Subgroups — #16784694

Order 3328.

570 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13130769	C₂	1	1	yes	maximal, characteristic
2	C₄×#88112	C₂	1	1	yes	maximal, characteristic
3	C₂×#11718245	C₂	1	4	yes	maximal
4	#13123233	C₂	1	1	yes	maximal, characteristic
5	C₂×#88142	—	2	1	no	—
6	C₂×#88142	—	2	1	no	—
7	C₂×#88142	C₂²	1	2	yes	—
8	C₂×#88112	—	2	1	no	—
9	C₂×#88112	C₂²	1	2	yes	—
10	C₂×#88112	C₂²	1	2	yes	—
11	C₂×C₄⋊#1057	—	2	1	no	—
12	C₂×C₄×#1057	—	2	1	no	—
13	C₁₃×#384	C₄	1	1	yes	characteristic
14	#11718637	—	2	1	no	—
15	#11718637	—	2	1	no	—
16	#11718637	C₄	1	2	yes	—
17	#11718596	C₄	1	1	yes	characteristic
18	C₄²×Dic₁₃	C₂²	1	1	yes	characteristic
19	#11718290	—	2	1	no	—
20	#11718245	—	2	8	no	—
21	C₂²×Dic₂₆	—	2	2	no	—
22	C₂²×#1057	—	2	2	no	—
23	C₂²×#1057	—	2	2	no	—
24	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
25	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
26	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
27	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄⋊Dic₁₃	—	2	1	no	—
29	C₂×C₄⋊Dic₁₃	—	2	1	no	—
30	C₂×C₄⋊Dic₁₃	C₂×C₄	1	2	yes	—
31	C₂×#6101	—	2	2	no	—
32	C₂×#6101	—	2	2	no	—
33	C₂×#6101	—	2	2	no	—
34	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄×Dic₁₃	D₄	1	1	yes	characteristic
36	C₂×C₄×Dic₁₃	D₄	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	1	yes	characteristic
40	#88142	—	2	4	no	—
41	#88142	—	2	4	no	—
42	#88142	—	4	4	no	—
43	#88142	—	4	2	no	—
44	#88142	—	4	2	no	—
45	#88112	—	2	4	no	—
46	#88112	—	2	2	no	—
47	#88112	—	4	2	no	—
48	#88112	D₄	1	4	yes	—
49	C₄⋊#1057	—	4	2	no	—
50	C₄×#1057	—	4	2	no	—
51	#9694	—	13	1	no	maximal, Sylow
52	C₂²×C₄×C₁₃	—	2	2	no	—
53	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
54	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
55	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
56	C₂²×Dic₁₃	—	2	2	no	—
57	C₂²×Dic₁₃	C₂²⋊C₄	1	1	yes	characteristic
58	C₂²×Dic₁₃	C₂²⋊C₄	1	1	yes	characteristic
59	C₂²×Dic₁₃	C₂×D₄	1	1	yes	characteristic
60	C₂²×Dic₁₃	C₂×D₄	1	1	yes	characteristic
61	C₂×Dic₂₆	—	4	4	no	—
62	C₂×Dic₂₆	—	4	2	no	—
63	C₂×Dic₂₆	—	4	2	no	—
64	C₂×Dic₂₆	—	4	4	no	—
65	C₂×#1057	—	2	4	no	—
66	C₂×#1057	—	4	2	no	—
67	C₂×#1057	—	4	2	no	—
68	C₂×#1057	—	4	2	no	—
69	C₂×#1057	—	4	2	no	—
70	C₂×#1057	—	4	2	no	—
71	C₁₃×C₄⋊C₄	—	2	4	no	—
72	C₁₃×C₄⋊C₄	—	4	2	no	—
73	C₁₃×C₄⋊C₄	—	4	2	no	—
74	C₄²×C₁₃	—	2	2	no	—
