Subgroups — #15881218

Order 3328.

1050 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718813	C₂	1	2	yes	maximal
2	#13117746	C₂	1	2	yes	maximal, complemented
3	#13115575	C₂	1	2	yes	maximal, complemented
4	#13115085	C₂	1	4	yes	maximal, complemented
5	#13115000	C₂	1	1	yes	maximal, characteristic, complemented
6	#13114846	C₂	1	2	yes	maximal, complemented
7	#13114703	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	#11718813	—	2	8	no	—
20	C₂×#88058	C₂²	1	2	yes	—
21	C₂×#88057	C₂²	1	2	yes	—
22	C₂×#88055	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	2	yes	—
59	#88058	—	4	2	no	—
60	#88057	—	2	4	no	—
61	#88055	—	2	4	no	—
62	C₄⋊#1055	—	2	2	no	—
63	#22672	—	13	1	no	maximal, Sylow
64	C₂⁴×C₁₃	—	2	2	no	—
65	C₂×D₄×C₁₃	—	2	2	no	—
66	C₂×D₄×C₁₃	—	2	2	no	—
67	C₂×D₄×C₁₃	—	2	2	no	—
68	C₂×D₄×C₁₃	C₂²×C₄	1	4	yes	—
69	C₂×D₄×C₁₃	C₂×D₄	1	4	yes	—
70	C₂×D₄×C₁₃	C₂×Q₈	1	4	yes	—
71	C₂²×C₄×C₁₃	—	2	2	no	—
72	C₂²×C₄×C₁₃	—	2	8	no	—
73	C₂²×C₄×C₁₃	—	8	4	no	—
74	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
77	C₂²×Dic₁₃	—	2	4	no	—
78	C₂²×Dic₁₃	—	4	8	no	—
79	C₂²×Dic₁₃	—	4	8	no	—
80	C₂²×Dic₁₃	—	8	4	no	—
81	C₂²×Dic₁₃	C₂×D₄	1	4	yes	—
82	C₁₃×D₈	C₄⋊C₄	1	16	yes	complemented
83	C₂×C₈×C₁₃	—	2	1	no	—
84	C₂×C₈×C₁₃	—	2	1	no	—
85	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
86	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
87	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
88	C₂×C₈×C₁₃	C₂×Q₈	1	2	yes	—
89	C₁₃×C₄⋊C₄	—	2	2	no	—
90	C₁₃×C₄⋊C₄	C₂²×C₄	1	4	yes	—
91	C₁₃×C₂²⋊C₄	—	2	8	no	—
92	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
93	C₄²×C₁₃	#41	1	2	yes	—
94	C₂²⋊Dic₁₃	—	4	8	no	—
95	C₄⋊Dic₁₃	—	2	4	no	—
96	C₄⋊Dic₁₃	—	2	4	no	—
97	C₄⋊Dic₁₃	—	4	2	no	—
98	C₄⋊Dic₁₃	—	8	4	no	—
99	C₄×Dic₁₃	—	2	2	no	—
100	C₄×Dic₁₃	—	2	2	no	—
101	C₂×#1055	—	2	2	no	—
102	C₂×#1055	—	2	2	no	—
103	C₂×#1055	—	2	2	no	—
104	C₂×#442	—	13	2	no	—
105	C₂×C₄×D₈	D₁₃	1	1	yes	characteristic
106	#2349	—	13	2	no	—
107	#1943	—	13	2	no	—
108	#1862	—	13	4	no	—
109	#1846	—	13	1	no	—
110	#1804	—	13	2	no	—
111	#1770	—	13	1	no	—
112	C₂³×C₁₃	—	2	4	no	—
113	C₂³×C₁₃	—	2	4	no	—
114	C₂³×C₁₃	—	2	4	no	—
115	C₂³×C₁₃	—	4	4	no	—
116	C₂³×C₁₃	—	4	4	no	—
117	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
