Subgroups — #15881332

Order 3328.

1062 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₈×Dic₁₃	C₂	1	2	yes	maximal
2	#13117746	C₂	1	2	yes	maximal, complemented
3	#13115570	C₂	1	2	yes	maximal, complemented
4	#13115099	C₂	1	4	yes	maximal, complemented
5	#13115000	C₂	1	1	yes	maximal, characteristic, complemented
6	#13114846	C₂	1	2	yes	maximal, complemented
7	C₈×C₄⋊Dic₁₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄×C₁₃×D₈	C₂	1	1	yes	maximal, characteristic
9	C₂²×C₁₃×D₈	C₂²	1	1	yes	characteristic
10	C₂×C₄×D₄×C₁₃	C₂²	1	2	yes	—
11	C₂×D₄×Dic₁₃	C₂²	1	4	yes	—
12	C₄×C₁₃×D₈	C₄	1	8	yes	complemented
13	C₂×C₁₃×#107	C₂²	1	1	yes	characteristic
14	C₂×C₁₃×#102	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
16	C₂×#88072	C₂²	1	4	yes	—
17	#11718873	—	2	2	no	—
18	C₂×#88071	—	2	2	no	—
19	D₈×Dic₁₃	—	2	8	no	—
20	C₂×#88058	C₂²	1	2	yes	—
21	C₂×#88057	C₂²	1	2	yes	—
22	C₂×C₈×Dic₁₃	C₂²	1	2	yes	—
23	#11718646	C₂²	1	4	yes	—
24	#11718611	—	2	4	no	—
25	#11718597	C₂²	1	2	yes	—
26	C₄×C₄⋊Dic₁₃	C₂²	1	1	yes	characteristic
27	C₂×C₄⋊#1055	C₂²	1	1	yes	characteristic
28	#11718142	C₂²	1	2	yes	—
29	#11718121	C₂²	1	4	yes	—
30	C₂²×D₄×C₁₃	C₂³	1	2	yes	—
31	C₂³×C₄×C₁₃	—	2	2	no	—
32	C₂³×Dic₁₃	—	2	4	no	—
33	C₂×C₁₃×D₈	C₂×C₄	1	4	yes	—
34	C₂×C₁₃×D₈	D₄	1	4	yes	—
35	C₂×C₁₃×D₈	Q₈	1	4	yes	—
36	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
37	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
38	C₄×D₄×C₁₃	—	2	4	no	—
39	C₄×D₄×C₁₃	C₂×C₄	1	8	yes	complemented
40	C₂×C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
41	C₂×C₁₃×C₂²⋊C₄	—	2	2	no	—
42	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
43	C₂×C₂²⋊Dic₁₃	—	2	4	no	—
44	D₄×Dic₁₃	—	2	8	no	—
45	D₄×Dic₁₃	—	4	4	no	—
46	C₂×C₄⋊Dic₁₃	—	2	4	no	—
47	C₂×C₄⋊Dic₁₃	D₄	1	2	yes	—
48	C₂×C₄⋊Dic₁₃	C₂³	1	4	yes	—
49	C₂×C₄×Dic₁₃	C₂³	1	2	yes	—
50	C₂²×#1055	C₂³	1	2	yes	—
51	C₁₃×#107	C₂×C₄	1	4	yes	complemented
52	C₁₃×#102	C₂×C₄	1	8	yes	complemented
53	C₄×C₈×C₁₃	C₂×C₄	1	4	yes	complemented
54	#88072	—	2	8	no	—
55	#88071	—	2	2	no	—
56	#88071	—	2	2	no	—
57	#88071	—	2	4	no	—
58	#88071	D₄	1	1	yes	characteristic
59	#88071	D₄	1	1	yes	characteristic
60	#88058	—	2	4	no	—
61	#88057	—	4	2	no	—
62	C₈×Dic₁₃	—	2	4	no	—
63	C₄⋊#1055	—	2	2	no	—
64	#22682	—	13	1	no	maximal, Sylow
