Subgroups — C₂×Q₈×#13050573 #36268788

Order 21248.

2154 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Q₈×D₈₃	C₂	1	1	yes	maximal, characteristic
2	Q₈×#13050573	C₂	1	16	yes	maximal, direct product factor
3	C₂×#19127831	C₂	1	3	yes	maximal, complemented
4	C₂×#19127828	C₂	1	3	yes	maximal, complemented
5	C₂×Q₈×C₈₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#19127723	C₂	1	3	yes	maximal
7	C₂×C₄×#13050573	C₂	1	3	yes	maximal
8	C₂×C₈×Q₈×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×Q₈×D₈₃	C₄	1	1	yes	characteristic
10	C₂²×#13050573	C₂²	1	3	yes	—
11	C₂²×#13050573	C₂²	1	1	yes	characteristic
12	C₂×Q₈×Dic₈₃	C₄	1	1	yes	characteristic
13	C₂×C₄×Dic₁₆₆	C₂²	1	3	yes	—
14	C₂×#14581757	C₄	1	3	yes	—
15	C₄×Q₈×D₈₃	C₄	1	8	yes	—
16	C₄×Q₈×D₈₃	C₂²	1	8	yes	—
17	C₂×C₄⋊C₄×D₈₃	C₄	1	3	yes	—
18	C₂×C₄²×D₈₃	C₂²	1	3	yes	—
19	#19127831	C₂²	1	24	yes	complemented
20	#19127828	C₂²	1	24	yes	complemented
21	Q₈×C₈₃⋊C₈	C₂²	1	8	yes	complemented
22	#19127723	C₂²	1	24	yes	—
23	C₂×C₄⋊(C₈₃⋊C₈)	C₂²	1	3	yes	—
24	C₂×Dic₈₃⋊C₈	C₂²	1	6	yes	—
25	C₄×#13050573	C₂²	1	24	yes	—
26	C₂×#14581702	C₂²	1	3	yes	—
27	C₂×#14581702	C₂²	1	3	yes	—
28	C₂×C₄×C₈₃⋊C₈	C₂²	1	3	yes	—
29	C₂×C₄×Q₈×C₈₃	C₂²	1	1	yes	characteristic
30	C₈×Q₈×C₈₃	C₂²	1	8	yes	complemented
31	C₂×C₄⋊C₈×C₈₃	C₂²	1	3	yes	—
32	C₂×C₄×C₈×C₈₃	C₂²	1	3	yes	—
33	C₂²×Dic₁₆₆	C₂×C₄	1	3	yes	—
34	C₂×Q₈×D₈₃	—	2	8	no	—
35	C₂×Q₈×D₈₃	C₂×C₄	1	8	yes	—
36	C₂²×C₄×D₈₃	C₂×C₄	1	3	yes	—
37	C₂²×C₄×D₈₃	C₂×C₄	1	3	yes	—
38	C₂²×C₄×D₈₃	C₂³	1	1	yes	characteristic
39	C₂×#13050573	—	2	12	no	—
40	C₂×#13050573	Q₈	1	8	yes	direct product factor
41	C₂×#13050573	C₂³	1	12	yes	—
42	C₂×#13050573	C₂³	1	4	yes	—
43	C₂²×C₈₃⋊C₈	C₂³	1	3	yes	—
44	C₂²×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
45	Q₈×Dic₈₃	C₂×C₄	1	8	yes	—
46	C₄×Dic₁₆₆	C₂×C₄	1	12	yes	—
47	C₄×Dic₁₆₆	C₂³	1	12	yes	—
48	#14581757	C₂×C₄	1	24	yes	—
49	C₂×C₄⋊Dic₈₃	C₂×C₄	1	3	yes	—
