Subgroups — C₁₃×#33843 #15720182

Order 3328.

668 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×#510	C₂	1	2	yes	maximal, complemented
2	C₁₃×#3285	C₂	1	2	yes	maximal, complemented
3	C₁₃×#2249	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₃×#2220	C₂	1	2	yes	maximal, complemented
5	C₁₃×#2211	C₂	1	2	yes	maximal, complemented
6	C₁₃×#2188	C₂	1	2	yes	maximal, complemented
7	C₁₃×#2186	C₂	1	2	yes	maximal, complemented
8	C₄×C₁₃×Q₃₂	C₂	1	1	yes	maximal, characteristic, complemented
9	C₁₃×#2169	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₁₃×#135	C₂²	1	2	yes	—
11	C₂²×C₁₃×Q₁₆	C₂²	1	2	yes	—
12	C₁₃×#548	—	2	2	no	—
13	C₁₃×#510	—	2	8	no	—
14	C₂×C₁₃×Q₃₂	C₂²	1	2	yes	—
15	C₂×C₁₃×Q₃₂	C₂²	1	1	yes	characteristic
16	C₂×C₁₃×Q₃₂	C₂²	1	1	yes	characteristic
17	C₂×C₁₃×SD₃₂	C₂²	1	2	yes	—
18	C₂×C₁₃×SD₃₂	C₂²	1	2	yes	—
19	C₂×C₁₃×D₁₆	C₂²	1	1	yes	characteristic
20	C₂×C₁₃×M₃₂	C₂²	1	2	yes	—
21	C₁₃×#495	C₂²	1	2	yes	—
22	C₁₃×#467	C₂²	1	2	yes	—
23	C₁₃×#466	C₂²	1	2	yes	—
24	C₁₃×#464	—	2	2	no	—
25	C₁₃×#452	—	2	2	no	—
26	C₁₃×#448	—	2	2	no	—
27	C₄×C₁₃×Q₁₆	C₂²	1	2	yes	—
28	C₈×D₄×C₁₃	C₂²	1	1	yes	characteristic
29	C₁₃×#366	C₂²	1	1	yes	characteristic
30	C₁₃×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
31	C₁₃×#358	C₂²	1	2	yes	—
32	C₁₃×#358	C₂²	1	2	yes	—
33	C₁₃×#357	C₂²	1	2	yes	—
34	C₁₃×C₂²⋊C₁₆	C₂²	1	2	yes	—
35	C₄×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
36	C₂×C₁₃×#41	—	2	2	no	—
37	C₂²×Q₈×C₁₃	—	2	2	no	—
38	C₁₃×#135	—	2	4	no	—
39	C₁₃×#135	—	4	2	no	—
40	C₂×C₁₃×Q₁₆	—	2	4	no	—
41	C₂×C₁₃×Q₁₆	—	2	2	no	—
42	C₂×C₁₃×Q₁₆	—	4	2	no	—
43	C₂×C₁₃×Q₁₆	C₂³	1	2	yes	—
44	C₂×C₁₃×Q₁₆	C₂³	1	2	yes	—
45	C₂×C₁₃×SD₁₆	—	2	2	no	—
46	C₂×C₁₃×SD₁₆	—	2	2	no	—
47	C₂×C₁₃×D₈	C₂³	1	2	yes	—
48	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
49	C₁₃×#124	—	2	2	no	—
50	C₁₃×#124	—	4	2	no	—
51	C₁₃×C₂²⋊Q₈	—	2	2	no	—
52	C₁₃×C₂²⋊Q₈	—	4	2	no	—
53	C₁₃×#121	—	2	2	no	—
54	C₁₃×#121	—	4	2	no	—
55	C₄×Q₈×C₁₃	—	2	2	no	—
56	C₄×D₄×C₁₃	—	4	2	no	—
57	C₄×D₄×C₁₃	D₄	1	1	yes	characteristic
58	C₁₃×Q₃₂	—	2	4	no	—
59	C₁₃×Q₃₂	—	4	1	no	—
60	C₁₃×Q₃₂	D₄	1	4	yes	complemented
61	C₁₃×SD₃₂	—	2	4	no	—
62	C₁₃×SD₃₂	—	2	4	no	—
63	C₁₃×D₁₆	—	4	1	no	—
64	C₁₃×M₃₂	—	2	4	no	—
