Subgroups — #15833435

Order 3328.

750 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13123548	C₂	1	2	yes	maximal, complemented
2	#13113721	C₂	1	4	yes	maximal, complemented
3	D₈×#1055	C₂	1	4	yes	maximal, complemented
4	#13113595	C₂	1	1	yes	maximal, characteristic, complemented
5	#13113287	C₂	1	2	yes	maximal, complemented
6	C₂×C₈×#1055	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₃×#3169	C₂	1	1	yes	maximal, characteristic
8	C₁₃×#532	C₂²	1	2	yes	—
9	C₂×#88202	—	2	2	no	—
10	C₁₃×#500	C₄	1	1	yes	characteristic
11	C₁₃×#493	C₄	1	1	yes	characteristic
12	C₁₃×#486	C₄	1	2	yes	—
13	C₁₃×#466	C₄	1	4	yes	—
14	C₄×C₁₃×D₈	C₂²	1	4	yes	—
15	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
16	#11718895	—	2	2	no	—
17	#11718673	—	2	2	no	—
18	#11718673	C₂²	1	4	yes	—
19	D₄×#1055	—	2	2	no	—
20	D₄×#1055	C₂²	1	4	yes	—
21	#11718585	C₂²	1	2	yes	—
22	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
23	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
24	D₄⋊#1055	C₂²	1	8	yes	—
25	#11718050	C₂²	1	4	yes	—
26	C₈×#1055	C₂²	1	2	yes	—
27	C₈×#1055	C₂²	1	2	yes	—
28	C₂×C₁₃×#41	—	2	2	no	—
29	C₂×C₁₃×D₈	C₂×C₄	1	4	yes	—
30	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
31	C₁₃×C₄⋊Q₈	C₂×C₄	1	2	yes	—
32	C₁₃×#127	C₂×C₄	1	2	yes	—
33	C₁₃×#124	—	2	2	no	—
34	C₁₃×#121	—	2	2	no	—
35	C₁₃×#121	C₂×C₄	1	4	yes	—
36	C₄×D₄×C₁₃	—	2	2	no	—
37	C₄×D₄×C₁₃	C₂³	1	4	yes	—
38	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
39	#88202	—	4	8	no	—
40	C₂×#6099	—	2	2	no	—
41	C₂×#6099	—	2	4	no	—
42	C₂²×#1055	—	2	1	no	—
43	C₂²×#1055	—	2	4	no	—
44	C₂²×#1055	D₄	1	1	yes	characteristic
45	C₂²×#1055	D₄	1	1	yes	characteristic
46	C₁₃×#107	C₂×C₄	1	4	yes	—
47	C₁₃×#102	C₂×C₄	1	8	yes	—
48	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
49	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
50	C₄×C₈×C₁₃	C₂³	1	2	yes	—
51	C₂²⋊#1055	—	2	4	no	—
52	C₂²⋊#1055	—	2	4	no	—
53	C₄⋊#1055	—	2	2	no	—
54	C₄⋊#1055	C₂³	1	4	yes	—
55	C₄×#1055	—	2	1	no	—
56	C₄×#1055	D₄	1	1	yes	characteristic
57	C₄×#1055	D₄	1	1	yes	characteristic
58	C₄×#1055	C₂³	1	2	yes	—
59	C₄×#1055	C₂³	1	1	yes	characteristic
60	C₄×#1055	C₂³	1	1	yes	characteristic
61	#17956	—	13	1	no	maximal, Sylow
62	C₁₃×#41	—	4	4	no	—
63	C₁₃×#41	—	4	4	no	—
64	C₂×Q₈×C₁₃	—	2	2	no	—
65	C₂×D₄×C₁₃	—	2	4	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	C₂²×C₄×C₁₃	—	2	4	no	—
70	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
