Subgroups — #35423408

Order 21248.

642 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	M₁₆×Dic₁₆₆	C₂	1	2	yes	maximal
2	C₂×C₈×Dic₁₆₆	C₂	1	1	yes	maximal, characteristic
3	#24908238	C₂	1	2	yes	maximal
4	#24908235	C₂	1	4	yes	maximal
5	#24907601	C₂	1	1	yes	maximal, characteristic
6	#24907599	C₂	1	2	yes	maximal
7	#24907364	C₂	1	2	yes	maximal
8	C₄×#108×C₈₃	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄×Dic₁₆₆	C₂²	1	1	yes	characteristic
10	#19127831	—	2	2	no	—
11	#19127829	—	2	4	no	—
12	C₈×Dic₁₆₆	—	2	2	no	—
13	C₈×Dic₁₆₆	C₄	1	4	yes	—
14	C₈×Dic₁₆₆	C₄	1	2	yes	—
15	C₈×Dic₁₆₆	C₄	1	2	yes	—
16	C₂×#14581727	C₂²	1	4	yes	—
17	#19127725	C₂²	1	2	yes	—
18	#19127724	C₂²	1	4	yes	—
19	C₂×C₄⋊(C₈₃⋊C₈)	C₂²	1	1	yes	characteristic
20	C₂×Dic₈₃⋊C₈	C₂²	1	2	yes	—
21	M₁₆×Dic₈₃	C₂²	1	4	yes	—
22	C₂×C₈×Dic₈₃	C₂²	1	2	yes	—
23	#19127466	C₂²	1	2	yes	—
24	#19127466	C₂²	1	2	yes	—
25	#19127465	C₂²	1	4	yes	—
26	C₂×#108×C₈₃	C₂²	1	2	yes	—
27	C₄×M₁₆×C₈₃	C₂²	1	2	yes	—
28	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
29	#341×C₈₃	C₂²	1	1	yes	characteristic
30	#341×C₈₃	C₂²	1	1	yes	characteristic
31	C₂²×Dic₁₆₆	Q₈	1	1	yes	characteristic
32	C₂×#13050574	C₂³	1	4	yes	—
33	C₂²×C₈₃⋊C₈	C₂³	1	2	yes	—
34	C₄×Dic₁₆₆	C₂×C₄	1	2	yes	—
35	C₄×Dic₁₆₆	C₂×C₄	1	1	yes	characteristic
36	C₄×Dic₁₆₆	C₂×C₄	1	1	yes	characteristic
37	C₄×Dic₁₆₆	D₄	1	1	yes	characteristic
38	C₄×Dic₁₆₆	D₄	1	2	yes	—
39	C₄×Dic₁₆₆	D₄	1	1	yes	characteristic
40	C₂×C₄⋊Dic₈₃	Q₈	1	1	yes	characteristic
41	C₂×Dic₈₃⋊C₄	Q₈	1	2	yes	—
42	C₂×C₄×Dic₈₃	C₂³	1	2	yes	—
43	#14581727	—	2	8	no	—
44	C₄⋊(C₈₃⋊C₈)	—	2	2	no	—
45	C₄⋊(C₈₃⋊C₈)	—	2	2	no	—
46	C₄⋊(C₈₃⋊C₈)	C₂×C₄	1	2	yes	—
47	C₄⋊(C₈₃⋊C₈)	C₂×C₄	1	2	yes	—
48	Dic₈₃⋊C₈	—	2	4	no	—
49	Dic₈₃⋊C₈	—	2	4	no	—
50	Dic₈₃⋊C₈	C₂×C₄	1	4	yes	—
51	Dic₈₃⋊C₈	C₂×C₄	1	4	yes	—
52	#14581702	—	2	4	no	—
53	C₈×Dic₈₃	—	2	4	no	—
54	C₈×Dic₈₃	C₂×C₄	1	4	yes	—
55	C₈×Dic₈₃	C₂×C₄	1	4	yes	—
56	C₂×M₁₆×C₈₃	C₂³	1	2	yes	—
