Subgroups — C₂²×D₁₃×C₄⋊Q₈ #16769607

Order 3328.

12190 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×Q₈×D₁₃	C₂	1	2	yes	maximal
2	C₂²×#88123	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×#88142	C₂	1	4	yes	maximal, complemented
4	C₂²×#88198	C₂	1	2	yes	maximal, complemented
5	C₂×D₁₃×C₄⋊Q₈	C₂	1	48	yes	maximal, direct product factor
6	C₂²×C₄⋊C₄×D₁₃	C₂	1	4	yes	maximal
7	C₂²×C₄²×D₁₃	C₂	1	1	yes	maximal, characteristic
8	C₂²×C₁₃×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂³×Q₈×C₁₃	C₂²	1	2	yes	—
10	C₂²×Q₈×D₁₃	—	2	32	no	—
11	C₂²×Q₈×D₁₃	C₂²	1	48	yes	—
12	C₂³×C₄×D₁₃	C₂²	1	4	yes	—
13	C₂³×C₄×D₁₃	C₂²	1	1	yes	characteristic
14	C₂³×C₄×D₁₃	C₂²	1	2	yes	—
15	C₂³×Dic₂₆	C₂²	1	2	yes	—
16	C₂³×Dic₂₆	C₂²	1	4	yes	—
17	C₂×C₁₃×C₄⋊Q₈	C₂²	1	24	yes	complemented
18	C₂²×C₁₃×C₄⋊C₄	C₂²	1	4	yes	—
19	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
20	C₂×#88198	C₂²	1	48	yes	complemented
21	C₂²×C₄⋊Dic₁₃	C₂²	1	4	yes	—
22	C₂²×#6101	C₂²	1	8	yes	—
23	C₂²×C₄×Dic₁₃	C₂²	1	1	yes	characteristic
24	C₂²×C₄×Dic₁₃	C₂²	1	2	yes	—
25	D₁₃×C₄⋊Q₈	C₂²	1	256	yes	direct product factor
26	C₂×C₄⋊C₄×D₁₃	C₂²	1	96	yes	—
27	C₂×#88142	C₂²	1	96	yes	complemented
28	C₂×C₄²×D₁₃	C₂²	1	24	yes	—
29	C₂×#88123	C₂²	1	24	yes	complemented
30	C₂⁴×D₁₃	C₂³	1	1	yes	characteristic
31	C₂²×Q₈×C₁₃	—	2	16	no	—
32	C₂²×Q₈×C₁₃	C₂³	1	24	yes	—
33	C₂³×C₄×C₁₃	C₂³	1	4	yes	—
34	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
35	C₂³×C₄×C₁₃	C₂³	1	2	yes	—
36	C₂³×Dic₁₃	C₂³	1	4	yes	—
37	C₂³×Dic₁₃	C₂³	1	1	yes	characteristic
38	C₂³×Dic₁₃	C₂³	1	2	yes	—
39	C₂×Q₈×D₁₃	—	2	384	no	—
40	C₂×Q₈×D₁₃	C₂³	1	128	yes	—
41	C₂²×C₄×D₁₃	—	2	32	no	—
42	C₂²×C₄×D₁₃	D₄	1	16	yes	—
43	C₂²×C₄×D₁₃	Q₈	1	32	yes	complemented
44	C₂²×C₄×D₁₃	C₂³	1	48	yes	—
45	C₂²×C₄×D₁₃	C₂³	1	12	yes	—
46	C₂²×C₄×D₁₃	C₂³	1	24	yes	—
47	C₂²×Dic₂₆	—	2	16	no	—
48	C₂²×Dic₂₆	—	2	32	no	—
49	C₂²×Dic₂₆	C₂³	1	24	yes	—
50	C₂²×Dic₂₆	C₂³	1	48	yes	—
51	C₁₃×C₄⋊Q₈	C₂³	1	64	yes	complemented
52	C₂×C₁₃×C₄⋊C₄	C₂³	1	48	yes	—
53	C₂×C₄²×C₁₃	C₂³	1	12	yes	—
54	#88198	C₂³	1	128	yes	complemented
55	C₂×C₄⋊Dic₁₃	C₂³	1	48	yes	—
