Subgroups — #15861571

Order 3328.

938 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718685	C₂	1	2	yes	maximal
2	#13114711	C₂	1	4	yes	maximal
3	C₄⋊C₄×#1055	C₂	1	2	yes	maximal
4	#13114689	C₂	1	1	yes	maximal, characteristic
5	#13114689	C₂	1	2	yes	maximal
6	C₄×#88043	C₂	1	2	yes	maximal
7	C₄×#88043	C₂	1	1	yes	maximal, characteristic
8	C₄²×Q₈×C₁₃	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄×Q₈×C₁₃	C₄	1	2	yes	—
10	C₂×C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic
11	C₄×C₁₃×C₄⋊C₄	C₄	1	4	yes	—
12	C₄×C₁₃×C₄⋊C₄	C₄	1	2	yes	—
13	C₄×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
14	C₄×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
15	C₄³×C₁₃	C₂²	1	2	yes	—
16	C₄³×C₁₃	C₂²	1	1	yes	characteristic
17	#11718685	C₄	1	16	yes	complemented
18	C₂×C₄⋊#1055	C₂²	1	2	yes	—
19	C₂×C₄⋊#1055	C₂²	1	4	yes	—
20	C₂×#88043	C₂²	1	2	yes	—
21	C₂×#88043	C₂²	1	1	yes	characteristic
22	C₂×#88043	C₂²	1	2	yes	—
23	C₂×#88043	C₂²	1	4	yes	—
24	C₂×#88043	C₂²	1	1	yes	characteristic
25	C₂×C₄×#1055	C₂²	1	4	yes	—
26	C₂×C₄×#1055	C₂²	1	2	yes	—
27	#11718116	C₂²	1	4	yes	—
28	#11718116	C₂²	1	2	yes	—
29	C₂²×Q₈×C₁₃	C₂×C₄	1	1	yes	characteristic
30	C₄×Q₈×C₁₃	—	2	8	no	—
31	C₄×Q₈×C₁₃	C₂×C₄	1	8	yes	—
32	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	4	yes	—
33	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
34	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
35	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
36	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
37	C₂×C₄²×C₁₃	C₂×C₄	1	4	yes	—
38	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
39	C₂×C₄²×C₁₃	C₂×C₄	1	4	yes	—
40	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
41	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
42	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
43	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
44	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
45	C₂²×#1055	C₂³	1	4	yes	—
46	C₂²×#1055	C₂³	1	2	yes	—
47	C₂²×#1055	C₂³	1	2	yes	—
48	C₁₃×#95	C₂×C₄	1	4	yes	—
49	C₁₃×#95	C₂×C₄	1	2	yes	—
50	C₄⋊#1055	C₂×C₄	1	16	yes	—
51	C₄⋊#1055	C₂×C₄	1	8	yes	—
52	#88043	—	2	4	no	—
53	#88043	—	2	2	no	—
54	#88043	—	2	4	no	—
55	#88043	C₂×C₄	1	16	yes	—
56	#88043	Q₈	1	4	yes	complemented
57	C₄×#1055	—	2	8	no	—
58	C₄×#1055	C₂×C₄	1	8	yes	—
59	#20520	—	13	1	no	maximal, Sylow
