Subgroups — C₄²×Q₈×D₁₃ #16021113

Order 3328.

3966 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Q₈×D₁₃	C₂	1	3	yes	maximal
2	C₄×Q₈×Dic₁₃	C₂	1	3	yes	maximal, complemented
3	C₄²×Dic₂₆	C₂	1	3	yes	maximal, complemented
4	C₄×#88141	C₂	1	9	yes	maximal, complemented
5	C₄×C₄⋊C₄×D₁₃	C₂	1	9	yes	maximal
6	C₄³×D₁₃	C₂	1	3	yes	maximal
7	C₄²×Q₈×C₁₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×Q₈×D₁₃	C₂²	1	1	yes	characteristic
9	C₂×C₄×Q₈×C₁₃	C₂²	1	3	yes	—
10	C₂×Q₈×Dic₁₃	C₂²	1	3	yes	—
11	C₄×Q₈×D₁₃	C₄	1	48	yes	direct product factor
12	C₂×C₄⋊C₄×D₁₃	C₂²	1	9	yes	—
13	C₂×#88141	C₂²	1	9	yes	—
14	C₂×C₄²×D₁₃	C₂²	1	9	yes	—
15	C₂×C₄²×D₁₃	C₂²	1	9	yes	—
16	C₂×C₄²×D₁₃	C₂²	1	1	yes	characteristic
17	C₂×C₄×Dic₂₆	C₂²	1	9	yes	—
18	C₄×C₁₃×C₄⋊C₄	C₂²	1	9	yes	—
19	C₄³×C₁₃	C₂²	1	3	yes	—
20	C₄⋊C₄×Dic₁₃	C₂²	1	18	yes	—
21	C₄×C₄⋊Dic₁₃	C₂²	1	9	yes	—
22	C₄×#6101	C₂²	1	18	yes	—
23	C₄²×Dic₁₃	C₂²	1	9	yes	—
24	C₄²×Dic₁₃	C₂²	1	9	yes	—
25	C₄²×Dic₁₃	C₂²	1	3	yes	—
26	D₁₃×#95	C₂²	1	6	yes	—
27	#11718310	C₂²	1	18	yes	—
28	C₂²×Q₈×C₁₃	C₂³	1	1	yes	characteristic
29	C₂×Q₈×D₁₃	C₂×C₄	1	24	yes	—
30	C₂²×C₄×D₁₃	C₂³	1	9	yes	—
31	C₂²×C₄×D₁₃	C₂³	1	3	yes	—
32	C₂²×C₄×D₁₃	C₂³	1	3	yes	—
33	C₂²×Dic₂₆	C₂³	1	3	yes	—
34	C₄×Q₈×C₁₃	C₂×C₄	1	24	yes	complemented
35	C₂×C₁₃×C₄⋊C₄	C₂³	1	9	yes	—
36	C₂×C₄²×C₁₃	C₂³	1	9	yes	—
37	C₂×C₄²×C₁₃	C₂³	1	9	yes	—
38	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
39	Q₈×Dic₁₃	C₂×C₄	1	24	yes	complemented
40	C₂×C₄⋊Dic₁₃	C₂³	1	9	yes	—
41	C₂×#6101	C₂³	1	18	yes	—
42	C₂×C₄×Dic₁₃	C₂³	1	18	yes	—
43	C₂×C₄×Dic₁₃	C₂³	1	9	yes	—
44	C₂×C₄×Dic₁₃	C₂³	1	9	yes	—
45	C₂×C₄×Dic₁₃	C₂³	1	9	yes	—
46	C₂×C₄×Dic₁₃	C₂³	1	9	yes	—
47	C₂×C₄×Dic₁₃	C₂³	1	3	yes	—
48	C₄⋊C₄×D₁₃	C₂×C₄	1	72	yes	—
49	#88141	C₂×C₄	1	72	yes	complemented
50	C₄²×D₁₃	—	2	36	no	—
51	C₄²×D₁₃	C₂×C₄	1	72	yes	—
52	C₄²×D₁₃	Q₈	1	8	yes	direct product factor
53	C₄×Dic₂₆	C₂×C₄	1	72	yes	complemented
54	C₁₃×#95	C₂³	1	6	yes	—
55	#88071	C₂³	1	18	yes	—
56	C₂×C₄²×Q₈	—	13	1	no	maximal, Sylow
57	C₂³×D₁₃	C₂⁴	1	1	yes	characteristic
58	C₂×Q₈×C₁₃	C₂²×C₄	1	12	yes	—
59	C₂²×C₄×C₁₃	C₂⁴	1	9	yes	—
60	C₂²×C₄×C₁₃	C₂⁴	1	3	yes	—
61	C₂²×C₄×C₁₃	C₂⁴	1	3	yes	—
62	C₂²×Dic₁₃	C₂⁴	1	9	yes	—
63	C₂²×Dic₁₃	C₂⁴	1	3	yes	—
64	C₂²×Dic₁₃	C₂⁴	1	3	yes	—
65	Q₈×D₁₃	C₄²	1	64	yes	direct product factor
66	C₂×C₄×D₁₃	—	2	36	no	—
67	C₂×C₄×D₁₃	C₂²×C₄	1	36	yes	—
68	C₂×C₄×D₁₃	C₂²×C₄	1	36	yes	—
69	C₂×C₄×D₁₃	C₂²×C₄	1	12	yes	—
70	C₂×C₄×D₁₃	C₂×Q₈	1	12	yes	—
71	C₂×C₄×D₁₃	#41	1	12	yes	—
