Subgroups — #16785029

Order 3328.

528 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	C₄×#88112	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊#1057	C₂	1	1	yes	maximal, characteristic
3	#13129672	C₂	1	2	yes	maximal
4	#13129672	C₂	1	2	yes	maximal
5	#13117407	C₂	1	1	yes	maximal, characteristic
6	C₂×#88142	C₄	1	1	yes	characteristic
7	C₂×#88141	—	2	1	no	—
8	C₂×#88112	—	2	1	no	—
9	C₂×C₄⋊#1057	C₂²	1	2	yes	—
10	C₂×C₄×#1057	—	2	1	no	—
11	C₂×C₄×#1057	C₂²	1	2	yes	—
12	C₁₃×#389	C₄	1	1	yes	characteristic
13	#11718637	C₄	1	1	yes	characteristic
14	C₄⋊C₄×Dic₁₃	C₂²	1	1	yes	characteristic
15	C₄⋊C₄×Dic₁₃	C₂²	1	1	yes	characteristic
16	#11718633	—	2	1	no	—
17	C₄²×Dic₁₃	C₂²	1	1	yes	characteristic
18	#11718589	C₄	1	1	yes	characteristic
19	#11718310	—	2	1	no	—
20	#11718307	—	2	1	no	—
21	#11718290	—	2	1	no	—
22	#11718290	—	2	1	no	—
23	C₂²×Dic₂₆	—	2	1	no	—
24	C₂²×#1057	—	2	2	no	—
25	C₂²×#1057	—	2	2	no	—
26	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
27	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
28	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄⋊Dic₁₃	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄⋊Dic₁₃	C₂×C₄	1	1	yes	characteristic
32	C₂×#6101	—	2	1	no	—
33	C₂×#6101	—	2	1	no	—
34	C₂×#6101	—	2	1	no	—
35	C₂×#6101	—	2	1	no	—
36	C₂×C₄×Dic₁₃	—	2	1	no	—
37	C₂×C₄×Dic₁₃	—	2	1	no	—
38	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄×Dic₁₃	D₄	1	1	yes	characteristic
40	C₂×C₄×Dic₁₃	D₄	1	1	yes	characteristic
41	C₂×C₄×Dic₁₃	D₄	1	1	yes	characteristic
42	C₂×C₄×Dic₁₃	D₄	1	1	yes	characteristic
43	C₂×C₄×Dic₁₃	C₂³	1	1	yes	characteristic
44	#88142	—	2	2	no	—
45	#88142	C₈	1	4	yes	complemented
46	#88141	—	4	4	no	—
47	#88112	—	2	4	no	—
48	C₄⋊#1057	—	2	2	no	—
49	C₄⋊#1057	Q₈	1	4	yes	complemented
50	C₄×#1057	—	2	4	no	—
51	C₄×#1057	—	4	2	no	—
52	C₁₃×#95	—	2	1	no	—
53	#88071	—	2	1	no	—
54	#88071	—	2	1	no	—
55	#88071	—	2	1	no	—
56	#9864	—	13	1	no	maximal, Sylow
57	C₂²×C₄×C₁₃	—	2	1	no	—
58	C₂²×C₄×C₁₃	—	2	1	no	—
59	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
