Subgroups — C₁₃×#33476 #15719815

Order 3328.

498 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#2960	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#2183	C₂	1	2	yes	maximal, complemented
3	C₁₃×#2183	C₂	1	4	yes	maximal, complemented
4	Q₈×C₁₃×C₁₆	C₂	1	2	yes	maximal, complemented
5	C₁₃×#2151	C₂	1	2	yes	maximal
6	C₂×C₁₃×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
7	C₁₃×#2108	C₂	1	2	yes	maximal
8	C₄×C₁₃×M₃₂	C₂	1	1	yes	maximal, characteristic
9	C₁₃×#533	C₄	1	1	yes	characteristic
10	C₂×C₁₃×M₃₂	C₂²	1	1	yes	characteristic
11	C₂×C₁₃×M₃₂	C₂²	1	1	yes	characteristic
12	C₂²×C₁₃×C₁₆	C₂²	1	2	yes	—
13	C₁₃×#446	C₄	1	2	yes	—
14	C₁₃×#446	C₄	1	2	yes	—
15	C₈×Q₈×C₁₃	C₂²	1	2	yes	—
16	C₈×Q₈×C₁₃	C₂²	1	2	yes	—
17	C₁₃×#424	C₄	1	2	yes	—
18	C₁₃×#423	C₄	1	1	yes	characteristic
19	C₁₃×#405	C₂²	1	2	yes	—
20	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
21	C₁₃×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
22	C₁₃×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
23	C₁₃×C₄⋊C₁₆	C₂²	1	2	yes	—
24	C₁₃×C₄⋊C₁₆	C₂²	1	4	yes	—
25	C₁₃×C₄⋊C₁₆	C₂²	1	4	yes	—
26	C₁₃×#346	C₂²	1	2	yes	—
27	C₁₃×#346	C₂²	1	4	yes	—
28	C₄×C₁₃×C₁₆	C₂²	1	2	yes	—
29	C₄×C₁₃×C₁₆	C₂²	1	4	yes	—
30	C₂×C₁₃×M₁₆	C₂×C₄	1	2	yes	—
31	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
33	C₁₃×C₄⋊Q₈	C₈	1	1	yes	characteristic
34	C₁₃×C₄⋊Q₈	C₈	1	1	yes	characteristic
35	C₁₃×#125	C₈	1	2	yes	—
36	C₁₃×C₂²⋊Q₈	C₈	1	2	yes	—
37	C₁₃×C₂²⋊Q₈	C₈	1	2	yes	—
38	C₄×Q₈×C₁₃	C₂×C₄	1	2	yes	—
39	C₄×Q₈×C₁₃	C₂×C₄	1	2	yes	—
40	C₁₃×#117	C₂×C₄	1	2	yes	—
41	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
42	C₁₃×M₃₂	—	2	2	no	—
43	C₁₃×M₃₂	Q₈	1	4	yes	complemented
44	C₂×C₁₃×C₁₆	—	2	4	no	—
45	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
46	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
47	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
48	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
49	C₁₃×C₄⋊C₈	C₂×C₄	1	2	yes	—
50	C₁₃×C₄⋊C₈	C₂×C₄	1	2	yes	—
51	C₁₃×C₄⋊C₈	C₂×C₄	1	4	yes	—
52	C₁₃×C₄⋊C₈	C₂×C₄	1	4	yes	—
53	C₁₃×#97	C₂×C₄	1	4	yes	—
54	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
55	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
56	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
57	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
58	C₄×C₈×C₁₃	C₂³	1	4	yes	—
59	#33476	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
60	C₂×Q₈×C₁₃	C₂×C₈	1	2	yes	—
61	C₂×Q₈×C₁₃	C₂×C₈	1	2	yes	—
62	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
