Subgroups — #16785027

Order 3328.

528 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₄⋊#1057	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊#1057	C₂	1	1	yes	maximal, characteristic
3	#13129672	C₂	1	4	yes	maximal
4	C₄⋊Q₈×Dic₁₃	C₂	1	1	yes	maximal, characteristic
5	C₂×C₁₃×C₄⋊Q₈	C₄	1	1	yes	characteristic
6	C₂×Q₈×Dic₁₃	—	2	1	no	—
7	C₂×C₄⋊#1057	—	2	1	no	—
8	C₂×C₄⋊#1057	C₂²	1	2	yes	—
9	C₂×C₄×#1057	—	2	1	no	—
10	C₂×C₄×#1057	C₂²	1	2	yes	—
11	#11718891	—	2	1	no	—
12	#11718646	—	2	1	no	—
13	#11718646	C₄	1	2	yes	—
14	C₄⋊C₄×Dic₁₃	—	2	1	no	—
15	C₄⋊C₄×Dic₁₃	C₂²	1	2	yes	—
16	#11718596	C₄	1	1	yes	characteristic
17	C₄²×Dic₁₃	C₂²	1	1	yes	characteristic
18	#11718290	—	2	1	no	—
19	#11718290	—	2	1	no	—
20	C₂²×Q₈×C₁₃	—	2	1	no	—
21	C₂²×#1057	—	2	2	no	—
22	C₂²×#1057	—	2	2	no	—
23	C₁₃×C₄⋊Q₈	—	2	2	no	—
24	C₁₃×C₄⋊Q₈	C₈	1	4	yes	complemented
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	Q₈×Dic₁₃	—	4	4	no	—
29	C₂×C₄⋊Dic₁₃	—	2	1	no	—
30	C₂×C₄⋊Dic₁₃	—	2	1	no	—
31	C₂×C₄⋊Dic₁₃	—	2	2	no	—
32	C₂×C₄⋊Dic₁₃	C₂×C₄	1	2	yes	—
33	C₂×C₄×Dic₁₃	—	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	C₄⋊#1057	—	2	4	no	—
41	C₄⋊#1057	—	2	2	no	—
42	C₄⋊#1057	Q₈	1	4	yes	complemented
43	C₄×#1057	—	2	4	no	—
44	C₄×#1057	—	4	2	no	—
45	#88071	—	2	2	no	—
46	#88071	—	2	2	no	—
47	#9864	—	13	1	no	maximal, Sylow
48	C₂×Q₈×C₁₃	—	2	4	no	—
49	C₂×Q₈×C₁₃	—	4	2	no	—
50	C₂×Q₈×C₁₃	—	4	2	no	—
51	C₂²×C₄×C₁₃	—	2	2	no	—
52	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
53	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
54	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
55	C₂²×Dic₁₃	—	2	2	no	—
56	C₂²×Dic₁₃	C₂²⋊C₄	1	1	yes	characteristic
57	C₂²×Dic₁₃	C₂²⋊C₄	1	1	yes	characteristic
58	C₂²×Dic₁₃	C₂×D₄	1	1	yes	characteristic
59	C₂²×Dic₁₃	C₂×D₄	1	1	yes	characteristic
60	C₂×#1057	—	2	4	no	—
61	C₂×#1057	—	2	4	no	—
62	C₂×#1057	—	4	2	no	—
63	C₂×#1057	—	4	2	no	—
64	C₂×#1057	—	4	2	no	—
65	C₂×#1057	—	4	2	no	—
66	C₁₃×C₄⋊C₄	—	2	4	no	—
67	C₁₃×C₄⋊C₄	—	2	2	no	—
68	C₁₃×C₄⋊C₄	C₂×C₈	1	4	yes	—
69	C₄²×C₁₃	C₂×C₈	1	2	yes	—
70	C₄²×C₁₃	M₁₆	1	2	yes	—
71	C₄⋊Dic₁₃	—	2	4	no	—
72	C₄⋊Dic₁₃	—	4	4	no	—