61	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
62	C₂²×Dic₁₃	—	2	1	no	—
63	C₂²×Dic₁₃	—	2	1	no	—
64	C₂²×Dic₁₃	C₂²⋊C₄	1	1	yes	characteristic
65	C₂²×Dic₁₃	C₂²⋊C₄	1	1	yes	characteristic
66	C₂²×Dic₁₃	C₂×D₄	1	1	yes	characteristic
67	C₂²×Dic₁₃	C₂×D₄	1	1	yes	characteristic
68	C₂×Dic₂₆	—	2	4	no	—
69	C₂×Dic₂₆	—	4	2	no	—
70	C₂×Dic₂₆	—	4	2	no	—
71	C₂×#1057	—	2	4	no	—
72	C₂×#1057	—	2	4	no	—
73	C₂×#1057	—	4	2	no	—
74	C₂×#1057	—	4	2	no	—
75	C₂×#1057	—	4	2	no	—
76	C₂×#1057	—	4	2	no	—
77	C₁₃×C₄⋊C₄	—	2	1	no	—
78	C₁₃×C₄⋊C₄	—	2	2	no	—
79	C₁₃×C₄⋊C₄	—	4	2	no	—
80	C₁₃×C₄⋊C₄	C₂×C₈	1	2	yes	—
81	C₄²×C₁₃	—	2	2	no	—
82	C₄⋊Dic₁₃	—	2	1	no	—
83	C₄⋊Dic₁₃	—	2	2	no	—
84	C₄⋊Dic₁₃	C₂×C₈	1	2	yes	—
85	#6101	—	2	4	no	—
86	#6101	—	4	2	no	—
87	#6101	—	4	2	no	—
88	#6101	—	4	2	no	—
89	C₄×Dic₁₃	—	2	2	no	—
90	C₄×Dic₁₃	—	2	2	no	—
91	C₄×Dic₁₃	—	4	1	no	—
92	C₄×Dic₁₃	—	4	2	no	—
93	C₄×Dic₁₃	—	4	2	no	—
94	C₄×Dic₁₃	C₂×C₈	1	2	yes	—
95	C₄×Dic₁₃	M₁₆	1	2	yes	—
96	C₄×Dic₁₃	SD₁₆	1	2	yes	—
97	C₄×Dic₁₃	Q₁₆	1	2	yes	—
98	C₄×Dic₁₃	C₂×Q₈	1	2	yes	—
99	C₄×Dic₁₃	#41	1	2	yes	—
100	C₄×C₄⋊Q₈	—	13	1	no	—
101	C₄×C₄⋊C₈	—	13	1	no	—
102	C₄×C₄⋊C₈	—	13	1	no	—
103	#1279	—	13	2	no	—
104	#1279	—	13	2	no	—
105	C₂³×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
106	C₂×C₄×C₁₃	—	2	1	no	—
107	C₂×C₄×C₁₃	—	2	1	no	—
108	C₂×C₄×C₁₃	—	2	2	no	—
109	C₂×C₄×C₁₃	—	2	1	no	—
110	C₂×C₄×C₁₃	—	2	1	no	—
111	C₂×C₄×C₁₃	—	2	1	no	—
112	C₂×C₄×C₁₃	—	4	1	no	—
113	C₂×C₄×C₁₃	—	4	1	no	—
114	C₂×C₄×C₁₃	—	4	1	no	—
115	C₂×C₄×C₁₃	—	4	1	no	—
116	C₂×C₄×C₁₃	—	4	1	no	—
117	C₂×C₄×C₁₃	#103	1	2	yes	—
118	C₂×C₄×C₁₃	C₄×Q₈	1	2	yes	—
119	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
120	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
121	C₂×C₄×C₁₃	#131	1	2	yes	—
122	C₂×Dic₁₃	—	2	1	no	—
123	C₂×Dic₁₃	—	2	1	no	—
124	C₂×Dic₁₃	—	2	1	no	—
125	C₂×Dic₁₃	—	2	1	no	—
126	C₂×Dic₁₃	—	2	2	no	—
127	C₂×Dic₁₃	—	4	1	no	—
128	C₂×Dic₁₃	—	4	1	no	—
129	C₂×Dic₁₃	—	4	1	no	—
130	C₂×Dic₁₃	—	4	1	no	—
131	C₂×Dic₁₃	—	4	1	no	—