72	C₂×Dic₂₆	C₂²×C₄	1	36	yes	—
73	C₁₃×C₄⋊C₄	C₂²×C₄	1	36	yes	—
74	C₄²×C₁₃	—	2	18	no	—
75	C₄²×C₁₃	C₂²×C₄	1	36	yes	—
76	C₄²×C₁₃	C₂×Q₈	1	4	yes	complemented
77	C₄⋊Dic₁₃	C₂²×C₄	1	36	yes	—
78	#6101	C₂²×C₄	1	72	yes	—
79	C₄×Dic₁₃	—	2	18	no	—
80	C₄×Dic₁₃	—	2	36	no	—
81	C₄×Dic₁₃	C₂²×C₄	1	36	yes	—
82	C₄×Dic₁₃	C₂²×C₄	1	36	yes	—
83	C₄×Dic₁₃	C₂²×C₄	1	36	yes	—
84	C₄×Dic₁₃	C₂×Q₈	1	12	yes	complemented
85	C₂²×C₄×Q₈	—	13	3	no	—
86	C₄²×Q₈	—	13	9	no	—
87	C₄²×Q₈	—	13	3	no	—
88	C₄²×Q₈	—	13	3	no	—
89	C₄²×Q₈	D₁₃	1	1	yes	characteristic, direct product factor
90	C₂×C₄×C₄⋊C₄	—	13	9	no	—
91	C₂×C₄³	—	13	3	no	—
92	C₂³×C₁₃	C₂⁵	1	1	yes	characteristic
93	C₂²×D₁₃	C₂³×C₄	1	6	yes	—
94	C₂²×D₁₃	C₂²×Q₈	1	2	yes	—
95	C₂²×D₁₃	C₂×#41	1	6	yes	—
96	Q₈×C₁₃	C₂×C₄²	1	16	yes	complemented
97	C₂×C₄×C₁₃	—	2	18	no	—
98	C₂×C₄×C₁₃	C₂³×C₄	1	18	yes	—
99	C₂×C₄×C₁₃	C₂³×C₄	1	6	yes	—
100	C₂×C₄×C₁₃	C₂³×C₄	1	18	yes	—
101	C₂×C₄×C₁₃	C₂²×Q₈	1	6	yes	—
102	C₂×C₄×C₁₃	C₂×#41	1	6	yes	—
103	C₂×Dic₁₃	—	2	18	no	—
104	C₂×Dic₁₃	C₂³×C₄	1	18	yes	—
105	C₂×Dic₁₃	C₂³×C₄	1	6	yes	—
106	C₂×Dic₁₃	C₂³×C₄	1	18	yes	—
107	C₂×Dic₁₃	C₂²×Q₈	1	6	yes	—
108	C₂×Dic₁₃	C₂×#41	1	6	yes	—
109	C₄×D₁₃	—	2	72	no	—
110	C₄×D₁₃	C₂×C₄²	1	48	yes	—
111	C₄×D₁₃	C₄×Q₈	1	48	yes	direct product factor
112	Dic₂₆	C₂×C₄²	1	48	yes	complemented
113	C₂³×Q₈	—	13	1	no	—
114	C₂×C₄×Q₈	—	13	3	no	—
115	C₂×C₄×Q₈	—	13	48	no	—
116	C₂×C₄×Q₈	—	13	9	no	—
117	C₂×C₄×Q₈	—	13	9	no	—
118	C₂×C₄×Q₈	C₂×D₁₃	1	3	yes	—
119	C₂²×C₄⋊C₄	—	13	9	no	—
120	C₂²×C₄²	—	13	9	no	—
121	C₂²×C₄²	—	13	1	no	—
122	C₂²×C₄²	—	13	9	no	—
123	C₄×C₄⋊C₄	—	13	18	no	—
124	C₄×C₄⋊C₄	—	13	18	no	—
125	C₄×C₄⋊C₄	—	13	9	no	—
126	C₄×C₄⋊C₄	—	13	18	no	—
127	C₄×C₄⋊C₄	C₂×D₁₃	1	9	yes	—
128	C₂×#95	—	13	6	no	—
129	C₄³	—	13	9	no	—
130	C₄³	—	13	9	no	—
131	C₄³	—	13	3	no	—
132	C₄³	C₂×D₁₃	1	3	yes	—
133	C₂²×C₁₃	C₂⁴×C₄	1	3	yes	—
134	C₂²×C₁₃	C₂³×Q₈	1	1	yes	characteristic
135	C₂²×C₁₃	C₂²×#41	1	3	yes	—
136	C₂×D₁₃	C₂²×C₄²	1	4	yes	—
137	C₂×D₁₃	C₂×C₄×Q₈	1	12	yes	—
138	C₂×D₁₃	C₄×#41	1	12	yes	—
139	C₄×C₁₃	—	2	18	no	—
140	C₄×C₁₃	C₂²×C₄²	1	12	yes	—
141	C₄×C₁₃	C₂×C₄×Q₈	1	12	yes	complemented
142	Dic₁₃	—	2	18	no	—
143	Dic₁₃	C₂²×C₄²	1	12	yes	—
144	Dic₁₃	C₂×C₄×Q₈	1	12	yes	complemented
145	C₂²×Q₈	—	13	24	no	—
146	C₂²×Q₈	—	13	3	no	—
147	C₂²×Q₈	C₂²×D₁₃	1	1	yes	characteristic
148	C₂³×C₄	—	13	3	no	—
149	C₂³×C₄	—	13	3	no	—
150	C₂³×C₄	—	13	9	no	—
151	C₄×Q₈	—	13	24	no	—
152	C₄×Q₈	—	13	72	no	—
153	C₄×Q₈	—	13	72	no	—
154	C₄×Q₈	C₄×D₁₃	1	24	yes	direct product factor