73	C₁₃×D₈	—	2	8	no	—
74	C₂×C₈×C₁₃	—	2	4	no	—
75	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
76	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
77	C₁₃×C₄⋊C₄	—	2	2	no	—
78	C₁₃×C₄⋊C₄	C₂²×C₄	1	4	yes	—
79	C₁₃×C₂²⋊C₄	—	2	4	no	—
80	C₁₃×C₂²⋊C₄	—	2	4	no	—
81	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
82	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
83	C₄²×C₁₃	#41	1	1	yes	characteristic
84	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
85	#6099	—	4	4	no	—
86	#6099	—	4	8	no	—
87	C₂×#1055	—	2	4	no	—
88	C₂×#1055	—	2	2	no	—
89	C₂×#1055	—	2	2	no	—
90	C₂×#1055	—	2	4	no	—
91	C₂×#1055	—	4	8	no	—
92	C₂×#1055	D₈	1	4	yes	complemented
93	C₂×#1055	C₂×D₄	1	1	yes	characteristic
94	C₂×#1055	C₂×D₄	1	1	yes	characteristic
95	C₂×#1055	C₂×D₄	1	1	yes	characteristic
96	C₂×#1055	C₂×D₄	1	1	yes	characteristic
97	#3169	D₁₃	1	1	yes	characteristic
98	#2931	—	13	2	no	—
99	#1589	—	13	4	no	—
100	C₈×D₈	—	13	4	no	—
101	#1567	—	13	1	no	—
102	#1492	—	13	2	no	—
103	C₂×C₈²	—	13	1	no	—
104	C₂³×C₁₃	—	2	4	no	—
105	C₂³×C₁₃	C₄×D₄	1	1	yes	characteristic
106	Q₈×C₁₃	—	4	4	no	—
107	D₄×C₁₃	—	2	8	no	—
108	D₄×C₁₃	—	4	8	no	—
109	D₄×C₁₃	—	4	4	no	—
110	D₄×C₁₃	—	4	4	no	—
111	C₂×C₄×C₁₃	—	2	2	no	—
112	C₂×C₄×C₁₃	—	2	2	no	—
113	C₂×C₄×C₁₃	—	2	4	no	—
114	C₂×C₄×C₁₃	—	4	4	no	—
115	C₂×C₄×C₁₃	—	4	4	no	—
116	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
117	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
118	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
119	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
120	C₂×C₄×C₁₃	#135	1	2	yes	—
121	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
122	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
123	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
124	C₈×C₁₃	—	2	2	no	—
125	C₈×C₁₃	#131	1	4	yes	complemented
126	#1055	—	2	4	no	—
127	#1055	—	2	1	no	—
128	#1055	—	2	1	no	—
129	#1055	—	4	4	no	—
130	#1055	C₂×D₈	1	2	yes	—
131	#1055	C₂×D₈	1	2	yes	—
132	C₂×#131	—	26	2	no	—
133	#532	C₂×D₁₃	1	2	yes	—
134	#500	Dic₁₃	1	1	yes	characteristic
135	#493	Dic₁₃	1	1	yes	characteristic
136	#486	Dic₁₃	1	2	yes	—
137	#466	Dic₁₃	1	4	yes	—
138	C₄×D₈	C₂×D₁₃	1	4	yes	—
139	#436	—	13	4	no	—
140	#436	—	26	2	no	—
141	C₈×D₄	—	13	4	no	—
142	C₈×D₄	—	26	2	no	—
143	#423	—	13	2	no	—
144	#408	—	26	2	no	—
145	C₂×C₄×C₈	—	13	1	no	—
146	C₂×C₄×C₈	—	13	1	no	—