56	C₂×#6101	C₂³	1	96	yes	—
57	C₂×C₄×Dic₁₃	C₂³	1	24	yes	—
58	C₂×C₄×Dic₁₃	C₂³	1	12	yes	—
59	C₄⋊C₄×D₁₃	C₂³	1	256	yes	—
60	#88142	C₂³	1	256	yes	complemented
61	C₄²×D₁₃	C₂³	1	64	yes	—
62	#88123	C₂³	1	64	yes	complemented
63	C₂³×C₄⋊Q₈	—	13	1	no	maximal, Sylow
64	C₂⁴×C₁₃	C₂⁴	1	1	yes	characteristic
65	C₂³×D₁₃	C₂×D₄	1	8	yes	—
66	C₂³×D₁₃	C₂×Q₈	1	16	yes	—
67	C₂³×D₁₃	C₂⁴	1	6	yes	—
68	C₂×Q₈×C₁₃	—	2	96	no	—
69	C₂×Q₈×C₁₃	C₂⁴	1	32	yes	—
70	C₂²×C₄×C₁₃	—	2	16	no	—
71	C₂²×C₄×C₁₃	C₂×D₄	1	8	yes	—
72	C₂²×C₄×C₁₃	C₂×Q₈	1	16	yes	complemented
73	C₂²×C₄×C₁₃	C₂⁴	1	24	yes	—
74	C₂²×C₄×C₁₃	C₂⁴	1	6	yes	—
75	C₂²×C₄×C₁₃	C₂⁴	1	12	yes	—
76	C₂²×Dic₁₃	—	2	16	no	—
77	C₂²×Dic₁₃	C₂×D₄	1	8	yes	—
78	C₂²×Dic₁₃	C₂×Q₈	1	16	yes	complemented
79	C₂²×Dic₁₃	C₂⁴	1	24	yes	—
80	C₂²×Dic₁₃	C₂⁴	1	6	yes	—
81	C₂²×Dic₁₃	C₂⁴	1	12	yes	—
82	Q₈×D₁₃	—	2	512	no	—
83	C₂×C₄×D₁₃	—	2	192	no	—
84	C₂×C₄×D₁₃	C₂×D₄	1	96	yes	—
85	C₂×C₄×D₁₃	C₂×Q₈	1	192	yes	complemented
86	C₂×C₄×D₁₃	C₂⁴	1	64	yes	—
87	C₂×C₄×D₁₃	C₂⁴	1	16	yes	—
88	C₂×C₄×D₁₃	C₂⁴	1	32	yes	—
89	C₂×Dic₂₆	—	2	96	no	—
90	C₂×Dic₂₆	—	2	192	no	—
91	C₂×Dic₂₆	C₂⁴	1	32	yes	—
92	C₂×Dic₂₆	C₂⁴	1	64	yes	—
93	C₁₃×C₄⋊C₄	C₂⁴	1	64	yes	—
94	C₄²×C₁₃	C₂⁴	1	16	yes	—
95	C₄⋊Dic₁₃	C₂⁴	1	64	yes	—
96	#6101	C₂⁴	1	128	yes	—
97	C₄×Dic₁₃	C₂⁴	1	32	yes	—
98	C₄×Dic₁₃	C₂⁴	1	16	yes	—
99	C₂⁴×Q₈	—	13	2	no	—
100	C₂²×C₄⋊Q₈	—	13	1	no	—
101	C₂²×C₄⋊Q₈	—	13	2	no	—
102	C₂²×C₄⋊Q₈	—	13	48	no	—
103	C₂²×C₄⋊Q₈	—	13	4	no	—
104	C₂²×C₄⋊Q₈	D₁₃	1	1	yes	characteristic, direct product factor
105	C₂³×C₄⋊C₄	—	13	4	no	—
106	C₂³×C₄²	—	13	1	no	—
107	C₂³×C₁₃	C₂²×D₄	1	4	yes	—
108	C₂³×C₁₃	C₂²×Q₈	1	8	yes	—
109	C₂³×C₁₃	C₂⁵	1	3	yes	—
110	C₂²×D₁₃	C₄⋊Q₈	1	64	yes	direct product factor
111	C₂²×D₁₃	C₂²×D₄	1	24	yes	—
112	C₂²×D₁₃	C₂²×Q₈	1	48	yes	—
113	C₂²×D₁₃	C₂⁵	1	4	yes	—
114	Q₈×C₁₃	—	2	64	no	—
115	C₂×C₄×C₁₃	—	2	48	no	—
116	C₂×C₄×C₁₃	C₂²×D₄	1	24	yes	—
117	C₂×C₄×C₁₃	C₂²×Q₈	1	48	yes	complemented
118	C₂×C₄×C₁₃	C₂⁵	1	16	yes	—
119	C₂×C₄×C₁₃	C₂⁵	1	4	yes	—
120	C₂×C₄×C₁₃	C₂⁵	1	8	yes	—
121	C₂×Dic₁₃	—	2	48	no	—
122	C₂×Dic₁₃	C₂²×D₄	1	24	yes	—