65	C₂⁴×C₁₃	—	2	2	no	—
66	C₂×D₄×C₁₃	—	2	2	no	—
67	C₂×D₄×C₁₃	—	2	2	no	—
68	C₂×D₄×C₁₃	—	2	2	no	—
69	C₂×D₄×C₁₃	C₂²×C₄	1	4	yes	—
70	C₂×D₄×C₁₃	C₂×D₄	1	4	yes	—
71	C₂×D₄×C₁₃	C₂×Q₈	1	4	yes	—
72	C₂²×C₄×C₁₃	—	2	2	no	—
73	C₂²×C₄×C₁₃	—	2	8	no	—
74	C₂²×C₄×C₁₃	—	8	4	no	—
75	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
78	C₂²×Dic₁₃	—	2	4	no	—
79	C₂²×Dic₁₃	—	4	8	no	—
80	C₂²×Dic₁₃	—	4	8	no	—
81	C₂²×Dic₁₃	—	8	4	no	—
82	C₂²×Dic₁₃	C₂×D₄	1	2	yes	—
83	C₂²×Dic₁₃	C₂×D₄	1	2	yes	—
84	C₁₃×D₈	C₄⋊C₄	1	16	yes	complemented
85	C₂×C₈×C₁₃	—	2	1	no	—
86	C₂×C₈×C₁₃	—	2	1	no	—
87	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
88	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
89	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
90	C₂×C₈×C₁₃	C₂×Q₈	1	2	yes	—
91	C₁₃×C₄⋊C₄	—	2	2	no	—
92	C₁₃×C₄⋊C₄	C₂²×C₄	1	4	yes	—
93	C₁₃×C₂²⋊C₄	—	2	8	no	—
94	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
95	C₄²×C₁₃	#41	1	2	yes	—
96	C₂²⋊Dic₁₃	—	4	8	no	—
97	C₄⋊Dic₁₃	—	2	4	no	—
98	C₄⋊Dic₁₃	—	2	4	no	—
99	C₄⋊Dic₁₃	—	2	4	no	—
100	C₄⋊Dic₁₃	—	8	4	no	—
101	C₄×Dic₁₃	—	2	2	no	—
102	C₄×Dic₁₃	—	2	2	no	—
103	C₂×#1055	—	2	2	no	—
104	C₂×#1055	—	2	2	no	—
105	C₂×#1055	—	2	2	no	—
106	C₂×C₄×D₈	—	13	2	no	—
107	C₂×C₄×D₈	D₁₃	1	1	yes	characteristic
108	#2349	—	13	2	no	—
109	#1942	—	13	2	no	—
110	#1864	—	13	4	no	—
111	#1846	—	13	1	no	—
112	#1804	—	13	2	no	—
113	C₈×C₄⋊C₄	—	13	1	no	—
114	C₂³×C₁₃	—	2	4	no	—
115	C₂³×C₁₃	—	2	4	no	—
116	C₂³×C₁₃	—	2	4	no	—
117	C₂³×C₁₃	—	4	4	no	—
118	C₂³×C₁₃	—	4	4	no	—
119	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
120	D₄×C₁₃	—	2	4	no	—
121	D₄×C₁₃	—	2	4	no	—
122	D₄×C₁₃	—	2	4	no	—
123	D₄×C₁₃	C₂×C₄⋊C₄	1	8	yes	—
124	C₂×C₄×C₁₃	—	2	2	no	—
125	C₂×C₄×C₁₃	—	2	2	no	—
126	C₂×C₄×C₁₃	—	2	4	no	—
127	C₂×C₄×C₁₃	—	4	4	no	—
128	C₂×C₄×C₁₃	—	8	8	no	—
129	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
130	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
131	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
132	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
133	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
134	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
135	C₂×C₄×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
136	C₂×C₄×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