155	C₂×C₄⋊C₄	—	13	18	no	—
156	C₂×C₄⋊C₄	—	13	9	no	—
157	C₂×C₄⋊C₄	—	13	72	no	—
158	C₂×C₄⋊C₄	C₂²×D₁₃	1	9	yes	—
159	C₂×C₄²	—	13	9	no	—
160	C₂×C₄²	—	13	9	no	—
161	C₂×C₄²	—	13	9	no	—
162	C₂×C₄²	—	13	3	no	—
163	C₂×C₄²	—	13	72	no	—
164	C₂×C₄²	—	13	8	no	—
165	C₂×C₄²	—	13	18	no	—
166	C₂×C₄²	—	13	9	no	—
167	C₂×C₄²	—	26	36	no	—
168	C₂×C₄²	C₂²×D₁₃	1	9	yes	—
169	C₂×C₄²	C₂²×D₁₃	1	9	yes	—
170	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
171	#95	—	13	18	no	—
172	#95	C₂²×D₁₃	1	6	yes	—
173	C₂×C₁₃	C₂³×C₄²	1	1	yes	characteristic
174	C₂×C₁₃	C₂²×C₄×Q₈	1	3	yes	—
175	C₂×C₁₃	C₂×C₄×#41	1	3	yes	—
176	D₁₃	C₄²×Q₈	1	8	yes	direct product factor
177	C₂⁴	—	13	1	no	—
178	C₂×Q₈	—	13	64	no	—
179	C₂×Q₈	—	13	36	no	—
180	C₂×Q₈	C₂×C₄×D₁₃	1	12	yes	—
181	C₂²×C₄	—	13	3	no	—
182	C₂²×C₄	—	13	3	no	—
183	C₂²×C₄	—	13	12	no	—
184	C₂²×C₄	—	13	36	no	—
185	C₂²×C₄	—	13	12	no	—
186	C₂²×C₄	—	13	12	no	—
187	C₂²×C₄	—	13	9	no	—
188	C₂²×C₄	—	13	36	no	—
189	C₂²×C₄	—	26	36	no	—
190	C₂²×C₄	C₂³×D₁₃	1	9	yes	—
191	C₂²×C₄	C₂³×D₁₃	1	3	yes	—
192	C₂²×C₄	C₂³×D₁₃	1	3	yes	—
193	C₄⋊C₄	—	13	72	no	—
194	C₄⋊C₄	—	13	36	no	—
195	C₄⋊C₄	C₂×C₄×D₁₃	1	36	yes	—
196	C₄²	—	2	18	no	—
197	C₄²	—	13	36	no	—
198	C₄²	—	13	36	no	—
199	C₄²	—	13	36	no	—
200	C₄²	—	13	12	no	—
201	C₄²	—	26	36	no	—
202	C₄²	—	26	18	no	—
203	C₄²	C₂×C₄×D₁₃	1	36	yes	—
204	C₄²	Q₈×D₁₃	1	4	yes	direct product factor
205	C₁₃	C₂×C₄²×Q₈	1	1	yes	characteristic, complemented, Sylow
206	C₂³	—	13	6	no	—
207	C₂³	—	13	2	no	—
208	C₂³	—	13	6	no	—
209	C₂³	C₂⁴×D₁₃	1	1	yes	characteristic
210	Q₈	—	13	48	no	—
211	Q₈	C₄²×D₁₃	1	16	yes	direct product factor
212	C₂×C₄	—	2	18	no	—
213	C₂×C₄	—	13	6	no	—
214	C₂×C₄	—	13	18	no	—
215	C₂×C₄	—	13	6	no	—
216	C₂×C₄	—	13	6	no	—
217	C₂×C₄	—	13	48	no	—
218	C₂×C₄	—	13	48	no	—
219	C₂×C₄	—	13	18	no	—
220	C₂×C₄	—	26	72	no	—
221	C₂×C₄	—	26	18	no	—
222	C₂×C₄	C₂²×C₄×D₁₃	1	18	yes	—
223	C₂×C₄	C₂²×C₄×D₁₃	1	18	yes	—
224	C₂×C₄	C₂²×C₄×D₁₃	1	6	yes	—
225	C₂×C₄	C₂×Q₈×D₁₃	1	6	yes	—
226	C₂×C₄	#41×D₁₃	1	6	yes	—
227	C₂²	—	13	4	no	—
228	C₂²	—	13	12	no	—
229	C₂²	—	13	12	no	—
230	C₂²	C₂³×C₄×D₁₃	1	3	yes	—
231	C₂²	C₂²×Q₈×D₁₃	1	1	yes	characteristic
232	C₂²	C₂×#41×D₁₃	1	3	yes	—
233	C₄	—	2	18	no	—
234	C₄	—	13	12	no	—
235	C₄	—	13	12	no	—
236	C₄	—	26	18	no	—
237	C₄	C₂×C₄²×D₁₃	1	12	yes	—
238	C₄	C₄×Q₈×D₁₃	1	12	yes	direct product factor
239	C₂	—	13	8	no	—
240	C₂	C₂²×C₄²×D₁₃	1	1	yes	characteristic
241	C₂	C₂×C₄×Q₈×D₁₃	1	3	yes	—
242	C₂	C₄×#41×D₁₃	1	3	yes	—

