Subgroups — #13128648

Order 1664.

1060 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#533	C₂	1	1	yes	maximal, characteristic, complemented
2	#11719433	C₂	1	2	yes	maximal, complemented
3	D₁₃×C₄⋊Q₈	C₂	1	2	yes	maximal, complemented
4	#11719394	C₂	1	4	yes	maximal, complemented
5	#11719340	C₂	1	4	yes	maximal, complemented
6	#11719337	C₂	1	4	yes	maximal, complemented
7	#11719284	C₂	1	2	yes	maximal, complemented
8	Q₈×D₅₂	C₂	1	2	yes	maximal, complemented
9	#11719272	C₂	1	2	yes	maximal, complemented
10	Q₈×Dic₂₆	C₂	1	2	yes	maximal, complemented
11	#11719228	C₂	1	2	yes	maximal
12	#11719224	C₂	1	2	yes	maximal, complemented
13	C₄×#6127	C₂	1	1	yes	maximal, characteristic
14	C₂×#88123	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂×Q₈×D₁₃	C₂²	1	4	yes	—
16	C₂×#6127	C₂²	1	1	yes	characteristic
17	C₂²×Dic₂₆	C₂²	1	2	yes	—
18	C₁₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
19	C₁₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
20	C₁₃×#125	C₂²	1	2	yes	—
21	C₁₃×C₂²⋊Q₈	C₂²	1	2	yes	—
22	C₁₃×C₂²⋊Q₈	C₂²	1	2	yes	—
23	C₄×Q₈×C₁₃	C₂²	1	2	yes	—
24	C₄×Q₈×C₁₃	C₂²	1	2	yes	—
25	C₁₃×#117	C₂²	1	2	yes	—
26	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
27	#88200	C₂²	1	4	yes	—
28	Q₈×Dic₁₃	C₂²	1	4	yes	—
29	#88198	C₂²	1	4	yes	—
30	C₄×C₂²⋊D₁₃	C₂²	1	2	yes	—
31	C₄×C₂²⋊D₁₃	C₂²	1	2	yes	—
32	#88180	C₂²	1	1	yes	characteristic
33	C₂×C₄⋊Dic₁₃	C₂²	1	2	yes	—
34	C₂×C₄⋊Dic₁₃	C₂²	1	2	yes	—
35	#88151	C₂²	1	4	yes	—
36	#88151	C₂²	1	8	yes	—
37	#88150	C₂²	1	8	yes	—
38	#88146	C₂²	1	4	yes	—
39	C₄⋊C₄×D₁₃	C₂²	1	4	yes	—
40	C₄⋊C₄×D₁₃	C₂²	1	4	yes	—
41	#88144	C₂²	1	4	yes	—
42	#88143	C₂²	1	8	yes	—
43	#88142	C₂²	1	4	yes	—
44	#88142	C₂²	1	4	yes	—
45	#88142	C₂²	1	8	yes	—
46	#88141	C₂²	1	4	yes	—
47	#88135	C₂²	1	8	yes	—
48	#88132	C₂²	1	8	yes	—
49	C₄×D₅₂	C₂²	1	1	yes	characteristic
50	C₄×D₅₂	C₂²	1	1	yes	characteristic
51	C₄×D₅₂	C₂²	1	4	yes	—
52	#88126	C₂²	1	2	yes	—
53	C₄²×D₁₃	C₂²	1	2	yes	—
54	#88124	C₂²	1	4	yes	—
55	#88123	C₂²	1	2	yes	—
56	#88123	C₂²	1	2	yes	—
57	#88123	C₂²	1	2	yes	—
58	#88123	C₂²	1	2	yes	—
59	#88123	C₂²	1	4	yes	—
60	C₄×Dic₂₆	C₂²	1	1	yes	characteristic
61	C₄×Dic₂₆	C₂²	1	1	yes	characteristic
62	C₄×Dic₂₆	C₂²	1	4	yes	—
63	C₂×Q₈×C₁₃	C₂³	1	2	yes	—
64	C₂×Q₈×C₁₃	C₂³	1	2	yes	—
