Subgroups — #16807268

Order 3328.

530 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13131253	C₂	1	4	yes	maximal, complemented
2	#13131251	C₂	1	2	yes	maximal, complemented
3	#13131250	C₂	1	2	yes	maximal, complemented
4	#13131169	C₂	1	1	yes	maximal, characteristic
5	#13131169	C₂	1	2	yes	maximal
6	#13131043	C₂	1	2	yes	maximal
7	C₄×#88098	C₂	1	1	yes	maximal, characteristic
8	C₈×Q₈×D₁₃	C₂	1	1	yes	maximal, characteristic
9	C₄×Q₈×D₁₃	C₄	1	1	yes	characteristic
10	C₂×#88098	C₂²	1	1	yes	characteristic
11	C₂×#88098	C₂²	1	2	yes	—
12	C₂×#88098	C₂²	1	1	yes	characteristic
13	C₈×Q₈×C₁₃	C₄	1	1	yes	characteristic
14	Q₈×#1055	C₂²	1	1	yes	characteristic
15	D₁₃×C₄⋊C₈	C₄	1	1	yes	characteristic
16	D₁₃×C₄⋊C₈	C₄	1	2	yes	—
17	#11718491	C₂²	1	1	yes	characteristic
18	#11718491	C₂²	1	2	yes	—
19	C₄×C₈×D₁₃	C₂²	1	1	yes	characteristic
20	C₄×C₈×D₁₃	C₂²	1	2	yes	—
21	C₈×Dic₂₆	C₄	1	1	yes	characteristic
22	C₈×Dic₂₆	C₄	1	2	yes	—
23	#11718260	C₂²	1	4	yes	—
24	#11718260	C₂²	1	2	yes	—
25	#11718258	C₂²	1	2	yes	—
26	#11718257	C₂²	1	4	yes	—
27	#11718231	C₂²	1	4	yes	—
28	C₄⋊#6092	C₂²	1	2	yes	—
29	C₄⋊#6092	C₂²	1	4	yes	—
30	C₄×#6092	C₂²	1	2	yes	—
31	C₂×Q₈×D₁₃	C₈	1	1	yes	characteristic
32	C₄×Q₈×C₁₃	C₂×C₄	1	1	yes	characteristic
33	Q₈×Dic₁₃	C₈	1	1	yes	characteristic
34	C₂×C₈×D₁₃	C₂×C₄	1	1	yes	characteristic
35	C₂×C₈×D₁₃	C₂×C₄	1	2	yes	—
36	C₂×C₈×D₁₃	C₂³	1	1	yes	characteristic
37	C₄⋊C₄×D₁₃	C₈	1	1	yes	characteristic
38	C₄⋊C₄×D₁₃	C₈	1	2	yes	—
39	#88141	C₈	1	2	yes	—
40	#88141	C₈	1	1	yes	characteristic
41	C₄²×D₁₃	C₂×C₄	1	1	yes	characteristic
42	C₄²×D₁₃	C₂×C₄	1	2	yes	—
43	C₄×Dic₂₆	C₂×C₄	1	2	yes	—
44	C₄×Dic₂₆	C₂×C₄	1	1	yes	characteristic
45	C₂×#6092	C₂³	1	2	yes	—
46	C₂×#6092	C₂³	1	4	yes	—
47	C₂×#6092	C₂³	1	2	yes	—
48	#88098	—	2	2	no	—
49	#88098	—	2	4	no	—
50	#88098	Q₈	1	4	yes	complemented
51	C₁₃×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
52	C₁₃×C₄⋊C₈	C₂×C₄	1	2	yes	—
53	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
54	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
55	#88054	C₂×C₄	1	2	yes	—
56	#88054	C₂×C₄	1	2	yes	—
57	#88054	C₂×C₄	1	2	yes	—
58	C₈×Dic₁₃	C₂×C₄	1	1	yes	characteristic
59	C₈×Dic₁₃	C₂×C₄	1	2	yes	—
60	C₈×Dic₁₃	C₂³	1	2	yes	—
61	C₈×Dic₁₃	C₂³	1	1	yes	characteristic
62	C₄⋊#1055	C₂×C₄	1	2	yes	—
63	C₄⋊#1055	C₂×C₄	1	1	yes	characteristic
64	C₄×#1055	C₂³	1	1	yes	characteristic
65	C₄×#1055	C₂³	1	2	yes	—
66	Q₈×M₃₂	—	13	1	no	maximal, Sylow