75	C₄⋊Dic₁₃	—	2	4	no	—
76	C₄⋊Dic₁₃	—	4	2	no	—
77	C₄⋊Dic₁₃	—	4	2	no	—
78	#6101	—	4	4	no	—
79	#6101	—	4	4	no	—
80	#6101	—	4	2	no	—
81	#6101	—	4	2	no	—
82	C₄×Dic₁₃	—	2	2	no	—
83	C₄×Dic₁₃	—	2	2	no	—
84	C₄×Dic₁₃	D₈	1	2	yes	—
85	C₄×Dic₁₃	SD₁₆	1	2	yes	—
86	C₄×Dic₁₃	SD₁₆	1	2	yes	—
87	C₄×Dic₁₃	SD₁₆	1	2	yes	—
88	C₄×Dic₁₃	Q₁₆	1	2	yes	—
89	C₄×Dic₁₃	Q₁₆	1	2	yes	—
90	C₄×Dic₁₃	C₂×D₄	1	2	yes	—
91	C₄×Dic₁₃	#41	1	2	yes	—
92	#2868	—	13	1	no	—
93	#1842	—	13	1	no	—
94	C₄×C₄⋊C₈	—	13	1	no	—
95	C₂×#332	—	13	4	no	—
96	C₂³×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
97	C₂×C₄×C₁₃	—	2	1	no	—
98	C₂×C₄×C₁₃	—	2	1	no	—
99	C₂×C₄×C₁₃	—	2	1	no	—
100	C₂×C₄×C₁₃	—	2	1	no	—
101	C₂×C₄×C₁₃	—	4	2	no	—
102	C₂×C₄×C₁₃	—	4	2	no	—
103	C₂×C₄×C₁₃	—	4	2	no	—
104	C₂×C₄×C₁₃	#101	1	2	yes	—
105	C₂×C₄×C₁₃	#102	1	2	yes	—
106	C₂×C₄×C₁₃	#103	1	2	yes	—
107	C₂×C₄×C₁₃	#103	1	2	yes	—
108	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
109	C₂×Dic₁₃	—	2	1	no	—
110	C₂×Dic₁₃	—	2	1	no	—
111	C₂×Dic₁₃	—	4	2	no	—
112	C₂×Dic₁₃	—	4	2	no	—
113	C₂×Dic₁₃	—	4	2	no	—
114	C₂×Dic₁₃	#101	1	2	yes	—
115	C₂×Dic₁₃	#102	1	2	yes	—
116	C₂×Dic₁₃	#103	1	2	yes	—
117	C₂×Dic₁₃	#103	1	2	yes	—
118	C₂×Dic₁₃	#121	1	2	yes	—
119	C₂×Dic₁₃	#124	1	1	yes	characteristic
120	C₂×Dic₁₃	#127	1	1	yes	characteristic
121	C₂×Dic₁₃	C₂×D₈	1	1	yes	characteristic
122	C₂×Dic₁₃	C₂×SD₁₆	1	1	yes	characteristic
123	C₂×Dic₁₃	C₂×SD₁₆	1	1	yes	characteristic
124	C₂×Dic₁₃	C₂×SD₁₆	1	1	yes	characteristic
125	C₂×Dic₁₃	C₂×Q₁₆	1	1	yes	characteristic
126	C₂×Dic₁₃	C₂×Q₁₆	1	1	yes	characteristic
127	C₂×Dic₁₃	AGL(1,ℤ/8)	1	2	yes	—
128	Dic₂₆	—	8	4	no	—
129	Dic₂₆	—	8	4	no	—
130	#1057	—	4	4	no	—
131	#1057	—	8	2	no	—
132	C₂×C₄⋊Q₈	—	13	2	no	—
133	C₂×C₄⋊Q₈	—	26	1	no	—
134	C₂×C₄⋊Q₈	—	26	1	no	—
135	C₂×C₄⋊C₈	—	13	2	no	—
136	C₂×C₄⋊C₈	—	13	2	no	—
137	C₂×C₄⋊C₈	—	26	1	no	—
138	C₂×C₄⋊C₈	—	26	1	no	—
139	C₂×C₄×C₈	—	26	1	no	—
140	#384	—	13	2	no	—
141	#384	—	13	1	no	—
142	#384	—	26	1	no	—
143	#384	—	26	1	no	—
144	#384	#261	1	1	yes	characteristic
145	C₄³	—	13	1	no	—
146	#336	—	26	1	no	—
147	#332	—	26	8	no	—
148	C₂²×C₁₃	#390	1	1	yes	characteristic
149	C₂²×C₁₃	C₂×#101	1	1	yes	characteristic
150	C₂²×C₁₃	C₂×#101	1	1	yes	characteristic
151	C₂²×C₁₃	C₂×#102	1	1	yes	characteristic
152	C₂²×C₁₃	C₂×#103	1	1	yes	characteristic
153	C₂²×C₁₃	C₂×#103	1	1	yes	characteristic
154	C₂²×C₁₃	#418	1	1	yes	characteristic
155	C₄×C₁₃	—	2	2	no	—
156	C₄×C₁₃	—	2	2	no	—
