Subgroups — C₂×C₈₃×#1956 #35249016

Order 21248.

1726 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄⋊Q₈×C₈₃	C₂	1	1	yes	maximal, characteristic
2	C₂²×#103×C₈₃	C₂	1	4	yes	maximal
3	C₂²×C₄×C₈×C₈₃	C₂	1	1	yes	maximal, characteristic
4	C₂×#384×C₈₃	C₂	1	1	yes	maximal, characteristic
5	C₈₃×#1956	C₂	1	8	yes	maximal, direct product factor
6	C₂³×Q₈×C₈₃	C₂²	1	2	yes	—
7	C₂³×C₈×C₈₃	C₂²	1	2	yes	—
8	C₂×C₄⋊Q₈×C₈₃	—	2	2	no	—
9	C₂×C₄⋊Q₈×C₈₃	—	2	4	no	—
10	C₂×C₄⋊Q₈×C₈₃	C₄	1	8	yes	complemented
11	C₂×C₄⋊Q₈×C₈₃	C₂²	1	4	yes	—
12	C₂²×C₄⋊C₄×C₈₃	—	2	2	no	—
13	C₂²×C₄⋊C₄×C₈₃	—	2	2	no	—
14	C₂²×C₄⋊C₄×C₈₃	C₂²	1	2	yes	—
15	C₂²×C₄²×C₈₃	C₂²	1	1	yes	characteristic
16	C₂×#103×C₈₃	—	2	16	no	—
17	C₂×#103×C₈₃	C₂²	1	16	yes	—
18	C₂×C₄×C₈×C₈₃	—	2	4	no	—
19	C₂×C₄×C₈×C₈₃	C₂²	1	4	yes	—
20	#384×C₈₃	—	2	2	no	—
21	#384×C₈₃	C₂²	1	4	yes	—
22	C₂²×Q₈×C₈₃	—	2	4	no	—
23	C₂²×Q₈×C₈₃	—	2	2	no	—
24	C₂²×Q₈×C₈₃	—	2	8	no	—
25	C₂²×Q₈×C₈₃	—	2	8	no	—
26	C₂²×Q₈×C₈₃	C₂×C₄	1	8	yes	—
27	C₂²×Q₈×C₈₃	C₂³	1	4	yes	—
28	C₂³×C₄×C₈₃	—	2	2	no	—
29	C₂³×C₄×C₈₃	—	2	2	no	—
30	C₂³×C₄×C₈₃	D₄	1	2	yes	—
31	C₂³×C₄×C₈₃	C₂³	1	1	yes	characteristic
32	C₂²×C₈×C₈₃	—	2	4	no	—
33	C₂²×C₈×C₈₃	—	2	4	no	—
34	C₂²×C₈×C₈₃	D₄	1	8	yes	—
35	C₂²×C₈×C₈₃	C₂³	1	4	yes	—
36	C₄⋊Q₈×C₈₃	—	2	16	no	—
37	C₄⋊Q₈×C₈₃	—	2	8	no	—
38	C₄⋊Q₈×C₈₃	C₂×C₄	1	16	yes	complemented
39	C₂×C₄⋊C₄×C₈₃	—	2	16	no	—
40	C₂×C₄⋊C₄×C₈₃	—	2	8	no	—
41	C₂×C₄⋊C₄×C₈₃	—	2	8	no	—
42	C₂×C₄⋊C₄×C₈₃	—	2	8	no	—
43	C₂×C₄⋊C₄×C₈₃	—	2	2	no	—
44	C₂×C₄⋊C₄×C₈₃	—	4	8	no	—
45	C₂×C₄⋊C₄×C₈₃	C₂³	1	4	yes	—
46	C₂×C₄²×C₈₃	C₂×C₄	1	4	yes	—
47	C₂×C₄²×C₈₃	D₄	1	2	yes	—
48	C₂×C₄²×C₈₃	D₄	1	4	yes	—
49	C₂×C₄²×C₈₃	C₂³	1	2	yes	—
50	#103×C₈₃	—	2	32	no	—
51	C₄×C₈×C₈₃	—	2	8	no	—
52	C₂⁴×C₈₃	C₂×D₄	1	1	yes	characteristic
53	C₂×Q₈×C₈₃	—	2	8	no	—
54	C₂×Q₈×C₈₃	—	2	4	no	—
55	C₂×Q₈×C₈₃	—	2	32	no	—
56	C₂×Q₈×C₈₃	—	2	8	no	—
57	C₂×Q₈×C₈₃	—	2	8	no	—
58	C₂×Q₈×C₈₃	—	4	16	no	—
59	C₂×Q₈×C₈₃	—	4	4	no	—
60	C₂×Q₈×C₈₃	—	4	4	no	—
61	C₂×Q₈×C₈₃	C₂²×C₄	1	8	yes	—