60	C₂×Q₈×C₁₃	—	2	4	no	—
61	C₂×Q₈×C₁₃	C₄²	1	4	yes	—
62	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
63	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
65	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
66	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
67	C₂²×C₄×C₁₃	C₂²×C₄	1	4	yes	—
68	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
70	C₁₃×C₄⋊C₄	—	2	8	no	—
71	C₁₃×C₄⋊C₄	C₄²	1	8	yes	—
72	C₁₃×C₄⋊C₄	C₄²	1	4	yes	—
73	C₁₃×C₄⋊C₄	M₁₆	1	8	yes	—
74	C₄²×C₁₃	—	2	8	no	—
75	C₄²×C₁₃	—	2	4	no	—
76	C₄²×C₁₃	—	2	4	no	—
77	C₄²×C₁₃	—	2	2	no	—
78	C₄²×C₁₃	—	4	4	no	—
79	C₄²×C₁₃	—	4	2	no	—
80	C₄²×C₁₃	C₂²×C₄	1	4	yes	—
81	C₄²×C₁₃	C₂²×C₄	1	8	yes	—
82	C₄²×C₁₃	C₂×Q₈	1	2	yes	—
83	C₄²×C₁₃	#41	1	2	yes	—
84	C₂×#1055	—	2	4	no	—
85	C₂×#1055	—	2	4	no	—
86	C₂×#1055	—	2	2	no	—
87	C₂×#1055	—	2	2	no	—
88	C₂×#1055	C₂²×C₄	1	4	yes	—
89	C₂×#1055	C₂²×C₄	1	8	yes	—
90	C₂×#1055	C₂²×C₄	1	4	yes	—
91	C₂×#1055	C₂×Q₈	1	4	yes	—
92	C₂×#1055	#41	1	4	yes	—
93	C₂×#446	—	13	2	no	—
94	C₄²×Q₈	D₁₃	1	1	yes	characteristic
95	#1771	—	13	4	no	—
96	C₈×C₄⋊C₄	—	13	2	no	—
97	#1767	—	13	2	no	—
98	#1767	—	13	1	no	—
99	C₄×#97	—	13	1	no	—
100	C₄×#97	—	13	2	no	—
101	C₂³×C₁₃	C₂³×C₄	1	1	yes	characteristic
102	Q₈×C₁₃	—	2	8	no	—
103	C₂×C₄×C₁₃	—	2	4	no	—
104	C₂×C₄×C₁₃	—	2	2	no	—
105	C₂×C₄×C₁₃	—	2	4	no	—
106	C₂×C₄×C₁₃	—	2	2	no	—
107	C₂×C₄×C₁₃	—	2	4	no	—
108	C₂×C₄×C₁₃	—	2	2	no	—
109	C₂×C₄×C₁₃	—	2	4	no	—
110	C₂×C₄×C₁₃	—	2	2	no	—
111	C₂×C₄×C₁₃	—	2	2	no	—
112	C₂×C₄×C₁₃	—	2	2	no	—
113	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
114	C₂×C₄×C₁₃	C₂×C₄²	1	4	yes	—
115	C₂×C₄×C₁₃	C₂×C₄²	1	4	yes	—
116	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
117	C₂×C₄×C₁₃	C₄×Q₈	1	4	yes	—
118	C₂×C₄×C₁₃	C₂×M₁₆	1	8	yes	—
119	C₂×C₄×C₁₃	C₂×M₁₆	1	4	yes	—
120	C₂×C₄×C₁₃	C₂³×C₄	1	2	yes	—
121	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
122	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
123	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
124	#1055	—	2	8	no	—
125	#1055	—	2	4	no	—
126	#1055	C₄×Q₈	1	8	yes	complemented
127	C₂×C₄×Q₈	Dic₁₃	1	2	yes	—
128	C₂×C₄×Q₈	C₂×D₁₃	1	1	yes	characteristic
129	#446	—	13	16	no	—
130	C₂×C₄⋊C₈	—	13	2	no	—
131	C₂×C₄⋊C₈	—	13	4	no	—
132	C₂×#97	—	13	4	no	—
133	C₂×#97	—	13	1	no	—
134	C₂×#97	—	13	1	no	—
135	C₂×#97	—	13	2	no	—
136	C₂×#97	—	13	2	no	—