157	C₄×C₁₃	—	8	2	no	—
158	C₄×C₁₃	#332	1	4	yes	complemented
159	Dic₁₃	—	2	2	no	—
160	Dic₁₃	—	8	2	no	—
161	Dic₁₃	#332	1	4	yes	complemented
162	Dic₁₃	#459	1	2	yes	—
163	Dic₁₃	#460	1	2	yes	—
164	Dic₁₃	#487	1	1	yes	characteristic
165	Dic₁₃	#487	1	1	yes	characteristic
166	Dic₁₃	#492	1	1	yes	characteristic
167	Dic₁₃	#494	1	1	yes	characteristic
168	C₂²×Q₈	—	26	2	no	—
169	C₂²×C₈	—	26	2	no	—
170	C₂²×C₈	—	26	2	no	—
171	C₄⋊Q₈	—	26	4	no	—
172	C₄⋊Q₈	—	26	4	no	—
173	C₄⋊Q₈	—	52	2	no	—
174	C₄⋊Q₈	—	52	2	no	—
175	C₄⋊Q₈	—	52	4	no	—
176	C₂×C₄⋊C₄	—	2	1	no	—
177	C₂×C₄⋊C₄	—	2	1	no	—
178	C₂×C₄⋊C₄	—	13	2	no	—
179	C₂×C₄⋊C₄	—	26	2	no	—
180	C₂×C₄⋊C₄	—	26	2	no	—
181	C₂×C₄⋊C₄	—	26	2	no	—
182	C₂×C₄⋊C₄	—	26	1	no	—
183	C₂×C₄⋊C₄	—	26	1	no	—
184	C₂×C₄⋊C₄	C₂×#261	1	2	yes	—
185	C₂×C₄²	—	13	1	no	—
186	C₂×C₄²	—	13	1	no	—
187	C₂×C₄²	—	13	1	no	—
188	C₂×C₄²	—	13	1	no	—
189	C₂×C₄²	—	13	1	no	—
190	C₂×C₄²	—	13	1	no	—
191	C₂×C₄²	C₂×#261	1	1	yes	characteristic
192	C₄⋊C₈	—	13	4	no	—
193	C₄⋊C₈	—	26	2	no	—
194	C₄⋊C₈	—	26	4	no	—
195	C₄⋊C₈	—	52	2	no	—
196	C₄⋊C₈	—	52	2	no	—
197	C₄×C₈	—	52	2	no	—
198	C₂×C₁₃	C₂×#332	1	1	yes	characteristic
199	C₂×C₁₃	C₂×#332	1	1	yes	characteristic
200	C₂×C₁₃	#1549	1	1	yes	characteristic
201	C₂×C₁₃	#1549	1	1	yes	characteristic
202	C₂×C₁₃	#1956	1	1	yes	characteristic
203	C₂×C₁₃	#1966	1	1	yes	characteristic
204	C₂×C₁₃	#1982	1	1	yes	characteristic
205	C₂×Q₈	—	52	4	no	—
206	C₂×Q₈	—	52	2	no	—
207	C₂×Q₈	—	52	2	no	—
208	C₂×Q₈	—	52	4	no	—
209	C₂²×C₄	—	2	2	no	—
210	C₂²×C₄	—	13	1	no	—
211	C₂²×C₄	—	13	1	no	—
212	C₂²×C₄	—	13	1	no	—
213	C₂²×C₄	—	13	1	no	—
214	C₂²×C₄	—	26	2	no	—
215	C₂²×C₄	C₂²⋊#261	1	1	yes	characteristic
216	C₂²×C₄	C₂²⋊#261	1	1	yes	characteristic
217	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
218	C₂×C₈	—	26	4	no	—
219	C₂×C₈	—	52	2	no	—
220	C₂×C₈	—	52	2	no	—
221	C₂×C₈	—	52	2	no	—
222	C₂×C₈	—	52	2	no	—
223	C₂×C₈	—	52	2	no	—
224	C₄⋊C₄	—	2	4	no	—
225	C₄⋊C₄	—	4	2	no	—
226	C₄⋊C₄	—	4	2	no	—
227	C₄⋊C₄	—	26	4	no	—
228	C₄⋊C₄	—	52	2	no	—
229	C₄⋊C₄	—	52	2	no	—
230	C₄⋊C₄	—	52	2	no	—
231	C₄⋊C₄	—	52	4	no	—
232	C₄⋊C₄	—	52	4	no	—
233	C₄⋊C₄	—	52	2	no	—
234	C₄²	—	2	2	no	—
235	C₄²	—	13	2	no	—
236	C₄²	—	13	2	no	—
237	C₄²	—	13	2	no	—
238	C₄²	—	13	2	no	—
239	C₄²	—	13	2	no	—
240	C₄²	—	13	2	no	—
241	C₄²	—	13	2	no	—
242	C₄²	—	13	2	no	—
243	C₄²	—	26	2	no	—
244	C₄²	—	26	2	no	—
245	C₁₃	#9694	1	1	yes	characteristic, complemented, Sylow