132	C₂×Dic₁₃	C₂²⋊C₈	1	2	yes	—
133	C₂×Dic₁₃	C₂²⋊C₈	1	2	yes	—
134	C₂×Dic₁₃	#103	1	2	yes	—
135	C₂×Dic₁₃	#104	1	2	yes	—
136	C₂×Dic₁₃	C₂²⋊Q₈	1	2	yes	—
137	C₂×Dic₁₃	#124	1	1	yes	characteristic
138	C₂×Dic₁₃	C₄⋊Q₈	1	1	yes	characteristic
139	C₂×Dic₁₃	C₂×SD₁₆	1	1	yes	characteristic
140	C₂×Dic₁₃	C₂×Q₁₆	1	1	yes	characteristic
141	C₂×Dic₁₃	AGL(1,ℤ/8)	1	2	yes	—
142	Dic₂₆	—	4	8	no	—
143	#1057	—	4	4	no	—
144	#1057	—	4	4	no	—
145	C₂×C₄⋊Q₈	—	13	1	no	—
146	C₂×C₄×Q₈	—	26	1	no	—
147	C₂×C₄⋊C₈	—	13	2	no	—
148	C₂×C₄⋊C₈	—	26	1	no	—
149	C₂×C₄×C₈	—	13	2	no	—
150	C₂×C₄×C₈	—	26	1	no	—
151	#391	—	26	1	no	—
152	#389	—	13	1	no	—
153	#389	—	26	1	no	—
154	#389	#261	1	1	yes	characteristic
155	#384	—	13	1	no	—
156	C₄×C₄⋊C₄	—	13	1	no	—
157	C₄×C₄⋊C₄	—	13	1	no	—
158	C₄×C₄⋊C₄	—	26	1	no	—
159	C₄³	—	13	1	no	—
160	#336	—	26	1	no	—
161	#336	—	26	1	no	—
162	C₂²×C₁₃	#391	1	1	yes	characteristic
163	C₂²×C₁₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
164	C₂²×C₁₃	#408	1	1	yes	characteristic
165	C₂²×C₁₃	C₂×#103	1	1	yes	characteristic
166	C₂²×C₁₃	#418	1	1	yes	characteristic
167	C₂²×C₁₃	C₂×#104	1	1	yes	characteristic
168	C₂²×C₁₃	#421	1	1	yes	characteristic
169	C₄×C₁₃	—	2	2	no	—
170	C₄×C₁₃	—	4	2	no	—
171	C₄×C₁₃	—	4	1	no	—
172	C₄×C₁₃	—	8	1	no	—
173	C₄×C₁₃	C₈×Q₈	1	2	yes	—
174	C₄×C₁₃	#446	1	2	yes	—
175	Dic₁₃	—	2	1	no	—
176	Dic₁₃	—	2	1	no	—
177	Dic₁₃	—	2	2	no	—
178	Dic₁₃	—	4	2	no	—
179	Dic₁₃	—	8	1	no	—
180	Dic₁₃	Q₈⋊C₈	1	4	yes	—
181	Dic₁₃	#476	1	2	yes	—
182	Dic₁₃	#477	1	2	yes	—
183	C₂²×Q₈	—	26	1	no	—
184	C₂²×C₈	—	26	2	no	—
185	C₂²×C₈	—	26	2	no	—
186	C₄⋊Q₈	—	13	4	no	—
187	C₄⋊Q₈	—	26	2	no	—
188	C₄×Q₈	—	52	4	no	—
189	C₂×C₄⋊C₄	—	2	1	no	—
190	C₂×C₄⋊C₄	—	13	1	no	—
191	C₂×C₄⋊C₄	—	13	1	no	—
192	C₂×C₄⋊C₄	—	26	1	no	—
193	C₂×C₄⋊C₄	—	26	1	no	—
194	C₂×C₄⋊C₄	—	26	1	no	—
195	C₂×C₄⋊C₄	—	26	1	no	—
196	C₂×C₄⋊C₄	C₂×#261	1	1	yes	characteristic
197	C₂×C₄⋊C₄	C₂×#261	1	1	yes	characteristic
198	C₂×C₄²	—	13	1	no	—
199	C₂×C₄²	—	13	1	no	—
200	C₂×C₄²	—	13	1	no	—
201	C₂×C₄²	—	13	1	no	—
202	C₂×C₄²	—	13	1	no	—
203	C₂×C₄²	—	13	1	no	—
