Subgroups — #15881349

Order 3328.

926 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Dic₅₂	C₂	1	1	yes	maximal, characteristic
2	C₂×#11718534	C₂	1	2	yes	maximal
3	#13117748	C₂	1	2	yes	maximal
4	#13115567	C₂	1	2	yes	maximal
5	#13115087	C₂	1	4	yes	maximal
6	#13114838	C₂	1	2	yes	maximal
7	C₈×C₄⋊Dic₁₃	C₂	1	1	yes	maximal, characteristic
8	C₁₃×#1845	C₂	1	1	yes	maximal, characteristic
9	C₂²×Dic₅₂	C₂²	1	1	yes	characteristic
10	C₂×#88141	C₂²	1	4	yes	—
11	C₂×C₄×Dic₂₆	C₂²	1	2	yes	—
12	C₂×C₁₃×#107	C₂²	1	2	yes	—
13	C₂×C₁₃×#107	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
15	C₁₃×#384	C₂²	1	2	yes	—
16	C₂×#88058	C₂²	1	1	yes	characteristic
17	C₂×#88056	C₂²	1	2	yes	—
18	C₂×C₈×Dic₁₃	C₂²	1	2	yes	—
19	#11718637	—	2	2	no	—
20	C₄⋊C₄×Dic₁₃	—	2	2	no	—
21	#11718634	C₂²	1	4	yes	—
22	#11718633	—	2	4	no	—
23	C₂×#88050	C₂²	1	4	yes	—
24	C₄×C₄⋊Dic₁₃	C₂²	1	1	yes	characteristic
25	C₂×C₄⋊#1055	C₂²	1	1	yes	characteristic
26	#11718534	—	2	8	no	—
27	C₄×Dic₅₂	C₄	1	8	yes	complemented
28	#11718142	C₂²	1	2	yes	—
29	#11718121	C₂²	1	4	yes	—
30	C₂²×Dic₂₆	C₂³	1	2	yes	—
31	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
32	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
33	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
34	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
35	C₂×C₁₃×C₄⋊C₄	C₂³	1	4	yes	—
36	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
37	C₂×C₄⋊Dic₁₃	D₄	1	1	yes	characteristic
38	C₂×C₄⋊Dic₁₃	D₄	1	1	yes	characteristic
39	C₂×C₄⋊Dic₁₃	C₂³	1	2	yes	—
40	C₂×#6101	—	2	2	no	—
41	C₂×#6101	—	2	4	no	—
42	C₂×#6101	—	2	4	no	—
43	C₂×C₄×Dic₁₃	—	2	2	no	—
44	C₂×C₄×Dic₁₃	—	2	4	no	—
45	C₂×C₄×Dic₁₃	C₂³	1	2	yes	—
46	C₂²×#1055	C₂³	1	2	yes	—
47	C₂×Dic₅₂	C₂×C₄	1	4	yes	—
48	C₂×Dic₅₂	D₄	1	4	yes	—
49	C₂×Dic₅₂	Q₈	1	4	yes	—
50	#88141	—	2	8	no	—
51	#88141	—	4	4	no	—
52	C₄×Dic₂₆	—	2	4	no	—
53	C₄×Dic₂₆	C₂×C₄	1	8	yes	—
54	C₁₃×#107	—	2	4	no	—
55	C₁₃×#107	—	4	2	no	—
56	C₄×C₈×C₁₃	C₂×C₄	1	4	yes	—
57	#88071	—	2	4	no	—
58	#88071	D₄	1	2	yes	—
59	#88058	C₂×C₄	1	4	yes	—
60	#88056	C₂×C₄	1	8	yes	—
61	C₈×Dic₁₃	—	2	4	no	—
62	#88050	—	2	8	no	—
63	C₄⋊#1055	—	2	2	no	—
64	C₄⋊C₄×Q₁₆	—	13	1	no	maximal, Sylow
65	C₂²×C₄×C₁₃	—	2	4	no	—
66	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
69	C₂²×Dic₁₃	—	2	2	no	—
70	C₂²×Dic₁₃	—	2	2	no	—
71	C₂²×Dic₁₃	C₂×D₄	1	2	yes	—
72	C₂²×Dic₁₃	C₂×D₄	1	2	yes	—
73	C₂×Dic₂₆	—	2	2	no	—
74	C₂×Dic₂₆	—	2	2	no	—
75	C₂×Dic₂₆	—	2	2	no	—
76	C₂×Dic₂₆	C₂²×C₄	1	4	yes	—
77	C₂×Dic₂₆	C₂×D₄	1	4	yes	—
78	C₂×Dic₂₆	C₂×Q₈	1	4	yes	—
79	C₂×C₈×C₁₃	—	2	1	no	—
80	C₂×C₈×C₁₃	—	2	1	no	—
81	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
82	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
83	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