118	D₄×C₁₃	—	2	4	no	—
119	D₄×C₁₃	—	2	4	no	—
120	D₄×C₁₃	—	2	4	no	—
121	D₄×C₁₃	C₂×C₄⋊C₄	1	8	yes	—
122	C₂×C₄×C₁₃	—	2	2	no	—
123	C₂×C₄×C₁₃	—	2	2	no	—
124	C₂×C₄×C₁₃	—	2	4	no	—
125	C₂×C₄×C₁₃	—	4	4	no	—
126	C₂×C₄×C₁₃	—	8	8	no	—
127	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
128	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
129	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
130	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
131	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
132	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
133	C₂×C₄×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
134	C₂×C₄×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
135	C₂×Dic₁₃	—	2	4	no	—
136	C₂×Dic₁₃	—	2	4	no	—
137	C₂×Dic₁₃	—	2	4	no	—
138	C₂×Dic₁₃	—	4	4	no	—
139	C₂×Dic₁₃	—	4	4	no	—
140	C₂×Dic₁₃	—	4	4	no	—
141	C₂×Dic₁₃	—	8	16	no	—
142	C₈×C₁₃	—	2	2	no	—
143	C₈×C₁₃	C₂×C₄⋊C₄	1	4	yes	—
144	#1055	—	4	2	no	—
145	C₂²×D₈	C₂×D₁₃	1	1	yes	characteristic
146	C₂×C₄×D₄	—	13	4	no	—
147	C₂×C₄×D₄	C₂×D₁₃	1	2	yes	—
148	#442	—	26	8	no	—
149	C₄×D₈	Dic₁₃	1	8	yes	complemented
150	C₂×#107	—	13	2	no	—
151	C₂×#107	C₂×D₁₃	1	1	yes	characteristic
152	C₂×#106	—	13	2	no	—
153	C₂×C₄⋊C₈	—	13	1	no	—
154	C₂×#102	—	13	4	no	—
155	C₂×#102	C₂×D₁₃	1	2	yes	—
156	C₂×#97	—	13	2	no	—
157	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
158	#389	—	13	4	no	—
159	#385	—	26	4	no	—
160	#382	—	13	2	no	—
161	#380	—	26	2	no	—
162	C₄×C₄⋊C₄	—	13	1	no	—
163	C₂×#95	—	26	2	no	—
164	#340	—	13	4	no	—
165	#336	—	13	2	no	—
166	C₂²×C₁₃	—	2	8	no	—
167	C₂²×C₁₃	—	4	8	no	—
168	C₂²×C₁₃	—	4	4	no	—
169	C₂²×C₁₃	—	4	4	no	—
170	C₂²×C₁₃	—	4	8	no	—
171	C₂²×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
172	C₂²×C₁₃	D₄²	1	1	yes	characteristic
173	C₂²×C₁₃	D₄×Q₈	1	1	yes	characteristic
174	C₂²×C₁₃	#575	1	2	yes	—
175	C₂²×C₁₃	#576	1	2	yes	—
176	C₄×C₁₃	—	2	2	no	—
177	C₄×C₁₃	—	2	2	no	—
178	C₄×C₁₃	—	4	4	no	—
179	C₄×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
180	C₄×C₁₃	#520	1	1	yes	characteristic
181	C₄×C₁₃	#547	1	1	yes	characteristic
182	C₄×C₁₃	#554	1	1	yes	characteristic
183	Dic₁₃	—	4	4	no	—
184	Dic₁₃	—	8	4	no	—
185	C₂²×D₄	C₂²×D₁₃	1	2	yes	—
186	C₂³×C₄	—	2	2	no	—
187	C₂³×C₄	—	26	4	no	—
188	C₂×D₈	Dic₂₆	1	4	yes	—
