Subgroups — C₁₃×#33630 #15719969

Order 3328.

542 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×#499	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×C₁₃×#135	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#2229	C₂	1	4	yes	maximal, complemented
4	C₁₃×#2228	C₂	1	4	yes	maximal, complemented
5	C₁₃×#2159	C₂	1	2	yes	maximal, complemented
6	C₁₃×#2150	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₃×#2139	C₂	1	2	yes	maximal
8	C₂×C₁₃×#135	C₂²	1	1	yes	characteristic
9	C₂×C₁₃×#125	—	2	1	no	—
10	C₄×C₁₃×#41	—	2	1	no	—
11	C₂×C₁₃×M₃₂	C₂²	1	2	yes	—
12	C₁₃×#499	—	2	2	no	—
13	C₁₃×#499	C₂²	1	4	yes	—
14	C₄×C₁₃×Q₁₆	C₂²	1	2	yes	—
15	C₄×C₁₃×SD₁₆	—	2	2	no	—
16	C₄×C₁₃×D₈	C₂²	1	2	yes	—
17	C₁₃×#427	C₂²	1	1	yes	characteristic
18	C₂×C₁₃×#107	C₂²	1	2	yes	—
19	C₂×C₁₃×#106	—	2	1	no	—
20	C₁₃×#416	—	2	1	no	—
21	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
22	C₁₃×#367	C₂²	1	4	yes	—
23	C₁₃×#366	C₂²	1	4	yes	—
24	C₁₃×C₄⋊C₁₆	C₂²	1	4	yes	—
25	C₁₃×#358	C₂²	1	4	yes	—
26	C₁₃×#357	C₂²	1	4	yes	—
27	C₂×C₁₃×#41	—	2	1	no	—
28	C₁₃×#135	—	2	2	no	—
29	C₁₃×#135	Q₈	1	4	yes	—
30	C₂×C₁₃×Q₁₆	C₂³	1	1	yes	characteristic
31	C₂×C₁₃×SD₁₆	—	2	1	no	—
32	C₂×C₁₃×D₈	C₂³	1	1	yes	characteristic
33	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
34	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
35	C₁₃×#125	—	2	4	no	—
36	C₁₃×#125	—	4	1	no	—
37	C₁₃×#125	—	4	1	no	—
38	C₄×Q₈×C₁₃	—	2	2	no	—
39	C₄×D₄×C₁₃	—	2	2	no	—
40	C₄×D₄×C₁₃	—	4	2	no	—
41	C₁₃×#117	—	2	1	no	—
42	C₁₃×#117	—	4	1	no	—
43	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
44	C₂×C₁₃×C₄⋊C₄	—	4	2	no	—
45	C₂×C₄²×C₁₃	—	4	1	no	—
46	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
47	C₁₃×M₃₂	—	2	4	no	—
48	C₂×C₁₃×C₁₆	C₂³	1	4	yes	—
49	C₁₃×#107	—	2	2	no	—
50	C₁₃×#107	C₂³	1	4	yes	—
51	C₁₃×#107	C₂³	1	2	yes	—
52	C₁₃×#106	—	2	4	no	—
53	C₁₃×#106	—	2	1	no	—
54	C₁₃×#103	—	2	2	no	—
55	C₁₃×#102	—	2	2	no	—
56	C₄×C₈×C₁₃	D₄	1	2	yes	—
57	C₄×C₈×C₁₃	C₂³	1	2	yes	—
58	#33630	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
59	C₁₃×#41	—	2	4	no	—
60	C₁₃×#41	—	4	2	no	—
61	C₂×Q₈×C₁₃	—	2	1	no	—
62	C₂×D₄×C₁₃	—	2	1	no	—
63	C₂×D₄×C₁₃	—	4	1	no	—
64	C₂²×C₄×C₁₃	—	4	1	no	—
65	C₂²×C₄×C₁₃	—	4	2	no	—
66	C₂²×C₄×C₁₃	—	4	2	no	—
67	C₂²×C₄×C₁₃	D₈	1	2	yes	—