84	C₂×C₈×C₁₃	C₂×Q₈	1	2	yes	—
85	C₁₃×C₄⋊C₄	—	2	4	no	—
86	C₁₃×C₄⋊C₄	—	2	4	no	—
87	C₁₃×C₄⋊C₄	—	4	4	no	—
88	C₁₃×C₄⋊C₄	—	8	2	no	—
89	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
90	C₄²×C₁₃	#41	1	2	yes	—
91	C₄⋊Dic₁₃	—	2	2	no	—
92	C₄⋊Dic₁₃	—	2	2	no	—
93	C₄⋊Dic₁₃	Q₁₆	1	4	yes	complemented
94	C₄⋊Dic₁₃	C₂²×C₄	1	4	yes	—
95	#6101	—	2	8	no	—
96	#6101	—	4	8	no	—
97	#6101	—	8	4	no	—
98	C₄×Dic₁₃	—	2	8	no	—
99	C₄×Dic₁₃	—	2	2	no	—
100	C₄×Dic₁₃	—	2	2	no	—
101	C₄×Dic₁₃	—	4	8	no	—
102	C₂×#1055	—	2	2	no	—
103	C₂×#1055	—	2	2	no	—
104	C₂×#1055	—	2	2	no	—
105	Dic₅₂	C₄⋊C₄	1	16	yes	complemented
106	C₂×C₄×Q₁₆	—	13	2	no	—
107	C₂×C₄×Q₁₆	—	13	1	no	—
108	Q₈×C₄⋊C₄	—	13	2	no	—
109	#1942	—	13	2	no	—
110	#1863	—	13	4	no	—
111	#1845	D₁₃	1	1	yes	characteristic
112	#1803	—	13	2	no	—
113	C₈×C₄⋊C₄	—	13	1	no	—
114	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
115	C₂×C₄×C₁₃	—	2	1	no	—
116	C₂×C₄×C₁₃	—	2	1	no	—
117	C₂×C₄×C₁₃	—	4	4	no	—
118	C₂×C₄×C₁₃	—	4	4	no	—
119	C₂×C₄×C₁₃	—	4	4	no	—
120	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
121	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
122	C₂×C₄×C₁₃	C₂×Q₁₆	1	2	yes	—
123	C₂×C₄×C₁₃	#135	1	2	yes	—
124	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
125	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
126	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
127	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
128	C₂×C₄×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
129	C₂×C₄×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
130	C₂×Dic₁₃	—	2	4	no	—
131	C₂×Dic₁₃	—	2	2	no	—
132	C₂×Dic₁₃	—	2	4	no	—
133	C₂×Dic₁₃	—	2	4	no	—
134	C₂×Dic₁₃	—	2	2	no	—
135	C₂×Dic₁₃	—	2	4	no	—
136	C₂×Dic₁₃	—	2	2	no	—
137	C₂×Dic₁₃	—	2	2	no	—
138	C₂×Dic₁₃	—	2	2	no	—
139	C₂×Dic₁₃	—	4	2	no	—
140	C₂×Dic₁₃	—	4	2	no	—
141	C₂×Dic₁₃	C₂×Q₁₆	1	4	yes	—
142	Dic₂₆	—	2	4	no	—
143	Dic₂₆	—	2	4	no	—
144	Dic₂₆	—	2	4	no	—
145	Dic₂₆	C₂×C₄⋊C₄	1	8	yes	—
146	C₈×C₁₃	—	2	2	no	—
147	C₈×C₁₃	C₂×C₄⋊C₄	1	4	yes	—
148	#1055	—	4	2	no	—
149	C₂²×Q₁₆	—	13	1	no	—
150	C₂×C₄×Q₈	—	13	4	no	—
151	C₂×C₄×Q₈	—	13	2	no	—
152	C₄×Q₁₆	—	13	8	no	—
153	C₄×Q₁₆	—	26	8	no	—
154	C₂×#107	—	13	1	no	—
155	C₂×#107	C₂×D₁₃	1	2	yes	—
156	C₂×#107	C₂×D₁₃	1	2	yes	—
157	C₂×C₄⋊C₈	—	13	1	no	—
158	C₂×#103	—	13	2	no	—
159	C₂×#103	—	13	4	no	—
160	C₂×C₄×C₈	—	13	2	no	—
161	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
162	#389	—	13	4	no	—
163	#389	—	26	4	no	—
164	#384	—	26	2	no	—
165	#384	C₂×D₁₃	1	2	yes	—
166	C₄×C₄⋊C₄	—	13	1	no	—
167	C₄×C₄⋊C₄	—	26	2	no	—
168	#340	—	13	4	no	—
169	#336	—	13	2	no	—
170	C₂²×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
171	C₂²×C₁₃	D₄²	1	1	yes	characteristic
172	C₂²×C₁₃	D₄×Q₈	1	1	yes	characteristic
