Subgroups — C₂×#24921273 #36312588

Order 21248.

6826 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#24921273	C₂	1	32	yes	maximal, direct product factor
2	C₂×Q₈×Dic₁₆₆	C₂	1	4	yes	maximal, complemented
3	C₂×#19128430	C₂	1	4	yes	maximal, complemented
4	C₂×#19128427	C₂	1	8	yes	maximal, complemented
5	C₂×#19128332	C₂	1	6	yes	maximal
6	C₂×#19128258	C₂	1	6	yes	maximal
7	C₂×#19128179	C₂	1	2	yes	maximal
8	C₂²×C₄×Q₈×C₈₃	C₂	1	1	yes	maximal, characteristic
9	Q₈×Dic₁₆₆	C₂²	1	64	yes	complemented
10	#19128430	C₂²	1	64	yes	complemented
11	#19128427	C₂²	1	128	yes	complemented
12	#19128332	C₂²	1	96	yes	—
13	#19128258	C₂²	1	96	yes	—
14	C₂×#14581792	C₂²	1	6	yes	—
15	C₂×#14581792	C₂²	1	6	yes	—
16	C₂×#14581791	C₂²	1	24	yes	—
17	C₂×#14581784	C₂²	1	12	yes	—
18	C₂×#14581783	C₂²	1	24	yes	—
19	C₂×#14581771	C₂²	1	12	yes	—
20	C₂×#14581771	C₂²	1	4	yes	—
21	#19128179	C₂²	1	32	yes	—
22	C₂×Q₈×Dic₈₃	C₂²	1	8	yes	—
23	C₂×Q₈×Dic₈₃	C₂²	1	8	yes	—
24	C₂×C₄×Dic₁₆₆	C₂²	1	12	yes	—
25	C₂×C₄×Dic₁₆₆	C₂²	1	12	yes	—
26	C₂×#14581751	C₂²	1	12	yes	—
27	C₂³×Q₈×C₈₃	C₂²	1	1	yes	characteristic
28	C₂×C₄×Q₈×C₈₃	C₂²	1	16	yes	—
29	C₂×C₄×Q₈×C₈₃	C₂²	1	4	yes	—
30	C₂×C₄×Q₈×C₈₃	C₂²	1	4	yes	—
31	C₂²×C₄⋊C₄×C₈₃	C₂²	1	3	yes	—
32	C₂²×C₄²×C₈₃	C₂²	1	3	yes	—
33	C₂²×Dic₁₆₆	C₂³	1	4	yes	—
34	C₂²×Dic₁₆₆	C₂³	1	12	yes	—
35	#14581792	C₂³	1	48	yes	—
36	#14581792	C₂³	1	48	yes	—
37	#14581791	C₂³	1	192	yes	—
38	#14581784	C₂³	1	96	yes	—
39	#14581783	C₂³	1	192	yes	—
40	#14581771	C₂³	1	96	yes	—
41	#14581771	C₂³	1	32	yes	—
42	Q₈×Dic₈₃	C₂³	1	64	yes	—
43	Q₈×Dic₈₃	C₂³	1	64	yes	—
44	C₄×Dic₁₆₆	C₂³	1	96	yes	—
45	C₄×Dic₁₆₆	C₂³	1	96	yes	—
46	#14581751	C₂³	1	96	yes	—
47	C₂×C₄⋊Dic₈₃	C₂³	1	12	yes	—
48	C₂×C₄⋊Dic₈₃	C₂³	1	12	yes	—
49	C₂×C₄⋊Dic₈₃	C₂³	1	4	yes	—
50	C₂×C₄⋊Dic₈₃	C₂³	1	12	yes	—
51	C₂×Dic₈₃⋊C₄	C₂³	1	8	yes	—
52	C₂×Dic₈₃⋊C₄	C₂³	1	24	yes	—
53	C₂×C₂²⋊Dic₈₃	C₂³	1	8	yes	—
54	C₂×C₄×Dic₈₃	C₂³	1	24	yes	—