65	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
66	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
67	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
68	C₂×C₂²⋊D₁₃	C₂³	1	2	yes	—
69	C₂²×Dic₁₃	C₂³	1	4	yes	—
70	Q₈×D₁₃	—	4	8	no	—
71	#6127	Q₈	1	8	yes	complemented
72	C₂×D₅₂	C₂³	1	1	yes	characteristic
73	C₂×C₄×D₁₃	C₂³	1	2	yes	—
74	C₂×C₄×D₁₃	C₂³	1	2	yes	—
75	C₂×C₄×D₁₃	C₂³	1	8	yes	—
76	C₂×C₄×D₁₃	C₂³	1	2	yes	—
77	C₂×Dic₂₆	—	4	4	no	—
78	C₂×Dic₂₆	C₂³	1	4	yes	—
79	C₂×Dic₂₆	C₂³	1	4	yes	—
80	C₂×Dic₂₆	C₂³	1	8	yes	—
81	C₂×Dic₂₆	C₂³	1	1	yes	characteristic
82	C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
83	C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
84	C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
85	C₁₃×C₄⋊C₄	C₂³	1	4	yes	—
86	C₁₃×C₄⋊C₄	C₂³	1	4	yes	—
87	C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
88	C₁₃×C₂²⋊C₄	C₂³	1	4	yes	—
89	C₄²×C₁₃	C₂³	1	1	yes	characteristic
90	C₄²×C₁₃	C₂³	1	1	yes	characteristic
91	C₄²×C₁₃	C₂³	1	4	yes	—
92	C₄²×C₁₃	C₂³	1	1	yes	characteristic
93	C₄²×C₁₃	C₂³	1	1	yes	characteristic
94	C₂²⋊Dic₁₃	C₂³	1	2	yes	—
95	#6103	C₂³	1	2	yes	—
96	#6103	C₂³	1	2	yes	—
97	#6103	C₂³	1	8	yes	—
98	C₄⋊Dic₁₃	C₂³	1	1	yes	characteristic
99	C₄⋊Dic₁₃	C₂³	1	4	yes	—
100	C₄⋊Dic₁₃	C₂³	1	1	yes	characteristic
101	C₄⋊Dic₁₃	C₂³	1	4	yes	—
102	C₄⋊Dic₁₃	C₂³	1	4	yes	—
103	C₄⋊Dic₁₃	C₂³	1	4	yes	—
104	C₄⋊Dic₁₃	C₂³	1	8	yes	—
105	#6101	C₂³	1	2	yes	—
106	#6101	C₂³	1	2	yes	—
107	#6101	C₂³	1	8	yes	—
108	#6101	C₂³	1	8	yes	—
109	#6101	C₂³	1	8	yes	—
110	C₄×Dic₁₃	C₂³	1	2	yes	—
111	C₄×Dic₁₃	C₂³	1	2	yes	—
112	C₄×Dic₁₃	C₂³	1	8	yes	—
113	C₄×Dic₁₃	C₂³	1	2	yes	—
114	#3503	—	13	1	no	maximal, Sylow
115	C₂³×C₁₃	C₂⁴	1	1	yes	characteristic
116	C₂²×D₁₃	C₂⁴	1	2	yes	—
117	Q₈×C₁₃	—	2	4	no	—
118	Q₈×C₁₃	—	2	4	no	—
119	C₂×C₄×C₁₃	—	2	2	no	—
120	C₂×C₄×C₁₃	—	2	2	no	—
121	C₂×C₄×C₁₃	C₂×Q₈	1	4	yes	complemented
122	C₂×C₄×C₁₃	C₂⁴	1	4	yes	—
123	C₂×C₄×C₁₃	C₂⁴	1	4	yes	—
124	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
125	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
126	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
127	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
128	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
129	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