73	C₄⋊Dic₁₃	—	4	2	no	—
74	C₄⋊Dic₁₃	—	4	2	no	—
75	C₄×Dic₁₃	—	2	2	no	—
76	C₄×Dic₁₃	—	2	2	no	—
77	C₄×Dic₁₃	—	2	2	no	—
78	C₄×Dic₁₃	—	4	1	no	—
79	C₄×Dic₁₃	—	4	4	no	—
80	C₄×Dic₁₃	SD₁₆	1	2	yes	—
81	C₄×Dic₁₃	Q₁₆	1	2	yes	—
82	C₄×Dic₁₃	C₂×Q₈	1	2	yes	—
83	C₄×Dic₁₃	#41	1	2	yes	—
84	C₄×C₄⋊Q₈	—	13	1	no	—
85	C₄×C₄⋊C₈	—	13	1	no	—
86	C₄×C₄⋊C₈	—	13	1	no	—
87	#1279	—	13	4	no	—
88	C₂³×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
89	Q₈×C₁₃	—	4	8	no	—
90	C₂×C₄×C₁₃	—	2	1	no	—
91	C₂×C₄×C₁₃	—	2	1	no	—
92	C₂×C₄×C₁₃	—	2	4	no	—
93	C₂×C₄×C₁₃	—	4	2	no	—
94	C₂×C₄×C₁₃	—	4	2	no	—
95	C₂×C₄×C₁₃	C₂²⋊C₈	1	2	yes	—
96	C₂×C₄×C₁₃	C₂²⋊C₈	1	2	yes	—
97	C₂×C₄×C₁₃	#103	1	2	yes	—
98	C₂×C₄×C₁₃	#104	1	2	yes	—
99	C₂×C₄×C₁₃	C₄×Q₈	1	2	yes	—
100	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
101	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
102	C₂×C₄×C₁₃	#131	1	2	yes	—
103	C₂×Dic₁₃	—	2	1	no	—
104	C₂×Dic₁₃	—	2	1	no	—
105	C₂×Dic₁₃	—	2	1	no	—
106	C₂×Dic₁₃	—	2	1	no	—
107	C₂×Dic₁₃	—	2	1	no	—
108	C₂×Dic₁₃	—	2	1	no	—
109	C₂×Dic₁₃	—	2	1	no	—
110	C₂×Dic₁₃	—	4	2	no	—
111	C₂×Dic₁₃	—	4	2	no	—
112	C₂×Dic₁₃	—	4	2	no	—
113	C₂×Dic₁₃	#103	1	2	yes	—
114	C₂×Dic₁₃	C₂²⋊Q₈	1	2	yes	—
115	C₂×Dic₁₃	#124	1	1	yes	characteristic
116	C₂×Dic₁₃	C₄⋊Q₈	1	1	yes	characteristic
117	C₂×Dic₁₃	C₂×SD₁₆	1	1	yes	characteristic
118	C₂×Dic₁₃	C₂×Q₁₆	1	1	yes	characteristic
119	C₂×Dic₁₃	AGL(1,ℤ/8)	1	2	yes	—
120	#1057	—	4	4	no	—
121	#1057	—	4	4	no	—
122	C₂×C₄⋊Q₈	#261	1	1	yes	characteristic
123	C₂×C₄×Q₈	—	26	1	no	—
124	C₂×C₄⋊C₈	—	13	2	no	—
125	C₂×C₄⋊C₈	—	26	1	no	—
126	C₂×C₄×C₈	—	13	2	no	—
127	C₂×C₄×C₈	—	26	1	no	—
128	#391	—	26	1	no	—
129	#389	—	13	2	no	—
130	#389	—	26	1	no	—
131	#384	—	13	1	no	—
132	C₄×C₄⋊C₄	—	13	2	no	—
133	C₄×C₄⋊C₄	—	26	1	no	—
134	C₄³	—	13	1	no	—
135	#336	—	26	1	no	—
136	#336	—	26	1	no	—
137	C₂²×C₁₃	#391	1	1	yes	characteristic
138	C₂²×C₁₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
139	C₂²×C₁₃	#408	1	1	yes	characteristic
140	C₂²×C₁₃	C₂×#103	1	1	yes	characteristic
141	C₂²×C₁₃	#418	1	1	yes	characteristic
142	C₂²×C₁₃	C₂×#104	1	1	yes	characteristic
143	C₂²×C₁₃	#421	1	1	yes	characteristic
144	C₄×C₁₃	—	2	2	no	—
