Subgroups — C₁₃×#13212 #15699551

Order 3328.

470 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×#494	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₃×#362	C₂	1	4	yes	maximal
3	C₄×C₁₃×M₃₂	C₂	1	1	yes	maximal, characteristic
4	C₁₃×#1848	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₁₃×Q₁₆	—	2	1	no	—
6	C₂²×C₁₃×Q₁₆	C₂²	1	2	yes	—
7	C₂×C₁₃×C₄⋊Q₈	—	2	1	no	—
8	C₂×C₁₃×M₃₂	C₂²	1	2	yes	—
9	C₁₃×#494	—	2	1	no	—
10	C₁₃×#494	—	2	1	no	—
11	C₁₃×#494	C₄	1	2	yes	—
12	C₁₃×#494	C₄	1	2	yes	—
13	C₂×C₁₃×#108	—	2	1	no	—
14	C₂×C₁₃×#108	C₂²	1	2	yes	—
15	C₁₃×#423	—	2	1	no	—
16	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
17	C₁₃×#362	—	2	8	no	—
18	C₁₃×#346	—	2	1	no	—
19	C₄×C₁₃×C₁₆	—	2	1	no	—
20	C₂²×Q₈×C₁₃	—	2	2	no	—
21	C₂×C₁₃×Q₁₆	—	2	2	no	—
22	C₂×C₁₃×Q₁₆	—	2	2	no	—
23	C₂×C₁₃×Q₁₆	—	2	4	no	—
24	C₂×C₁₃×Q₁₆	—	2	2	no	—
25	C₂×C₁₃×Q₁₆	—	2	4	no	—
26	C₂×C₁₃×Q₁₆	—	4	2	no	—
27	C₂×C₁₃×Q₁₆	—	4	2	no	—
28	C₂×C₁₃×Q₁₆	C₂×C₄	1	4	yes	—
29	C₂×C₁₃×M₁₆	—	2	2	no	—
30	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
31	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
32	C₁₃×C₄⋊Q₈	—	2	2	no	—
33	C₁₃×C₄⋊Q₈	—	2	2	no	—
34	C₁₃×C₄⋊Q₈	—	4	2	no	—
35	C₂×C₁₃×C₄⋊C₄	—	4	2	no	—
36	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
37	C₁₃×M₃₂	—	2	2	no	—
38	C₁₃×M₃₂	D₄	1	4	yes	—
39	C₂×C₁₃×C₁₆	—	2	2	no	—
40	C₁₃×#108	—	2	4	no	—
41	C₁₃×#108	—	4	2	no	—
42	C₁₃×C₄⋊C₈	—	4	2	no	—
43	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
44	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
45	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
46	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
47	#13212	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
48	C₂×Q₈×C₁₃	—	2	4	no	—
49	C₂×Q₈×C₁₃	—	4	2	no	—
50	C₂×Q₈×C₁₃	—	4	2	no	—
51	C₂×Q₈×C₁₃	—	4	2	no	—
52	C₂×Q₈×C₁₃	—	4	2	no	—
53	C₂×Q₈×C₁₃	—	4	2	no	—
54	C₂²×C₄×C₁₃	—	4	2	no	—
55	C₂²×C₄×C₁₃	SD₁₆	1	2	yes	—
56	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
57	C₁₃×Q₁₆	—	2	8	no	—
58	C₁₃×Q₁₆	—	4	4	no	—
59	C₁₃×Q₁₆	—	4	4	no	—
60	C₁₃×Q₁₆	—	4	4	no	—
61	C₁₃×M₁₆	—	4	2	no	—
62	C₁₃×M₁₆	—	4	2	no	—
63	C₂×C₈×C₁₃	—	4	2	no	—
64	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
65	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
66	C₂×C₈×C₁₃	D₈	1	2	yes	—
67	C₂×C₈×C₁₃	SD₁₆	1	2	yes	—
68	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
70	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
71	C₂×C₈×C₁₃	#41	1	2	yes	—
72	C₁₃×C₄⋊C₄	—	4	4	no	—
73	C₁₃×C₄⋊C₄	—	4	4	no	—
74	C₄²×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
75	C₄²×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
76	C₄²×C₁₃	D₈	1	2	yes	—
77	C₁₃×C₁₆	—	2	4	no	—
78	C₂×#494	C₂×C₁₃	1	1	yes	characteristic