123	C₂×Dic₁₃	C₂²×Q₈	1	48	yes	complemented
124	C₂×Dic₁₃	C₂⁵	1	16	yes	—
125	C₂×Dic₁₃	C₂⁵	1	4	yes	—
126	C₂×Dic₁₃	C₂⁵	1	8	yes	—
127	C₄×D₁₃	—	2	128	no	—
128	C₄×D₁₃	C₂²×D₄	1	64	yes	—
129	C₄×D₁₃	C₂²×Q₈	1	128	yes	complemented
130	Dic₂₆	—	2	64	no	—
131	Dic₂₆	—	2	128	no	—
132	C₂³×Q₈	—	13	48	no	—
133	C₂³×Q₈	—	13	4	no	—
134	C₂³×Q₈	—	13	2	no	—
135	C₂³×Q₈	—	26	32	no	—
136	C₂³×Q₈	C₂×D₁₃	1	2	yes	—
137	C₂⁴×C₄	—	13	1	no	—
138	C₂⁴×C₄	—	13	2	no	—
139	C₂⁴×C₄	—	13	4	no	—
140	C₂×C₄⋊Q₈	—	13	48	no	—
141	C₂×C₄⋊Q₈	—	13	256	no	—
142	C₂×C₄⋊Q₈	—	13	96	no	—
143	C₂×C₄⋊Q₈	—	13	24	no	—
144	C₂×C₄⋊Q₈	C₂×D₁₃	1	24	yes	direct product factor
145	C₂²×C₄⋊C₄	—	13	8	no	—
146	C₂²×C₄⋊C₄	—	13	4	no	—
147	C₂²×C₄⋊C₄	—	13	96	no	—
148	C₂²×C₄⋊C₄	C₂×D₁₃	1	4	yes	—
149	C₂²×C₄²	—	13	1	no	—
150	C₂²×C₄²	—	13	2	no	—
151	C₂²×C₄²	—	13	24	no	—
152	C₂²×C₄²	C₂×D₁₃	1	1	yes	characteristic
153	C₂²×C₁₃	C₂×C₄⋊Q₈	1	16	yes	complemented
154	C₂²×C₁₃	C₂³×D₄	1	6	yes	—
155	C₂²×C₁₃	C₂³×Q₈	1	12	yes	—
156	C₂²×C₁₃	C₂⁶	1	1	yes	characteristic
157	C₂×D₁₃	C₂×C₄⋊Q₈	1	96	yes	direct product factor
158	C₂×D₁₃	C₂³×D₄	1	8	yes	—
159	C₂×D₁₃	C₂³×Q₈	1	16	yes	—
160	C₄×C₁₃	—	2	16	no	—
161	C₄×C₁₃	C₂³×D₄	1	8	yes	—
162	C₄×C₁₃	C₂³×Q₈	1	16	yes	complemented
163	Dic₁₃	—	2	16	no	—
164	Dic₁₃	C₂³×D₄	1	8	yes	—
165	Dic₁₃	C₂³×Q₈	1	16	yes	complemented
166	C₂⁵	—	13	1	no	—
167	C₂²×Q₈	—	2	16	no	—
168	C₂²×Q₈	—	13	128	no	—
169	C₂²×Q₈	—	13	48	no	—
170	C₂²×Q₈	—	13	24	no	—
171	C₂²×Q₈	—	26	32	no	—
172	C₂²×Q₈	—	26	384	no	—
173	C₂²×Q₈	—	26	16	no	—
174	C₂²×Q₈	C₂²×D₁₃	1	24	yes	—
175	C₂³×C₄	—	13	2	no	—
176	C₂³×C₄	—	13	1	no	—
177	C₂³×C₄	—	13	16	no	—
178	C₂³×C₄	—	13	32	no	—
179	C₂³×C₄	—	13	24	no	—
180	C₂³×C₄	—	13	12	no	—
181	C₂³×C₄	—	13	4	no	—
182	C₂³×C₄	—	13	48	no	—
183	C₂³×C₄	—	26	32	no	—
184	C₂³×C₄	C₂²×D₁₃	1	4	yes	—
185	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
186	C₂³×C₄	C₂²×D₁₃	1	2	yes	—
187	C₄⋊Q₈	—	13	128	no	—
188	C₄⋊Q₈	—	13	256	no	—
189	C₄⋊Q₈	—	13	64	no	—
190	C₄⋊Q₈	C₂²×D₁₃	1	64	yes	direct product factor
191	C₂×C₄⋊C₄	—	13	96	no	—
192	C₂×C₄⋊C₄	—	13	48	no	—
193	C₂×C₄⋊C₄	—	13	256	no	—