130	C₂²⋊D₁₃	C₂×Q₈	1	8	yes	complemented
131	C₂×Dic₁₃	—	4	4	no	—
132	C₂×Dic₁₃	C₂⁴	1	2	yes	—
133	C₂×Dic₁₃	C₂⁴	1	2	yes	—
134	C₂×Dic₁₃	C₂⁴	1	8	yes	—
135	C₂×Dic₁₃	C₂⁴	1	2	yes	—
136	D₅₂	C₂×Q₈	1	4	yes	complemented
137	C₄×D₁₃	—	2	4	no	—
138	C₄×D₁₃	—	2	4	no	—
139	C₄×D₁₃	—	4	8	no	—
140	C₄×D₁₃	C₂×Q₈	1	8	yes	complemented
141	Dic₂₆	—	2	8	no	—
142	Dic₂₆	—	4	4	no	—
143	Dic₂₆	—	4	8	no	—
144	Dic₂₆	C₂×Q₈	1	4	yes	complemented
145	Q₈²	—	13	2	no	—
146	#557	—	13	2	no	—
147	#554	—	13	4	no	—
148	#554	—	13	2	no	—
149	D₄×Q₈	—	13	4	no	—
150	D₄×Q₈	—	13	2	no	—
151	#544	—	13	4	no	—
152	#544	—	13	2	no	—
153	#533	—	13	2	no	—
154	#533	D₁₃	1	1	yes	characteristic, complemented
155	C₂×C₄⋊Q₈	—	13	2	no	—
156	C₂×C₄⋊Q₈	—	13	1	no	—
157	#520	—	13	2	no	—
158	C₄×#41	—	13	1	no	—
159	C₂²×C₁₃	—	2	1	no	—
160	C₂²×C₁₃	—	2	1	no	—
161	C₂²×C₁₃	C₂²×Q₈	1	2	yes	—
162	C₂²×C₁₃	C₂⁵	1	1	yes	characteristic
163	C₂×D₁₃	—	2	2	no	—
164	C₂×D₁₃	—	2	2	no	—
165	C₂×D₁₃	C₂²×Q₈	1	4	yes	—
166	C₄×C₁₃	—	2	1	no	—
167	C₄×C₁₃	—	2	4	no	—
168	C₄×C₁₃	—	2	1	no	—
169	C₄×C₁₃	—	2	4	no	—
170	C₄×C₁₃	C₂²×Q₈	1	2	yes	—
171	C₄×C₁₃	C₂²×Q₈	1	2	yes	—
172	C₄×C₁₃	H(5,𝔽₂)	1	2	yes	—
173	C₄×C₁₃	2₋¹⁺⁴	1	2	yes	—
174	Dic₁₃	—	2	8	no	—
175	Dic₁₃	—	2	2	no	—
176	Dic₁₃	—	2	2	no	—
177	Dic₁₃	C₂²×Q₈	1	4	yes	—
178	C₂×#41	—	13	1	no	—
179	C₂²×Q₈	—	13	4	no	—
180	C₂²×Q₈	—	13	2	no	—
181	C₄⋊Q₈	—	13	4	no	—
182	C₄⋊Q₈	—	13	4	no	—
183	C₄⋊Q₈	—	13	4	no	—
184	C₄⋊Q₈	—	13	8	no	—
185	C₄⋊Q₈	—	13	2	no	—
186	C₄⋊Q₈	—	13	2	no	—
187	C₄⋊Q₈	—	13	2	no	—
188	C₄⋊Q₈	—	13	2	no	—
189	C₄⋊Q₈	—	13	4	no	—
190	C₄⋊Q₈	C₂×D₁₃	1	1	yes	characteristic
191	C₄⋊Q₈	C₂×D₁₃	1	1	yes	characteristic
192	#125	—	13	4	no	—
193	#125	—	13	8	no	—
194	#125	—	13	4	no	—
195	#125	C₂×D₁₃	1	2	yes	—
196	C₂²⋊Q₈	—	13	4	no	—
197	C₂²⋊Q₈	—	13	4	no	—
198	C₂²⋊Q₈	—	13	8	no	—
199	C₂²⋊Q₈	—	13	8	no	—
200	C₂²⋊Q₈	—	13	8	no	—
201	C₂²⋊Q₈	C₂×D₁₃	1	2	yes	—
202	C₂²⋊Q₈	C₂×D₁₃	1	2	yes	—
203	C₄×Q₈	—	13	4	no	—
204	C₄×Q₈	—	13	4	no	—
205	C₄×Q₈	—	13	1	no	—
206	C₄×Q₈	—	13	1	no	—
207	C₄×Q₈	—	13	4	no	—
208	C₄×Q₈	C₂×D₁₃	1	2	yes	—
209	C₄×Q₈	C₂×D₁₃	1	2	yes	—
210	C₄×D₄	—	13	2	no	—
211	C₄×D₄	—	13	2	no	—
212	C₄×D₄	—	13	8	no	—
213	C₄×D₄	—	13	1	no	—
214	C₄×D₄	—	13	1	no	—
215	C₄×D₄	—	13	4	no	—
216	#117	—	13	1	no	—
217	#117	—	13	4	no	—
218	#117	—	13	2	no	—
219	#117	C₂×D₁₃	1	2	yes	—