55	C₂²×Q₈×C₈₃	Q₈	1	16	yes	complemented
56	C₂²×Q₈×C₈₃	C₂³	1	4	yes	—
57	C₂²×Q₈×C₈₃	C₂³	1	8	yes	—
58	C₂³×C₄×C₈₃	C₂³	1	1	yes	characteristic
59	C₂³×C₄×C₈₃	C₂³	1	3	yes	—
60	C₂³×C₄×C₈₃	C₂³	1	3	yes	—
61	C₄×Q₈×C₈₃	C₂³	1	32	yes	—
62	C₄×Q₈×C₈₃	C₂³	1	32	yes	—
63	C₂×C₄⋊C₄×C₈₃	C₂³	1	24	yes	—
64	C₂×C₄⋊C₄×C₈₃	C₂³	1	12	yes	—
65	C₂×C₄²×C₈₃	C₂³	1	6	yes	—
66	C₂×C₄²×C₈₃	C₂³	1	6	yes	—
67	C₂×C₄²×C₈₃	C₂³	1	24	yes	—
68	C₂×Dic₁₆₆	—	2	16	no	—
69	C₂×Dic₁₆₆	—	4	24	no	—
70	C₂×Dic₁₆₆	C₂⁴	1	16	yes	—
71	C₂×Dic₁₆₆	C₂⁴	1	48	yes	—
72	C₂²×Dic₈₃	C₂⁴	1	8	yes	—
73	C₂²×Dic₈₃	C₂⁴	1	8	yes	—
74	C₄⋊Dic₈₃	C₂⁴	1	48	yes	—
75	C₄⋊Dic₈₃	C₂⁴	1	16	yes	—
76	C₄⋊Dic₈₃	C₂⁴	1	48	yes	—
77	C₄⋊Dic₈₃	C₂⁴	1	48	yes	—
78	Dic₈₃⋊C₄	C₂⁴	1	96	yes	—
79	Dic₈₃⋊C₄	C₂⁴	1	32	yes	—
80	C₂²⋊Dic₈₃	C₂⁴	1	32	yes	—
81	C₄×Dic₈₃	C₂⁴	1	96	yes	—
82	C₂⁴×C₈₃	C₂⁴	1	1	yes	characteristic
83	C₂×Q₈×C₈₃	C₂×Q₈	1	32	yes	—
84	C₂×Q₈×C₈₃	C₂×Q₈	1	64	yes	complemented
85	C₂×Q₈×C₈₃	C₂⁴	1	16	yes	—
86	C₂²×C₄×C₈₃	—	2	4	no	—
87	C₂²×C₄×C₈₃	—	4	6	no	—
88	C₂²×C₄×C₈₃	C₂×Q₈	1	24	yes	—
89	C₂²×C₄×C₈₃	C₂⁴	1	4	yes	—
90	C₂²×C₄×C₈₃	C₂⁴	1	6	yes	—
91	C₂²×C₄×C₈₃	C₂⁴	1	1	yes	characteristic
92	C₂²×C₄×C₈₃	C₂⁴	1	12	yes	—
93	C₂²×C₄×C₈₃	C₂⁴	1	6	yes	—
94	C₂²×C₄×C₈₃	C₂⁴	1	12	yes	—
95	C₂²×C₄×C₈₃	C₂⁴	1	1	yes	characteristic
96	C₄⋊C₄×C₈₃	C₂⁴	1	48	yes	—
97	C₄²×C₈₃	C₂⁴	1	24	yes	—
98	C₄²×C₈₃	C₂⁴	1	24	yes	—
99	C₂³×C₈₃	—	2	2	no	—
100	C₂³×C₈₃	—	2	2	no	—
101	C₂³×C₈₃	C₂²×Q₈	1	4	yes	—
102	C₂³×C₈₃	C₂⁵	1	2	yes	—
103	C₂³×C₈₃	C₂⁵	1	1	yes	characteristic
104	Q₈×C₈₃	C₂²×Q₈	1	64	yes	—
105	C₂×C₄×C₈₃	—	2	2	no	—
106	C₂×C₄×C₈₃	—	2	12	no	—
107	C₂×C₄×C₈₃	—	2	8	no	—
108	C₂×C₄×C₈₃	—	4	12	no	—
109	C₂×C₄×C₈₃	C₂²×Q₈	1	24	yes	—
110	C₂×C₄×C₈₃	C₂²×Q₈	1	48	yes	—
111	C₂×C₄×C₈₃	2₋¹⁺⁴	1	4	yes	—
112	C₂×C₄×C₈₃	C₂⁵	1	2	yes	—
113	C₂×C₄×C₈₃	C₂⁵	1	12	yes	—
114	C₂×C₄×C₈₃	C₂⁵	1	12	yes	—
115	C₂×C₄×C₈₃	C₂⁵	1	2	yes	—
116	C₂×Dic₈₃	—	2	16	no	—
117	C₂×Dic₈₃	—	2	16	no	—
118	C₂×Dic₈₃	C₂⁵	1	16	yes	—
119	C₂×Dic₈₃	C₂⁵	1	16	yes	—
