes	—
185	C₂×C₈	C₂×D₃₃₂	1	1	yes	characteristic
186	C₂×C₈	C₂×D₃₃₂	1	1	yes	characteristic
187	C₂×C₈	C₂×D₃₃₂	1	1	yes	characteristic
188	C₂×C₈	#13050590	1	4	yes	—
189	C₂×C₈	C₂³×D₈₃	1	1	yes	characteristic
190	C₄⋊C₄	—	332	4	no	—
191	C₄⋊C₄	—	332	4	no	—
192	C₄⋊C₄	—	332	4	no	—
193	C₄⋊C₄	—	332	2	no	—
194	C₄⋊C₄	—	166	4	no	—
195	C₂²⋊C₄	—	332	2	no	—
196	C₄²	—	332	2	no	—
197	C₄²	D₆₆₄	1	2	yes	—
198	C₄²	C₂×D₃₃₂	1	1	yes	characteristic
199	C₄²	C₂×D₃₃₂	1	1	yes	characteristic
200	C₁₆	—	2	4	no	—
201	C₂³	C₂×D₆₆₄	1	1	yes	characteristic
202	Q₈	—	332	4	no	—
203	Q₈	—	332	4	no	—
204	Q₈	—	664	4	no	—
205	Q₈	—	664	4	no	—
206	C₂×C₄	—	2	4	no	—
207	C₂×C₄	—	332	4	no	—
208	C₂×C₄	—	332	4	no	—
209	C₂×C₄	—	664	4	no	—
210	C₂×C₄	#14581780	1	2	yes	—
211	C₂×C₄	C₂×D₆₆₄	1	2	yes	—
212	C₂×C₄	C₂×D₆₆₄	1	2	yes	—
213	C₂×C₄	C₂×D₆₆₄	1	1	yes	characteristic
214	C₂×C₄	C₂×#13050579	1	2	yes	—
215	C₂×C₄	C₂²×D₃₃₂	1	1	yes	characteristic
216	C₈	#14581780	1	2	yes	—
217	C₈	C₂×#13050579	1	2	yes	—
218	C₈	C₂×#13050579	1	2	yes	—
219	C₈	C₂×#13050590	1	2	yes	—
220	C₂²	—	2	2	no	—
221	C₂²	#19128048	1	2	yes	—
222	C₂²	C₂²×D₆₆₄	1	1	yes	characteristic
223	C₄	—	2	2	no	—
224	C₄	—	664	4	no	—
225	C₄	—	664	4	no	—
226	C₄	#19128048	1	2	yes	—
227	C₄	#19128144	1	4	yes	—
228	C₄	C₂×#14581780	1	1	yes	characteristic
229	C₄	C₂²×#13050579	1	1	yes	characteristic
230	C₂	—	2	2	no	—
231	C₂	C₂×#19128048	1	1	yes	characteristic
232	C₂	C₂×#19128144	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂²×D₆₆₄
commutator, cosocle	C₂×C₈×C₈₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₈₃
Fitting	C₄×M₃₂×C₈₃	C₂
socle	C₂²×C₈₃	C₂²×D₈
prime radical	C₂³×C₈₃	C₂×D₈
hypercenter	C₄×M₃₂	D₈₃
hypocenter	C₈₃	#33705
LCS	G ▷ C₂×C₈×C₈₃ ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×M₃₂

Complements

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

N	H ≅ G/N	type
3. #24909709	C₂	semidirect
4. #24909691	C₂	semidirect
123. C₈₃	#33705	semidirect

Maximal subgroups

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

subgroup	index
#24917977	2
C₂×#19128081	2
#24909709	2
#24909691	2
#24909184	2
C₄×M₃₂×C₈₃	2
#33705	83