57	C₂×M₁₆×C₈₃	C₂³	1	2	yes	—
58	C₂²×C₈×C₈₃	C₂³	1	1	yes	characteristic
59	C₂²×C₈×C₈₃	C₂³	1	1	yes	characteristic
60	C₂×C₄²×C₈₃	C₂³	1	1	yes	characteristic
61	#108×C₈₃	—	2	4	no	—
62	#97×C₈₃	—	2	2	no	—
63	C₄×C₈×C₈₃	—	2	2	no	—
64	C₄×C₈×C₈₃	C₂×C₄	1	2	yes	—
65	C₄×C₈×C₈₃	C₂×C₄	1	2	yes	—
66	C₂×Dic₁₆₆	—	2	4	no	—
67	C₂×Dic₁₆₆	C₄⋊C₄	1	1	yes	characteristic
68	C₂×Dic₁₆₆	C₄⋊C₄	1	2	yes	—
69	C₂×Dic₁₆₆	C₄⋊C₄	1	1	yes	characteristic
70	C₂²×Dic₈₃	C₂×Q₈	1	2	yes	—
71	C₂²×Dic₈₃	C₂×Q₈	1	2	yes	—
72	#13050574	—	2	4	no	—
73	#13050574	—	2	4	no	—
74	C₂×C₈₃⋊C₈	—	2	4	no	—
75	C₂×C₈₃⋊C₈	—	2	4	no	—
76	C₂×C₈₃⋊C₈	C₂²×C₄	1	4	yes	—
77	C₄⋊Dic₈₃	C₄⋊C₄	1	2	yes	—
78	C₄⋊Dic₈₃	C₄⋊C₄	1	1	yes	characteristic
79	C₄⋊Dic₈₃	C₄⋊C₄	1	1	yes	characteristic
80	Dic₈₃⋊C₄	C₄⋊C₄	1	2	yes	—
81	Dic₈₃⋊C₄	C₄⋊C₄	1	2	yes	—
82	Dic₈₃⋊C₄	C₄⋊C₄	1	2	yes	—
83	Dic₈₃⋊C₄	C₄⋊C₄	1	2	yes	—
84	C₄×Dic₈₃	C₂²×C₄	1	2	yes	—
85	C₄×Dic₈₃	C₂²×C₄	1	2	yes	—
86	C₄×Dic₈₃	C₂×D₄	1	2	yes	—
87	C₄×Dic₈₃	C₂×D₄	1	2	yes	—
88	C₂²×C₄×C₈₃	C₂×Q₈	1	1	yes	characteristic
89	C₂²×C₄×C₈₃	C₂×Q₈	1	1	yes	characteristic
90	C₂²×C₄×C₈₃	C₂⁴	1	1	yes	characteristic
91	M₁₆×C₈₃	—	2	2	no	—
92	M₁₆×C₈₃	—	2	2	no	—
93	M₁₆×C₈₃	—	2	2	no	—
94	M₁₆×C₈₃	#41	1	4	yes	—
95	C₂×C₈×C₈₃	—	2	2	no	—
96	C₂×C₈×C₈₃	—	2	2	no	—
97	C₂×C₈×C₈₃	—	2	2	no	—
98	C₂×C₈×C₈₃	—	2	2	no	—
99	C₂×C₈×C₈₃	C₂²×C₄	1	2	yes	—
100	C₂×C₈×C₈₃	C₂²×C₄	1	2	yes	—
101	C₄²×C₈₃	C₂²×C₄	1	1	yes	characteristic
102	C₄²×C₈₃	C₂²×C₄	1	1	yes	characteristic
103	C₄²×C₈₃	C₂×D₄	1	1	yes	characteristic
104	C₄²×C₈₃	C₂×D₄	1	1	yes	characteristic
105	C₂³×C₈₃	C₂²×Q₈	1	1	yes	characteristic
106	C₂×C₄×C₈₃	—	2	2	no	—
107	C₂×C₄×C₈₃	—	4	1	no	—
108	C₂×C₄×C₈₃	C₂×C₄⋊C₄	1	1	yes	characteristic
109	C₂×C₄×C₈₃	C₂×C₄⋊C₄	1	1	yes	characteristic
110	C₂×C₄×C₈₃	C₂×C₄⋊C₄	1	1	yes	characteristic
111	C₂×C₄×C₈₃	C₂×C₄⋊C₄	1	1	yes	characteristic
112	C₂×C₄×C₈₃	C₂³×C₄	1	1	yes	characteristic
113	C₂×C₄×C₈₃	C₂²×D₄	1	1	yes	characteristic
114	C₂×C₄×C₈₃	C₂×#41	1	2	yes	—