137	C₂×Dic₁₃	—	2	2	no	—
138	C₂×Dic₁₃	—	2	2	no	—
139	C₂×Dic₁₃	—	2	2	no	—
140	C₂×Dic₁₃	—	2	2	no	—
141	C₂×Dic₁₃	—	2	2	no	—
142	C₂×Dic₁₃	—	4	4	no	—
143	C₂×Dic₁₃	—	4	4	no	—
144	C₂×Dic₁₃	—	4	4	no	—
145	C₂×Dic₁₃	—	8	16	no	—
146	C₂×Dic₁₃	#135	1	4	yes	—
147	C₈×C₁₃	—	2	2	no	—
148	C₈×C₁₃	C₂×C₄⋊C₄	1	4	yes	—
149	#1055	—	4	2	no	—
150	C₂²×D₈	C₂×D₁₃	1	1	yes	characteristic
151	C₂×C₄×D₄	—	13	4	no	—
152	C₂×C₄×D₄	C₂×D₁₃	1	2	yes	—
153	C₄×D₈	—	26	8	no	—
154	C₄×D₈	Dic₁₃	1	8	yes	complemented
155	C₂×#107	—	13	2	no	—
156	C₂×#107	C₂×D₁₃	1	1	yes	characteristic
157	C₂×#106	—	13	2	no	—
158	C₂×C₄⋊C₈	—	13	1	no	—
159	C₂×#102	—	13	4	no	—
160	C₂×#102	C₂×D₁₃	1	2	yes	—
161	C₂×C₄×C₈	—	13	2	no	—
162	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
163	#389	—	13	4	no	—
164	#385	—	26	4	no	—
165	#382	—	13	2	no	—
166	#380	—	26	2	no	—
167	C₄×C₄⋊C₄	—	13	1	no	—
168	C₂×#95	—	26	2	no	—
169	#340	—	13	4	no	—
170	#336	—	13	2	no	—
171	C₂²×C₁₃	—	2	8	no	—
172	C₂²×C₁₃	—	4	8	no	—
173	C₂²×C₁₃	—	4	4	no	—
174	C₂²×C₁₃	—	4	4	no	—
175	C₂²×C₁₃	—	4	8	no	—
176	C₂²×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
177	C₂²×C₁₃	D₄²	1	1	yes	characteristic
178	C₂²×C₁₃	D₄×Q₈	1	1	yes	characteristic
179	C₂²×C₁₃	C₂×#135	1	2	yes	—
180	C₂²×C₁₃	#576	1	2	yes	—
181	C₄×C₁₃	—	2	2	no	—
182	C₄×C₁₃	—	2	2	no	—
183	C₄×C₁₃	—	4	4	no	—
184	C₄×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
185	C₄×C₁₃	#520	1	1	yes	characteristic
186	C₄×C₁₃	#547	1	1	yes	characteristic
187	C₄×C₁₃	#554	1	1	yes	characteristic
188	Dic₁₃	—	2	4	no	—
189	Dic₁₃	—	4	2	no	—
190	Dic₁₃	—	8	4	no	—
191	C₂²×D₄	C₂²×D₁₃	1	2	yes	—
192	C₂³×C₄	—	2	2	no	—
193	C₂³×C₄	—	26	4	no	—
194	C₂×D₈	Dic₂₆	1	4	yes	—
195	C₂×D₈	D₅₂	1	4	yes	—
196	C₂×D₈	C₂×Dic₁₃	1	4	yes	—
197	C₂²×C₈	—	13	2	no	—
198	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
199	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
200	C₄×D₄	—	2	4	no	—
201	C₄×D₄	—	26	8	no	—
202	C₄×D₄	—	52	4	no	—
203	C₄×D₄	C₂×Dic₁₃	1	8	yes	complemented
204	C₂×C₄⋊C₄	—	13	2	no	—
205	C₂×C₄⋊C₄	—	13	4	no	—
206	C₂×C₄⋊C₄	—	26	4	no	—
207	C₂×C₄⋊C₄	C₂²×D₁₃	1	2	yes	—
208	C₂×C₂²⋊C₄	—	2	2	no	—
209	C₂×C₂²⋊C₄	—	26	4	no	—
210	C₂×C₄²	—	13	2	no	—
211	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
212	#107	—	26	4	no	—