50	C₂×Dic₈₃⋊C₄	C₂×C₄	1	6	yes	—
51	C₄⋊C₄×D₈₃	C₂×C₄	1	24	yes	—
52	C₂×C₄×Dic₈₃	C₂×C₄	1	3	yes	—
53	C₂×C₄×Dic₈₃	C₂×C₄	1	3	yes	—
54	C₂×C₄×Dic₈₃	C₂³	1	3	yes	—
55	C₄²×D₈₃	C₂×C₄	1	12	yes	—
56	C₄²×D₈₃	C₂³	1	12	yes	—
57	C₄⋊(C₈₃⋊C₈)	C₂³	1	12	yes	—
58	Dic₈₃⋊C₈	C₂³	1	24	yes	—
59	#14581702	C₂³	1	12	yes	—
60	#14581702	C₂³	1	12	yes	—
61	C₄×C₈₃⋊C₈	C₂³	1	12	yes	—
62	C₂²×Q₈×C₈₃	C₂×C₄	1	1	yes	characteristic
63	C₂²×C₈×C₈₃	C₂³	1	3	yes	—
64	C₂²×C₈×C₈₃	C₂³	1	1	yes	characteristic
65	C₄×Q₈×C₈₃	C₂×C₄	1	4	yes	—
66	C₄×Q₈×C₈₃	C₂³	1	4	yes	—
67	C₂×C₄⋊C₄×C₈₃	C₂×C₄	1	3	yes	—
68	C₂×C₄²×C₈₃	C₂³	1	3	yes	—
69	C₄⋊C₈×C₈₃	C₂³	1	12	yes	—
70	C₄×C₈×C₈₃	C₂³	1	12	yes	—
71	C₂³×D₈₃	C₂²×C₄	1	1	yes	characteristic
72	C₂×Dic₁₆₆	—	2	12	no	—
73	C₂×Dic₁₆₆	C₂²×C₄	1	12	yes	—
74	Q₈×D₈₃	—	2	32	no	—
75	C₂²×Dic₈₃	C₂²×C₄	1	3	yes	—
76	C₂²×Dic₈₃	C₂²×C₄	1	3	yes	—
77	C₂²×Dic₈₃	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₄×D₈₃	—	2	12	no	—
79	C₂×C₄×D₈₃	—	4	6	no	—
80	C₂×C₄×D₈₃	C₂²×C₄	1	12	yes	—
81	C₂×C₄×D₈₃	C₂²×C₄	1	12	yes	—
82	C₂×C₄×D₈₃	C₂²×C₄	1	2	yes	—
83	C₂×C₄×D₈₃	C₂×Q₈	1	4	yes	—
84	C₂×C₄×D₈₃	#41	1	4	yes	—
85	C₂×C₄×D₈₃	C₂⁴	1	2	yes	—
86	#13050573	—	2	24	no	—
87	#13050573	C₂×Q₈	1	16	yes	direct product factor
88	C₂×C₈₃⋊C₈	—	2	6	no	—
89	C₂×C₈₃⋊C₈	C₂×Q₈	1	4	yes	complemented
90	C₂×C₈₃⋊C₈	C₂⁴	1	6	yes	—
91	C₂×C₈₃⋊C₈	C₂⁴	1	2	yes	—
92	C₄⋊Dic₈₃	C₂²×C₄	1	12	yes	—
93	Dic₈₃⋊C₄	C₂²×C₄	1	24	yes	—
94	C₄×Dic₈₃	C₂²×C₄	1	6	yes	—
95	C₄×Dic₈₃	C₂²×C₄	1	12	yes	—
96	C₄×Dic₈₃	C₂²×C₄	1	12	yes	—
97	C₄×Dic₈₃	C₂⁴	1	6	yes	—
98	C₂×Q₈×C₈₃	M₁₆	1	8	yes	complemented
99	C₂×Q₈×C₈₃	C₂²×C₄	1	4	yes	—
100	C₂²×C₄×C₈₃	C₂²×C₄	1	3	yes	—
101	C₂²×C₄×C₈₃	C₂²×C₄	1	3	yes	—
102	C₂²×C₄×C₈₃	C₂⁴	1	1	yes	characteristic
103	C₂×C₈×C₈₃	—	2	6	no	—
104	C₂×C₈×C₈₃	C₂×Q₈	1	4	yes	complemented
105	C₂×C₈×C₈₃	C₂⁴	1	6	yes	—
106	C₂×C₈×C₈₃	C₂⁴	1	2	yes	—
107	C₄⋊C₄×C₈₃	C₂²×C₄	1	12	yes	—
108	C₄²×C₈₃	C₂²×C₄	1	6	yes	—
109	C₄²×C₈₃	C₂⁴	1	6	yes	—