189	C₂×D₈	D₅₂	1	4	yes	—
190	C₂×D₈	C₂×Dic₁₃	1	4	yes	—
191	C₂²×C₈	—	13	2	no	—
192	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
193	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
194	C₄×D₄	—	2	4	no	—
195	C₄×D₄	—	26	8	no	—
196	C₄×D₄	—	52	4	no	—
197	C₄×D₄	C₂×Dic₁₃	1	8	yes	complemented
198	C₂×C₄⋊C₄	—	13	4	no	—
199	C₂×C₄⋊C₄	—	13	2	no	—
200	C₂×C₄⋊C₄	—	26	4	no	—
201	C₂×C₄⋊C₄	C₂²×D₁₃	1	2	yes	—
202	C₂×C₂²⋊C₄	—	2	2	no	—
203	C₂×C₂²⋊C₄	—	26	4	no	—
204	C₂×C₄²	—	13	2	no	—
205	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
206	#107	—	52	2	no	—
207	#107	C₂×Dic₁₃	1	4	yes	complemented
208	#106	—	26	4	no	—
209	C₄⋊C₈	—	26	2	no	—
210	#102	—	26	8	no	—
211	#102	C₂×Dic₁₃	1	8	yes	complemented
212	#97	—	26	4	no	—
213	C₄×C₈	C₂×Dic₁₃	1	4	yes	complemented
214	#95	—	13	2	no	—
215	#95	—	26	4	no	—
216	#95	—	26	2	no	—
217	#95	—	26	2	no	—
218	C₂×C₁₃	—	4	8	no	—
219	C₂×C₁₃	D₄×C₄⋊C₄	1	1	yes	characteristic
220	C₂×C₁₃	#2943	1	1	yes	characteristic
221	C₂×C₁₃	#2948	1	1	yes	characteristic
222	C₂×C₁₃	#3273	1	2	yes	—
223	C₂×C₁₃	#3389	1	2	yes	—
224	C₂⁴	—	2	2	no	—
225	C₂×D₄	—	2	2	no	—
226	C₂×D₄	—	2	2	no	—
227	C₂×D₄	—	2	2	no	—
228	C₂×D₄	C₂×Dic₂₆	1	4	yes	—
229	C₂×D₄	C₂×D₅₂	1	4	yes	—
230	C₂×D₄	C₂²×Dic₁₃	1	4	yes	—
231	C₂²×C₄	—	2	2	no	—
232	C₂²×C₄	—	2	8	no	—
233	C₂²×C₄	—	8	4	no	—
234	C₂²×C₄	—	13	4	no	—
235	C₂²×C₄	—	26	4	no	—
236	C₂²×C₄	—	52	8	no	—
237	C₂²×C₄	—	52	8	no	—
238	C₂²×C₄	—	104	4	no	—
239	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
240	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
241	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
242	D₈	C₄⋊Dic₁₃	1	16	yes	complemented
243	C₂×C₈	—	2	1	no	—
244	C₂×C₈	—	2	1	no	—
245	C₂×C₈	—	26	2	no	—
246	C₂×C₈	—	26	2	no	—
247	C₂×C₈	—	26	2	no	—
248	C₂×C₈	C₂×Dic₂₆	1	2	yes	—
249	C₂×C₈	C₂×D₅₂	1	2	yes	—
250	C₂×C₈	C₂²×Dic₁₃	1	2	yes	—
251	C₂×C₈	C₂²×Dic₁₃	1	2	yes	—
252	C₄⋊C₄	—	2	2	no	—
253	C₄⋊C₄	—	26	4	no	—
254	C₄⋊C₄	—	26	4	no	—
255	C₄⋊C₄	—	52	2	no	—
256	C₄⋊C₄	—	104	4	no	—
257	C₄⋊C₄	C₂²×Dic₁₃	1	4	yes	—
258	C₂²⋊C₄	—	2	8	no	—
259	C₂²⋊C₄	—	52	8	no	—
260	C₄²	—	26	2	no	—
261	C₄²	—	26	2	no	—
262	C₄²	#6129	1	2	yes	—
263	C₄²	C₂²×Dic₁₃	1	2	yes	—
264	C₁₃	#22672	1	1	yes	characteristic, complemented, Sylow
265	C₂³	—	2	4	no	—
266	C₂³	—	2	4	no	—