67	C₂×Q₈×C₁₃	C₂×C₈	1	1	yes	characteristic
68	Q₈×D₁₃	—	2	4	no	—
69	C₂×C₄×D₁₃	C₂×C₈	1	1	yes	characteristic
70	C₂×C₄×D₁₃	C₂×C₈	1	2	yes	—
71	C₂×C₄×D₁₃	C₂×C₈	1	1	yes	characteristic
72	C₂×C₄×D₁₃	C₂×C₈	1	2	yes	—
73	C₂×C₄×D₁₃	C₂²×C₄	1	1	yes	characteristic
74	C₂×Dic₂₆	C₂×C₈	1	2	yes	—
75	C₂×Dic₂₆	C₂×C₈	1	1	yes	characteristic
76	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
77	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
79	C₁₃×C₄⋊C₄	C₂×C₈	1	2	yes	—
80	C₁₃×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
81	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
82	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
83	C₄⋊Dic₁₃	C₂×C₈	1	2	yes	—
84	C₄⋊Dic₁₃	C₂×C₈	1	1	yes	characteristic
85	#6101	C₂×C₈	1	2	yes	—
86	#6101	C₂×C₈	1	2	yes	—
87	#6101	C₂×C₈	1	2	yes	—
88	C₄×Dic₁₃	C₂×C₈	1	1	yes	characteristic
89	C₄×Dic₁₃	C₂×C₈	1	2	yes	—
90	C₄×Dic₁₃	C₂×C₈	1	2	yes	—
91	C₄×Dic₁₃	C₂×C₈	1	1	yes	characteristic
92	C₄×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
93	C₄×Dic₁₃	C₂²×C₄	1	2	yes	—
94	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
95	C₂×#1055	C₂²×C₄	1	2	yes	—
96	C₂×#1055	C₂⁴	1	1	yes	characteristic
97	C₈×D₁₃	—	2	2	no	—
98	C₈×D₁₃	—	2	4	no	—
99	C₈×D₁₃	C₂×Q₈	1	2	yes	—
100	C₈×D₁₃	#41	1	2	yes	—
101	#6092	—	2	2	no	—
102	#6092	—	2	4	no	—
103	#6092	C₂×Q₈	1	4	yes	complemented
104	C₂×C₈×Q₈	—	13	1	no	—
105	#2183	—	13	2	no	—
106	#2183	—	13	4	no	—
107	Q₈×C₁₆	—	13	2	no	—
108	#2150	—	13	1	no	—
109	#2150	—	13	2	no	—
110	C₄×M₃₂	—	13	1	no	—
111	C₄×M₃₂	—	13	2	no	—
112	C₂²×D₁₃	C₂²×C₈	1	1	yes	characteristic
113	Q₈×C₁₃	—	2	2	no	—
114	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
115	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
116	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
117	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
118	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
119	C₂×Dic₁₃	C₂²×C₈	1	2	yes	—
120	C₂×Dic₁₃	C₂²×C₈	1	1	yes	characteristic
121	C₂×Dic₁₃	C₂²×C₈	1	1	yes	characteristic
122	C₂×Dic₁₃	C₂²×C₈	1	1	yes	characteristic
123	C₂×Dic₁₃	C₂²×C₈	1	2	yes	—
124	C₄×D₁₃	—	2	2	no	—
125	C₄×D₁₃	—	2	4	no	—
126	C₄×D₁₃	—	4	1	no	—
127	C₄×D₁₃	—	4	2	no	—
128	C₄×D₁₃	C₄×Q₈	1	2	yes	—
129	C₄×D₁₃	#131	1	2	yes	—
130	Dic₂₆	—	2	4	no	—
131	Dic₂₆	M₃₂	1	4	yes	complemented
132	C₈×C₁₃	—	2	2	no	—
133	C₈×C₁₃	—	2	1	no	—
134	C₈×C₁₃	C₄×Q₈	1	2	yes	—
135	#1055	—	2	1	no	—
136	#1055	—	2	2	no	—
137	#1055	C₂²×Q₈	1	1	yes	characteristic