110	C₂³×C₈₃	C₂³×C₄	1	1	yes	characteristic
111	C₂²×D₈₃	—	2	2	no	—
112	C₂²×D₈₃	—	2	2	no	—
113	C₂²×D₈₃	C₄×Q₈	1	4	yes	—
114	C₂²×D₈₃	C₂³×C₄	1	2	yes	—
115	Q₈×C₈₃	C₂×M₁₆	1	16	yes	complemented
116	C₂×C₄×C₈₃	—	2	6	no	—
117	C₂×C₄×C₈₃	C₂×M₁₆	1	12	yes	—
118	C₂×C₄×C₈₃	C₂³×C₄	1	6	yes	—
119	C₂×C₄×C₈₃	C₂³×C₄	1	1	yes	characteristic
120	C₂×C₄×C₈₃	C₂³×C₄	1	6	yes	—
121	C₂×C₄×C₈₃	C₂²×Q₈	1	2	yes	—
122	C₂×C₄×C₈₃	C₂×#41	1	2	yes	—
123	C₂×C₄×C₈₃	C₂⁵	1	1	yes	characteristic
124	C₂×Dic₈₃	—	2	6	no	—
125	C₂×Dic₈₃	—	2	6	no	—
126	C₂×Dic₈₃	C₄×Q₈	1	4	yes	—
127	C₂×Dic₈₃	C₂³×C₄	1	6	yes	—
128	C₂×Dic₈₃	C₂³×C₄	1	6	yes	—
129	C₂×Dic₈₃	C₂³×C₄	1	2	yes	—
130	C₄×D₈₃	—	2	24	no	—
131	C₄×D₈₃	—	4	12	no	—
132	C₄×D₈₃	C₄×Q₈	1	8	yes	—
133	C₄×D₈₃	C₂²×Q₈	1	4	yes	—
134	C₄×D₈₃	C₂×#41	1	4	yes	—
135	Dic₁₆₆	—	2	24	no	—
136	C₈×C₈₃	—	2	6	no	—
137	C₈×C₈₃	C₂²×Q₈	1	4	yes	complemented
138	C₈₃⋊C₈	—	2	6	no	—
139	C₈₃⋊C₈	C₂²×Q₈	1	4	yes	complemented
140	C₂²×C₈₃	C₂×C₄×Q₈	1	2	yes	—
141	C₂²×C₈₃	C₂²×M₁₆	1	2	yes	—
142	C₂²×C₈₃	#568	1	2	yes	—
143	C₂²×C₈₃	C₂⁴×C₄	1	1	yes	characteristic
144	C₂×D₈₃	—	2	8	no	—
145	C₂×D₈₃	—	2	2	no	—
146	C₂×D₈₃	—	2	2	no	—
147	C₂×D₈₃	C₂×C₄×Q₈	1	4	yes	—
148	C₄×C₈₃	—	2	6	no	—
149	C₄×C₈₃	C₂×C₄×Q₈	1	2	yes	—
150	C₄×C₈₃	C₂²×M₁₆	1	12	yes	—
151	C₄×C₈₃	C₂³×Q₈	1	1	yes	characteristic
152	C₄×C₈₃	C₂²×#41	1	1	yes	characteristic
153	Dic₈₃	—	2	6	no	—
154	Dic₈₃	—	2	6	no	—
155	Dic₈₃	C₂×C₄×Q₈	1	4	yes	—
156	C₂×Q₈×M₁₆	—	83	1	no	maximal, Sylow
157	C₂×C₈₃	Q₈×M₁₆	1	4	yes	complemented
158	C₂×C₈₃	C₂²×C₄×Q₈	1	1	yes	characteristic
159	C₂×C₈₃	C₂³×M₁₆	1	1	yes	characteristic
160	C₂×C₈₃	C₂×#568	1	1	yes	characteristic
161	D₈₃	—	2	4	no	—
162	C₂²×C₄×Q₈	—	83	1	no	—
163	Q₈×M₁₆	—	83	16	no	—
164	C₂×#446	—	83	3	no	—
165	C₂×#446	—	83	3	no	—
166	C₂×C₈×Q₈	—	83	1	no	—
167	C₂×C₈×Q₈	D₈₃	1	1	yes	characteristic, complemented
168	C₂×#423	—	83	3	no	—
169	C₂×C₄×M₁₆	—	83	3	no	—