62	C₂²×C₄×C₈₃	—	2	8	no	—
63	C₂²×C₄×C₈₃	—	2	4	no	—
64	C₂²×C₄×C₈₃	—	2	4	no	—
65	C₂²×C₄×C₈₃	—	4	4	no	—
66	C₂²×C₄×C₈₃	—	4	4	no	—
67	C₂²×C₄×C₈₃	—	4	4	no	—
68	C₂²×C₄×C₈₃	—	4	8	no	—
69	C₂²×C₄×C₈₃	C₂²⋊C₄	1	8	yes	—
70	C₂²×C₄×C₈₃	SD₁₆	1	8	yes	—
71	C₂²×C₄×C₈₃	Q₁₆	1	8	yes	—
72	C₂²×C₄×C₈₃	C₂²×C₄	1	2	yes	—
73	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₈₃	C₂×D₄	1	4	yes	—
75	C₂²×C₄×C₈₃	C₂×D₄	1	2	yes	—
76	C₂²×C₄×C₈₃	C₂×D₄	1	4	yes	—
77	C₂²×C₄×C₈₃	#41	1	4	yes	—
78	C₂²×C₄×C₈₃	C₂⁴	1	1	yes	characteristic
79	C₂×C₈×C₈₃	—	2	16	no	—
80	C₂×C₈×C₈₃	—	2	4	no	—
81	C₂×C₈×C₈₃	—	2	4	no	—
82	C₂×C₈×C₈₃	C₂×D₄	1	8	yes	—
83	C₄⋊C₄×C₈₃	—	2	32	no	—
84	C₄⋊C₄×C₈₃	—	2	8	no	—
85	C₄⋊C₄×C₈₃	—	4	4	no	—
86	C₄⋊C₄×C₈₃	—	4	16	no	—
87	C₄²×C₈₃	C₂²⋊C₄	1	8	yes	—
88	C₄²×C₈₃	C₂²×C₄	1	4	yes	—
89	C₄²×C₈₃	C₂×D₄	1	4	yes	—
90	C₂³×C₈₃	C₂×C₂²⋊C₄	1	2	yes	—
91	C₂³×C₈₃	#124	1	2	yes	—
92	C₂³×C₈₃	#127	1	2	yes	—
93	C₂³×C₈₃	C₂×SD₁₆	1	4	yes	—
94	C₂³×C₈₃	C₂×Q₁₆	1	4	yes	—
95	C₂³×C₈₃	C₂²×D₄	1	1	yes	characteristic
96	Q₈×C₈₃	—	2	16	no	—
97	Q₈×C₈₃	—	4	8	no	—
98	Q₈×C₈₃	—	4	16	no	—
99	C₂×C₄×C₈₃	—	2	8	no	—
100	C₂×C₄×C₈₃	—	4	4	no	—
101	C₂×C₄×C₈₃	—	4	4	no	—
102	C₂×C₄×C₈₃	—	4	16	no	—
103	C₂×C₄×C₈₃	—	4	4	no	—
104	C₂×C₄×C₈₃	—	4	8	no	—
105	C₂×C₄×C₈₃	—	8	8	no	—
106	C₂×C₄×C₈₃	#103	1	32	yes	complemented
107	C₂×C₄×C₈₃	C₂×C₂²⋊C₄	1	2	yes	—
108	C₂×C₄×C₈₃	C₂×C₂²⋊C₄	1	8	yes	—
109	C₂×C₄×C₈₃	C₄×D₄	1	8	yes	—
110	C₂×C₄×C₈₃	#121	1	8	yes	—
111	C₂×C₄×C₈₃	#124	1	2	yes	—
112	C₂×C₄×C₈₃	#127	1	2	yes	—
113	C₂×C₄×C₈₃	C₂×SD₁₆	1	8	yes	—
114	C₂×C₄×C₈₃	C₂×Q₁₆	1	8	yes	—
115	C₂×C₄×C₈₃	C₂³×C₄	1	1	yes	characteristic
116	C₂×C₄×C₈₃	C₂²×D₄	1	2	yes	—
117	C₂×C₄×C₈₃	C₂²×D₄	1	1	yes	characteristic
118	C₂×C₄×C₈₃	C₂×#41	1	2	yes	—
119	C₈×C₈₃	—	2	8	no	—
120	C₂²×C₈₃	#390	1	4	yes	—
121	C₂²×C₈₃	C₂×#103	1	8	yes	—
122	C₂²×C₈₃	#487	1	8	yes	—
123	C₂²×C₈₃	#492	1	4	yes	—
124	C₂²×C₈₃	#494	1	4	yes	—
125	C₂²×C₈₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