63	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
64	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
65	C₁₃×M₁₆	—	2	4	no	—
66	C₂×C₈×C₁₃	—	2	2	no	—
67	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
68	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
70	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
71	C₂×C₈×C₁₃	C₂×Q₈	1	2	yes	—
72	C₂×C₈×C₁₃	#41	1	2	yes	—
73	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
74	C₁₃×C₄⋊C₄	C₂×C₈	1	2	yes	—
75	C₁₃×C₄⋊C₄	C₂×C₈	1	2	yes	—
76	C₁₃×C₄⋊C₄	C₂×C₈	1	4	yes	—
77	C₁₃×C₄⋊C₄	C₂×C₈	1	4	yes	—
78	C₁₃×C₄⋊C₄	C₂×C₈	1	2	yes	—
79	C₁₃×C₄⋊C₄	C₂×C₈	1	2	yes	—
80	C₁₃×C₂²⋊C₄	C₂×C₈	1	4	yes	—
81	C₄²×C₁₃	C₂×C₈	1	1	yes	characteristic
82	C₄²×C₁₃	C₂×C₈	1	1	yes	characteristic
83	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
84	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
85	C₄²×C₁₃	C₂²×C₄	1	4	yes	—
86	C₁₃×C₁₆	—	2	2	no	—
87	C₁₃×C₁₆	—	2	2	no	—
88	C₁₃×C₁₆	—	2	2	no	—
89	C₁₃×C₁₆	C₂×Q₈	1	4	yes	—
90	#2960	C₂×C₁₃	1	1	yes	characteristic
91	#2183	C₂×C₁₃	1	2	yes	complemented
92	#2183	C₂×C₁₃	1	4	yes	complemented
93	Q₈×C₁₆	C₂×C₁₃	1	2	yes	complemented
94	#2151	C₂×C₁₃	1	2	yes	—
95	C₂×C₄⋊C₁₆	C₂×C₁₃	1	1	yes	characteristic
96	#2108	C₂×C₁₃	1	2	yes	—
97	C₄×M₃₂	C₂×C₁₃	1	1	yes	characteristic
98	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
99	Q₈×C₁₃	—	2	4	no	—
100	Q₈×C₁₃	—	4	2	no	—
101	C₂×C₄×C₁₃	—	2	2	no	—
102	C₂×C₄×C₁₃	—	4	1	no	—
103	C₂×C₄×C₁₃	C₄×Q₈	1	2	yes	—
104	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
105	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
106	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
107	C₂×C₄×C₁₃	C₂²×C₈	1	4	yes	—
108	C₂×C₄×C₁₃	C₂²×C₈	1	4	yes	—
109	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
110	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
111	C₂×C₄×C₁₃	#131	1	2	yes	—
112	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
113	C₈×C₁₃	—	2	1	no	—
114	C₈×C₁₃	—	2	1	no	—
115	C₈×C₁₃	—	2	4	no	—
116	C₈×C₁₃	C₄×Q₈	1	2	yes	—
117	C₈×C₁₃	C₂²×Q₈	1	1	yes	characteristic
118	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
119	#533	C₄×C₁₃	1	1	yes	characteristic
120	C₂×M₃₂	C₂²×C₁₃	1	1	yes	characteristic
121	C₂×M₃₂	C₂²×C₁₃	1	1	yes	characteristic
122	C₂²×C₁₆	C₂²×C₁₃	1	2	yes	—
123	#446	C₄×C₁₃	1	2	yes	—
124	#446	C₄×C₁₃	1	2	yes	—
125	C₈×Q₈	C₂²×C₁₃	1	2	yes	—
126	C₈×Q₈	C₂²×C₁₃	1	2	yes	—
127	#424	C₄×C₁₃	1	2	yes	—
128	#423	C₄×C₁₃	1	1	yes	characteristic
129	#405	C₂²×C₁₃	1	2	yes	—