170	C₈₃	C₂×Q₈×M₁₆	1	1	yes	characteristic, complemented, Sylow
171	C₂³×Q₈	—	83	1	no	—
172	C₂²×M₁₆	—	83	1	no	—
173	C₂²×M₁₆	—	83	3	no	—
174	C₂×C₄×Q₈	—	83	1	no	—
175	C₂×C₄×Q₈	—	83	8	no	—
176	C₂×C₄×Q₈	—	83	8	no	—
177	C₂×C₄×Q₈	—	83	3	no	—
178	C₂×C₄×Q₈	—	83	3	no	—
179	C₂×C₄×Q₈	C₂×D₈₃	1	1	yes	characteristic
180	C₂²×C₄⋊C₄	—	83	3	no	—
181	C₂²×C₄²	—	83	3	no	—
182	#446	—	83	24	no	—
183	#446	—	83	24	no	—
184	C₈×Q₈	—	83	8	no	—
185	C₈×Q₈	C₂×D₈₃	1	8	yes	complemented
186	#423	—	83	24	no	—
187	C₂×C₄⋊C₈	—	83	6	no	—
188	C₂×C₄⋊C₈	—	83	3	no	—
189	C₂×C₄⋊C₈	C₂×D₈₃	1	3	yes	—
190	C₄×M₁₆	—	83	24	no	—
191	C₂×#97	—	83	3	no	—
192	C₂×#97	—	83	3	no	—
193	C₂×C₄×C₈	—	83	3	no	—
194	C₂×C₄×C₈	C₂×D₈₃	1	3	yes	—
195	C₂²×Q₈	—	83	8	no	—
196	C₂²×Q₈	—	83	3	no	—
197	C₂²×Q₈	—	166	8	no	—
198	C₂²×Q₈	C₄×D₈₃	1	1	yes	characteristic
199	C₂³×C₄	—	83	3	no	—
200	C₂³×C₄	—	83	1	no	—
201	C₂³×C₄	—	83	3	no	—
202	C₂×M₁₆	—	83	8	no	—
203	C₂×M₁₆	—	83	4	no	—
204	C₂×M₁₆	—	83	12	no	—
205	C₂×M₁₆	—	166	12	no	—
206	C₂²×C₈	—	83	1	no	—
207	C₂²×C₈	—	83	3	no	—
208	C₂²×C₈	C₂²×D₈₃	1	3	yes	—
209	C₂²×C₈	C₂²×D₈₃	1	1	yes	characteristic
210	C₄×Q₈	—	83	8	no	—
211	C₄×Q₈	—	83	24	no	—
212	C₄×Q₈	—	83	12	no	—
213	C₄×Q₈	—	83	12	no	—
214	C₄×Q₈	C₄×D₈₃	1	4	yes	—
215	C₄×Q₈	C₂²×D₈₃	1	4	yes	—
216	C₂×C₄⋊C₄	—	83	6	no	—
217	C₂×C₄⋊C₄	—	83	3	no	—
218	C₂×C₄⋊C₄	—	83	24	no	—
219	C₂×C₄⋊C₄	C₄×D₈₃	1	3	yes	—
220	C₂×C₄²	—	83	3	no	—
221	C₂×C₄²	—	83	3	no	—
222	C₂×C₄²	—	83	3	no	—
223	C₂×C₄²	—	83	12	no	—
224	C₂×C₄²	—	83	12	no	—
225	C₂×C₄²	C₂²×D₈₃	1	3	yes	—
226	C₄⋊C₈	—	83	24	no	—
227	C₄⋊C₈	—	83	12	no	—
228	C₄⋊C₈	C₂²×D₈₃	1	12	yes	—
229	#97	—	83	12	no	—
230	#97	—	83	12	no	—
231	C₄×C₈	—	83	12	no	—
232	C₄×C₈	C₂²×D₈₃	1	12	yes	—
233	C₂⁴	—	83	1	no	—
234	C₂×Q₈	—	83	12	no	—
235	C₂×Q₈	—	166	32	no	—
236	C₂×Q₈	—	166	12	no	—
237	C₂×Q₈	#13050573	1	8	yes	direct product factor
238	C₂×Q₈	C₂×C₄×D₈₃	1	4	yes	—
239	C₂²×C₄	—	332	6	no	—
240	C₂²×C₄	—	83	12	no	—