68	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
69	C₁₃×Q₁₆	—	2	1	no	—
70	C₁₃×Q₁₆	C₂×Q₈	1	2	yes	—
71	C₁₃×SD₁₆	—	2	4	no	—
72	C₁₃×D₈	—	2	1	no	—
73	C₁₃×D₈	C₂×Q₈	1	2	yes	—
74	C₂×C₈×C₁₃	—	2	2	no	—
75	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
76	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
77	C₂×C₈×C₁₃	C₂×Q₈	1	2	yes	—
78	C₂×C₈×C₁₃	#41	1	2	yes	—
79	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
80	C₁₃×C₄⋊C₄	—	2	2	no	—
81	C₁₃×C₄⋊C₄	—	2	4	no	—
82	C₁₃×C₄⋊C₄	—	4	2	no	—
83	C₁₃×C₄⋊C₄	—	4	4	no	—
84	C₁₃×C₄⋊C₄	—	4	4	no	—
85	C₁₃×C₄⋊C₄	—	4	2	no	—
86	C₁₃×C₂²⋊C₄	—	4	1	no	—
87	C₁₃×C₂²⋊C₄	—	4	2	no	—
88	C₄²×C₁₃	—	4	1	no	—
89	C₄²×C₁₃	—	4	2	no	—
90	C₄²×C₁₃	D₈	1	1	yes	characteristic
91	C₄²×C₁₃	D₈	1	1	yes	characteristic
92	C₄²×C₁₃	C₂×D₄	1	2	yes	—
93	C₁₃×C₁₆	—	2	4	no	—
94	C₂×#499	C₂×C₁₃	1	1	yes	characteristic, complemented
95	C₄×#135	C₂×C₁₃	1	1	yes	characteristic
96	#2229	C₂×C₁₃	1	4	yes	complemented
97	#2228	C₂×C₁₃	1	4	yes	complemented
98	#2159	C₂×C₁₃	1	2	yes	complemented
99	#2150	C₂×C₁₃	1	1	yes	characteristic, complemented
100	#2139	C₂×C₁₃	1	2	yes	—
101	C₂³×C₁₃	—	4	1	no	—
102	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
103	Q₈×C₁₃	—	2	2	no	—
104	Q₈×C₁₃	—	4	1	no	—
105	D₄×C₁₃	—	2	2	no	—
106	D₄×C₁₃	—	4	4	no	—
107	D₄×C₁₃	—	4	1	no	—
108	C₂×C₄×C₁₃	—	4	2	no	—
109	C₂×C₄×C₁₃	—	4	1	no	—
110	C₂×C₄×C₁₃	—	4	4	no	—
111	C₂×C₄×C₁₃	—	4	2	no	—
112	C₂×C₄×C₁₃	—	4	4	no	—
113	C₂×C₄×C₁₃	—	8	4	no	—
114	C₂×C₄×C₁₃	—	8	4	no	—
115	C₂×C₄×C₁₃	C₂²⋊Q₈	1	2	yes	—
116	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
117	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
118	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
119	C₂×C₄×C₁₃	#137	1	2	yes	—
120	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
121	C₈×C₁₃	—	2	2	no	—
122	C₈×C₁₃	C₂²⋊Q₈	1	2	yes	—
123	C₈×C₁₃	C₂²×Q₈	1	1	yes	characteristic
124	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
125	C₂×#135	C₂²×C₁₃	1	1	yes	characteristic
126	C₂×#125	—	2	1	no	—
127	C₄×#41	—	2	1	no	—
128	C₂×M₃₂	C₂²×C₁₃	1	2	yes	—
129	#499	—	2	2	no	—
130	#499	C₂²×C₁₃	1	4	yes	—
131	C₄×Q₁₆	C₂²×C₁₃	1	2	yes	—
132	C₄×SD₁₆	—	2	2	no	—
133	C₄×D₈	C₂²×C₁₃	1	2	yes	—
134	#427	C₂²×C₁₃	1	1	yes	characteristic
135	C₂×#107	C₂²×C₁₃	1	2	yes	—