120	Dic₁₆₆	—	2	32	no	—
121	Dic₁₆₆	—	4	48	no	—
122	C₂²×C₈₃	—	2	2	no	—
123	C₂²×C₈₃	—	2	2	no	—
124	C₂²×C₈₃	—	2	8	no	—
125	C₂²×C₈₃	C₂³×Q₈	1	4	yes	—
126	C₂²×C₈₃	C₂³×Q₈	1	2	yes	—
127	C₂²×C₈₃	C₂×2₋¹⁺⁴	1	2	yes	—
128	C₂²×C₈₃	#585	1	2	yes	—
129	C₂²×C₈₃	C₂⁶	1	1	yes	characteristic
130	C₄×C₈₃	—	2	2	no	—
131	C₄×C₈₃	—	2	12	no	—
132	C₄×C₈₃	C₂³×Q₈	1	24	yes	—
133	C₄×C₈₃	C₂×2₋¹⁺⁴	1	4	yes	—
134	Dic₈₃	—	2	16	no	—
135	Dic₈₃	—	2	16	no	—
136	C₂×#3502	—	83	1	no	maximal, Sylow
137	C₂×C₈₃	—	2	4	no	—
138	C₂×C₈₃	#3502	1	4	yes	complemented
139	C₂×C₈₃	C₂⁴×Q₈	1	1	yes	characteristic
140	C₂×C₈₃	C₂²×2₋¹⁺⁴	1	1	yes	characteristic
141	C₂×C₈₃	C₂×#585	1	1	yes	characteristic
142	#3502	—	83	32	no	—
143	C₂×Q₈²	—	83	4	no	—
144	C₂×#557	—	83	8	no	—
145	C₂×#557	—	83	4	no	—
146	C₂×#544	—	83	6	no	—
147	C₂×#533	—	83	6	no	—
148	C₂×#519	—	83	2	no	—
149	C₂²×C₄×Q₈	D₈₃	1	1	yes	characteristic
150	C₈₃	C₂×#3502	1	1	yes	characteristic, complemented, Sylow
151	C₂³×Q₈	C₂×D₈₃	1	1	yes	characteristic
152	Q₈²	—	83	64	no	—
153	#557	—	83	128	no	—
154	#557	—	83	64	no	—
155	#544	—	83	96	no	—
156	#533	—	83	96	no	—
157	C₂×C₄⋊Q₈	—	83	24	no	—
158	C₂×C₄⋊Q₈	—	83	6	no	—
159	C₂×C₄⋊Q₈	—	83	6	no	—
160	C₂×#125	—	83	24	no	—
161	C₂×#125	—	83	12	no	—
162	C₂×C₂²⋊Q₈	—	83	4	no	—
163	C₂×C₂²⋊Q₈	—	83	12	no	—
164	#519	—	83	32	no	—
165	C₂×C₄×Q₈	—	83	8	no	—
166	C₂×C₄×Q₈	—	83	8	no	—
167	C₂×C₄×Q₈	—	83	12	no	—
168	C₂×C₄×Q₈	—	83	12	no	—
169	C₂×C₄×Q₈	C₂×D₈₃	1	4	yes	—
170	C₂×C₄×Q₈	C₂×D₈₃	1	4	yes	—
171	C₂×C₄×Q₈	C₂×D₈₃	1	16	yes	—
172	C₂×#117	—	83	12	no	—
173	C₂²×C₄⋊C₄	C₂×D₈₃	1	3	yes	—
174	C₂²×C₄²	C₂×D₈₃	1	3	yes	—
175	C₂²×Q₈	—	83	12	no	—
176	C₂²×Q₈	—	83	4	no	—
177	C₂²×Q₈	Dic₁₆₆	1	16	yes	complemented
178	C₂²×Q₈	C₂²×D₈₃	1	4	yes	—
179	C₂²×Q₈	C₂²×D₈₃	1	8	yes	—
180	C₂³×C₄	C₂²×D₈₃	1	1	yes	characteristic
181	C₂³×C₄	C₂²×D₈₃	1	3	yes	—
182	C₂³×C₄	C₂²×D₈₃	1	3	yes	—
183	C₄⋊Q₈	—	83	192	no	—
184	C₄⋊Q₈	—	83	48	no	—
185	C₄⋊Q₈	—	83	48	no	—
186	#125	—	83	192	no	—