241	C₂²×C₄	—	83	1	no	—
242	C₂²×C₄	—	83	2	no	—
243	C₂²×C₄	—	83	4	no	—
244	C₂²×C₄	—	83	4	no	—
245	C₂²×C₄	—	83	2	no	—
246	C₂²×C₄	—	83	3	no	—
247	C₂²×C₄	—	83	3	no	—
248	C₂²×C₄	—	83	12	no	—
249	C₂²×C₄	—	166	12	no	—
250	C₂²×C₄	C₂×C₄×D₈₃	1	3	yes	—
251	C₂²×C₄	C₂×C₄×D₈₃	1	3	yes	—
252	C₂²×C₄	C₂³×D₈₃	1	1	yes	characteristic
253	M₁₆	—	83	16	no	—
254	M₁₆	—	166	24	no	—
255	C₂×C₈	—	2	6	no	—
256	C₂×C₈	—	83	4	no	—
257	C₂×C₈	—	83	2	no	—
258	C₂×C₈	—	83	6	no	—
259	C₂×C₈	—	166	6	no	—
260	C₂×C₈	Q₈×D₈₃	1	4	yes	complemented
261	C₂×C₈	C₂³×D₈₃	1	6	yes	—
262	C₂×C₈	C₂³×D₈₃	1	2	yes	—
263	C₄⋊C₄	—	83	24	no	—
264	C₄⋊C₄	—	83	12	no	—
265	C₄⋊C₄	C₂×C₄×D₈₃	1	12	yes	—
266	C₄²	—	83	12	no	—
267	C₄²	—	83	12	no	—
268	C₄²	—	83	6	no	—
269	C₄²	—	83	6	no	—
270	C₄²	C₂×C₄×D₈₃	1	6	yes	—
271	C₄²	C₂³×D₈₃	1	6	yes	—
272	C₂³	—	83	4	no	—
273	C₂³	—	83	2	no	—
274	C₂³	—	166	2	no	—
275	C₂³	—	166	2	no	—
276	C₂³	C₂²×C₄×D₈₃	1	1	yes	characteristic
277	Q₈	—	166	24	no	—
278	Q₈	C₂×#13050573	1	16	yes	direct product factor
279	C₂×C₄	—	2	6	no	—
280	C₂×C₄	—	332	12	no	—
281	C₂×C₄	—	83	4	no	—
282	C₂×C₄	—	83	2	no	—
283	C₂×C₄	—	83	8	no	—
284	C₂×C₄	—	83	4	no	—
285	C₂×C₄	—	83	4	no	—
286	C₂×C₄	—	83	6	no	—
287	C₂×C₄	—	83	6	no	—
288	C₂×C₄	—	166	6	no	—
289	C₂×C₄	—	166	6	no	—
290	C₂×C₄	—	166	24	no	—
291	C₂×C₄	C₂×#13050573	1	12	yes	—
292	C₂×C₄	C₂²×C₄×D₈₃	1	6	yes	—
293	C₂×C₄	C₂²×C₄×D₈₃	1	1	yes	characteristic
294	C₂×C₄	C₂²×C₄×D₈₃	1	6	yes	—
295	C₂×C₄	C₂×Q₈×D₈₃	1	2	yes	—
296	C₂×C₄	#41×D₈₃	1	2	yes	—
297	C₂×C₄	C₂⁴×D₈₃	1	1	yes	characteristic
298	C₈	—	2	6	no	—
299	C₈	—	83	4	no	—
300	C₈	—	166	6	no	—
301	C₈	C₂×Q₈×D₈₃	1	4	yes	complemented
302	C₂²	—	83	4	no	—
303	C₂²	—	166	2	no	—
304	C₂²	—	166	2	no	—
305	C₂²	—	166	8	no	—
306	C₂²	C₄×Q₈×D₈₃	1	2	yes	—
307	C₂²	C₂²×#13050573	1	2	yes	—
308	C₂²	#19128466	1	2	yes	—
309	C₂²	C₂³×C₄×D₈₃	1	1	yes	characteristic
310	C₄	—	2	6	no	—
311	C₄	—	83	4	no	—
312	C₄	—	166	6	no	—