126	C₂²×C₈₃	C₂×#124	1	1	yes	characteristic
127	C₂²×C₈₃	C₂×#127	1	1	yes	characteristic
128	C₂²×C₈₃	C₂²×SD₁₆	1	2	yes	—
129	C₂²×C₈₃	C₂²×Q₁₆	1	2	yes	—
130	C₄×C₈₃	—	4	8	no	—
131	C₄×C₈₃	—	8	4	no	—
132	C₄×C₈₃	#390	1	4	yes	—
133	C₄×C₈₃	C₂×#103	1	16	yes	complemented
134	C₄×C₈₃	C₂×C₄×D₄	1	2	yes	—
135	C₄×C₈₃	C₂×#121	1	2	yes	—
136	C₂×#1956	C₈₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
137	C₂×C₈₃	#1956	1	8	yes	direct product factor
138	C₂×C₈₃	C₂×#390	1	1	yes	characteristic
139	C₂×C₈₃	C₂²×#103	1	2	yes	—
140	C₂×C₈₃	C₂×#487	1	2	yes	—
141	C₂×C₈₃	C₂×#492	1	1	yes	characteristic
142	C₂×C₈₃	C₂×#494	1	1	yes	characteristic
143	C₂²×C₄⋊Q₈	C₂×C₈₃	1	1	yes	characteristic
144	C₂²×#103	C₂×C₈₃	1	4	yes	—
145	C₂²×C₄×C₈	C₂×C₈₃	1	1	yes	characteristic
146	C₂×#384	C₂×C₈₃	1	1	yes	characteristic
147	#1956	C₂×C₈₃	1	8	yes	direct product factor
148	C₈₃	C₂×#1956	1	1	yes	characteristic, direct product factor, Sylow
149	C₂³×Q₈	C₂²×C₈₃	1	2	yes	—
150	C₂³×C₈	C₂²×C₈₃	1	2	yes	—
151	C₂×C₄⋊Q₈	—	2	2	no	—
152	C₂×C₄⋊Q₈	—	2	4	no	—
153	C₂×C₄⋊Q₈	C₄×C₈₃	1	8	yes	complemented
154	C₂×C₄⋊Q₈	C₂²×C₈₃	1	4	yes	—
155	C₂²×C₄⋊C₄	—	2	2	no	—
156	C₂²×C₄⋊C₄	—	2	2	no	—
157	C₂²×C₄⋊C₄	C₂²×C₈₃	1	2	yes	—
158	C₂²×C₄²	C₂²×C₈₃	1	1	yes	characteristic
159	C₂×#103	—	2	16	no	—
160	C₂×#103	C₂²×C₈₃	1	16	yes	—
161	C₂×C₄×C₈	—	2	4	no	—
162	C₂×C₄×C₈	C₂²×C₈₃	1	4	yes	—
163	#384	—	2	2	no	—
164	#384	C₂²×C₈₃	1	4	yes	—
165	C₂²×Q₈	—	2	4	no	—
166	C₂²×Q₈	—	2	2	no	—
167	C₂²×Q₈	—	2	8	no	—
168	C₂²×Q₈	—	2	8	no	—
169	C₂²×Q₈	C₂×C₄×C₈₃	1	8	yes	—
170	C₂²×Q₈	C₂³×C₈₃	1	4	yes	—
171	C₂³×C₄	—	2	2	no	—
172	C₂³×C₄	—	2	2	no	—
173	C₂³×C₄	D₄×C₈₃	1	2	yes	—
174	C₂³×C₄	C₂³×C₈₃	1	1	yes	characteristic
175	C₂²×C₈	—	2	4	no	—
176	C₂²×C₈	—	2	4	no	—
177	C₂²×C₈	D₄×C₈₃	1	8	yes	—
178	C₂²×C₈	C₂³×C₈₃	1	4	yes	—
179	C₄⋊Q₈	—	2	16	no	—
180	C₄⋊Q₈	—	2	8	no	—
181	C₄⋊Q₈	C₂×C₄×C₈₃	1	16	yes	complemented
182	C₂×C₄⋊C₄	—	2	16	no	—
183	C₂×C₄⋊C₄	—	2	8	no	—
184	C₂×C₄⋊C₄	—	2	8	no	—
185	C₂×C₄⋊C₄	—	2	8	no	—