187	#125	—	83	96	no	—
188	C₂²⋊Q₈	—	83	32	no	—
189	C₂²⋊Q₈	—	83	96	no	—
190	C₄×Q₈	—	83	64	no	—
191	C₄×Q₈	—	83	64	no	—
192	C₄×Q₈	—	83	96	no	—
193	C₄×Q₈	—	83	96	no	—
194	C₄×Q₈	C₂²×D₈₃	1	32	yes	—
195	C₄×Q₈	C₂²×D₈₃	1	32	yes	—
196	#117	—	83	96	no	—
197	C₂×C₄⋊C₄	—	83	12	no	—
198	C₂×C₄⋊C₄	—	83	12	no	—
199	C₂×C₄⋊C₄	—	83	8	no	—
200	C₂×C₄⋊C₄	—	83	4	no	—
201	C₂×C₄⋊C₄	—	83	12	no	—
202	C₂×C₄⋊C₄	—	83	24	no	—
203	C₂×C₄⋊C₄	C₂²×D₈₃	1	12	yes	—
204	C₂×C₄⋊C₄	C₂²×D₈₃	1	24	yes	—
205	C₂×C₂²⋊C₄	—	83	8	no	—
206	C₂×C₄²	—	83	24	no	—
207	C₂×C₄²	C₂²×D₈₃	1	6	yes	—
208	C₂×C₄²	C₂²×D₈₃	1	6	yes	—
209	C₂×C₄²	C₂²×D₈₃	1	24	yes	—
210	C₂⁴	C₂³×D₈₃	1	1	yes	characteristic
211	C₂×Q₈	—	332	24	no	—
212	C₂×Q₈	—	83	48	no	—
213	C₂×Q₈	—	83	16	no	—
214	C₂×Q₈	—	166	16	no	—
215	C₂×Q₈	C₂×Dic₁₆₆	1	32	yes	—
216	C₂×Q₈	C₂×Dic₁₆₆	1	64	yes	complemented
217	C₂×Q₈	C₂³×D₈₃	1	16	yes	—
218	C₂²×C₄	—	2	4	no	—
219	C₂²×C₄	—	4	6	no	—
220	C₂²×C₄	—	83	8	no	—
221	C₂²×C₄	—	83	8	no	—
222	C₂²×C₄	C₂×Dic₁₆₆	1	24	yes	—
223	C₂²×C₄	C₂³×D₈₃	1	4	yes	—
224	C₂²×C₄	C₂³×D₈₃	1	1	yes	characteristic
225	C₂²×C₄	C₂³×D₈₃	1	6	yes	—
226	C₂²×C₄	C₂³×D₈₃	1	12	yes	—
227	C₂²×C₄	C₂³×D₈₃	1	12	yes	—
228	C₂²×C₄	C₂³×D₈₃	1	6	yes	—
229	C₂²×C₄	C₂³×D₈₃	1	1	yes	characteristic
230	C₄⋊C₄	—	83	48	no	—
231	C₄⋊C₄	—	83	48	no	—
232	C₄⋊C₄	—	83	32	no	—
233	C₄⋊C₄	—	83	16	no	—
234	C₄⋊C₄	—	83	48	no	—
235	C₄⋊C₄	—	83	96	no	—
236	C₄⋊C₄	C₂³×D₈₃	1	48	yes	—
237	C₂²⋊C₄	—	83	32	no	—
238	C₄²	—	83	96	no	—
239	C₄²	C₂³×D₈₃	1	24	yes	—
240	C₄²	C₂³×D₈₃	1	24	yes	—
241	C₂³	—	2	2	no	—
242	C₂³	—	2	2	no	—
243	C₂³	C₂²×Dic₁₆₆	1	4	yes	—
244	C₂³	C₂⁴×D₈₃	1	2	yes	—
245	C₂³	C₂⁴×D₈₃	1	1	yes	characteristic
246	Q₈	—	332	48	no	—
247	Q₈	—	166	32	no	—
248	Q₈	C₂²×Dic₁₆₆	1	64	yes	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	2	12	no	—
251	C₂×C₄	—	2	8	no	—
252	C₂×C₄	—	4	12	no	—
253	C₂×C₄	—	83	16	no	—
254	C₂×C₄	—	83	16	no	—
255	C₂×C₄	—	166	16	no	—
256	C₂×C₄	—	166	16	no	—