145	C₄×C₁₃	—	4	4	no	—
146	C₄×C₁₃	Q₈⋊C₈	1	4	yes	—
147	C₄×C₁₃	C₈×Q₈	1	2	yes	—
148	C₄×C₁₃	#446	1	2	yes	—
149	Dic₁₃	—	2	1	no	—
150	Dic₁₃	—	2	1	no	—
151	Dic₁₃	—	2	2	no	—
152	Dic₁₃	—	4	1	no	—
153	Dic₁₃	—	8	2	no	—
154	Dic₁₃	#476	1	2	yes	—
155	Dic₁₃	#477	1	2	yes	—
156	C₂²×Q₈	—	2	1	no	—
157	C₂²×C₈	—	26	2	no	—
158	C₂²×C₈	—	26	2	no	—
159	C₄⋊Q₈	—	2	2	no	—
160	C₄⋊Q₈	#1057	1	4	yes	complemented
161	C₄×Q₈	—	52	4	no	—
162	C₂×C₄⋊C₄	—	2	1	no	—
163	C₂×C₄⋊C₄	—	13	2	no	—
164	C₂×C₄⋊C₄	—	26	2	no	—
165	C₂×C₄⋊C₄	—	26	1	no	—
166	C₂×C₄⋊C₄	—	26	1	no	—
167	C₂×C₄⋊C₄	C₂×#261	1	2	yes	—
168	C₂×C₄²	—	13	1	no	—
169	C₂×C₄²	—	13	1	no	—
170	C₂×C₄²	—	13	1	no	—
171	C₂×C₄²	—	13	1	no	—
172	C₂×C₄²	—	13	1	no	—
173	C₂×C₄²	—	13	1	no	—
174	C₂×C₄²	—	26	2	no	—
175	C₂×C₄²	C₂×#261	1	1	yes	characteristic
176	C₄⋊C₈	—	13	4	no	—
177	C₄⋊C₈	—	26	2	no	—
178	C₄⋊C₈	—	26	4	no	—
179	C₄×C₈	—	26	4	no	—
180	C₄×C₈	—	52	2	no	—
181	#95	—	26	2	no	—
182	#95	—	26	2	no	—
183	C₂×C₁₃	C₂×Q₈⋊C₈	1	1	yes	characteristic
184	C₂×C₁₃	#1478	1	1	yes	characteristic
185	C₂×C₁₃	#1481	1	1	yes	characteristic
186	C₂×C₁₃	#1484	1	1	yes	characteristic
187	C₂×C₁₃	#1984	1	1	yes	characteristic
188	C₂×C₁₃	#1986	1	1	yes	characteristic
189	C₂×C₁₃	#1987	1	1	yes	characteristic
190	C₂×Q₈	—	2	4	no	—
191	C₂×Q₈	—	4	2	no	—
192	C₂×Q₈	—	4	2	no	—
193	C₂²×C₄	—	2	2	no	—
194	C₂²×C₄	—	13	1	no	—
195	C₂²×C₄	—	13	1	no	—
196	C₂²×C₄	—	13	1	no	—
197	C₂²×C₄	—	13	1	no	—
198	C₂²×C₄	—	26	2	no	—
199	C₂²×C₄	C₂²⋊#261	1	1	yes	characteristic
200	C₂²×C₄	C₂²⋊#261	1	1	yes	characteristic
201	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
202	C₂×C₈	—	26	4	no	—
203	C₂×C₈	—	26	4	no	—
204	C₂×C₈	—	52	2	no	—
205	C₂×C₈	—	52	2	no	—
206	C₂×C₈	—	52	2	no	—
207	C₂×C₈	—	52	2	no	—
208	C₄⋊C₄	—	2	4	no	—
209	C₄⋊C₄	—	2	2	no	—
210	C₄⋊C₄	—	26	4	no	—
211	C₄⋊C₄	—	52	2	no	—
212	C₄⋊C₄	—	52	4	no	—
213	C₄⋊C₄	—	52	2	no	—
214	C₄⋊C₄	C₂×#1057	1	4	yes	—
215	C₄²	—	13	2	no	—
216	C₄²	—	13	2	no	—
217	C₄²	—	13	2	no	—
218	C₄²	—	13	2	no	—
219	C₄²	—	26	2	no	—
220	C₄²	—	26	2	no	—
221	C₄²	—	26	2	no	—
222	C₄²	—	52	1	no	—
223	C₄²	—	52	4	no	—
224	C₄²	C₂×#1057	1	2	yes	—
225	C₄²	#6122	1	2	yes	—