194	C₂×C₄⋊C₄	C₂²×D₁₃	1	48	yes	—
195	C₂×C₄²	—	13	12	no	—
196	C₂×C₄²	—	13	24	no	—
197	C₂×C₄²	—	13	64	no	—
198	C₂×C₄²	C₂²×D₁₃	1	12	yes	—
199	C₂×C₁₃	C₂²×C₄⋊Q₈	1	12	yes	complemented
200	C₂×C₁₃	C₂⁴×D₄	1	1	yes	characteristic
201	C₂×C₁₃	C₂⁴×Q₈	1	2	yes	—
202	D₁₃	C₂²×C₄⋊Q₈	1	16	yes	direct product factor
203	C₂⁴	—	13	8	no	—
204	C₂⁴	—	13	16	no	—
205	C₂⁴	—	13	6	no	—
206	C₂⁴	C₂³×D₁₃	1	1	yes	characteristic
207	C₂×Q₈	—	2	96	no	—
208	C₂×Q₈	—	13	64	no	—
209	C₂×Q₈	—	13	32	no	—
210	C₂×Q₈	—	26	192	no	—
211	C₂×Q₈	—	26	512	no	—
212	C₂×Q₈	—	26	96	no	—
213	C₂×Q₈	C₂³×D₁₃	1	32	yes	—
214	C₂²×C₄	—	2	16	no	—
215	C₂²×C₄	—	13	8	no	—
216	C₂²×C₄	—	13	16	no	—
217	C₂²×C₄	—	13	12	no	—
218	C₂²×C₄	—	13	6	no	—
219	C₂²×C₄	—	13	96	no	—
220	C₂²×C₄	—	13	192	no	—
221	C₂²×C₄	—	13	32	no	—
222	C₂²×C₄	—	13	16	no	—
223	C₂²×C₄	—	13	24	no	—
224	C₂²×C₄	—	13	64	no	—
225	C₂²×C₄	—	26	192	no	—
226	C₂²×C₄	—	26	16	no	—
227	C₂²×C₄	D₄×D₁₃	1	8	yes	—
228	C₂²×C₄	Q₈×D₁₃	1	16	yes	complemented
229	C₂²×C₄	C₂³×D₁₃	1	24	yes	—
230	C₂²×C₄	C₂³×D₁₃	1	6	yes	—
231	C₂²×C₄	C₂³×D₁₃	1	12	yes	—
232	C₄⋊C₄	—	13	128	no	—
233	C₄⋊C₄	—	13	64	no	—
234	C₄⋊C₄	C₂³×D₁₃	1	64	yes	—
235	C₄²	—	13	16	no	—
236	C₄²	—	13	32	no	—
237	C₄²	C₂³×D₁₃	1	16	yes	—
238	C₁₃	C₂³×C₄⋊Q₈	1	1	yes	characteristic, complemented, Sylow
239	C₂³	—	13	64	no	—
240	C₂³	—	13	24	no	—
241	C₂³	—	13	48	no	—
242	C₂³	—	13	4	no	—
243	C₂³	C₂×D₄×D₁₃	1	4	yes	—
244	C₂³	C₂×Q₈×D₁₃	1	8	yes	—
245	C₂³	C₂⁴×D₁₃	1	3	yes	—
246	Q₈	—	2	64	no	—
247	Q₈	—	26	128	no	—
248	Q₈	—	26	64	no	—
249	C₂×C₄	—	2	48	no	—
250	C₂×C₄	—	13	24	no	—
251	C₂×C₄	—	13	48	no	—
252	C₂×C₄	—	13	8	no	—
253	C₂×C₄	—	13	4	no	—
254	C₂×C₄	—	13	64	no	—
255	C₂×C₄	—	13	128	no	—
256	C₂×C₄	—	13	16	no	—
257	C₂×C₄	—	26	128	no	—
258	C₂×C₄	—	26	48	no	—
259	C₂×C₄	C₂×D₄×D₁₃	1	24	yes	—
260	C₂×C₄	C₂×Q₈×D₁₃	1	48	yes	complemented
261	C₂×C₄	C₂⁴×D₁₃	1	16	yes	—
262	C₂×C₄	C₂⁴×D₁₃	1	4	yes	—
263	C₂×C₄	C₂⁴×D₁₃	1	8	yes	—
264	C₂²	—	13	8	no	—
265	C₂²	—	13	16	no	—
266	C₂²	—	13	96	no	—
267	C₂²	D₁₃×C₄⋊Q₈	1	16	yes	direct product factor
268	C₂²	C₂²×D₄×D₁₃	1	6	yes	—
269	C₂²	C₂²×Q₈×D₁₃	1	12	yes	—