186	C₂×C₄⋊C₄	—	2	2	no	—
187	C₂×C₄⋊C₄	—	4	8	no	—
188	C₂×C₄⋊C₄	C₂³×C₈₃	1	4	yes	—
189	C₂×C₄²	C₂×C₄×C₈₃	1	4	yes	—
190	C₂×C₄²	D₄×C₈₃	1	2	yes	—
191	C₂×C₄²	D₄×C₈₃	1	4	yes	—
192	C₂×C₄²	C₂³×C₈₃	1	2	yes	—
193	#103	—	2	32	no	—
194	C₄×C₈	—	2	8	no	—
195	C₂⁴	C₂×D₄×C₈₃	1	1	yes	characteristic
196	C₂×Q₈	—	2	8	no	—
197	C₂×Q₈	—	2	4	no	—
198	C₂×Q₈	—	2	32	no	—
199	C₂×Q₈	—	2	8	no	—
200	C₂×Q₈	—	2	8	no	—
201	C₂×Q₈	—	4	16	no	—
202	C₂×Q₈	—	4	4	no	—
203	C₂×Q₈	—	4	4	no	—
204	C₂×Q₈	C₂²×C₄×C₈₃	1	8	yes	—
205	C₂²×C₄	—	2	8	no	—
206	C₂²×C₄	—	2	4	no	—
207	C₂²×C₄	—	2	4	no	—
208	C₂²×C₄	—	4	4	no	—
209	C₂²×C₄	—	4	4	no	—
210	C₂²×C₄	—	4	4	no	—
211	C₂²×C₄	—	4	8	no	—
212	C₂²×C₄	C₂²⋊C₄×C₈₃	1	8	yes	—
213	C₂²×C₄	SD₁₆×C₈₃	1	8	yes	—
214	C₂²×C₄	Q₁₆×C₈₃	1	8	yes	—
215	C₂²×C₄	C₂²×C₄×C₈₃	1	2	yes	—
216	C₂²×C₄	C₂×D₄×C₈₃	1	1	yes	characteristic
217	C₂²×C₄	C₂×D₄×C₈₃	1	4	yes	—
218	C₂²×C₄	C₂×D₄×C₈₃	1	2	yes	—
219	C₂²×C₄	C₂×D₄×C₈₃	1	4	yes	—
220	C₂²×C₄	#41×C₈₃	1	4	yes	—
221	C₂²×C₄	C₂⁴×C₈₃	1	1	yes	characteristic
222	C₂×C₈	—	2	16	no	—
223	C₂×C₈	—	2	4	no	—
224	C₂×C₈	—	2	4	no	—
225	C₂×C₈	C₂×D₄×C₈₃	1	8	yes	—
226	C₄⋊C₄	—	2	32	no	—
227	C₄⋊C₄	—	2	8	no	—
228	C₄⋊C₄	—	4	4	no	—
229	C₄⋊C₄	—	4	16	no	—
230	C₄²	C₂²⋊C₄×C₈₃	1	8	yes	—
231	C₄²	C₂²×C₄×C₈₃	1	4	yes	—
232	C₄²	C₂×D₄×C₈₃	1	4	yes	—
233	C₂³	C₂×C₂²⋊C₄×C₈₃	1	2	yes	—
234	C₂³	#124×C₈₃	1	2	yes	—
235	C₂³	#127×C₈₃	1	2	yes	—
236	C₂³	C₂×SD₁₆×C₈₃	1	4	yes	—
237	C₂³	C₂×Q₁₆×C₈₃	1	4	yes	—
238	C₂³	C₂²×D₄×C₈₃	1	1	yes	characteristic
239	Q₈	—	2	16	no	—
240	Q₈	—	4	8	no	—
241	Q₈	—	4	16	no	—
242	C₂×C₄	—	2	8	no	—
243	C₂×C₄	—	4	4	no	—
244	C₂×C₄	—	4	4	no	—
245	C₂×C₄	—	4	16	no	—
246	C₂×C₄	—	4	4	no	—
247	C₂×C₄	—	4	8	no	—
248	C₂×C₄	—	8	8	no	—
249	C₂×C₄	#103×C₈₃	1	32	yes	complemented
250	C₂×C₄	C₂×C₂²⋊C₄×C₈₃	1	2	yes	—
251	C₂×C₄	C₂×C₂²⋊C₄×C₈₃	1	8	yes	—
252	C₂×C₄	C₄×D₄×C₈₃	1	8	yes	—
253	C₂×C₄	#121×C₈₃	1	8	yes	—
254	C₂×C₄	#124×C₈₃	1	2	yes	—
255	C₂×C₄	#127×C₈₃	1	2	yes	—
256	C₂×C₄	C₂×SD₁₆×C₈₃	1	8	yes	—