204	C₂×C₄²	—	26	1	no	—
205	C₂×C₄²	—	26	1	no	—
206	C₂×C₄²	C₂×#261	1	1	yes	characteristic
207	C₄⋊C₈	—	13	4	no	—
208	C₄⋊C₈	—	26	2	no	—
209	C₄⋊C₈	—	26	4	no	—
210	C₄×C₈	—	26	4	no	—
211	C₄×C₈	—	52	2	no	—
212	#95	—	2	1	no	—
213	#95	—	26	1	no	—
214	#95	—	26	1	no	—
215	#95	—	26	1	no	—
216	C₂×C₁₃	C₂×Q₈⋊C₈	1	1	yes	characteristic
217	C₂×C₁₃	#1478	1	1	yes	characteristic
218	C₂×C₁₃	#1481	1	1	yes	characteristic
219	C₂×C₁₃	#1484	1	1	yes	characteristic
220	C₂×C₁₃	#1984	1	1	yes	characteristic
221	C₂×C₁₃	#1986	1	1	yes	characteristic
222	C₂×C₁₃	#1987	1	1	yes	characteristic
223	C₂×Q₈	—	26	4	no	—
224	C₂×Q₈	—	52	2	no	—
225	C₂×Q₈	—	52	2	no	—
226	C₂²×C₄	—	2	1	no	—
227	C₂²×C₄	—	2	1	no	—
228	C₂²×C₄	—	13	1	no	—
229	C₂²×C₄	—	13	1	no	—
230	C₂²×C₄	—	13	1	no	—
231	C₂²×C₄	—	13	1	no	—
232	C₂²×C₄	—	26	1	no	—
233	C₂²×C₄	—	26	1	no	—
234	C₂²×C₄	C₂²⋊#261	1	1	yes	characteristic
235	C₂²×C₄	C₂²⋊#261	1	1	yes	characteristic
236	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
237	C₂×C₈	—	26	4	no	—
238	C₂×C₈	—	26	4	no	—
239	C₂×C₈	—	52	2	no	—
240	C₂×C₈	—	52	2	no	—
241	C₂×C₈	—	52	2	no	—
242	C₂×C₈	—	52	2	no	—
243	C₄⋊C₄	—	2	1	no	—
244	C₄⋊C₄	—	2	2	no	—
245	C₄⋊C₄	—	4	2	no	—
246	C₄⋊C₄	—	13	2	no	—
247	C₄⋊C₄	—	26	4	no	—
248	C₄⋊C₄	—	26	1	no	—
249	C₄⋊C₄	—	26	2	no	—
250	C₄⋊C₄	—	52	2	no	—
251	C₄⋊C₄	—	52	2	no	—
252	C₄⋊C₄	—	52	2	no	—
253	C₄⋊C₄	D₁₃⋊C₈	1	2	yes	—
254	C₄²	—	2	2	no	—
255	C₄²	—	13	2	no	—
256	C₄²	—	13	2	no	—
257	C₄²	—	13	2	no	—
258	C₄²	—	13	2	no	—
259	C₄²	—	13	2	no	—
260	C₄²	—	13	2	no	—
261	C₄²	—	26	2	no	—
262	C₄²	—	26	2	no	—
263	C₄²	—	52	1	no	—
264	C₄²	—	52	2	no	—
265	C₄²	—	52	2	no	—
266	C₁₃	#9864	1	1	yes	characteristic, complemented, Sylow
267	C₂³	C₂×C₂²⋊#261	1	1	yes	characteristic
268	Q₈	—	52	8	no	—
269	C₂×C₄	—	2	1	no	—
270	C₂×C₄	—	2	1	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	4	1	no	—
276	C₂×C₄	—	4	1	no	—
277	C₂×C₄	—	4	1	no	—
278	C₂×C₄	—	4	1	no	—
279	C₂×C₄	—	4	1	no	—
280	C₂×C₄	—	13	2	no	—
281	C₂×C₄	—	13	2	no	—
282	C₂×C₄	—	13	1	no	—
283	C₂×C₄	—	13	2	no	—
284	C₂×C₄	—	13	1	no	—