270	C₂²	C₂⁵×D₁₃	1	1	yes	characteristic
271	C₄	—	2	16	no	—
272	C₄	—	13	8	no	—
273	C₄	—	13	16	no	—
274	C₄	—	26	16	no	—
275	C₄	C₂²×D₄×D₁₃	1	8	yes	—
276	C₄	C₂²×Q₈×D₁₃	1	16	yes	complemented
277	C₂	—	13	16	no	—
278	C₂	C₂×D₁₃×C₄⋊Q₈	1	12	yes	direct product factor
279	C₂	C₂³×D₄×D₁₃	1	1	yes	characteristic
280	C₂	C₂³×Q₈×D₁₃	1	2	yes	—

	subgroup	G/H
center	C₂⁴	C₂³×D₁₃
commutator, cosocle	C₂²×C₁₃	C₂⁶
Frattini	C₂²	C₂⁵×D₁₃
Fitting	C₂²×C₁₃×C₄⋊Q₈	C₂
socle	C₂⁴×C₁₃	C₂⁴
prime radical	C₂⁴×D₁₃	C₂³
hypercenter	C₂²×C₄⋊Q₈	D₁₃
hypocenter	C₁₃	C₂³×C₄⋊Q₈
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
2. C₂²×#88123	C₂	semidirect
3. C₂²×#88142	C₂	semidirect
4. C₂²×#88198	C₂	semidirect
5. C₂×D₁₃×C₄⋊Q₈	C₂	direct (N × H)
8. C₂²×C₁₃×C₄⋊Q₈	C₂	semidirect
17. C₂×C₁₃×C₄⋊Q₈	C₂²	semidirect
20. C₂×#88198	C₂²	semidirect
25. D₁₃×C₄⋊Q₈	C₂²	direct (N × H)
27. C₂×#88142	C₂²	semidirect
29. C₂×#88123	C₂²	semidirect
43. C₂²×C₄×D₁₃	Q₈	semidirect
51. C₁₃×C₄⋊Q₈	C₂³	semidirect
54. #88198	C₂³	semidirect
60. #88142	C₂³	semidirect
62. #88123	C₂³	semidirect
72. C₂²×C₄×C₁₃	C₂×Q₈	semidirect
78. C₂²×Dic₁₃	C₂×Q₈	semidirect
85. C₂×C₄×D₁₃	C₂×Q₈	semidirect
104. C₂²×C₄⋊Q₈	D₁₃	direct (N × H)
110. C₂²×D₁₃	C₄⋊Q₈	direct (N × H)
117. C₂×C₄×C₁₃	C₂²×Q₈	semidirect
123. C₂×Dic₁₃	C₂²×Q₈	semidirect
129. C₄×D₁₃	C₂²×Q₈	semidirect
144. C₂×C₄⋊Q₈	C₂×D₁₃	direct (N × H)
153. C₂²×C₁₃	C₂×C₄⋊Q₈	semidirect
157. C₂×D₁₃	C₂×C₄⋊Q₈	direct (N × H)
162. C₄×C₁₃	C₂³×Q₈	semidirect
165. Dic₁₃	C₂³×Q₈	semidirect
190. C₄⋊Q₈	C₂²×D₁₃	direct (N × H)
199. C₂×C₁₃	C₂²×C₄⋊Q₈	semidirect
202. D₁₃	C₂²×C₄⋊Q₈	direct (N × H)
228. C₂²×C₄	Q₈×D₁₃	semidirect
238. C₁₃	C₂³×C₄⋊Q₈	semidirect
260. C₂×C₄	C₂×Q₈×D₁₃	semidirect
267. C₂²	D₁₃×C₄⋊Q₈	direct (N × H)
276. C₄	C₂²×Q₈×D₁₃	semidirect
278. C₂	C₂×D₁₃×C₄⋊Q₈	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×Q₈×D₁₃	2
C₂²×#88123	2
C₂²×#88142	2
C₂²×#88198	2
C₂×D₁₃×C₄⋊Q₈	2
C₂²×C₄⋊C₄×D₁₃	2
C₂²×C₄²×D₁₃	2
C₂²×C₁₃×C₄⋊Q₈	2
C₂³×C₄⋊Q₈	13

Subgroups — C22×D13×C4⋊Q8 #16769607

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×D₁₃×C₄⋊Q₈ #16769607