147	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
148	#335	—	13	4	no	—
149	D₄⋊C₈	—	13	8	no	—
150	C₈²	—	13	2	no	—
151	C₈²	—	13	2	no	—
152	C₂²×C₁₃	—	2	1	no	—
153	C₂²×C₁₃	—	2	1	no	—
154	C₂²×C₁₃	—	4	4	no	—
155	C₂²×C₁₃	—	4	4	no	—
156	C₂²×C₁₃	—	4	4	no	—
157	C₂²×C₁₃	C₄×D₈	1	2	yes	—
158	C₂²×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
159	C₄×C₁₃	—	2	1	no	—
160	C₄×C₁₃	—	2	1	no	—
161	C₄×C₁₃	—	4	4	no	—
162	C₄×C₁₃	C₄×D₈	1	2	yes	—
163	C₄×C₁₃	#443	1	2	yes	—
164	C₄×C₁₃	C₂×#131	1	2	yes	—
165	C₄×C₁₃	C₂²×D₈	1	1	yes	characteristic
166	C₄×C₁₃	C₂×#135	1	1	yes	characteristic
167	C₂×#41	—	2	2	no	—
168	C₂×D₈	C₂×Dic₁₃	1	4	yes	—
169	#131	—	52	8	no	—
170	C₂×M₁₆	—	26	4	no	—
171	C₂×M₁₆	—	26	2	no	—
172	C₂²×C₈	—	13	1	no	—
173	C₂²×C₈	—	13	1	no	—
174	C₂²×C₈	—	26	4	no	—
175	C₂²×C₈	—	26	1	no	—
176	C₂²×C₈	C₂×Dic₁₃	1	2	yes	—
177	C₄⋊Q₈	C₂×Dic₁₃	1	2	yes	—
178	#127	C₂×Dic₁₃	1	2	yes	—
179	#124	—	2	2	no	—
180	#121	—	2	2	no	—
181	#121	C₂×Dic₁₃	1	4	yes	—
182	C₄×D₄	—	2	2	no	—
183	C₄×D₄	C₂²×D₁₃	1	4	yes	—
184	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
185	#107	C₂×Dic₁₃	1	4	yes	—
186	C₄⋊C₈	—	13	4	no	—
187	C₄⋊C₈	—	26	2	no	—
188	#102	C₂×Dic₁₃	1	8	yes	—
189	C₂²⋊C₈	—	26	4	no	—
190	C₂²⋊C₈	—	26	4	no	—
191	C₄×C₈	—	13	1	no	—
192	C₄×C₈	—	13	1	no	—
193	C₄×C₈	—	13	1	no	—
194	C₄×C₈	—	13	1	no	—
195	C₄×C₈	—	13	2	no	—
196	C₄×C₈	—	26	1	no	—
197	C₄×C₈	C₂×Dic₁₃	1	1	yes	characteristic
198	C₄×C₈	C₂×Dic₁₃	1	1	yes	characteristic
199	C₄×C₈	C₂²×D₁₃	1	2	yes	—
200	C₂×C₁₃	—	2	2	no	—
201	C₂×C₁₃	—	8	4	no	—
202	C₂×C₁₃	#2931	1	1	yes	characteristic
203	C₂×C₁₃	C₂×C₄×D₈	1	1	yes	characteristic
204	C₂×C₁₃	C₂×#443	1	1	yes	characteristic
205	#41	—	4	4	no	—
206	#41	—	4	4	no	—
207	C₂×Q₈	—	2	2	no	—
208	C₂×D₄	—	2	4	no	—
209	C₂×D₄	—	2	2	no	—
210	C₂×D₄	—	2	2	no	—
211	C₂×D₄	C₂²×Dic₁₃	1	4	yes	—
212	C₂²×C₄	—	2	4	no	—
213	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
214	C₂²×C₄	#6129	1	1	yes	characteristic
215	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
216	D₈	—	2	8	no	—
217	M₁₆	—	52	8	no	—
218	M₁₆	—	52	4	no	—
219	C₂×C₈	—	2	4	no	—
220	C₂×C₈	—	13	4	no	—
221	C₂×C₈	—	13	1	no	—
222	C₂×C₈	—	13	1	no	—
223	C₂×C₈	—	13	1	no	—
224	C₂×C₈	—	13	1	no	—
225	C₂×C₈	—	26	4	no	—
226	C₂×C₈	—	26	2	no	—
227	C₂×C₈	—	26	2	no	—
228	C₂×C₈	—	26	4	no	—
229	C₂×C₈	—	52	8	no	—
230	C₂×C₈	C₂²×Dic₁₃	1	2	yes	—
231	C₂×C₈	C₂²×Dic₁₃	1	2	yes	—
232	C₄⋊C₄	—	2	2	no	—
233	C₄⋊C₄	C₂²×Dic₁₃	1	4	yes	—
234	C₂²⋊C₄	—	2	4	no	—