65	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
66	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
67	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
68	C₁₃×#107	C₂³	1	2	yes	—
69	C₁₃×C₄⋊C₈	D₄	1	1	yes	characteristic
70	C₁₃×#103	—	2	2	no	—
71	C₁₃×#103	—	2	2	no	—
72	C₁₃×#102	—	2	2	no	—
73	C₁₃×C₂²⋊C₈	D₄	1	2	yes	—
74	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
75	#33843	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
76	C₁₃×#41	—	4	4	no	—
77	C₁₃×#41	—	8	2	no	—
78	C₂×Q₈×C₁₃	—	2	2	no	—
79	C₂×Q₈×C₁₃	—	2	2	no	—
80	C₂×Q₈×C₁₃	—	4	4	no	—
81	C₂×Q₈×C₁₃	—	4	2	no	—
82	C₂×Q₈×C₁₃	—	8	2	no	—
83	C₂×D₄×C₁₃	—	2	2	no	—
84	C₂×D₄×C₁₃	—	4	2	no	—
85	C₂×D₄×C₁₃	—	4	2	no	—
86	C₂×D₄×C₁₃	D₈	1	1	yes	characteristic
87	C₂²×C₄×C₁₃	—	4	2	no	—
88	C₂²×C₄×C₁₃	—	4	2	no	—
89	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
90	C₁₃×Q₁₆	—	2	2	no	—
91	C₁₃×Q₁₆	—	2	2	no	—
92	C₁₃×Q₁₆	—	2	2	no	—
93	C₁₃×Q₁₆	—	4	4	no	—
94	C₁₃×Q₁₆	C₂×D₄	1	4	yes	—
95	C₁₃×SD₁₆	—	4	4	no	—
96	C₁₃×SD₁₆	—	8	2	no	—
97	C₁₃×D₈	—	2	2	no	—
98	C₁₃×D₈	—	2	2	no	—
99	C₂×C₈×C₁₃	—	2	2	no	—
100	C₂×C₈×C₁₃	—	2	2	no	—
101	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
102	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
103	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
104	C₁₃×C₄⋊C₄	—	2	2	no	—
105	C₁₃×C₄⋊C₄	—	4	2	no	—
106	C₁₃×C₄⋊C₄	—	4	2	no	—
107	C₁₃×C₄⋊C₄	D₈	1	1	yes	characteristic
108	C₁₃×C₂²⋊C₄	—	4	2	no	—
109	C₁₃×C₂²⋊C₄	—	4	2	no	—
110	C₁₃×C₂²⋊C₄	—	4	2	no	—
111	C₁₃×C₂²⋊C₄	—	4	2	no	—
112	C₁₃×C₂²⋊C₄	D₈	1	2	yes	—
113	C₄²×C₁₃	—	4	2	no	—
114	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
115	C₁₃×C₁₆	—	2	2	no	—
116	C₁₃×C₁₆	—	2	1	no	—
117	C₁₃×C₁₆	C₂×D₄	1	2	yes	—
118	C₂×#510	C₂×C₁₃	1	2	yes	complemented
119	#3285	C₂×C₁₃	1	2	yes	complemented
120	#2249	C₂×C₁₃	1	1	yes	characteristic, complemented
121	#2220	C₂×C₁₃	1	2	yes	complemented
122	#2211	C₂×C₁₃	1	2	yes	complemented
123	#2188	C₂×C₁₃	1	2	yes	complemented
124	#2186	C₂×C₁₃	1	2	yes	complemented
125	C₄×Q₃₂	C₂×C₁₃	1	1	yes	characteristic, complemented
126	#2169	C₂×C₁₃	1	1	yes	characteristic, complemented
127	C₂³×C₁₃	—	4	2	no	—
128	C₂³×C₁₃	C₂×D₈	1	2	yes	—
129	Q₈×C₁₃	—	2	4	no	—