257	C₂×C₄	C₂²×Dic₁₆₆	1	24	yes	—
258	C₂×C₄	C₂²×Dic₁₆₆	1	48	yes	—
259	C₂×C₄	#14581840	1	4	yes	—
260	C₂×C₄	C₂⁴×D₈₃	1	2	yes	—
261	C₂×C₄	C₂⁴×D₈₃	1	12	yes	—
262	C₂×C₄	C₂⁴×D₈₃	1	12	yes	—
263	C₂×C₄	C₂⁴×D₈₃	1	2	yes	—
264	C₂²	—	2	2	no	—
265	C₂²	—	2	2	no	—
266	C₂²	—	2	8	no	—
267	C₂²	C₂³×Dic₁₆₆	1	4	yes	—
268	C₂²	C₂³×Dic₁₆₆	1	2	yes	—
269	C₂²	C₂×#14581840	1	2	yes	—
270	C₂²	#19128537	1	2	yes	—
271	C₂²	C₂⁵×D₈₃	1	1	yes	characteristic
272	C₄	—	2	2	no	—
273	C₄	—	2	12	no	—
274	C₄	—	166	16	no	—
275	C₄	—	166	16	no	—
276	C₄	C₂³×Dic₁₆₆	1	24	yes	—
277	C₄	C₂×#14581840	1	4	yes	—
278	C₂	—	2	4	no	—
279	C₂	#24921273	1	4	yes	direct product factor
280	C₂	C₂⁴×Dic₁₆₆	1	1	yes	characteristic
281	C₂	C₂²×#14581840	1	1	yes	characteristic
282	C₂	C₂×#19128537	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂⁴×D₈₃
commutator, cosocle	C₂²×C₈₃	C₂⁶
Frattini	C₂²	C₂⁵×D₈₃
Fitting	C₂²×C₄×Q₈×C₈₃	C₂
socle	C₂³×C₈₃	C₂⁵
prime radical	C₂⁴×C₈₃	C₂⁴
hypercenter	C₂²×C₄×Q₈	D₈₃
hypocenter	C₈₃	C₂×#3502
LCS	G ▷ C₂²×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄×Q₈

Complements

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

N	H ≅ G/N	type
1. #24921273	C₂	direct (N × H)
2. C₂×Q₈×Dic₁₆₆	C₂	semidirect
3. C₂×#19128430	C₂	semidirect
4. C₂×#19128427	C₂	semidirect
9. Q₈×Dic₁₆₆	C₂²	semidirect
10. #19128430	C₂²	semidirect
11. #19128427	C₂²	semidirect
55. C₂²×Q₈×C₈₃	Q₈	semidirect
84. C₂×Q₈×C₈₃	C₂×Q₈	semidirect
138. C₂×C₈₃	#3502	semidirect
150. C₈₃	C₂×#3502	semidirect
177. C₂²×Q₈	Dic₁₆₆	semidirect
216. C₂×Q₈	C₂×Dic₁₆₆	semidirect
279. C₂	#24921273	direct (N × H)

Maximal subgroups

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

subgroup	index
#24921273	2
C₂×Q₈×Dic₁₆₆	2
C₂×#19128430	2
C₂×#19128427	2
C₂×#19128332	2
C₂×#19128258	2
C₂×#19128179	2
C₂²×C₄×Q₈×C₈₃	2
C₂×#3502	83

Subgroups — C2×#24921273 #36312588

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#24921273 #36312588