213	#107	C₂×Dic₁₃	1	4	yes	complemented
214	#106	—	52	2	no	—
215	C₄⋊C₈	—	26	2	no	—
216	#102	—	26	8	no	—
217	#102	C₂×Dic₁₃	1	8	yes	complemented
218	C₄×C₈	—	26	4	no	—
219	C₄×C₈	C₂×Dic₁₃	1	4	yes	complemented
220	#95	—	13	1	no	—
221	#95	—	13	1	no	—
222	#95	—	26	4	no	—
223	#95	—	26	2	no	—
224	#95	—	26	2	no	—
225	C₂×C₁₃	—	4	8	no	—
226	C₂×C₁₃	D₄×C₄⋊C₄	1	1	yes	characteristic
227	C₂×C₁₃	C₄×#135	1	1	yes	characteristic
228	C₂×C₁₃	#2948	1	1	yes	characteristic
229	C₂×C₁₃	#3280	1	2	yes	—
230	C₂×C₁₃	#3380	1	2	yes	—
231	C₂⁴	—	2	2	no	—
232	C₂×D₄	—	2	2	no	—
233	C₂×D₄	—	2	2	no	—
234	C₂×D₄	—	2	2	no	—
235	C₂×D₄	C₂×Dic₂₆	1	4	yes	—
236	C₂×D₄	C₂×D₅₂	1	4	yes	—
237	C₂×D₄	C₂²×Dic₁₃	1	4	yes	—
238	C₂²×C₄	—	2	2	no	—
239	C₂²×C₄	—	2	8	no	—
240	C₂²×C₄	—	8	4	no	—
241	C₂²×C₄	—	13	2	no	—
242	C₂²×C₄	—	13	2	no	—
243	C₂²×C₄	—	26	4	no	—
244	C₂²×C₄	—	52	8	no	—
245	C₂²×C₄	—	52	8	no	—
246	C₂²×C₄	—	104	4	no	—
247	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
248	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
249	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
250	D₈	C₄⋊Dic₁₃	1	16	yes	complemented
251	C₂×C₈	—	2	1	no	—
252	C₂×C₈	—	2	1	no	—
253	C₂×C₈	—	26	2	no	—
254	C₂×C₈	—	26	2	no	—
255	C₂×C₈	—	26	2	no	—
256	C₂×C₈	C₂×Dic₂₆	1	2	yes	—
257	C₂×C₈	C₂×D₅₂	1	2	yes	—
258	C₂×C₈	C₂²×Dic₁₃	1	2	yes	—
259	C₂×C₈	C₂²×Dic₁₃	1	2	yes	—
260	C₄⋊C₄	—	2	2	no	—
261	C₄⋊C₄	—	26	4	no	—
262	C₄⋊C₄	—	26	4	no	—
263	C₄⋊C₄	—	26	4	no	—
264	C₄⋊C₄	—	104	4	no	—
265	C₄⋊C₄	C₂²×Dic₁₃	1	4	yes	—
266	C₂²⋊C₄	—	2	8	no	—
267	C₂²⋊C₄	—	52	8	no	—
268	C₄²	—	26	2	no	—
269	C₄²	—	26	2	no	—
270	C₄²	#6129	1	2	yes	—
271	C₄²	C₂²×Dic₁₃	1	2	yes	—
272	C₁₃	#22682	1	1	yes	characteristic, complemented, Sylow
273	C₂³	—	2	4	no	—
274	C₂³	—	2	4	no	—
275	C₂³	—	2	4	no	—
276	C₂³	—	4	4	no	—
277	C₂³	—	4	4	no	—
278	C₂³	C₂×D₄×D₁₃	1	1	yes	characteristic
279	D₄	—	2	4	no	—
280	D₄	—	2	4	no	—
281	D₄	—	2	4	no	—
282	D₄	C₂×C₄⋊Dic₁₃	1	8	yes	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	—	2	4	no	—
286	C₂×C₄	—	4	4	no	—
287	C₂×C₄	—	8	8	no	—
288	C₂×C₄	—	13	4	no	—
289	C₂×C₄	—	26	2	no	—
290	C₂×C₄	—	26	2	no	—
291	C₂×C₄	—	26	2	no	—
292	C₂×C₄	—	26	2	no	—
293	C₂×C₄	—	26	2	no	—
294	C₂×C₄	—	52	4	no	—
295	C₂×C₄	—	52	4	no	—
296	C₂×C₄	—	52	4	no	—
297	C₂×C₄	—	104	16	no	—