130	Q₈×C₁₃	—	4	2	no	—
131	Q₈×C₁₃	—	4	2	no	—
132	Q₈×C₁₃	—	4	2	no	—
133	Q₈×C₁₃	—	8	4	no	—
134	D₄×C₁₃	—	4	1	no	—
135	D₄×C₁₃	—	4	2	no	—
136	D₄×C₁₃	—	4	2	no	—
137	D₄×C₁₃	—	8	4	no	—
138	D₄×C₁₃	—	8	4	no	—
139	C₂×C₄×C₁₃	—	2	2	no	—
140	C₂×C₄×C₁₃	—	2	2	no	—
141	C₂×C₄×C₁₃	—	4	2	no	—
142	C₂×C₄×C₁₃	—	4	2	no	—
143	C₂×C₄×C₁₃	—	4	2	no	—
144	C₂×C₄×C₁₃	—	8	4	no	—
145	C₂×C₄×C₁₃	—	8	4	no	—
146	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
147	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
148	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
149	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
150	C₈×C₁₃	—	2	2	no	—
151	C₈×C₁₃	—	2	1	no	—
152	C₈×C₁₃	C₂²×D₄	1	1	yes	characteristic
153	C₈×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
154	C₂×#135	C₂²×C₁₃	1	2	yes	—
155	C₂²×Q₁₆	C₂²×C₁₃	1	2	yes	—
156	#548	—	2	2	no	—
157	#510	—	2	8	no	—
158	C₂×Q₃₂	C₂²×C₁₃	1	2	yes	—
159	C₂×Q₃₂	C₂²×C₁₃	1	1	yes	characteristic
160	C₂×Q₃₂	C₂²×C₁₃	1	1	yes	characteristic
161	C₂×SD₃₂	C₂²×C₁₃	1	2	yes	—
162	C₂×SD₃₂	C₂²×C₁₃	1	2	yes	—
163	C₂×D₁₆	C₂²×C₁₃	1	1	yes	characteristic
164	C₂×M₃₂	C₂²×C₁₃	1	2	yes	—
165	#495	C₂²×C₁₃	1	2	yes	—
166	#467	C₂²×C₁₃	1	2	yes	—
167	#466	C₂²×C₁₃	1	2	yes	—
168	#464	—	2	2	no	—
169	#452	—	2	2	no	—
170	#448	—	2	2	no	—
171	C₄×Q₁₆	C₂²×C₁₃	1	2	yes	—
172	C₈×D₄	C₂²×C₁₃	1	1	yes	characteristic
173	#366	C₂²×C₁₃	1	1	yes	characteristic
174	C₄⋊C₁₆	C₂²×C₁₃	1	1	yes	characteristic
175	#358	C₂²×C₁₃	1	2	yes	—
176	#358	C₂²×C₁₃	1	2	yes	—
177	#357	C₂²×C₁₃	1	2	yes	—
178	C₂²⋊C₁₆	C₂²×C₁₃	1	2	yes	—
179	C₄×C₁₆	C₂²×C₁₃	1	1	yes	characteristic
180	C₂²×C₁₃	—	2	2	no	—
181	C₂²×C₁₃	—	2	2	no	—
182	C₂²×C₁₃	—	2	2	no	—
183	C₂²×C₁₃	—	8	2	no	—
184	C₂²×C₁₃	—	8	2	no	—
185	C₂²×C₁₃	—	8	2	no	—
186	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
187	C₄×C₁₃	—	2	1	no	—
188	C₄×C₁₃	—	2	1	no	—
189	C₄×C₁₃	—	2	2	no	—
190	C₄×C₁₃	—	4	4	no	—
191	C₄×C₁₃	—	8	2	no	—
192	C₄×C₁₃	—	8	2	no	—
193	C₄×C₁₃	D₄²	1	1	yes	characteristic
194	C₄×C₁₃	#576	1	1	yes	characteristic
195	C₂×#41	—	2	2	no	—
196	C₂²×Q₈	—	2	2	no	—
197	#135	—	2	4	no	—
198	#135	—	4	2	no	—
199	C₂×Q₁₆	—	2	4	no	—
200	C₂×Q₁₆	—	2	2	no	—
201	C₂×Q₁₆	—	4	2	no	—
202	C₂×Q₁₆	C₂³×C₁₃	1	2	yes	—