115	C₂×C₄×C₈₃	2₋¹⁺⁴	1	2	yes	—
116	C₂×Dic₈₃	—	2	4	no	—
117	C₂×Dic₈₃	—	4	2	no	—
118	C₂×Dic₈₃	C₂×C₄⋊C₄	1	2	yes	—
119	C₂×Dic₈₃	C₂×C₄⋊C₄	1	2	yes	—
120	C₂×Dic₈₃	C₂×C₄⋊C₄	1	2	yes	—
121	C₂×Dic₈₃	C₂×C₄⋊C₄	1	2	yes	—
122	Dic₁₆₆	—	2	2	no	—
123	Dic₁₆₆	—	2	4	no	—
124	Dic₁₆₆	#108	1	4	yes	complemented
125	C₈×C₈₃	—	2	2	no	—
126	C₈×C₈₃	—	2	2	no	—
127	C₈×C₈₃	—	2	4	no	—
128	C₈×C₈₃	—	4	2	no	—
129	C₈₃⋊C₈	—	2	4	no	—
130	C₈₃⋊C₈	—	4	4	no	—
131	C₂²×C₈₃	—	2	1	no	—
132	C₂²×C₈₃	—	2	1	no	—
133	C₂²×C₈₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
134	C₂²×C₈₃	#557	1	2	yes	—
135	C₄×C₈₃	—	2	1	no	—
136	C₄×C₈₃	—	2	1	no	—
137	C₄×C₈₃	—	2	1	no	—
138	C₄×C₈₃	C₂×#108	1	2	yes	—
139	C₄×C₈₃	C₄×#41	1	1	yes	characteristic
140	C₄×C₈₃	#519	1	1	yes	characteristic
141	C₄×C₈₃	#548	1	2	yes	—
142	Dic₈₃	—	2	2	no	—
143	Dic₈₃	—	2	2	no	—
144	Dic₈₃	—	2	2	no	—
145	Dic₈₃	C₂×#108	1	4	yes	—
146	#22746	—	83	1	no	maximal, Sylow
147	C₂×C₈₃	—	2	2	no	—
148	C₂×C₈₃	#2351	1	1	yes	characteristic
149	C₂×C₈₃	C₂²×#108	1	1	yes	characteristic
150	C₂×C₈₃	#2916	1	1	yes	characteristic
151	Q₈×M₁₆	—	83	2	no	—
152	C₂×C₈×Q₈	—	83	1	no	—
153	#1950	—	83	4	no	—
154	#1950	—	83	2	no	—
155	#1837	—	83	1	no	—
156	#1837	—	83	2	no	—
157	C₄×#108	—	83	2	no	—
158	C₄×#108	D₈₃	1	1	yes	characteristic
159	C₈₃	#22746	1	1	yes	characteristic, complemented, Sylow
160	C₂×C₄×Q₈	—	83	1	no	—
161	#446	—	166	2	no	—
162	#446	—	166	4	no	—
163	C₈×Q₈	—	83	2	no	—
164	C₈×Q₈	—	83	4	no	—
165	C₈×Q₈	—	83	2	no	—
166	C₈×Q₈	—	166	2	no	—
167	C₂×#108	—	83	4	no	—
168	C₂×#108	C₂×D₈₃	1	2	yes	—
169	#423	—	83	4	no	—
170	#423	—	83	2	no	—
171	C₂×C₄⋊C₈	—	83	2	no	—
172	C₂×C₄⋊C₈	—	83	1	no	—
173	C₄×M₁₆	—	83	4	no	—
174	C₄×M₁₆	C₂×D₈₃	1	2	yes	—
175	C₂×C₄×C₈	—	83	2	no	—
176	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
177	#341	—	83	4	no	—
178	#341	—	83	2	no	—
179	#341	—	83	2	no	—
180	#341	C₂×D₈₃	1	1	yes	characteristic
181	#341	C₂×D₈₃	1	1	yes	characteristic