138	#1055	C₂×#41	1	1	yes	characteristic
139	C₂×C₄×Q₈	—	13	1	no	—
140	C₂×M₃₂	—	13	1	no	—
141	C₂×M₃₂	—	13	1	no	—
142	C₂×M₃₂	—	13	2	no	—
143	C₈×Q₈	—	13	1	no	—
144	C₈×Q₈	—	13	1	no	—
145	C₈×Q₈	—	13	2	no	—
146	C₈×Q₈	—	13	2	no	—
147	C₈×Q₈	—	13	1	no	—
148	C₈×Q₈	#261	1	1	yes	characteristic
149	C₂×C₄⋊C₈	—	13	2	no	—
150	C₂×C₄⋊C₈	—	13	1	no	—
151	C₂×C₄×C₈	—	13	2	no	—
152	C₂×C₄×C₈	—	13	1	no	—
153	C₄⋊C₁₆	—	13	2	no	—
154	C₄⋊C₁₆	—	13	4	no	—
155	C₄⋊C₁₆	—	13	4	no	—
156	C₄⋊C₁₆	—	13	2	no	—
157	#346	—	13	4	no	—
158	#346	—	13	2	no	—
159	C₄×C₁₆	—	13	2	no	—
160	C₄×C₁₆	—	13	4	no	—
161	C₂²×C₁₃	C₂³×C₈	1	1	yes	characteristic
162	C₂×D₁₃	—	2	1	no	—
163	C₂×D₁₃	—	2	1	no	—
164	C₂×D₁₃	C₈×Q₈	1	2	yes	—
165	C₄×C₁₃	—	2	2	no	—
166	C₄×C₁₃	—	2	1	no	—
167	C₄×C₁₃	—	2	2	no	—
168	C₄×C₁₃	C₂×M₃₂	1	2	yes	—
169	C₄×C₁₃	C₂×C₄×Q₈	1	1	yes	characteristic
170	C₄×C₁₃	C₂×#131	1	1	yes	characteristic
171	Dic₁₃	—	2	1	no	—
172	Dic₁₃	—	2	2	no	—
173	Dic₁₃	—	2	1	no	—
174	Dic₁₃	C₈×Q₈	1	2	yes	—
175	Dic₁₃	C₂×M₃₂	1	4	yes	—
176	C₂²×Q₈	—	13	1	no	—
177	C₂²×C₈	—	13	1	no	—
178	C₂²×C₈	—	13	1	no	—
179	C₂²×C₈	—	13	2	no	—
180	C₄×Q₈	—	13	1	no	—
181	C₄×Q₈	—	13	1	no	—
182	C₄×Q₈	—	13	2	no	—
183	C₄×Q₈	—	13	1	no	—
184	C₄×Q₈	—	13	2	no	—
185	C₄×Q₈	C₂×#261	1	1	yes	characteristic
186	C₂×C₄⋊C₄	—	13	2	no	—
187	C₂×C₄⋊C₄	—	13	1	no	—
188	C₂×C₄²	—	13	2	no	—
189	C₂×C₄²	—	13	1	no	—
190	M₃₂	—	13	4	no	—
191	M₃₂	—	26	2	no	—
192	M₃₂	—	26	4	no	—
193	C₂×C₁₆	—	13	2	no	—
194	C₂×C₁₆	—	13	2	no	—
195	C₂×C₁₆	—	13	4	no	—
196	C₄⋊C₈	—	13	2	no	—
197	C₄⋊C₈	—	13	2	no	—
198	C₄⋊C₈	—	13	2	no	—
199	C₄⋊C₈	—	13	2	no	—
200	C₄⋊C₈	—	13	1	no	—
201	C₄⋊C₈	C₂×#261	1	1	yes	characteristic
202	C₄⋊C₈	C₂×#261	1	2	yes	—
203	C₄×C₈	—	13	2	no	—
204	C₄×C₈	—	13	1	no	—
205	C₄×C₈	—	13	2	no	—
206	C₄×C₈	—	13	1	no	—
207	C₄×C₈	—	13	1	no	—
208	C₄×C₈	—	13	2	no	—
209	C₄×C₈	C₂×#261	1	2	yes	—
210	C₄×C₈	C₂×#261	1	1	yes	characteristic
211	C₂×C₁₃	C₂×C₈×Q₈	1	1	yes	characteristic
212	C₂×C₁₃	C₂²×M₃₂	1	1	yes	characteristic
213	C₂×C₁₃	#3407	1	1	yes	characteristic
214	D₁₃	—	2	2	no	—
215	C₂×Q₈	—	13	1	no	—
216	C₂×Q₈	—	13	2	no	—
217	C₂×Q₈	—	26	4	no	—
218	C₂×Q₈	D₁₃⋊C₈	1	1	yes	characteristic
219	C₂²×C₄	—	13	1	no	—
220	C₂²×C₄	—	13	2	no	—
221	C₂²×C₄	—	13	1	no	—
222	C₂²×C₄	—	13	2	no	—
223	C₂²×C₄	—	13	1	no	—
224	C₂×C₈	—	13	1	no	—
225	C₂×C₈	—	13	2	no	—
226	C₂×C₈	—	13	2	no	—
227	C₂×C₈	—	13	1	no	—
228	C₂×C₈	—	13	2	no	—
229	C₂×C₈	—	26	4	no	—
230	C₂×C₈	—	26	2	no	—
231	C₂×C₈	C₂²×#261	1	2	yes	—
232	C₂×C₈	C₂²×#261	1	1	yes	characteristic