137	C₂×C₄×C₈	—	13	2	no	—
138	C₂×C₄×C₈	—	13	4	no	—
139	C₄×C₄⋊C₄	Dic₁₃	1	4	yes	—
140	C₄×C₄⋊C₄	Dic₁₃	1	2	yes	—
141	C₄×C₄⋊C₄	C₂×D₁₃	1	1	yes	characteristic
142	C₄×C₄⋊C₄	C₂×D₁₃	1	2	yes	—
143	C₄³	C₂×D₁₃	1	2	yes	—
144	C₄³	C₂×D₁₃	1	1	yes	characteristic
145	#336	—	13	4	no	—
146	#336	—	13	2	no	—
147	C₂²×C₁₃	C₂²×C₄²	1	1	yes	characteristic
148	C₂²×C₁₃	C₂×C₄×Q₈	1	1	yes	characteristic
149	C₂²×C₁₃	C₄×#41	1	1	yes	characteristic
150	C₂²×C₁₃	C₂²×M₁₆	1	2	yes	—
151	C₂²×C₁₃	#568	1	2	yes	—
152	C₄×C₁₃	—	2	4	no	—
153	C₄×C₁₃	—	2	2	no	—
154	C₄×C₁₃	—	2	4	no	—
155	C₄×C₁₃	—	2	4	no	—
156	C₄×C₁₃	—	2	8	no	—
157	C₄×C₁₃	—	4	2	no	—
158	C₄×C₁₃	C₄×M₁₆	1	4	yes	—
159	C₄×C₁₃	C₂×C₄×Q₈	1	2	yes	—
160	C₄×C₁₃	C₄×#41	1	2	yes	—
161	C₂²×Q₈	C₂×Dic₁₃	1	1	yes	characteristic
162	C₂²×C₈	—	13	2	no	—
163	C₂²×C₈	—	13	2	no	—
164	C₂²×C₈	—	13	4	no	—
165	C₄×Q₈	—	2	8	no	—
166	C₄×Q₈	C₄×D₁₃	1	8	yes	—
167	C₂×C₄⋊C₄	C₂×Dic₁₃	1	4	yes	—
168	C₂×C₄⋊C₄	C₂×Dic₁₃	1	2	yes	—
169	C₂×C₄⋊C₄	C₂×Dic₁₃	1	2	yes	—
170	C₂×C₄⋊C₄	C₂×Dic₁₃	1	1	yes	characteristic
171	C₂×C₄²	C₂×Dic₁₃	1	2	yes	—
172	C₂×C₄²	C₂×Dic₁₃	1	4	yes	—
173	C₂×C₄²	C₂×Dic₁₃	1	2	yes	—
174	C₂×C₄²	C₂×Dic₁₃	1	4	yes	—
175	C₂×C₄²	C₂²×D₁₃	1	2	yes	—
176	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
177	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
178	C₂×C₄²	C₂²×D₁₃	1	2	yes	—
179	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
180	C₄⋊C₈	—	13	8	no	—
181	C₄⋊C₈	—	13	16	no	—
182	#97	—	13	16	no	—
183	#97	—	13	4	no	—
184	#97	—	26	4	no	—
185	#97	—	26	2	no	—
186	#97	—	26	4	no	—
187	C₄×C₈	—	13	8	no	—
188	C₄×C₈	—	26	8	no	—
189	#95	C₂×Dic₁₃	1	4	yes	—
190	#95	C₂×Dic₁₃	1	2	yes	—
191	C₂×C₁₃	C₄²×Q₈	1	1	yes	characteristic
192	C₂×C₁₃	C₂×C₄×M₁₆	1	1	yes	characteristic
193	C₂×C₁₃	#2875	1	1	yes	characteristic
194	C₂×C₁₃	Q₈×M₁₆	1	2	yes	—
195	C₂×C₁₃	#2970	1	2	yes	—
196	C₂×Q₈	—	2	4	no	—
197	C₂×Q₈	C₄×Dic₁₃	1	4	yes	—
198	C₂²×C₄	C₂²×Dic₁₃	1	2	yes	—
199	C₂²×C₄	C₂²×Dic₁₃	1	4	yes	—
200	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
201	C₂²×C₄	C₂²×Dic₁₃	1	2	yes	—
202	C₂²×C₄	C₂²×Dic₁₃	1	2	yes	—
203	C₂²×C₄	C₂²×Dic₁₃	1	2	yes	—
204	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
205	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
206	C₂×C₈	—	13	4	no	—
207	C₂×C₈	—	13	4	no	—
208	C₂×C₈	—	13	4	no	—
209	C₂×C₈	—	13	4	no	—
210	C₂×C₈	—	13	8	no	—
211	C₂×C₈	—	26	2	no	—