130	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
131	C₄⋊C₁₆	C₂²×C₁₃	1	1	yes	characteristic
132	C₄⋊C₁₆	C₂²×C₁₃	1	1	yes	characteristic
133	C₄⋊C₁₆	C₂²×C₁₃	1	2	yes	—
134	C₄⋊C₁₆	C₂²×C₁₃	1	4	yes	—
135	C₄⋊C₁₆	C₂²×C₁₃	1	4	yes	—
136	#346	C₂²×C₁₃	1	2	yes	—
137	#346	C₂²×C₁₃	1	4	yes	—
138	C₄×C₁₆	C₂²×C₁₃	1	2	yes	—
139	C₄×C₁₆	C₂²×C₁₃	1	4	yes	—
140	C₂²×C₁₃	—	2	1	no	—
141	C₂²×C₁₃	—	2	1	no	—
142	C₂²×C₁₃	C₈×Q₈	1	2	yes	—
143	C₂²×C₁₃	C₂³×C₈	1	1	yes	characteristic
144	C₄×C₁₃	—	2	1	no	—
145	C₄×C₁₃	—	2	4	no	—
146	C₄×C₁₃	—	2	1	no	—
147	C₄×C₁₃	—	2	1	no	—
148	C₄×C₁₃	—	2	4	no	—
149	C₄×C₁₃	C₈×Q₈	1	2	yes	—
150	C₄×C₁₃	#504	1	2	yes	—
151	C₄×C₁₃	C₂×C₄×Q₈	1	1	yes	characteristic
152	C₄×C₁₃	C₂×#131	1	1	yes	characteristic
153	C₂×M₁₆	C₂×C₄×C₁₃	1	2	yes	—
154	C₂²×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
155	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
156	C₄⋊Q₈	C₈×C₁₃	1	1	yes	characteristic
157	C₄⋊Q₈	C₈×C₁₃	1	1	yes	characteristic
158	#125	C₈×C₁₃	1	2	yes	—
159	C₂²⋊Q₈	C₈×C₁₃	1	2	yes	—
160	C₂²⋊Q₈	C₈×C₁₃	1	2	yes	—
161	C₄×Q₈	C₂×C₄×C₁₃	1	2	yes	—
162	C₄×Q₈	C₂×C₄×C₁₃	1	2	yes	—
163	#117	C₂×C₄×C₁₃	1	2	yes	—
164	C₂×C₄²	C₂×C₄×C₁₃	1	1	yes	characteristic
165	M₃₂	—	2	2	no	—
166	M₃₂	Q₈×C₁₃	1	4	yes	complemented
167	C₂×C₁₆	—	2	4	no	—
168	C₂×C₁₆	C₂³×C₁₃	1	2	yes	—
169	C₂×C₁₆	C₂³×C₁₃	1	2	yes	—
170	C₂×C₁₆	C₂³×C₁₃	1	2	yes	—
171	C₂×C₁₆	C₂³×C₁₃	1	2	yes	—
172	C₄⋊C₈	C₂×C₄×C₁₃	1	2	yes	—
173	C₄⋊C₈	C₂×C₄×C₁₃	1	2	yes	—
174	C₄⋊C₈	C₂×C₄×C₁₃	1	4	yes	—
175	C₄⋊C₈	C₂×C₄×C₁₃	1	4	yes	—
176	#97	C₂×C₄×C₁₃	1	4	yes	—
177	C₄×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
178	C₄×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
179	C₄×C₈	C₂³×C₁₃	1	1	yes	characteristic
180	C₄×C₈	C₂³×C₁₃	1	4	yes	—
181	C₄×C₈	C₂³×C₁₃	1	1	yes	characteristic
182	C₂×C₁₃	—	2	2	no	—
183	C₂×C₁₃	C₂×C₈×Q₈	1	1	yes	characteristic
184	C₂×C₁₃	C₂×#504	1	1	yes	characteristic
185	C₂×C₁₃	#3407	1	1	yes	characteristic
186	C₂×Q₈	C₂×C₈×C₁₃	1	2	yes	—
187	C₂×Q₈	C₂×C₈×C₁₃	1	2	yes	—
188	C₂²×C₄	C₂×C₈×C₁₃	1	1	yes	characteristic
189	C₂²×C₄	C₂×C₈×C₁₃	1	1	yes	characteristic
190	C₂²×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
191	M₁₆	—	2	4	no	—
192	C₂×C₈	—	2	2	no	—
193	C₂×C₈	C₂²×C₄×C₁₃	1	1	yes	characteristic
194	C₂×C₈	C₂²×C₄×C₁₃	1	1	yes	characteristic
195	C₂×C₈	C₂²×C₄×C₁₃	1	4	yes	—
196	C₂×C₈	C₂²×C₄×C₁₃	1	1	yes	characteristic
197	C₂×C₈	C₂×Q₈×C₁₃	1	2	yes	—
198	C₂×C₈	C₁₃×#41	1	2	yes	—
199	C₂×C₈	C₂⁴×C₁₃	1	1	yes	characteristic
200	C₄⋊C₄	C₂×C₈×C₁₃	1	2	yes	—
201	C₄⋊C₄	C₂×C₈×C₁₃	1	2	yes	—