233	C₂×C₈	C₂²×#261	1	1	yes	characteristic
234	C₄⋊C₄	—	13	2	no	—
235	C₄⋊C₄	—	13	2	no	—
236	C₄⋊C₄	—	13	2	no	—
237	C₄⋊C₄	—	13	2	no	—
238	C₄⋊C₄	—	13	1	no	—
239	C₄⋊C₄	D₁₃⋊C₈	1	2	yes	—
240	C₄⋊C₄	D₁₃⋊C₈	1	1	yes	characteristic
241	C₄²	—	13	2	no	—
242	C₄²	—	13	2	no	—
243	C₄²	—	13	1	no	—
244	C₄²	—	13	1	no	—
245	C₄²	—	13	2	no	—
246	C₄²	—	13	1	no	—
247	C₄²	C₂²×#261	1	2	yes	—
248	C₄²	C₂²×#261	1	1	yes	characteristic
249	C₁₆	—	13	4	no	—
250	C₁₆	—	26	2	no	—
251	C₁₆	—	26	4	no	—
252	C₁₃	Q₈×M₃₂	1	1	yes	characteristic, complemented, Sylow
253	C₂³	—	13	1	no	—
254	Q₈	—	2	2	no	—
255	Q₈	—	13	4	no	—
256	Q₈	—	26	4	no	—
257	C₂×C₄	—	13	1	no	—
258	C₂×C₄	—	13	2	no	—
259	C₂×C₄	—	13	2	no	—
260	C₂×C₄	—	13	2	no	—
261	C₂×C₄	—	13	2	no	—
262	C₂×C₄	—	13	1	no	—
263	C₂×C₄	—	13	1	no	—
264	C₂×C₄	—	26	4	no	—
265	C₂×C₄	—	26	2	no	—
266	C₂×C₄	—	52	2	no	—
267	C₂×C₄	—	52	1	no	—
268	C₂×C₄	C₂×D₁₃⋊C₈	1	2	yes	—
269	C₂×C₄	C₂×D₁₃⋊C₈	1	1	yes	characteristic
270	C₂×C₄	C₂×D₁₃⋊C₈	1	2	yes	—
271	C₂×C₄	C₂×D₁₃⋊C₈	1	1	yes	characteristic
272	C₂×C₄	C₂³×#261	1	1	yes	characteristic
273	C₈	—	2	2	no	—
274	C₈	—	2	1	no	—
275	C₈	—	13	1	no	—
276	C₈	—	13	1	no	—
277	C₈	—	26	1	no	—
278	C₈	—	26	2	no	—
279	C₈	Q₈×#261	1	2	yes	—
280	C₂²	—	13	2	no	—
281	C₂²	—	26	1	no	—
282	C₂²	—	26	1	no	—
283	C₂²	C₂²×D₁₃⋊C₈	1	1	yes	characteristic
284	C₄	—	2	2	no	—
285	C₄	—	2	2	no	—
286	C₄	—	2	1	no	—
287	C₄	—	13	4	no	—
288	C₄	—	13	2	no	—
289	C₄	—	26	1	no	—
290	C₄	—	26	2	no	—
291	C₄	—	26	1	no	—
292	C₄	D₁₃⋊M₃₂	1	2	yes	—
293	C₄	C₂×Q₈×#261	1	1	yes	characteristic
294	C₄	D₁₃⋊#131	1	1	yes	characteristic
295	C₂	—	26	2	no	—
296	C₂	Q₈×D₁₃⋊C₈	1	1	yes	characteristic
297	C₂	C₂×D₁₃⋊M₃₂	1	1	yes	characteristic
298	C₂	#13131804	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×#261
commutator, socle	C₂²×C₁₃	C₂³×C₈
Fitting	C₈×Q₈×C₁₃	C₄
prime radical	C₂²×D₁₃	C₂²×C₈
cosocle	C₂×#1055	C₂⁴
hypercenter	C₈×Q₈	#261
hypocenter	C₁₃	Q₈×M₃₂
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₈×Q₈

Complements

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

N	H ≅ G/N	type
1. #13131253	C₂	semidirect
2. #13131251	C₂	semidirect
3. #13131250	C₂	semidirect
50. #88098	Q₈	semidirect
103. #6092	C₂×Q₈	semidirect
131. Dic₂₆	M₃₂	semidirect
252. C₁₃	Q₈×M₃₂	semidirect

Maximal subgroups

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

subgroup	index
#13131253	2
#13131251	2
#13131250	2
#13131169	2
#13131043	2
C₄×#88098	2
C₈×Q₈×D₁₃	2
Q₈×M₃₂	13