173	C₂²×C₁₃	C₂²×Q₁₆	1	1	yes	characteristic
174	C₂²×C₁₃	C₂×#135	1	1	yes	characteristic
175	C₂²×C₁₃	#576	1	1	yes	characteristic
176	C₂²×C₁₃	#578	1	1	yes	characteristic
177	C₄×C₁₃	—	2	2	no	—
178	C₄×C₁₃	—	8	4	no	—
179	C₄×C₁₃	C₄×Q₁₆	1	4	yes	complemented
180	C₄×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
181	C₄×C₁₃	#520	1	1	yes	characteristic
182	C₄×C₁₃	#547	1	1	yes	characteristic
183	C₄×C₁₃	#554	1	1	yes	characteristic
184	Dic₁₃	—	2	8	no	—
185	Dic₁₃	—	2	4	no	—
186	Dic₁₃	—	4	4	no	—
187	Dic₁₃	—	4	2	no	—
188	C₂²×Q₈	—	13	2	no	—
189	C₂×Q₁₆	—	13	4	no	—
190	C₂×Q₁₆	—	13	4	no	—
191	C₂×Q₁₆	—	13	4	no	—
192	C₂²×C₈	—	13	2	no	—
193	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
194	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
195	C₄×Q₈	—	13	8	no	—
196	C₄×Q₈	—	26	4	no	—
197	C₄×Q₈	—	26	8	no	—
198	C₄×Q₈	—	52	4	no	—
199	C₂×C₄⋊C₄	—	2	2	no	—
200	C₂×C₄⋊C₄	—	2	2	no	—
201	C₂×C₄⋊C₄	—	13	1	no	—
202	C₂×C₄⋊C₄	—	13	2	no	—
203	C₂×C₄⋊C₄	—	13	1	no	—
204	C₂×C₄⋊C₄	—	26	2	no	—
205	C₂×C₄⋊C₄	—	26	4	no	—
206	C₂×C₄⋊C₄	—	26	4	no	—
207	C₂×C₄⋊C₄	C₂²×D₁₃	1	4	yes	—
208	C₂×C₄²	—	13	2	no	—
209	C₂×C₄²	—	26	2	no	—
210	C₂×C₄²	—	26	4	no	—
211	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
212	#107	—	2	4	no	—
213	#107	—	4	2	no	—
214	#107	—	13	4	no	—
215	C₄⋊C₈	—	26	2	no	—
216	#103	—	13	8	no	—
217	#103	—	26	8	no	—
218	C₄×C₈	—	26	4	no	—
219	C₄×C₈	C₄×D₁₃	1	4	yes	—
220	#95	—	13	2	no	—
221	#95	—	26	4	no	—
222	C₂×C₁₃	D₄×C₄⋊C₄	1	1	yes	characteristic
223	C₂×C₁₃	C₂×C₄×Q₁₆	1	1	yes	characteristic
224	C₂×C₁₃	#2951	1	1	yes	characteristic
225	C₂×C₁₃	D₄×Q₁₆	1	1	yes	characteristic
226	C₂×C₁₃	#3287	1	1	yes	characteristic
227	C₂×C₁₃	Q₈×Q₁₆	1	1	yes	characteristic
228	C₂×C₁₃	#3383	1	1	yes	characteristic
229	C₂×Q₈	—	13	4	no	—
230	C₂×Q₈	—	13	4	no	—
231	C₂×Q₈	—	13	4	no	—
232	C₂×Q₈	—	26	2	no	—
233	C₂×Q₈	—	26	2	no	—
234	C₂×Q₈	—	26	2	no	—
235	C₂²×C₄	—	2	4	no	—
236	C₂²×C₄	—	13	2	no	—
237	C₂²×C₄	—	13	2	no	—
238	C₂²×C₄	—	26	2	no	—
239	C₂²×C₄	—	26	2	no	—
240	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
241	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
242	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
243	Q₁₆	—	13	16	no	—
244	C₂×C₈	—	2	1	no	—
245	C₂×C₈	—	2	1	no	—
246	C₂×C₈	—	26	2	no	—
247	C₂×C₈	—	26	2	no	—
248	C₂×C₈	—	26	2	no	—
249	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
250	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
251	C₂×C₈	D₄×D₁₃	1	2	yes	—
252	C₂×C₈	Q₈×D₁₃	1	2	yes	—
253	C₄⋊C₄	—	2	4	no	—
254	C₄⋊C₄	—	2	4	no	—
255	C₄⋊C₄	—	4	4	no	—
256	C₄⋊C₄	—	8	2	no	—
257	C₄⋊C₄	—	13	4	no	—
258	C₄⋊C₄	—	13	4	no	—
259	C₄⋊C₄	—	26	8	no	—
260	C₄⋊C₄	—	26	2	no	—
261	C₄⋊C₄	—	26	2	no	—
262	C₄⋊C₄	—	52	8	no	—
263	C₄⋊C₄	—	104	4	no	—
264	C₄²	—	26	8	no	—
265	C₄²	—	26	2	no	—