79	C₂×#362	C₂×C₁₃	1	4	yes	—
80	C₄×M₃₂	C₂×C₁₃	1	1	yes	characteristic
81	#1848	C₂×C₁₃	1	1	yes	characteristic
82	C₂³×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
83	Q₈×C₁₃	—	4	4	no	—
84	Q₈×C₁₃	—	4	4	no	—
85	Q₈×C₁₃	—	8	2	no	—
86	Q₈×C₁₃	—	8	2	no	—
87	C₂×C₄×C₁₃	—	2	4	no	—
88	C₂×C₄×C₁₃	—	4	4	no	—
89	C₂×C₄×C₁₃	—	8	2	no	—
90	C₂×C₄×C₁₃	—	8	2	no	—
91	C₂×C₄×C₁₃	#102	1	2	yes	—
92	C₂×C₄×C₁₃	#102	1	2	yes	—
93	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
94	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
95	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
96	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
97	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
98	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
99	C₈×C₁₃	—	8	2	no	—
100	C₈×C₁₃	#102	1	2	yes	—
101	C₈×C₁₃	#102	1	2	yes	—
102	C₈×C₁₃	C₄×D₄	1	2	yes	—
103	C₈×C₁₃	#121	1	2	yes	—
104	C₂²×Q₁₆	—	2	1	no	—
105	C₂²×Q₁₆	C₂²×C₁₃	1	2	yes	—
106	C₂×C₄⋊Q₈	—	2	1	no	—
107	C₂×M₃₂	C₂²×C₁₃	1	2	yes	—
108	#494	—	2	1	no	—
109	#494	—	2	1	no	—
110	#494	C₄×C₁₃	1	2	yes	—
111	#494	C₄×C₁₃	1	2	yes	—
112	C₂×#108	—	2	1	no	—
113	C₂×#108	C₂²×C₁₃	1	2	yes	—
114	#423	—	2	1	no	—
115	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
116	#362	—	2	8	no	—
117	#346	—	2	1	no	—
118	C₄×C₁₆	—	2	1	no	—
119	C₂²×C₁₃	—	2	2	no	—
120	C₂²×C₁₃	C₂×#102	1	1	yes	characteristic
121	C₂²×C₁₃	#486	1	1	yes	characteristic
122	C₂²×C₁₃	#492	1	1	yes	characteristic
123	C₄×C₁₃	—	2	2	no	—
124	C₄×C₁₃	—	8	4	no	—
125	C₄×C₁₃	#362	1	4	yes	complemented
126	C₄×C₁₃	#390	1	1	yes	characteristic
127	C₄×C₁₃	C₂×#102	1	1	yes	characteristic
128	C₄×C₁₃	#486	1	1	yes	characteristic
129	C₄×C₁₃	#493	1	1	yes	characteristic
130	C₂²×Q₈	—	2	2	no	—
131	C₂×Q₁₆	—	2	2	no	—
132	C₂×Q₁₆	—	2	2	no	—
133	C₂×Q₁₆	—	2	4	no	—
134	C₂×Q₁₆	—	2	2	no	—
135	C₂×Q₁₆	—	2	4	no	—
136	C₂×Q₁₆	—	4	2	no	—
137	C₂×Q₁₆	—	4	2	no	—
138	C₂×Q₁₆	C₂×C₄×C₁₃	1	4	yes	—
139	C₂×M₁₆	—	2	2	no	—
140	C₂²×C₈	D₄×C₁₃	1	1	yes	characteristic
141	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
142	C₄⋊Q₈	—	2	2	no	—
143	C₄⋊Q₈	—	2	2	no	—
144	C₄⋊Q₈	—	4	2	no	—
145	C₂×C₄⋊C₄	—	4	2	no	—
146	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
147	M₃₂	—	2	2	no	—
148	M₃₂	D₄×C₁₃	1	4	yes	—
149	C₂×C₁₆	—	2	2	no	—
150	#108	—	2	4	no	—
151	#108	—	4	2	no	—
152	C₄⋊C₈	—	4	2	no	—
153	C₄×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
154	C₄×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
155	C₄×C₈	D₄×C₁₃	1	1	yes	characteristic
156	C₄×C₈	D₄×C₁₃	1	1	yes	characteristic
157	C₂×C₁₃	—	2	1	no	—
158	C₂×C₁₃	—	2	1	no	—
159	C₂×C₁₃	#1957	1	1	yes	characteristic
160	C₂×C₁₃	C₂×#362	1	2	yes	—
161	C₂×Q₈	—	2	4	no	—
162	C₂×Q₈	—	4	2	no	—
