Subgroups — #15881930

Order 3328.

622 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×#6099	C₂	1	2	yes	maximal
2	#13115612	C₂	1	4	yes	maximal
3	#13115612	C₂	1	2	yes	maximal
4	#13114977	C₂	1	2	yes	maximal
5	#13114974	C₂	1	1	yes	maximal, characteristic
6	#13114742	C₂	1	2	yes	maximal
7	C₄×#88059	C₂	1	1	yes	maximal, characteristic
8	C₂×C₁₃×#446	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic
10	C₁₃×#446	C₄	1	4	yes	—
11	C₁₃×#446	C₄	1	4	yes	—
12	C₂×C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
13	C₂×C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
14	C₂×C₁₃×#97	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
16	C₂×#88059	C₂²	1	2	yes	—
17	C₂×#88059	C₂²	1	4	yes	—
18	#11718685	—	2	2	no	—
19	#11718685	—	2	4	no	—
20	Q₈×#1055	—	2	2	no	—
21	#11718585	C₂²	1	2	yes	—
22	#11718585	C₂²	1	4	yes	—
23	C₄×#6099	C₂²	1	2	yes	—
24	C₄×#6099	C₂²	1	4	yes	—
25	#11718144	C₂²	1	2	yes	—
26	#11718144	C₂²	1	2	yes	—
27	#11718144	C₂²	1	1	yes	characteristic
28	#11718144	C₂²	1	4	yes	—
29	#11718144	C₂²	1	1	yes	characteristic
30	C₂²×Q₈×C₁₃	Q₈	1	1	yes	characteristic
31	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
32	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
33	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
34	C₄×Q₈×C₁₃	C₂×C₄	1	2	yes	—
35	C₄×Q₈×C₁₃	C₂×C₄	1	2	yes	—
36	C₄×Q₈×C₁₃	D₄	1	2	yes	—
37	C₄×Q₈×C₁₃	D₄	1	2	yes	—
38	C₂×C₁₃×C₄⋊C₄	Q₈	1	1	yes	characteristic
39	C₂×C₁₃×C₄⋊C₄	Q₈	1	2	yes	—
40	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
41	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
42	C₂×#6099	C₂³	1	4	yes	—
43	C₂×#6099	C₂³	1	2	yes	—
44	C₂×#6099	C₂³	1	2	yes	—
45	C₁₃×C₄⋊C₈	C₂×C₄	1	2	yes	—
46	C₁₃×C₄⋊C₈	C₂×C₄	1	2	yes	—
47	C₁₃×C₄⋊C₈	C₂×C₄	1	8	yes	—
48	C₁₃×#97	C₂×C₄	1	8	yes	—
49	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
50	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
51	#88059	—	2	4	no	—
52	#88059	—	2	8	no	—
53	C₄⋊#1055	—	2	2	no	—
54	C₄⋊#1055	—	2	4	no	—
55	C₄⋊#1055	—	2	4	no	—
56	C₄⋊#1055	—	2	2	no	—
57	#88043	—	2	4	no	—
58	#88043	—	2	2	no	—
59	C₄×#1055	—	2	2	no	—
60	C₄×#1055	—	2	4	no	—
61	#22747	—	13	1	no	maximal, Sylow
62	C₂×Q₈×C₁₃	—	4	2	no	—
63	C₂×Q₈×C₁₃	C₄⋊C₄	1	2	yes	—
64	C₂×Q₈×C₁₃	C₄⋊C₄	1	2	yes	—
65	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
66	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
67	C₂²×C₄×C₁₃	C₂×Q₈	1	2	yes	—
68	C₂²×C₄×C₁₃	C₂×Q₈	1	2	yes	—
69	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
70	C₂×C₈×C₁₃	—	2	2	no	—
71	C₂×C₈×C₁₃	—	2	4	no	—
72	C₂×C₈×C₁₃	—	2	2	no	—
73	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
74	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
75	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
76	C₁₃×C₄⋊C₄	C₄⋊C₄	1	4	yes	—
77	C₁₃×C₄⋊C₄	C₄⋊C₄	1	4	yes	—
78	C₁₃×C₄⋊C₄	C₄⋊C₄	1	2	yes	—
79	C₁₃×C₄⋊C₄	C₄⋊C₄	1	2	yes	—
80	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
81	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
82	C₄²×C₁₃	C₂²×C₄	1	4	yes	—
83	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
84	C₄²×C₁₃	C₂×D₄	1	4	yes	—
85	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