182	C₂²×Q₈	—	83	1	no	—
183	C₂×M₁₆	—	83	4	no	—
184	C₂×M₁₆	C₂²×D₈₃	1	2	yes	—
185	C₂×M₁₆	C₂²×D₈₃	1	2	yes	—
186	C₂²×C₈	—	83	2	no	—
187	C₂²×C₈	C₂²×D₈₃	1	1	yes	characteristic
188	C₂²×C₈	C₂²×D₈₃	1	1	yes	characteristic
189	C₄×Q₈	—	83	1	no	—
190	C₄×Q₈	—	83	2	no	—
191	C₄×Q₈	—	83	1	no	—
192	C₄×Q₈	—	83	1	no	—
193	C₄×Q₈	—	83	2	no	—
194	C₄×Q₈	—	83	1	no	—
195	C₂×C₄⋊C₄	—	83	2	no	—
196	C₂×C₄⋊C₄	—	83	1	no	—
197	C₂×C₄²	—	83	2	no	—
198	C₂×C₄²	C₂²×D₈₃	1	1	yes	characteristic
199	#108	—	2	4	no	—
200	#108	—	166	8	no	—
201	C₄⋊C₈	—	83	4	no	—
202	C₄⋊C₈	—	83	4	no	—
203	C₄⋊C₈	—	83	2	no	—
204	C₄⋊C₈	—	83	2	no	—
205	C₄⋊C₈	—	166	4	no	—
206	C₄⋊C₈	—	166	4	no	—
207	C₄⋊C₈	—	166	2	no	—
208	C₄⋊C₈	—	166	2	no	—
209	#97	—	2	2	no	—
210	#97	—	166	4	no	—
211	C₄×C₈	—	2	2	no	—
212	C₄×C₈	—	83	4	no	—
213	C₄×C₈	—	83	4	no	—
214	C₄×C₈	—	166	4	no	—
215	C₄×C₈	C₄×D₈₃	1	2	yes	—
216	C₄×C₈	C₄×D₈₃	1	2	yes	—
217	C₂×Q₈	—	83	1	no	—
218	C₂×Q₈	—	83	2	no	—
219	C₂×Q₈	—	83	1	no	—
220	C₂×Q₈	—	166	4	no	—
221	C₂²×C₄	—	83	2	no	—
222	C₂²×C₄	—	83	2	no	—
223	C₂²×C₄	Q₈×D₈₃	1	1	yes	characteristic
224	C₂²×C₄	Q₈×D₈₃	1	1	yes	characteristic
225	C₂²×C₄	C₂³×D₈₃	1	1	yes	characteristic
226	M₁₆	—	2	2	no	—
227	M₁₆	—	2	2	no	—
228	M₁₆	—	2	2	no	—
229	M₁₆	—	166	4	no	—
230	M₁₆	—	166	4	no	—
231	M₁₆	#13050590	1	4	yes	—
232	C₂×C₈	—	2	2	no	—
233	C₂×C₈	—	2	2	no	—
234	C₂×C₈	—	2	2	no	—
235	C₂×C₈	—	2	2	no	—
236	C₂×C₈	—	83	4	no	—
237	C₂×C₈	—	166	4	no	—
238	C₂×C₈	—	166	4	no	—
239	C₂×C₈	C₂×C₄×D₈₃	1	2	yes	—
240	C₂×C₈	C₂×C₄×D₈₃	1	2	yes	—
241	C₄⋊C₄	—	83	2	no	—
242	C₄⋊C₄	—	83	2	no	—
243	C₄⋊C₄	—	83	2	no	—
244	C₄⋊C₄	—	83	2	no	—
245	C₄⋊C₄	—	83	2	no	—
246	C₄⋊C₄	—	83	1	no	—
247	C₄⋊C₄	—	83	1	no	—
248	C₄²	—	83	2	no	—
249	C₄²	—	83	2	no	—
250	C₄²	—	83	2	no	—
251	C₄²	—	83	2	no	—
252	C₄²	C₂×C₄×D₈₃	1	1	yes	characteristic
253	C₄²	C₂×C₄×D₈₃	1	1	yes	characteristic
254	C₄²	D₄×D₈₃	1	1	yes	characteristic
255	C₄²	D₄×D₈₃	1	1	yes	characteristic
256	C₂³	C₂×Q₈×D₈₃	1	1	yes	characteristic