220	C₂×C₄⋊C₄	—	13	2	no	—
221	C₂×C₄⋊C₄	—	13	2	no	—
222	C₂×C₄⋊C₄	—	13	4	no	—
223	C₂×C₄⋊C₄	—	13	4	no	—
224	C₂×C₄²	—	13	2	no	—
225	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
226	C₂×C₁₃	—	2	2	no	—
227	C₂×C₁₃	C₂³×Q₈	1	1	yes	characteristic
228	C₂×C₁₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
229	C₂×C₁₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
230	D₁₃	—	2	4	no	—
231	#41	—	13	8	no	—
232	C₂×Q₈	—	13	1	no	—
233	C₂×Q₈	—	13	4	no	—
234	C₂×Q₈	—	13	8	no	—
235	C₂×Q₈	—	13	4	no	—
236	C₂×Q₈	—	52	4	no	—
237	C₂×Q₈	—	52	8	no	—
238	C₂×Q₈	C₂²×D₁₃	1	2	yes	—
239	C₂×Q₈	C₂²×D₁₃	1	2	yes	—
240	C₂×D₄	—	13	2	no	—
241	C₂×D₄	—	13	1	no	—
242	C₂²×C₄	—	13	2	no	—
243	C₂²×C₄	—	13	4	no	—
244	C₂²×C₄	—	13	2	no	—
245	C₂²×C₄	—	13	2	no	—
246	C₂²×C₄	—	13	8	no	—
247	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
248	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
249	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
250	C₄⋊C₄	—	13	4	no	—
251	C₄⋊C₄	—	13	4	no	—
252	C₄⋊C₄	—	13	8	no	—
253	C₄⋊C₄	—	13	4	no	—
254	C₄⋊C₄	—	13	8	no	—
255	C₄⋊C₄	—	13	4	no	—
256	C₄⋊C₄	—	13	2	no	—
257	C₄⋊C₄	—	13	1	no	—
258	C₄⋊C₄	—	13	2	no	—
259	C₄⋊C₄	—	13	1	no	—
260	C₄⋊C₄	—	13	8	no	—
261	C₄⋊C₄	—	13	8	no	—
262	C₄⋊C₄	C₂²×D₁₃	1	2	yes	—
263	C₄⋊C₄	C₂²×D₁₃	1	2	yes	—
264	C₄⋊C₄	C₂²×D₁₃	1	2	yes	—
265	C₄⋊C₄	C₂²×D₁₃	1	2	yes	—
266	C₄⋊C₄	C₂²×D₁₃	1	4	yes	—
267	C₄⋊C₄	C₂²×D₁₃	1	4	yes	—
268	C₂²⋊C₄	—	13	2	no	—
269	C₂²⋊C₄	—	13	2	no	—
270	C₂²⋊C₄	—	13	2	no	—
271	C₂²⋊C₄	—	13	8	no	—
272	C₂²⋊C₄	C₂²×D₁₃	1	4	yes	—
273	C₄²	—	13	2	no	—
274	C₄²	—	13	2	no	—
275	C₄²	—	13	2	no	—
276	C₄²	—	13	8	no	—
277	C₄²	C₂²×D₁₃	1	1	yes	characteristic
278	C₄²	C₂²×D₁₃	1	1	yes	characteristic
279	C₄²	C₂²×D₁₃	1	1	yes	characteristic
280	C₄²	C₂²×D₁₃	1	1	yes	characteristic
281	C₄²	C₂²×D₁₃	1	4	yes	—
282	C₁₃	#3503	1	1	yes	characteristic, complemented, Sylow
283	C₂³	—	13	2	no	—
284	C₂³	C₂³×D₁₃	1	1	yes	characteristic
285	Q₈	—	2	4	no	—
286	Q₈	—	2	4	no	—
287	Q₈	—	13	4	no	—
288	Q₈	—	26	8	no	—
289	Q₈	—	52	4	no	—
290	Q₈	—	52	8	no	—
291	D₄	—	13	8	no	—
292	D₄	—	13	4	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	13	2	no	—
296	C₂×C₄	—	13	8	no	—
297	C₂×C₄	—	13	2	no	—
298	C₂×C₄	—	13	2	no	—
299	C₂×C₄	—	13	8	no	—
300	C₂×C₄	—	26	4	no	—