226	C₁₃	#9864	1	1	yes	characteristic, complemented, Sylow
227	C₂³	C₂×C₂²⋊#261	1	1	yes	characteristic
228	Q₈	—	4	8	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	4	no	—
232	C₂×C₄	—	4	2	no	—
233	C₂×C₄	—	4	2	no	—
234	C₂×C₄	—	13	2	no	—
235	C₂×C₄	—	13	1	no	—
236	C₂×C₄	—	13	1	no	—
237	C₂×C₄	—	13	2	no	—
238	C₂×C₄	—	13	2	no	—
239	C₂×C₄	—	13	1	no	—
240	C₂×C₄	—	13	1	no	—
241	C₂×C₄	—	26	1	no	—
242	C₂×C₄	—	26	1	no	—
243	C₂×C₄	—	26	1	no	—
244	C₂×C₄	—	26	1	no	—
245	C₂×C₄	—	26	1	no	—
246	C₂×C₄	—	26	1	no	—
247	C₂×C₄	—	26	1	no	—
248	C₂×C₄	—	52	2	no	—
249	C₂×C₄	—	52	2	no	—
250	C₂×C₄	—	52	2	no	—
251	C₂×C₄	#88117	1	2	yes	—
252	C₂×C₄	#88118	1	2	yes	—
253	C₂×C₄	C₂²⋊#1057	1	2	yes	—
254	C₂×C₄	C₂²⋊#1057	1	2	yes	—
255	C₂×C₄	#88240	1	2	yes	—
256	C₂×C₄	Q₈×#261	1	2	yes	—
257	C₂×C₄	C₂²×#1057	1	1	yes	characteristic
258	C₂×C₄	C₂×#6122	1	1	yes	characteristic
259	C₈	—	52	4	no	—
260	C₈	—	52	4	no	—
261	C₂²	#11719138	1	1	yes	characteristic
262	C₂²	#11719140	1	1	yes	characteristic
263	C₂²	C₂×#88117	1	1	yes	characteristic
264	C₂²	C₂×#88118	1	1	yes	characteristic
265	C₂²	#11719159	1	1	yes	characteristic
266	C₂²	(C₂×#261)⋊Q₈	1	1	yes	characteristic
267	C₂²	C₂×C₂²⋊#1057	1	1	yes	characteristic
268	C₄	—	2	2	no	—
269	C₄	—	4	4	no	—
270	C₄	—	13	2	no	—
271	C₄	—	13	2	no	—
272	C₄	—	26	1	no	—
273	C₄	—	26	1	no	—
274	C₄	—	26	2	no	—
275	C₄	—	52	1	no	—
276	C₄	—	104	2	no	—
277	C₄	Q₈⋊#1057	1	4	yes	—
278	C₄	Q₈×#1057	1	2	yes	—
279	C₄	#11719158	1	2	yes	—
280	C₂	C₂×Q₈⋊#1057	1	1	yes	characteristic
281	C₂	#13130135	1	1	yes	characteristic
282	C₂	#13130146	1	1	yes	characteristic
283	C₂	#13130157	1	1	yes	characteristic
284	C₂	#13131005	1	1	yes	characteristic
285	C₂	#13131011	1	1	yes	characteristic
286	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	C₂×C₁₃×C₄⋊Q₈	C₄
socle, prime radical	C₂³×C₁₃	C₂×C₂²⋊C₄
cosocle	C₂×C₄×Dic₁₃	C₂³
hypercenter	C₂×C₄⋊Q₈	#261
hypocenter	C₁₃	#9864
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
24. C₁₃×C₄⋊Q₈	C₈	semidirect
42. C₄⋊#1057	Q₈	semidirect
160. C₄⋊Q₈	#1057	semidirect
226. 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×C₄⋊#1057	2
#13129672	2
C₄⋊Q₈×Dic₁₃	2
#9864	13