203	C₂×Q₁₆	C₂³×C₁₃	1	2	yes	—
204	C₂×SD₁₆	—	2	2	no	—
205	C₂×SD₁₆	—	2	2	no	—
206	C₂×D₈	C₂³×C₁₃	1	2	yes	—
207	C₂²×C₈	C₂³×C₁₃	1	2	yes	—
208	#124	—	2	2	no	—
209	#124	—	4	2	no	—
210	C₂²⋊Q₈	—	2	2	no	—
211	C₂²⋊Q₈	—	4	2	no	—
212	#121	—	2	2	no	—
213	#121	—	4	2	no	—
214	C₄×Q₈	—	2	2	no	—
215	C₄×D₄	—	4	2	no	—
216	C₄×D₄	D₄×C₁₃	1	1	yes	characteristic
217	Q₃₂	—	2	4	no	—
218	Q₃₂	—	4	1	no	—
219	Q₃₂	D₄×C₁₃	1	4	yes	complemented
220	SD₃₂	—	2	4	no	—
221	SD₃₂	—	2	4	no	—
222	D₁₆	—	4	1	no	—
223	M₃₂	—	2	4	no	—
224	C₂×C₁₆	C₂³×C₁₃	1	1	yes	characteristic
225	C₂×C₁₆	C₂³×C₁₃	1	2	yes	—
226	C₂×C₁₆	C₂³×C₁₃	1	1	yes	characteristic
227	#107	C₂³×C₁₃	1	2	yes	—
228	C₄⋊C₈	D₄×C₁₃	1	1	yes	characteristic
229	#103	—	2	2	no	—
230	#103	—	2	2	no	—
231	#102	—	2	2	no	—
232	C₂²⋊C₈	D₄×C₁₃	1	2	yes	—
233	C₄×C₈	C₂³×C₁₃	1	1	yes	characteristic
234	C₂×C₁₃	—	4	2	no	—
235	C₂×C₁₃	—	16	2	no	—
236	C₂×C₁₃	D₄×D₈	1	1	yes	characteristic
237	C₂×C₁₃	#3414	1	1	yes	characteristic
238	C₂×C₁₃	#3417	1	1	yes	characteristic
239	#41	—	4	4	no	—
240	#41	—	8	2	no	—
241	C₂×Q₈	—	2	2	no	—
242	C₂×Q₈	—	2	2	no	—
243	C₂×Q₈	—	4	4	no	—
244	C₂×Q₈	—	4	2	no	—
245	C₂×Q₈	—	8	2	no	—
246	C₂×D₄	—	2	2	no	—
247	C₂×D₄	—	4	2	no	—
248	C₂×D₄	—	4	2	no	—
249	C₂×D₄	C₁₃×D₈	1	1	yes	characteristic
250	C₂²×C₄	—	4	2	no	—
251	C₂²×C₄	—	4	2	no	—
252	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
253	Q₁₆	—	2	2	no	—
254	Q₁₆	—	2	2	no	—
255	Q₁₆	—	2	2	no	—
256	Q₁₆	—	4	4	no	—
257	Q₁₆	C₂×D₄×C₁₃	1	4	yes	—
258	SD₁₆	—	4	4	no	—
259	SD₁₆	—	8	2	no	—
260	D₈	—	2	2	no	—
261	D₈	—	2	2	no	—
262	C₂×C₈	—	2	2	no	—
263	C₂×C₈	—	2	2	no	—
264	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
265	C₂×C₈	C₂×D₄×C₁₃	1	2	yes	—
266	C₂×C₈	C₂⁴×C₁₃	1	1	yes	characteristic
267	C₄⋊C₄	—	2	2	no	—
268	C₄⋊C₄	—	4	2	no	—
269	C₄⋊C₄	—	4	2	no	—
270	C₄⋊C₄	C₁₃×D₈	1	1	yes	characteristic
271	C₂²⋊C₄	—	4	2	no	—
272	C₂²⋊C₄	—	4	2	no	—
273	C₂²⋊C₄	—	4	2	no	—
274	C₂²⋊C₄	—	4	2	no	—
275	C₂²⋊C₄	C₁₃×D₈	1	2	yes	—
276	C₄²	—	4	2	no	—
277	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
278	C₁₆	—	2	2	no	—
279	C₁₆	—	2	1	no	—
280	C₁₆	C₂×D₄×C₁₃	1	2	yes	—
281	C₁₃	#33843	1	1	yes	characteristic, direct product factor, Sylow
282	C₂³	—	4	2	no	—
283	C₂³	C₂×C₁₃×D₈	1	2	yes	—