86	#6099	—	2	2	no	—
87	#6099	—	2	4	no	—
88	#6099	—	2	2	no	—
89	#6099	—	2	4	no	—
90	#6099	—	2	2	no	—
91	#6099	#41	1	4	yes	—
92	C₂×#1055	—	2	2	no	—
93	C₂×#1055	—	2	4	no	—
94	C₂×#1055	—	2	2	no	—
95	Q₈×M₁₆	—	13	2	no	—
96	C₂×#446	D₁₃	1	1	yes	characteristic
97	#1950	—	13	4	no	—
98	#1950	—	13	2	no	—
99	#1838	—	13	2	no	—
100	#1837	—	13	1	no	—
101	#1778	—	13	2	no	—
102	C₄×#108	—	13	1	no	—
103	C₂³×C₁₃	C₂²×Q₈	1	1	yes	characteristic
104	Q₈×C₁₃	—	2	4	no	—
105	Q₈×C₁₃	—	4	2	no	—
106	C₂×C₄×C₁₃	—	2	2	no	—
107	C₂×C₄×C₁₃	—	4	1	no	—
108	C₂×C₄×C₁₃	—	4	2	no	—
109	C₂×C₄×C₁₃	—	4	2	no	—
110	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
111	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
112	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	4	yes	—
113	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
114	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	4	yes	—
115	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
116	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
117	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
118	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
119	C₂×C₄×C₁₃	2₋¹⁺⁴	1	2	yes	—
120	C₈×C₁₃	—	2	2	no	—
121	C₈×C₁₃	—	2	4	no	—
122	C₈×C₁₃	—	2	2	no	—
123	#1055	—	2	4	no	—
124	#1055	—	4	4	no	—
125	#1055	—	4	2	no	—
126	C₂×C₄×Q₈	C₂×D₁₃	1	1	yes	characteristic
127	#446	—	26	2	no	—
128	#446	—	26	4	no	—
129	#446	Dic₁₃	1	4	yes	—
130	#446	Dic₁₃	1	4	yes	—
131	C₈×Q₈	—	26	2	no	—
132	C₂×#108	—	13	2	no	—
133	C₂×#108	—	13	4	no	—
134	#423	—	13	2	no	—
135	#423	—	13	4	no	—
136	C₂×C₄⋊C₈	C₂×D₁₃	1	1	yes	characteristic
137	C₂×C₄⋊C₈	C₂×D₁₃	1	2	yes	—
138	C₄×M₁₆	—	13	2	no	—
139	C₄×M₁₆	—	13	4	no	—
140	C₂×#97	C₂×D₁₃	1	2	yes	—
141	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
142	#341	—	13	2	no	—
143	#341	—	13	2	no	—
144	#341	—	13	1	no	—
145	#341	—	13	4	no	—
146	#341	—	13	1	no	—
147	C₂²×C₁₃	—	2	1	no	—
148	C₂²×C₁₃	—	2	1	no	—
149	C₂²×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
150	C₂²×C₁₃	#557	1	2	yes	—
151	C₄×C₁₃	—	2	1	no	—
152	C₄×C₁₃	—	2	4	no	—
153	C₄×C₁₃	—	2	1	no	—
154	C₄×C₁₃	—	2	1	no	—
155	C₄×C₁₃	—	2	4	no	—
156	C₄×C₁₃	#430	1	2	yes	—
157	C₄×C₁₃	C₄×#41	1	1	yes	characteristic
158	C₄×C₁₃	#519	1	1	yes	characteristic
159	C₄×C₁₃	#548	1	2	yes	—
160	C₂²×Q₈	Dic₂₆	1	1	yes	characteristic
161	C₂×M₁₆	—	13	2	no	—
162	C₂×M₁₆	—	13	4	no	—
163	C₂×M₁₆	—	13	2	no	—
164	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
165	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
166	C₂²×C₈	C₂²×D₁₃	1	2	yes	—
167	C₄×Q₈	D₅₂	1	2	yes	—
168	C₄×Q₈	D₅₂	1	2	yes	—
169	C₄×Q₈	C₂×Dic₁₃	1	2	yes	—
170	C₄×Q₈	C₂×Dic₁₃	1	2	yes	—
171	C₂×C₄⋊C₄	Dic₂₆	1	1	yes	characteristic
172	C₂×C₄⋊C₄	Dic₂₆	1	2	yes	—
173	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
174	C₂×C₄²	C₂²×D₁₃	1	2	yes	—
175	#108	—	26	4	no	—
176	#108	—	26	8	no	—
177	C₄⋊C₈	—	26	2	no	—
178	C₄⋊C₈	—	26	4	no	—
179	C₄⋊C₈	—	26	4	no	—
180	C₄⋊C₈	—	26	2	no	—
181	C₄⋊C₈	C₂×Dic₁₃	1	2	yes	—
182	C₄⋊C₈	C₂×Dic₁₃	1	2	yes	—
183	C₄⋊C₈	C₂×Dic₁₃	1	8	yes	—
184	#97	—	26	4	no	—
185	#97	—	26	2	no	—
186	#97	C₂×Dic₁₃	1	8	yes	—
187	C₄×C₈	—	26	2	no	—