301	C₂×C₄	—	26	4	no	—
302	C₂×C₄	—	52	8	no	—
303	C₂×C₄	—	52	4	no	—
304	C₂×C₄	Q₈×D₁₃	1	4	yes	complemented
305	C₂×C₄	C₂³×D₁₃	1	4	yes	—
306	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
307	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
308	C₂×C₄	C₂³×D₁₃	1	4	yes	—
309	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
310	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
311	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
312	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
313	C₂²	—	2	1	no	—
314	C₂²	—	2	1	no	—
315	C₂²	—	13	4	no	—
316	C₂²	—	26	2	no	—
317	C₂²	—	26	2	no	—
318	C₂²	C₂×Q₈×D₁₃	1	2	yes	—
319	C₂²	C₂⁴×D₁₃	1	1	yes	characteristic
320	C₄	—	2	1	no	—
321	C₄	—	2	4	no	—
322	C₄	—	2	1	no	—
323	C₄	—	2	4	no	—
324	C₄	—	13	4	no	—
325	C₄	—	26	2	no	—
326	C₄	—	26	8	no	—
327	C₄	—	26	2	no	—
328	C₄	C₂×Q₈×D₁₃	1	2	yes	—
329	C₄	C₂×Q₈×D₁₃	1	2	yes	—
330	C₄	#88256	1	2	yes	—
331	C₄	#88257	1	2	yes	—
332	C₂	—	2	2	no	—
333	C₂	—	26	4	no	—
334	C₂	C₂²×Q₈×D₁₃	1	1	yes	characteristic
335	C₂	C₂×#88256	1	1	yes	characteristic
336	C₂	C₂×#88257	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₁₃
commutator, socle, cosocle	C₂²×C₁₃	C₂⁵
Fitting	C₁₃×#533	C₂
prime radical	C₂×#6127	C₂²
hypercenter	#533	D₁₃
hypocenter	C₁₃	#3503
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ #533

Complements

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

N	H ≅ G/N	type
1. C₁₃×#533	C₂	semidirect
2. #11719433	C₂	semidirect
3. D₁₃×C₄⋊Q₈	C₂	semidirect
4. #11719394	C₂	semidirect
5. #11719340	C₂	semidirect
6. #11719337	C₂	semidirect
7. #11719284	C₂	semidirect
8. Q₈×D₅₂	C₂	semidirect
9. #11719272	C₂	semidirect
10. Q₈×Dic₂₆	C₂	semidirect
12. #11719224	C₂	semidirect
14. C₂×#88123	C₂	semidirect
71. #6127	Q₈	semidirect
121. C₂×C₄×C₁₃	C₂×Q₈	semidirect
130. C₂²⋊D₁₃	C₂×Q₈	semidirect
136. D₅₂	C₂×Q₈	semidirect
140. C₄×D₁₃	C₂×Q₈	semidirect
144. Dic₂₆	C₂×Q₈	semidirect
154. #533	D₁₃	semidirect
282. C₁₃	#3503	semidirect
304. C₂×C₄	Q₈×D₁₃	semidirect

Maximal subgroups

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

subgroup	index
C₁₃×#533	2
#11719433	2
D₁₃×C₄⋊Q₈	2
#11719394	2
#11719340	2
#11719337	2
#11719284	2
Q₈×D₅₂	2
#11719272	2
Q₈×Dic₂₆	2
#11719228	2
#11719224	2
C₄×#6127	2
C₂×#88123	2
#3503	13