257	C₂×C₄	C₂×Q₁₆×C₈₃	1	8	yes	—
258	C₂×C₄	C₂³×C₄×C₈₃	1	1	yes	characteristic
259	C₂×C₄	C₂²×D₄×C₈₃	1	2	yes	—
260	C₂×C₄	C₂²×D₄×C₈₃	1	1	yes	characteristic
261	C₂×C₄	C₂×#41×C₈₃	1	2	yes	—
262	C₈	—	2	8	no	—
263	C₂²	#390×C₈₃	1	4	yes	—
264	C₂²	C₂×#103×C₈₃	1	8	yes	—
265	C₂²	#487×C₈₃	1	8	yes	—
266	C₂²	#492×C₈₃	1	4	yes	—
267	C₂²	#494×C₈₃	1	4	yes	—
268	C₂²	C₂²×C₂²⋊C₄×C₈₃	1	1	yes	characteristic
269	C₂²	C₂×#124×C₈₃	1	1	yes	characteristic
270	C₂²	C₂×#127×C₈₃	1	1	yes	characteristic
271	C₂²	C₂²×SD₁₆×C₈₃	1	2	yes	—
272	C₂²	C₂²×Q₁₆×C₈₃	1	2	yes	—
273	C₄	—	4	8	no	—
274	C₄	—	8	4	no	—
275	C₄	#390×C₈₃	1	4	yes	—
276	C₄	C₂×#103×C₈₃	1	16	yes	complemented
277	C₄	C₂×C₄×D₄×C₈₃	1	2	yes	—
278	C₄	C₂×#121×C₈₃	1	2	yes	—
279	C₂	C₈₃×#1956	1	8	yes	direct product factor
280	C₂	C₂×#390×C₈₃	1	1	yes	characteristic
281	C₂	C₂²×#103×C₈₃	1	2	yes	—
282	C₂	C₂×#487×C₈₃	1	2	yes	—
283	C₂	C₂×#492×C₈₃	1	1	yes	characteristic
284	C₂	C₂×#494×C₈₃	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂⁴×C₈₃	C₂×D₄
commutator	C₂×C₄	C₂³×C₄×C₈₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₈₃
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₈₃ ◁ C₂²×C₄²×C₈₃ ◁ G

Complements

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

N	H ≅ G/N	type
5. C₈₃×#1956	C₂	direct (N × H)
10. C₂×C₄⋊Q₈×C₈₃	C₄	semidirect
38. C₄⋊Q₈×C₈₃	C₂×C₄	semidirect
106. C₂×C₄×C₈₃	#103	semidirect
133. C₄×C₈₃	C₂×#103	semidirect
136. C₂×#1956	C₈₃	direct (N × H)
137. C₂×C₈₃	#1956	direct (N × H)
147. #1956	C₂×C₈₃	direct (N × H)
148. C₈₃	C₂×#1956	direct (N × H)
153. C₂×C₄⋊Q₈	C₄×C₈₃	semidirect
181. C₄⋊Q₈	C₂×C₄×C₈₃	semidirect
249. C₂×C₄	#103×C₈₃	semidirect
276. C₄	C₂×#103×C₈₃	semidirect
279. C₂	C₈₃×#1956	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₄⋊Q₈×C₈₃	2
C₂²×#103×C₈₃	2
C₂²×C₄×C₈×C₈₃	2
C₂×#384×C₈₃	2
C₈₃×#1956	2
C₂×#1956	83

Subgroups — C2×C83×#1956 #35249016

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₈₃×#1956 #35249016