266	C₄²	—	26	2	no	—
267	C₄²	—	52	8	no	—
268	C₄²	C₂×C₄×D₁₃	1	2	yes	—
269	C₄²	#6131	1	2	yes	—
270	C₁₃	C₄⋊C₄×Q₁₆	1	1	yes	characteristic, complemented, Sylow
271	C₂³	C₂×D₄×D₁₃	1	1	yes	characteristic
272	Q₈	—	13	8	no	—
273	Q₈	—	26	4	no	—
274	Q₈	—	26	4	no	—
275	Q₈	—	26	4	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	4	4	no	—
279	C₂×C₄	—	4	4	no	—
280	C₂×C₄	—	4	4	no	—
281	C₂×C₄	—	13	4	no	—
282	C₂×C₄	—	26	4	no	—
283	C₂×C₄	—	26	4	no	—
284	C₂×C₄	—	26	2	no	—
285	C₂×C₄	—	26	2	no	—
286	C₂×C₄	—	26	4	no	—
287	C₂×C₄	—	26	2	no	—
288	C₂×C₄	—	26	2	no	—
289	C₂×C₄	—	26	4	no	—
290	C₂×C₄	—	26	2	no	—
291	C₂×C₄	—	52	2	no	—
292	C₂×C₄	—	52	2	no	—
293	C₂×C₄	#88147	1	2	yes	—
294	C₂×C₄	#88147	1	2	yes	—
295	C₂×C₄	#88166	1	2	yes	—
296	C₂×C₄	Q₁₆×D₁₃	1	2	yes	—
297	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
298	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
299	C₂×C₄	#88251	1	1	yes	characteristic
300	C₂×C₄	C₂×Q₈×D₁₃	1	1	yes	characteristic
301	C₂×C₄	C₂×#6131	1	1	yes	characteristic
302	C₂×C₄	#88254	1	1	yes	characteristic
303	C₈	—	2	2	no	—
304	C₈	—	52	2	no	—
305	C₈	C₄⋊C₄×D₁₃	1	4	yes	—
306	C₂²	C₂×#88147	1	1	yes	characteristic
307	C₂²	#11719422	1	1	yes	characteristic
308	C₂²	#11719432	1	1	yes	characteristic
309	C₂²	C₂×#88166	1	1	yes	characteristic
310	C₂²	#11719462	1	1	yes	characteristic
311	C₂²	C₂×Q₁₆×D₁₃	1	1	yes	characteristic
312	C₂²	#11719471	1	1	yes	characteristic
313	C₄	—	2	2	no	—
314	C₄	—	8	4	no	—
315	C₄	—	26	8	no	—
316	C₄	—	26	4	no	—
317	C₄	—	52	4	no	—
318	C₄	—	52	2	no	—
319	C₄	#11718534	1	4	yes	complemented
320	C₄	C₂×C₄⋊C₄×D₁₃	1	1	yes	characteristic
321	C₄	#11719209	1	1	yes	characteristic
322	C₄	#11719431	1	1	yes	characteristic
323	C₄	#11719435	1	1	yes	characteristic
324	C₂	#13117736	1	1	yes	characteristic
325	C₂	C₂×#11718534	1	1	yes	characteristic
326	C₂	#13123710	1	1	yes	characteristic
327	C₂	#13126586	1	1	yes	characteristic
328	C₂	#13126594	1	1	yes	characteristic
329	C₂	#13127719	1	1	yes	characteristic
330	C₂	#13127729	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×D₄×D₁₃
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₁₃
Fitting	C₁₃×#1845	C₂
socle, prime radical	C₂³×C₁₃	C₂²×D₄
cosocle	C₂²×C₄×C₁₃	C₂⁴
hypercenter	#1845	D₁₃
hypocenter	C₁₃	C₄⋊C₄×Q₁₆
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1845

Complements

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

N	H ≅ G/N	type
27. C₄×Dic₅₂	C₄	semidirect
93. C₄⋊Dic₁₃	Q₁₆	semidirect
105. Dic₅₂	C₄⋊C₄	semidirect
179. C₄×C₁₃	C₄×Q₁₆	semidirect
270. C₁₃	C₄⋊C₄×Q₁₆	semidirect
319. C₄	#11718534	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×Dic₅₂	2
C₂×#11718534	2
#13117748	2
#13115567	2
#13115087	2
#13114838	2
C₈×C₄⋊Dic₁₃	2
C₁₃×#1845	2
C₄⋊C₄×Q₁₆	13