188	C₄×C₈	—	26	4	no	—
189	C₄×C₈	C₂×Dic₁₃	1	2	yes	—
190	C₄×C₈	C₂×Dic₁₃	1	2	yes	—
191	C₂×C₁₃	—	2	2	no	—
192	C₂×C₁₃	#2351	1	1	yes	characteristic
193	C₂×C₁₃	C₂×#430	1	1	yes	characteristic
194	C₂×C₁₃	#2916	1	1	yes	characteristic
195	C₂×Q₈	—	4	2	no	—
196	C₂×Q₈	C₄⋊Dic₁₃	1	2	yes	—
197	C₂×Q₈	C₄⋊Dic₁₃	1	2	yes	—
198	C₂²×C₄	C₂×Dic₂₆	1	1	yes	characteristic
199	C₂²×C₄	C₂×Dic₂₆	1	1	yes	characteristic
200	C₂²×C₄	C₂×Dic₂₆	1	2	yes	—
201	C₂²×C₄	C₂×Dic₂₆	1	2	yes	—
202	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
203	M₁₆	—	13	4	no	—
204	M₁₆	—	26	2	no	—
205	M₁₆	—	26	4	no	—
206	M₁₆	—	26	2	no	—
207	M₁₆	—	26	2	no	—
208	M₁₆	—	26	4	no	—
209	C₂×C₈	—	2	2	no	—
210	C₂×C₈	—	2	2	no	—
211	C₂×C₈	—	2	4	no	—
212	C₂×C₈	—	26	2	no	—
213	C₂×C₈	—	26	2	no	—
214	C₂×C₈	—	26	4	no	—
215	C₂×C₈	C₂²×Dic₁₃	1	2	yes	—
216	C₂×C₈	C₂²×Dic₁₃	1	2	yes	—
217	C₂×C₈	C₂²×Dic₁₃	1	4	yes	—
218	C₄⋊C₄	C₄⋊Dic₁₃	1	4	yes	—
219	C₄⋊C₄	C₄⋊Dic₁₃	1	2	yes	—
220	C₄⋊C₄	C₄⋊Dic₁₃	1	4	yes	—
221	C₄⋊C₄	C₄⋊Dic₁₃	1	2	yes	—
222	C₄²	C₂×D₅₂	1	1	yes	characteristic
223	C₄²	C₂×D₅₂	1	4	yes	—
224	C₄²	C₂×D₅₂	1	1	yes	characteristic
225	C₄²	C₂²×Dic₁₃	1	1	yes	characteristic
226	C₄²	C₂²×Dic₁₃	1	1	yes	characteristic
227	C₄²	C₂²×Dic₁₃	1	4	yes	—
228	C₁₃	#22747	1	1	yes	characteristic, complemented, Sylow
229	C₂³	C₂²×Dic₂₆	1	1	yes	characteristic
230	Q₈	—	2	4	no	—
231	Q₈	—	4	2	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	4	1	no	—
234	C₂×C₄	—	4	2	no	—
235	C₂×C₄	—	4	2	no	—
236	C₂×C₄	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
237	C₂×C₄	C₂×C₄⋊Dic₁₃	1	4	yes	—
238	C₂×C₄	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
239	C₂×C₄	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
240	C₂×C₄	C₂×C₄⋊Dic₁₃	1	4	yes	—
241	C₂×C₄	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
242	C₂×C₄	C₂²×D₅₂	1	1	yes	characteristic
243	C₂×C₄	#88254	1	2	yes	—
244	C₂×C₄	#41×D₁₃	1	2	yes	—
245	C₂×C₄	C₂³×Dic₁₃	1	1	yes	characteristic
246	C₈	—	2	2	no	—
247	C₈	—	2	2	no	—
248	C₈	—	2	4	no	—
249	C₈	—	26	4	no	—
250	C₈	—	52	2	no	—
251	C₈	—	52	4	no	—
252	C₂²	—	2	1	no	—
253	C₂²	—	2	1	no	—
254	C₂²	#11719274	1	2	yes	—
255	C₂²	C₂²×C₄⋊Dic₁₃	1	1	yes	characteristic
256	C₄	—	2	1	no	—
257	C₄	—	2	4	no	—
258	C₄	—	2	1	no	—
259	C₄	—	2	4	no	—
260	C₄	—	2	1	no	—
261	C₄	#11718786	1	2	yes	—
262	C₄	#11719281	1	2	yes	—
263	C₄	#11719517	1	1	yes	characteristic
264	C₄	#41×Dic₁₃	1	1	yes	characteristic
265	C₂	—	2	2	no	—
266	C₂	#13117757	1	1	yes	characteristic
267	C₂	C₂×#11718786	1	1	yes	characteristic
268	C₂	#13123447	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₅₂
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₁₃
Fitting	C₂×C₁₃×#446	C₂
socle	C₂²×C₁₃	C₂²×C₄⋊C₄
prime radical	C₂³×C₁₃	C₂²×Q₈
cosocle	C₂²×C₄×C₁₃	C₂⁴
hypercenter	C₂×#446	D₁₃
hypocenter	C₁₃	#22747
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×Q₈ ◁ C₂×#446

Complements

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

N	H ≅ G/N	type
228. C₁₃	#22747	semidirect

Maximal subgroups

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

subgroup	index
Q₈×#6099	2
#13115612	2
#13114977	2
#13114974	2
#13114742	2
C₄×#88059	2
C₂×C₁₃×#446	2
#22747	13