136	C₂×#106	—	2	1	no	—
137	#416	—	2	1	no	—
138	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
139	#367	C₂²×C₁₃	1	4	yes	—
140	#366	C₂²×C₁₃	1	4	yes	—
141	C₄⋊C₁₆	C₂²×C₁₃	1	4	yes	—
142	#358	C₂²×C₁₃	1	4	yes	—
143	#357	C₂²×C₁₃	1	4	yes	—
144	C₂²×C₁₃	—	2	2	no	—
145	C₂²×C₁₃	—	4	2	no	—
146	C₂²×C₁₃	—	8	2	no	—
147	C₂²×C₁₃	#474	1	2	yes	—
148	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
149	C₄×C₁₃	—	2	2	no	—
150	C₄×C₁₃	—	2	2	no	—
151	C₄×C₁₃	—	4	2	no	—
152	C₄×C₁₃	—	8	2	no	—
153	C₄×C₁₃	—	8	4	no	—
154	C₄×C₁₃	#474	1	2	yes	—
155	C₄×C₁₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
156	C₄×C₁₃	C₂×#137	1	1	yes	characteristic
157	C₂×#41	—	2	1	no	—
158	#135	—	2	2	no	—
159	#135	Q₈×C₁₃	1	4	yes	—
160	C₂×Q₁₆	C₂³×C₁₃	1	1	yes	characteristic
161	C₂×SD₁₆	—	2	1	no	—
162	C₂×D₈	C₂³×C₁₃	1	1	yes	characteristic
163	C₂²×C₈	D₄×C₁₃	1	1	yes	characteristic
164	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
165	#125	—	2	4	no	—
166	#125	—	4	1	no	—
167	#125	—	4	1	no	—
168	C₄×Q₈	—	2	2	no	—
169	C₄×D₄	—	2	2	no	—
170	C₄×D₄	—	4	2	no	—
171	#117	—	2	1	no	—
172	#117	—	4	1	no	—
173	C₂×C₄⋊C₄	—	2	2	no	—
174	C₂×C₄⋊C₄	—	4	2	no	—
175	C₂×C₄²	—	4	1	no	—
176	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
177	M₃₂	—	2	4	no	—
178	C₂×C₁₆	C₂³×C₁₃	1	4	yes	—
179	#107	—	2	2	no	—
180	#107	C₂³×C₁₃	1	4	yes	—
181	#107	C₂³×C₁₃	1	2	yes	—
182	#106	—	2	4	no	—
183	#106	—	2	1	no	—
184	#103	—	2	2	no	—
185	#102	—	2	2	no	—
186	C₄×C₈	D₄×C₁₃	1	2	yes	—
187	C₄×C₈	C₂³×C₁₃	1	2	yes	—
188	C₂×C₁₃	—	2	2	no	—
189	C₂×C₁₃	—	8	2	no	—
190	C₂×C₁₃	C₂×#474	1	1	yes	characteristic
191	C₂×C₁₃	#3414	1	2	yes	—
192	#41	—	2	4	no	—
193	#41	—	4	2	no	—
194	C₂×Q₈	—	2	1	no	—
195	C₂×D₄	—	2	1	no	—
196	C₂×D₄	—	4	1	no	—
197	C₂²×C₄	—	4	1	no	—
198	C₂²×C₄	—	4	2	no	—
199	C₂²×C₄	—	4	2	no	—
200	C₂²×C₄	C₁₃×D₈	1	2	yes	—
201	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
202	Q₁₆	—	2	1	no	—
203	Q₁₆	C₂×Q₈×C₁₃	1	2	yes	—
204	SD₁₆	—	2	4	no	—
205	D₈	—	2	1	no	—
206	D₈	C₂×Q₈×C₁₃	1	2	yes	—
207	C₂×C₈	—	2	2	no	—
208	C₂×C₈	C₂×D₄×C₁₃	1	2	yes	—
209	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
210	C₂×C₈	C₂×Q₈×C₁₃	1	2	yes	—
211	C₂×C₈	C₁₃×#41	1	2	yes	—
212	C₂×C₈	C₂⁴×C₁₃	1	1	yes	characteristic
213	C₄⋊C₄	—	2	2	no	—
214	C₄⋊C₄	—	2	4	no	—
215	C₄⋊C₄	—	4	2	no	—