163	C₂×Q₈	—	4	2	no	—
164	C₂×Q₈	—	4	2	no	—
165	C₂×Q₈	—	4	2	no	—
166	C₂×Q₈	—	4	2	no	—
167	C₂²×C₄	—	4	2	no	—
168	C₂²×C₄	C₁₃×SD₁₆	1	2	yes	—
169	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
170	Q₁₆	—	2	8	no	—
171	Q₁₆	—	4	4	no	—
172	Q₁₆	—	4	4	no	—
173	Q₁₆	—	4	4	no	—
174	M₁₆	—	4	2	no	—
175	M₁₆	—	4	2	no	—
176	C₂×C₈	—	4	2	no	—
177	C₂×C₈	C₁₃×C₂²⋊C₄	1	1	yes	characteristic
178	C₂×C₈	C₁₃×C₂²⋊C₄	1	1	yes	characteristic
179	C₂×C₈	C₁₃×D₈	1	2	yes	—
180	C₂×C₈	C₁₃×SD₁₆	1	2	yes	—
181	C₂×C₈	C₂²×C₄×C₁₃	1	1	yes	characteristic
182	C₂×C₈	C₂×D₄×C₁₃	1	2	yes	—
183	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
184	C₂×C₈	C₁₃×#41	1	2	yes	—
185	C₄⋊C₄	—	4	4	no	—
186	C₄⋊C₄	—	4	4	no	—
187	C₄²	C₁₃×C₂²⋊C₄	1	1	yes	characteristic
188	C₄²	C₁₃×C₂²⋊C₄	1	1	yes	characteristic
189	C₄²	C₁₃×D₈	1	2	yes	—
190	C₁₆	—	2	4	no	—
191	C₁₃	#13212	1	1	yes	characteristic, direct product factor, Sylow
192	C₂³	C₂×C₁₃×SD₁₆	1	1	yes	characteristic
193	Q₈	—	4	4	no	—
194	Q₈	—	4	4	no	—
195	Q₈	—	8	2	no	—
196	Q₈	—	8	2	no	—
197	C₂×C₄	—	2	4	no	—
198	C₂×C₄	—	4	4	no	—
199	C₂×C₄	—	8	2	no	—
200	C₂×C₄	—	8	2	no	—
201	C₂×C₄	C₁₃×#102	1	2	yes	—
202	C₂×C₄	C₁₃×#102	1	2	yes	—
203	C₂×C₄	C₂×C₁₃×C₂²⋊C₄	1	1	yes	characteristic
204	C₂×C₄	C₁₃×#124	1	1	yes	characteristic
205	C₂×C₄	C₁₃×#127	1	1	yes	characteristic
206	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
207	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
208	C₂×C₄	C₂×C₁₃×SD₁₆	1	1	yes	characteristic
209	C₈	—	8	2	no	—
210	C₈	C₁₃×#102	1	2	yes	—
211	C₈	C₁₃×#102	1	2	yes	—
212	C₈	C₄×D₄×C₁₃	1	2	yes	—
213	C₈	C₁₃×#121	1	2	yes	—
214	C₂²	—	2	2	no	—
215	C₂²	C₂×C₁₃×#102	1	1	yes	characteristic
216	C₂²	C₁₃×#486	1	1	yes	characteristic
217	C₂²	C₁₃×#492	1	1	yes	characteristic
218	C₄	—	2	2	no	—
219	C₄	—	8	4	no	—
220	C₄	C₁₃×#362	1	4	yes	complemented
221	C₄	C₁₃×#390	1	1	yes	characteristic
222	C₄	C₂×C₁₃×#102	1	1	yes	characteristic
223	C₄	C₁₃×#486	1	1	yes	characteristic
224	C₄	C₁₃×#493	1	1	yes	characteristic
225	C₂	—	2	1	no	—
226	C₂	—	2	1	no	—
227	C₂	C₁₃×#1957	1	1	yes	characteristic
228	C₂	C₂×C₁₃×#362	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₃	C₂×#102
commutator	C₂×C₈	C₂²×C₄×C₁₃
Frattini, cosocle	C₂²×C₈	C₂³×C₁₃
prime radical	C₂³×C₁₃	C₂×SD₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃ ◁ C₂×C₄²×C₁₃ ◁ C₂×C₄×C₈×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
47. #13212	C₁₃	direct (N × H)
125. C₄×C₁₃	#362	semidirect
191. C₁₃	#13212	direct (N × H)
220. C₄	C₁₃×#362	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₁₃×#494	2
C₂×C₁₃×#362	2
C₄×C₁₃×M₃₂	2
C₁₃×#1848	2
#13212	13

Subgroups — C13×#13212 #15699551

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#13212 #15699551