298	C₂×C₄	D₄×Dic₁₃	1	2	yes	—
299	C₂×C₄	D₄×Dic₁₃	1	2	yes	—
300	C₂×C₄	C₂²×Dic₂₆	1	1	yes	characteristic
301	C₂×C₄	C₂²×D₅₂	1	1	yes	characteristic
302	C₂×C₄	C₂×#6129	1	1	yes	characteristic
303	C₂×C₄	#88256	1	1	yes	characteristic
304	C₂×C₄	#88257	1	1	yes	characteristic
305	C₂×C₄	C₂³×Dic₁₃	1	1	yes	characteristic
306	C₈	—	2	2	no	—
307	C₈	—	52	2	no	—
308	C₈	C₂×C₄⋊Dic₁₃	1	4	yes	—
309	C₂²	—	2	8	no	—
310	C₂²	—	4	8	no	—
311	C₂²	—	4	4	no	—
312	C₂²	—	4	4	no	—
313	C₂²	—	4	8	no	—
314	C₂²	D₄×Dic₂₆	1	1	yes	characteristic
315	C₂²	D₄×D₅₂	1	1	yes	characteristic
316	C₂²	#11719462	1	2	yes	—
317	C₂²	D₁₃×#135	1	2	yes	—
318	C₂²	C₂×D₄×Dic₁₃	1	1	yes	characteristic
319	C₄	—	2	2	no	—
320	C₄	—	2	2	no	—
321	C₄	—	4	4	no	—
322	C₄	—	26	4	no	—
323	C₄	—	52	2	no	—
324	C₄	—	104	4	no	—
325	C₄	#11719248	1	1	yes	characteristic
326	C₄	#11719260	1	1	yes	characteristic
327	C₄	C₂²×C₄⋊Dic₁₃	1	1	yes	characteristic
328	C₄	#11719532	1	1	yes	characteristic
329	C₂	—	4	8	no	—
330	C₂	D₄×C₄⋊Dic₁₃	1	1	yes	characteristic
331	C₂	#135×Dic₁₃	1	1	yes	characteristic
332	C₂	#13123688	1	1	yes	characteristic
333	C₂	#13126522	1	2	yes	—
334	C₂	#13127703	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄×D₁₃
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₁₃
Fitting	C₂×C₄×C₁₃×D₈	C₂
socle	C₂³×C₁₃	C₂²×D₄
prime radical	C₂²×C₁₃×D₈	C₂²
cosocle	C₂²×C₄×C₁₃	C₂⁴
hypercenter	C₂×C₄×D₈	D₁₃
hypocenter	C₁₃	#22682
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×D₈

Complements

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

N	H ≅ G/N	type
2. #13117746	C₂	semidirect
3. #13115570	C₂	semidirect
4. #13115099	C₂	semidirect
5. #13115000	C₂	semidirect
6. #13114846	C₂	semidirect
7. C₈×C₄⋊Dic₁₃	C₂	semidirect
12. C₄×C₁₃×D₈	C₄	semidirect
39. C₄×D₄×C₁₃	C₂×C₄	semidirect
51. C₁₃×#107	C₂×C₄	semidirect
52. C₁₃×#102	C₂×C₄	semidirect
53. C₄×C₈×C₁₃	C₂×C₄	semidirect
84. C₁₃×D₈	C₄⋊C₄	semidirect
154. C₄×D₈	Dic₁₃	semidirect
203. C₄×D₄	C₂×Dic₁₃	semidirect
213. #107	C₂×Dic₁₃	semidirect
217. #102	C₂×Dic₁₃	semidirect
219. C₄×C₈	C₂×Dic₁₃	semidirect
250. D₈	C₄⋊Dic₁₃	semidirect
272. C₁₃	#22682	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₈×Dic₁₃	2
#13117746	2
#13115570	2
#13115099	2
#13115000	2
#13114846	2
C₈×C₄⋊Dic₁₃	2
C₂×C₄×C₁₃×D₈	2
#22682	13