267	C₂³	—	2	4	no	—
268	C₂³	—	4	4	no	—
269	C₂³	—	4	4	no	—
270	C₂³	C₂×D₄×D₁₃	1	1	yes	characteristic
271	D₄	—	2	4	no	—
272	D₄	—	2	4	no	—
273	D₄	—	2	4	no	—
274	D₄	C₂×C₄⋊Dic₁₃	1	8	yes	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	—	2	4	no	—
278	C₂×C₄	—	4	4	no	—
279	C₂×C₄	—	8	8	no	—
280	C₂×C₄	—	26	4	no	—
281	C₂×C₄	—	26	4	no	—
282	C₂×C₄	—	26	4	no	—
283	C₂×C₄	—	52	4	no	—
284	C₂×C₄	—	52	4	no	—
285	C₂×C₄	—	52	4	no	—
286	C₂×C₄	—	104	16	no	—
287	C₂×C₄	D₄×Dic₁₃	1	2	yes	—
288	C₂×C₄	D₄×Dic₁₃	1	2	yes	—
289	C₂×C₄	C₂²×Dic₂₆	1	1	yes	characteristic
290	C₂×C₄	C₂²×D₅₂	1	1	yes	characteristic
291	C₂×C₄	C₂×#6129	1	1	yes	characteristic
292	C₂×C₄	#88256	1	1	yes	characteristic
293	C₂×C₄	#88257	1	1	yes	characteristic
294	C₂×C₄	C₂³×Dic₁₃	1	1	yes	characteristic
295	C₈	—	2	2	no	—
296	C₈	—	52	2	no	—
297	C₈	C₂×C₄⋊Dic₁₃	1	4	yes	—
298	C₂²	—	2	8	no	—
299	C₂²	—	4	8	no	—
300	C₂²	—	4	4	no	—
301	C₂²	—	4	4	no	—
302	C₂²	—	4	8	no	—
303	C₂²	D₄×Dic₂₆	1	1	yes	characteristic
304	C₂²	D₄×D₅₂	1	1	yes	characteristic
305	C₂²	#11719462	1	2	yes	—
306	C₂²	#11719473	1	2	yes	—
307	C₂²	C₂×D₄×Dic₁₃	1	1	yes	characteristic
308	C₄	—	2	2	no	—
309	C₄	—	2	2	no	—
310	C₄	—	4	4	no	—
311	C₄	—	52	4	no	—
312	C₄	—	104	4	no	—
313	C₄	#11719248	1	1	yes	characteristic
314	C₄	#11719260	1	1	yes	characteristic
315	C₄	C₂²×C₄⋊Dic₁₃	1	1	yes	characteristic
316	C₄	#11719532	1	1	yes	characteristic
317	C₂	—	4	8	no	—
318	C₂	D₄×C₄⋊Dic₁₃	1	1	yes	characteristic
319	C₂	#13123637	1	1	yes	characteristic
320	C₂	#13123688	1	1	yes	characteristic
321	C₂	#13126446	1	2	yes	—
322	C₂	#13127792	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	#22672
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. #13115575	C₂	semidirect
4. #13115085	C₂	semidirect
5. #13115000	C₂	semidirect
6. #13114846	C₂	semidirect
7. #13114703	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
82. C₁₃×D₈	C₄⋊C₄	semidirect
149. C₄×D₈	Dic₁₃	semidirect
197. C₄×D₄	C₂×Dic₁₃	semidirect
207. #107	C₂×Dic₁₃	semidirect
211. #102	C₂×Dic₁₃	semidirect
213. C₄×C₈	C₂×Dic₁₃	semidirect
242. D₈	C₄⋊Dic₁₃	semidirect
264. C₁₃	#22672	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718813	2
#13117746	2
#13115575	2
#13115085	2
#13115000	2
#13114846	2
#13114703	2
C₂×C₄×C₁₃×D₈	2
#22672	13