285	C₂×C₄	—	13	2	no	—
286	C₂×C₄	—	13	2	no	—
287	C₂×C₄	—	13	2	no	—
288	C₂×C₄	—	13	1	no	—
289	C₂×C₄	—	13	1	no	—
290	C₂×C₄	—	26	1	no	—
291	C₂×C₄	—	26	1	no	—
292	C₂×C₄	—	26	1	no	—
293	C₂×C₄	—	26	1	no	—
294	C₂×C₄	—	26	2	no	—
295	C₂×C₄	—	52	1	no	—
296	C₂×C₄	—	52	1	no	—
297	C₂×C₄	—	52	1	no	—
298	C₂×C₄	—	52	1	no	—
299	C₂×C₄	—	52	1	no	—
300	C₂×C₄	#88117	1	2	yes	—
301	C₂×C₄	C₂×D₁₃⋊C₈	1	1	yes	characteristic
302	C₂×C₄	D₁₃⋊M₁₆	1	1	yes	characteristic
303	C₂×C₄	#88238	1	2	yes	—
304	C₂×C₄	Q₈×#261	1	2	yes	—
305	C₈	—	52	4	no	—
306	C₈	—	52	4	no	—
307	C₂²	D₁₃⋊(C₂²⋊C₈)	1	1	yes	characteristic
308	C₂²	#11719138	1	1	yes	characteristic
309	C₂²	#11719146	1	1	yes	characteristic
310	C₂²	C₂×#88117	1	1	yes	characteristic
311	C₂²	(C₂×#261)⋊Q₈	1	1	yes	characteristic
312	C₂²	D₁₃⋊#104	1	1	yes	characteristic
313	C₂²	#11719166	1	1	yes	characteristic
314	C₄	—	2	2	no	—
315	C₄	—	4	2	no	—
316	C₄	—	4	1	no	—
317	C₄	—	8	1	no	—
318	C₄	—	13	2	no	—
319	C₄	—	13	2	no	—
320	C₄	—	13	4	no	—
321	C₄	—	26	1	no	—
322	C₄	—	26	1	no	—
323	C₄	—	26	2	no	—
324	C₄	—	52	2	no	—
325	C₄	—	104	1	no	—
326	C₄	#11719074	1	2	yes	—
327	C₄	#11719078	1	2	yes	—
328	C₂	D₁₃⋊(Q₈⋊C₈)	1	1	yes	characteristic
329	C₂	#13130136	1	1	yes	characteristic
330	C₂	#13130144	1	1	yes	characteristic
331	C₂	#13130155	1	1	yes	characteristic
332	C₂	#13131007	1	1	yes	characteristic
333	C₂	#13131013	1	1	yes	characteristic
334	C₂	#13131014	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊#261
commutator	C₂×C₄×C₁₃	C₂²×C₈
Frattini	C₂²×C₄	C₂²×#261
Fitting	C₁₃×#389	C₄
socle, prime radical	C₂³×C₁₃	C₂×C₂²⋊C₄
cosocle	C₂×C₄×Dic₁₃	C₂³
hypercenter	#389	#261
hypocenter	C₁₃	#9864
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #389

Complements

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

N	H ≅ G/N	type
45. #88142	C₈	semidirect
49. C₄⋊#1057	Q₈	semidirect
266. C₁₃	#9864	semidirect

Maximal subgroups

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

subgroup	index
C₄×#88112	2
C₄×C₄⋊#1057	2
#13129672	2
#13117407	2
#9864	13