	subgroup	G/H
center	C₂×C₄²	C₂²×D₁₃
commutator	C₂×C₁₃	C₂³×C₄²
Frattini	C₂³	C₂⁴×D₁₃
Fitting	C₄²×Q₈×C₁₃	C₂
socle, cosocle	C₂³×C₁₃	C₂⁵
prime radical	C₂³×D₁₃	C₂⁴
hypercenter	C₄²×Q₈	D₁₃
hypocenter	C₁₃	C₂×C₄²×Q₈
LCS	G ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄² ◁ C₄²×Q₈

Complements

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

N	H ≅ G/N	type
2. C₄×Q₈×Dic₁₃	C₂	semidirect
3. C₄²×Dic₂₆	C₂	semidirect
4. C₄×#88141	C₂	semidirect
7. C₄²×Q₈×C₁₃	C₂	semidirect
11. C₄×Q₈×D₁₃	C₄	direct (N × H)
34. C₄×Q₈×C₁₃	C₂×C₄	semidirect
39. Q₈×Dic₁₃	C₂×C₄	semidirect
49. #88141	C₂×C₄	semidirect
52. C₄²×D₁₃	Q₈	direct (N × H)
53. C₄×Dic₂₆	C₂×C₄	semidirect
65. Q₈×D₁₃	C₄²	direct (N × H)
76. C₄²×C₁₃	C₂×Q₈	semidirect
84. C₄×Dic₁₃	C₂×Q₈	semidirect
89. C₄²×Q₈	D₁₃	direct (N × H)
96. Q₈×C₁₃	C₂×C₄²	semidirect
111. C₄×D₁₃	C₄×Q₈	direct (N × H)
112. Dic₂₆	C₂×C₄²	semidirect
141. C₄×C₁₃	C₂×C₄×Q₈	semidirect
144. Dic₁₃	C₂×C₄×Q₈	semidirect
154. C₄×Q₈	C₄×D₁₃	direct (N × H)
176. D₁₃	C₄²×Q₈	direct (N × H)
204. C₄²	Q₈×D₁₃	direct (N × H)
205. C₁₃	C₂×C₄²×Q₈	semidirect
211. Q₈	C₄²×D₁₃	direct (N × H)
238. C₄	C₄×Q₈×D₁₃	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×D₁₃	2
C₄×Q₈×Dic₁₃	2
C₄²×Dic₂₆	2
C₄×#88141	2
C₄×C₄⋊C₄×D₁₃	2
C₄³×D₁₃	2
C₄²×Q₈×C₁₃	2
C₂×C₄²×Q₈	13

Subgroups — C42×Q8×D13 #16021113

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄²×Q₈×D₁₃ #16021113