313	C₄	—	166	6	no	—
314	C₄	C₄×Q₈×D₈₃	1	2	yes	—
315	C₄	C₂²×#13050573	1	12	yes	—
316	C₄	C₂²×Q₈×D₈₃	1	1	yes	characteristic
317	C₄	C₂×#41×D₈₃	1	1	yes	characteristic
318	C₂	—	166	4	no	—
319	C₂	Q₈×#13050573	1	4	yes	direct product factor
320	C₂	C₂×C₄×Q₈×D₈₃	1	1	yes	characteristic
321	C₂	C₂³×#13050573	1	1	yes	characteristic
322	C₂	C₂×#19128466	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂³×D₈₃
commutator	C₂²×C₈₃	C₂⁴×C₄
Frattini	C₂×C₄	C₂⁴×D₈₃
Fitting	C₂×C₈×Q₈×C₈₃	C₂
socle	C₂³×C₈₃	C₂³×C₄
prime radical	C₂³×D₈₃	C₂²×C₄
cosocle	C₂×C₄×C₈₃	C₂⁵
hypercenter	C₂×C₈×Q₈	D₈₃
hypocenter	C₈₃	C₂×Q₈×M₁₆
LCS	G ▷ C₂²×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₈×Q₈

Complements

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

N	H ≅ G/N	type
2. Q₈×#13050573	C₂	direct (N × H)
3. C₂×#19127831	C₂	semidirect
4. C₂×#19127828	C₂	semidirect
5. C₂×Q₈×C₈₃⋊C₈	C₂	semidirect
8. C₂×C₈×Q₈×C₈₃	C₂	semidirect
19. #19127831	C₂²	semidirect
20. #19127828	C₂²	semidirect
21. Q₈×C₈₃⋊C₈	C₂²	semidirect
30. C₈×Q₈×C₈₃	C₂²	semidirect
40. C₂×#13050573	Q₈	direct (N × H)
87. #13050573	C₂×Q₈	direct (N × H)
89. C₂×C₈₃⋊C₈	C₂×Q₈	semidirect
98. C₂×Q₈×C₈₃	M₁₆	semidirect
104. C₂×C₈×C₈₃	C₂×Q₈	semidirect
115. Q₈×C₈₃	C₂×M₁₆	semidirect
137. C₈×C₈₃	C₂²×Q₈	semidirect
139. C₈₃⋊C₈	C₂²×Q₈	semidirect
157. C₂×C₈₃	Q₈×M₁₆	semidirect
167. C₂×C₈×Q₈	D₈₃	semidirect
170. C₈₃	C₂×Q₈×M₁₆	semidirect
185. C₈×Q₈	C₂×D₈₃	semidirect
237. C₂×Q₈	#13050573	direct (N × H)
260. C₂×C₈	Q₈×D₈₃	semidirect
278. Q₈	C₂×#13050573	direct (N × H)
301. C₈	C₂×Q₈×D₈₃	semidirect
319. C₂	Q₈×#13050573	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₄×Q₈×D₈₃	2
Q₈×#13050573	2
C₂×#19127831	2
C₂×#19127828	2
C₂×Q₈×C₈₃⋊C₈	2
C₂×#19127723	2
C₂×C₄×#13050573	2
C₂×C₈×Q₈×C₈₃	2
C₂×Q₈×M₁₆	83

Subgroups — C2×Q8×#13050573 #36268788

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×Q₈×#13050573 #36268788