284	Q₈	—	2	4	no	—
285	Q₈	—	4	2	no	—
286	Q₈	—	4	2	no	—
287	Q₈	—	4	2	no	—
288	Q₈	—	8	4	no	—
289	D₄	—	4	1	no	—
290	D₄	—	4	2	no	—
291	D₄	—	4	2	no	—
292	D₄	—	8	4	no	—
293	D₄	—	8	4	no	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	4	2	no	—
299	C₂×C₄	—	8	4	no	—
300	C₂×C₄	—	8	4	no	—
301	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
302	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
303	C₂×C₄	C₂×C₁₃×D₈	1	2	yes	—
304	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
305	C₈	—	2	2	no	—
306	C₈	—	2	1	no	—
307	C₈	C₂²×D₄×C₁₃	1	1	yes	characteristic
308	C₈	C₁₃×H(5,𝔽₂)	1	1	yes	characteristic
309	C₂²	—	2	2	no	—
310	C₂²	—	2	2	no	—
311	C₂²	—	2	2	no	—
312	C₂²	—	8	2	no	—
313	C₂²	—	8	2	no	—
314	C₂²	—	8	2	no	—
315	C₂²	C₂²×C₁₃×D₈	1	1	yes	characteristic
316	C₄	—	2	2	no	—
317	C₄	—	2	1	no	—
318	C₄	—	2	1	no	—
319	C₄	—	4	4	no	—
320	C₄	—	8	2	no	—
321	C₄	—	8	2	no	—
322	C₄	D₄²×C₁₃	1	1	yes	characteristic
323	C₄	C₁₃×#576	1	1	yes	characteristic
324	C₂	—	4	2	no	—
325	C₂	—	16	2	no	—
326	C₂	D₄×C₁₃×D₈	1	1	yes	characteristic
327	C₂	C₁₃×#3414	1	1	yes	characteristic
328	C₂	C₁₃×#3417	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₁₃
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃ ◁ C₄×D₄×C₁₃ ◁ C₈×D₄×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₁₃×#510	C₂	semidirect
2. C₁₃×#3285	C₂	semidirect
3. C₁₃×#2249	C₂	semidirect
4. C₁₃×#2220	C₂	semidirect
5. C₁₃×#2211	C₂	semidirect
6. C₁₃×#2188	C₂	semidirect
7. C₁₃×#2186	C₂	semidirect
8. C₄×C₁₃×Q₃₂	C₂	semidirect
9. C₁₃×#2169	C₂	semidirect
60. C₁₃×Q₃₂	D₄	semidirect
75. #33843	C₁₃	direct (N × H)
118. C₂×#510	C₂×C₁₃	semidirect
119. #3285	C₂×C₁₃	semidirect
120. #2249	C₂×C₁₃	semidirect
121. #2220	C₂×C₁₃	semidirect
122. #2211	C₂×C₁₃	semidirect
123. #2188	C₂×C₁₃	semidirect
124. #2186	C₂×C₁₃	semidirect
125. C₄×Q₃₂	C₂×C₁₃	semidirect
126. #2169	C₂×C₁₃	semidirect
219. Q₃₂	D₄×C₁₃	semidirect
281. C₁₃	#33843	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×#510	2
C₁₃×#3285	2
C₁₃×#2249	2
C₁₃×#2220	2
C₁₃×#2211	2
C₁₃×#2188	2
C₁₃×#2186	2
C₄×C₁₃×Q₃₂	2
C₁₃×#2169	2
#33843	13

Subgroups — C13×#33843 #15720182

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#33843 #15720182