216	C₄⋊C₄	—	4	4	no	—
217	C₄⋊C₄	—	4	4	no	—
218	C₄⋊C₄	—	4	2	no	—
219	C₂²⋊C₄	—	4	1	no	—
220	C₂²⋊C₄	—	4	2	no	—
221	C₄²	—	4	1	no	—
222	C₄²	—	4	2	no	—
223	C₄²	C₁₃×D₈	1	1	yes	characteristic
224	C₄²	C₁₃×D₈	1	1	yes	characteristic
225	C₄²	C₂×D₄×C₁₃	1	2	yes	—
226	C₁₆	—	2	4	no	—
227	C₁₃	#33630	1	1	yes	characteristic, direct product factor, Sylow
228	C₂³	—	4	1	no	—
229	C₂³	C₂×C₁₃×D₈	1	1	yes	characteristic
230	Q₈	—	2	2	no	—
231	Q₈	—	4	1	no	—
232	D₄	—	2	2	no	—
233	D₄	—	4	4	no	—
234	D₄	—	4	1	no	—
235	C₂×C₄	—	4	2	no	—
236	C₂×C₄	—	4	1	no	—
237	C₂×C₄	—	4	4	no	—
238	C₂×C₄	—	4	2	no	—
239	C₂×C₄	—	4	4	no	—
240	C₂×C₄	—	8	4	no	—
241	C₂×C₄	—	8	4	no	—
242	C₂×C₄	C₁₃×C₂²⋊Q₈	1	2	yes	—
243	C₂×C₄	C₂×C₁₃×D₈	1	2	yes	—
244	C₂×C₄	C₂×C₁₃×D₈	1	2	yes	—
245	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
246	C₂×C₄	C₁₃×#137	1	2	yes	—
247	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
248	C₈	—	2	2	no	—
249	C₈	C₁₃×C₂²⋊Q₈	1	2	yes	—
250	C₈	C₂²×Q₈×C₁₃	1	1	yes	characteristic
251	C₈	C₂×C₁₃×#41	1	1	yes	characteristic
252	C₂²	—	2	2	no	—
253	C₂²	—	4	2	no	—
254	C₂²	—	8	2	no	—
255	C₂²	C₁₃×#474	1	2	yes	—
256	C₂²	C₂²×C₁₃×D₈	1	1	yes	characteristic
257	C₄	—	2	2	no	—
258	C₄	—	2	2	no	—
259	C₄	—	4	2	no	—
260	C₄	—	8	2	no	—
261	C₄	—	8	4	no	—
262	C₄	C₁₃×#474	1	2	yes	—
263	C₄	C₂×C₁₃×C₂²⋊Q₈	1	1	yes	characteristic
264	C₄	C₂×C₁₃×#137	1	1	yes	characteristic
265	C₂	—	2	2	no	—
266	C₂	—	8	2	no	—
267	C₂	C₂×C₁₃×#474	1	1	yes	characteristic
268	C₂	C₁₃×#3414	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₃	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₁₃
prime radical	C₂×C₁₃×#135	C₂²
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
1. C₂×C₁₃×#499	C₂	semidirect
3. C₁₃×#2229	C₂	semidirect
4. C₁₃×#2228	C₂	semidirect
5. C₁₃×#2159	C₂	semidirect
6. C₁₃×#2150	C₂	semidirect
58. #33630	C₁₃	direct (N × H)
94. C₂×#499	C₂×C₁₃	semidirect
96. #2229	C₂×C₁₃	semidirect
97. #2228	C₂×C₁₃	semidirect
98. #2159	C₂×C₁₃	semidirect
99. #2150	C₂×C₁₃	semidirect
227. C₁₃	#33630	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×C₁₃×#499	2
C₄×C₁₃×#135	2
C₁₃×#2229	2
C₁₃×#2228	2
C₁₃×#2159	2
C₁₃×#2150	2
C₁₃×#2139	2
#33630	13

Subgroups — C13×#33630 #15719969

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#33630 #15719969