246	C₂³	C₂×C₂²⋊#261	1	1	yes	characteristic
247	Q₈	—	104	4	no	—
248	Q₈	—	104	4	no	—
249	C₂×C₄	—	2	1	no	—
250	C₂×C₄	—	2	1	no	—
251	C₂×C₄	—	2	1	no	—
252	C₂×C₄	—	2	1	no	—
253	C₂×C₄	—	4	2	no	—
254	C₂×C₄	—	4	2	no	—
255	C₂×C₄	—	4	2	no	—
256	C₂×C₄	—	13	2	no	—
257	C₂×C₄	—	13	1	no	—
258	C₂×C₄	—	13	1	no	—
259	C₂×C₄	—	13	2	no	—
260	C₂×C₄	—	13	1	no	—
261	C₂×C₄	—	13	1	no	—
262	C₂×C₄	—	13	2	no	—
263	C₂×C₄	—	13	1	no	—
264	C₂×C₄	—	13	1	no	—
265	C₂×C₄	—	13	2	no	—
266	C₂×C₄	—	13	1	no	—
267	C₂×C₄	—	13	1	no	—
268	C₂×C₄	—	13	2	no	—
269	C₂×C₄	—	13	2	no	—
270	C₂×C₄	—	26	1	no	—
271	C₂×C₄	—	26	1	no	—
272	C₂×C₄	—	52	2	no	—
273	C₂×C₄	—	52	2	no	—
274	C₂×C₄	—	52	2	no	—
275	C₂×C₄	#88113	1	2	yes	—
276	C₂×C₄	#88115	1	2	yes	—
277	C₂×C₄	#88117	1	2	yes	—
278	C₂×C₄	#88117	1	2	yes	—
279	C₂×C₄	D₄×#261	1	2	yes	—
280	C₈	—	52	4	no	—
281	C₈	—	104	2	no	—
282	C₂²	C₂×#88113	1	1	yes	characteristic
283	C₂²	C₂×#88115	1	1	yes	characteristic
284	C₂²	#11719138	1	1	yes	characteristic
285	C₂²	#11719151	1	1	yes	characteristic
286	C₂²	C₂×#88117	1	1	yes	characteristic
287	C₂²	C₂×#88117	1	1	yes	characteristic
288	C₂²	D₁₃⋊#101	1	1	yes	characteristic
289	C₄	—	2	2	no	—
290	C₄	—	2	2	no	—
291	C₄	—	8	2	no	—
292	C₄	—	13	4	no	—
293	C₄	—	13	1	no	—
294	C₄	—	13	1	no	—
295	C₄	—	13	1	no	—
296	C₄	—	13	1	no	—
297	C₄	—	13	2	no	—
298	C₄	—	13	2	no	—
299	C₄	—	26	2	no	—
300	C₄	—	104	2	no	—
301	C₄	#11718245	1	4	yes	complemented
302	C₂	C₂×#11718245	1	1	yes	characteristic
303	C₂	D₁₃⋊#332	1	1	yes	characteristic
304	C₂	#13130351	1	1	yes	characteristic
305	C₂	#13130352	1	1	yes	characteristic
306	C₂	#13130963	1	1	yes	characteristic
307	C₂	#13130979	1	1	yes	characteristic
308	C₂	#13131003	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊#261
commutator	C₂²×C₄×C₁₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×#261
Fitting	C₁₃×#384	C₄
socle, prime radical	C₂³×C₁₃	C₂×C₂²⋊C₄
cosocle	C₂×C₄×Dic₁₃	C₂³
hypercenter	#384	#261
hypocenter	C₁₃	#9694
LCS	G ▷ C₂²×C₄×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

Complements

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

N	H ≅ G/N	type
158. C₄×C₁₃	#332	semidirect
161. Dic₁₃	#332	semidirect
245. C₁₃	#9694	semidirect
301. C₄	#11718245	semidirect

Maximal subgroups

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

subgroup	index
#13130769	2
C₄×#88112	2
C₂×#11718245	2
#13123233	2
#9694	13