202	C₄⋊C₄	C₂×C₈×C₁₃	1	4	yes	—
203	C₄⋊C₄	C₂×C₈×C₁₃	1	4	yes	—
204	C₄⋊C₄	C₂×C₈×C₁₃	1	2	yes	—
205	C₄⋊C₄	C₂×C₈×C₁₃	1	2	yes	—
206	C₂²⋊C₄	C₂×C₈×C₁₃	1	4	yes	—
207	C₄²	C₂×C₈×C₁₃	1	1	yes	characteristic
208	C₄²	C₂×C₈×C₁₃	1	1	yes	characteristic
209	C₄²	C₂²×C₄×C₁₃	1	1	yes	characteristic
210	C₄²	C₂²×C₄×C₁₃	1	1	yes	characteristic
211	C₄²	C₂²×C₄×C₁₃	1	4	yes	—
212	C₁₆	—	2	2	no	—
213	C₁₆	—	2	2	no	—
214	C₁₆	—	2	2	no	—
215	C₁₆	C₂×Q₈×C₁₃	1	4	yes	—
216	C₁₃	#33476	1	1	yes	characteristic, direct product factor, Sylow
217	C₂³	C₂²×C₈×C₁₃	1	1	yes	characteristic
218	Q₈	—	2	4	no	—
219	Q₈	—	4	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	4	1	no	—
222	C₂×C₄	C₄×Q₈×C₁₃	1	2	yes	—
223	C₂×C₄	C₂²×C₈×C₁₃	1	1	yes	characteristic
224	C₂×C₄	C₂²×C₈×C₁₃	1	1	yes	characteristic
225	C₂×C₄	C₂²×C₈×C₁₃	1	1	yes	characteristic
226	C₂×C₄	C₂²×C₈×C₁₃	1	4	yes	—
227	C₂×C₄	C₂²×C₈×C₁₃	1	4	yes	—
228	C₂×C₄	C₂²×C₈×C₁₃	1	1	yes	characteristic
229	C₂×C₄	C₂²×C₈×C₁₃	1	1	yes	characteristic
230	C₂×C₄	C₁₃×#131	1	2	yes	—
231	C₂×C₄	C₂³×C₄×C₁₃	1	1	yes	characteristic
232	C₈	—	2	1	no	—
233	C₈	—	2	1	no	—
234	C₈	—	2	4	no	—
235	C₈	C₄×Q₈×C₁₃	1	2	yes	—
236	C₈	C₂²×Q₈×C₁₃	1	1	yes	characteristic
237	C₈	C₂×C₁₃×#41	1	1	yes	characteristic
238	C₂²	—	2	1	no	—
239	C₂²	—	2	1	no	—
240	C₂²	C₈×Q₈×C₁₃	1	2	yes	—
241	C₂²	C₂³×C₈×C₁₃	1	1	yes	characteristic
242	C₄	—	2	1	no	—
243	C₄	—	2	4	no	—
244	C₄	—	2	1	no	—
245	C₄	—	2	1	no	—
246	C₄	—	2	4	no	—
247	C₄	C₈×Q₈×C₁₃	1	2	yes	—
248	C₄	C₁₃×#504	1	2	yes	—
249	C₄	C₂×C₄×Q₈×C₁₃	1	1	yes	characteristic
250	C₄	C₂×C₁₃×#131	1	1	yes	characteristic
251	C₂	—	2	2	no	—
252	C₂	C₂×C₈×Q₈×C₁₃	1	1	yes	characteristic
253	C₂	C₂×C₁₃×#504	1	1	yes	characteristic
254	C₂	C₁₃×#3407	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈×C₁₃	C₂⁴
commutator	C₂²	C₂³×C₈×C₁₃
Frattini, cosocle	C₂×C₈	C₂⁴×C₁₃
socle	C₂²×C₁₃	C₂³×C₈
prime radical	C₂³×C₁₃	C₂²×C₈
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₁₃×#2183	C₂	semidirect
3. C₁₃×#2183	C₂	semidirect
4. Q₈×C₁₃×C₁₆	C₂	semidirect
43. C₁₃×M₃₂	Q₈	semidirect
59. #33476	C₁₃	direct (N × H)
91. #2183	C₂×C₁₃	semidirect
92. #2183	C₂×C₁₃	semidirect
93. Q₈×C₁₆	C₂×C₁₃	semidirect
166. M₃₂	Q₈×C₁₃	semidirect
216. C₁₃	#33476	direct (N × H)

Maximal subgroups

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

subgroup	index
C₁₃×#2960	2
C₁₃×#2183	2
Q₈×C₁₃×C₁₆	2
C₁₃×#2151	2
C₂×C₁₃×C₄⋊C₁₆	2
C₁₃×#2108	2
C₄×C₁₃×M₃₂	2
#33476	13

Subgroups — C13×#33476 #15719815

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#33476 #15719815