212	C₂×C₈	—	26	4	no	—
213	C₂×C₈	—	26	4	no	—
214	C₂×C₈	—	26	2	no	—
215	C₄⋊C₄	—	2	8	no	—
216	C₄⋊C₄	#6099	1	8	yes	—
217	C₄⋊C₄	C₄×Dic₁₃	1	8	yes	—
218	C₄⋊C₄	C₄×Dic₁₃	1	4	yes	—
219	C₄²	—	2	8	no	—
220	C₄²	—	2	4	no	—
221	C₄²	—	2	4	no	—
222	C₄²	—	2	2	no	—
223	C₄²	—	4	4	no	—
224	C₄²	—	4	2	no	—
225	C₄²	C₂×C₄×D₁₃	1	4	yes	—
226	C₄²	C₂×C₄×D₁₃	1	8	yes	—
227	C₄²	Q₈×D₁₃	1	2	yes	—
228	C₄²	#6131	1	2	yes	—
229	C₁₃	#20520	1	1	yes	characteristic, complemented, Sylow
230	C₂³	C₂³×Dic₁₃	1	1	yes	characteristic
231	Q₈	—	2	8	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	4	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	4	no	—
236	C₂×C₄	—	2	4	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	4	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	C₂×#6099	1	8	yes	—
243	C₂×C₄	C₂×#6099	1	4	yes	—
244	C₂×C₄	C₂×C₄×Dic₁₃	1	2	yes	—
245	C₂×C₄	C₂×C₄×Dic₁₃	1	4	yes	—
246	C₂×C₄	C₂×C₄×Dic₁₃	1	4	yes	—
247	C₂×C₄	C₂×C₄×Dic₁₃	1	2	yes	—
248	C₂×C₄	Q₈×Dic₁₃	1	4	yes	—
249	C₂×C₄	C₂²×C₄×D₁₃	1	2	yes	—
250	C₂×C₄	C₂×Q₈×D₁₃	1	1	yes	characteristic
251	C₂×C₄	C₂×#6131	1	1	yes	characteristic
252	C₂×C₄	#41×D₁₃	1	2	yes	—
253	C₈	—	13	8	no	—
254	C₈	—	26	4	no	—
255	C₈	—	26	8	no	—
256	C₂²	C₂²×#6099	1	2	yes	—
257	C₂²	C₂²×C₄×Dic₁₃	1	1	yes	characteristic
258	C₂²	C₂×Q₈×Dic₁₃	1	1	yes	characteristic
259	C₂²	#11719524	1	2	yes	—
260	C₂²	#41×Dic₁₃	1	1	yes	characteristic
261	C₄	—	2	8	no	—
262	C₄	—	2	4	no	—
263	C₄	—	2	2	no	—
264	C₄	—	2	4	no	—
265	C₄	—	2	4	no	—
266	C₄	—	4	2	no	—
267	C₄	C₄×#6099	1	4	yes	—
268	C₄	C₄×Q₈×D₁₃	1	2	yes	—
269	C₄	C₄×#6131	1	2	yes	—
270	C₂	C₄×Q₈×Dic₁₃	1	1	yes	characteristic
271	C₂	C₂×C₄×#6099	1	1	yes	characteristic
272	C₂	#13123264	1	1	yes	characteristic
273	C₂	Q₈×#6099	1	2	yes	—
274	C₂	#13123850	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₁₃
commutator	C₂²×C₁₃	C₂²×C₄²
Fitting	C₄²×Q₈×C₁₃	C₂
socle, prime radical	C₂³×C₁₃	C₂³×C₄
cosocle	C₂²×C₄×C₁₃	C₂⁴
hypercenter	C₄²×Q₈	D₁₃
hypocenter	C₁₃	#20520
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄²×Q₈

Complements

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

N	H ≅ G/N	type
17. #11718685	C₄	semidirect
56. #88043	Q₈	semidirect
126. #1055	C₄×Q₈	semidirect
229. C₁₃	#20520	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718685	2
#13114711	2
C₄⋊C₄×#1055	2
#13114689	2
C₄×#88043	2
C₄²×Q₈×C₁₃	2
#20520	13