257	Q₈	—	83	4	no	—
258	Q₈	—	166	4	no	—
259	Q₈	—	166	2	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	4	1	no	—
262	C₂×C₄	—	332	2	no	—
263	C₂×C₄	—	83	2	no	—
264	C₂×C₄	—	83	2	no	—
265	C₂×C₄	—	83	2	no	—
266	C₂×C₄	—	83	2	no	—
267	C₂×C₄	—	166	4	no	—
268	C₂×C₄	C₄⋊C₄×D₈₃	1	1	yes	characteristic
269	C₂×C₄	C₄⋊C₄×D₈₃	1	1	yes	characteristic
270	C₂×C₄	C₄⋊C₄×D₈₃	1	1	yes	characteristic
271	C₂×C₄	C₄⋊C₄×D₈₃	1	1	yes	characteristic
272	C₂×C₄	C₂²×C₄×D₈₃	1	1	yes	characteristic
273	C₂×C₄	C₂×D₄×D₈₃	1	1	yes	characteristic
274	C₂×C₄	C₂×#13050590	1	2	yes	—
275	C₂×C₄	#14581839	1	2	yes	—
276	C₈	—	2	4	no	—
277	C₈	—	2	2	no	—
278	C₈	—	2	2	no	—
279	C₈	—	4	2	no	—
280	C₈	—	332	4	no	—
281	C₈	—	166	4	no	—
282	C₂²	—	2	1	no	—
283	C₂²	—	2	1	no	—
284	C₂²	C₂×C₄⋊C₄×D₈₃	1	1	yes	characteristic
285	C₂²	#19128429	1	2	yes	—
286	C₄	—	2	1	no	—
287	C₄	—	2	1	no	—
288	C₄	—	2	1	no	—
289	C₄	—	83	4	no	—
290	C₄	—	166	2	no	—
291	C₄	—	166	2	no	—
292	C₄	—	166	2	no	—
293	C₄	#108×D₈₃	1	2	yes	—
294	C₄	C₄×#13050590	1	1	yes	characteristic
295	C₄	#19128178	1	1	yes	characteristic
296	C₄	#19128360	1	2	yes	—
297	C₂	—	2	2	no	—
298	C₂	#24910383	1	1	yes	characteristic
299	C₂	C₂×#108×D₈₃	1	1	yes	characteristic
300	C₂	#24916071	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₈₃
commutator	C₂×C₄×C₈₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₈₃
Fitting	C₄×#108×C₈₃	C₂
socle	C₂²×C₈₃	C₂²×C₄⋊C₄
prime radical	C₂³×C₈₃	C₂²×Q₈
cosocle	C₂²×C₄×C₈₃	C₂⁴
hypercenter	C₄×#108	D₈₃
hypocenter	C₈₃	#22746
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×#108

Complements

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

N	H ≅ G/N	type
124. Dic₁₆₆	#108	semidirect
159. C₈₃	#22746	semidirect

Maximal subgroups

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

subgroup	index
M₁₆×Dic₁₆₆	2
C₂×C₈×Dic₁₆₆	2
#24908238	2
#24908235	2
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