235	C₂²⋊C₄	—	2	4	no	—
236	C₄²	D₄×D₁₃	1	1	yes	characteristic
237	C₄²	#6129	1	1	yes	characteristic
238	C₄²	C₂²×Dic₁₃	1	1	yes	characteristic
239	C₄²	C₂³×D₁₃	1	1	yes	characteristic
240	C₁₃	#17956	1	1	yes	characteristic, complemented, Sylow
241	C₂³	—	2	4	no	—
242	C₂³	D₄×Dic₁₃	1	1	yes	characteristic
243	Q₈	—	4	4	no	—
244	D₄	—	2	8	no	—
245	D₄	—	4	8	no	—
246	D₄	—	4	4	no	—
247	D₄	—	4	4	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	2	4	no	—
251	C₂×C₄	—	4	4	no	—
252	C₂×C₄	—	4	4	no	—
253	C₂×C₄	D₈×D₁₃	1	2	yes	—
254	C₂×C₄	#88166	1	2	yes	—
255	C₂×C₄	D₄×Dic₁₃	1	1	yes	characteristic
256	C₂×C₄	D₄×Dic₁₃	1	1	yes	characteristic
257	C₂×C₄	D₄×Dic₁₃	1	1	yes	characteristic
258	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
259	C₂×C₄	C₂×#6129	1	1	yes	characteristic
260	C₂×C₄	C₂³×Dic₁₃	1	1	yes	characteristic
261	C₈	—	2	2	no	—
262	C₈	—	13	2	no	—
263	C₈	—	13	2	no	—
264	C₈	—	26	4	no	—
265	C₈	—	26	1	no	—
266	C₈	—	26	1	no	—
267	C₈	—	52	4	no	—
268	C₈	#88202	1	4	yes	complemented
269	C₂²	—	2	1	no	—
270	C₂²	—	2	1	no	—
271	C₂²	—	4	4	no	—
272	C₂²	—	4	4	no	—
273	C₂²	—	4	4	no	—
274	C₂²	D₈×Dic₁₃	1	2	yes	—
275	C₂²	C₂×D₄×Dic₁₃	1	1	yes	characteristic
276	C₄	—	2	1	no	—
277	C₄	—	2	1	no	—
278	C₄	—	4	4	no	—
279	C₄	D₈×Dic₁₃	1	2	yes	—
280	C₄	#11718857	1	2	yes	—
281	C₄	C₂×D₈×D₁₃	1	1	yes	characteristic
282	C₄	C₂×#88166	1	1	yes	characteristic
283	C₄	C₂×#88202	1	2	yes	—
284	C₂	—	2	2	no	—
285	C₂	—	8	4	no	—
286	C₂	#13123548	1	1	yes	characteristic
287	C₂	C₂×D₈×Dic₁₃	1	1	yes	characteristic
288	C₂	C₂×#11718857	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₁₃
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₄²	C₂³×D₁₃
Fitting, prime radical	C₁₃×#3169	C₂
socle	C₂²×C₁₃	C₂×C₄×D₄
cosocle	C₄²×C₁₃	C₂⁴
hypercenter	#3169	D₁₃
hypocenter	C₁₃	#17956
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3169

Complements

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

N	H ≅ G/N	type
1. #13123548	C₂	semidirect
2. #13113721	C₂	semidirect
3. D₈×#1055	C₂	semidirect
4. #13113595	C₂	semidirect
5. #13113287	C₂	semidirect
6. C₂×C₈×#1055	C₂	semidirect
92. C₂×#1055	D₈	semidirect
125. C₈×C₁₃	#131	semidirect
240. C₁₃	#17956	semidirect
268. C₈	#88202	semidirect

Maximal subgroups

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

subgroup	index
#13123548	2
#13113721	2
D₈×#1055	2
#13113595	2
#13113287	2
C₂×C₈×#1055	2
C₁₃×#3169	2
#17956	13