Subgroups — #15863354

Order 3328.

1102 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718782	C₂	1	4	yes	maximal
2	#13117243	C₂	1	1	yes	maximal, characteristic
3	C₂×#11718597	C₂	1	1	yes	maximal, characteristic
4	#13115004	C₂	1	4	yes	maximal, complemented
5	#13115000	C₂	1	4	yes	maximal, complemented
6	C₂×C₄×C₁₃×M₁₆	C₂	1	1	yes	maximal, characteristic
7	C₂²×C₁₃×M₁₆	C₂²	1	2	yes	—
8	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
9	C₂×#88180	C₂²	1	2	yes	—
10	C₂²×C₄⋊Dic₁₃	C₂²	1	2	yes	—
11	C₄×C₁₃×M₁₆	C₄	1	8	yes	complemented
12	C₂×C₁₃×#97	C₂²	1	2	yes	—
13	C₂×C₄×C₈×C₁₃	C₂²	1	2	yes	—
14	#11718873	—	2	2	no	—
15	C₂×#88071	—	2	2	no	—
16	#11718782	—	2	16	no	—
17	C₂×#88058	C₂²	1	8	yes	—
18	C₂×#88057	C₂²	1	8	yes	—
19	#11718597	—	2	2	no	—
20	#11718597	C₂²	1	2	yes	—
21	#11718597	C₂²	1	2	yes	—
22	#11718597	C₂²	1	2	yes	—
23	#11718596	C₂²	1	2	yes	—
24	C₄×C₄⋊Dic₁₃	—	2	2	no	—
25	C₂³×C₄×C₁₃	D₄	1	2	yes	—
26	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
27	C₂³×Dic₁₃	—	2	2	no	—
28	C₂×C₁₃×M₁₆	C₂×C₄	1	8	yes	—
29	C₂×C₁₃×M₁₆	D₄	1	8	yes	—
30	C₂×C₁₃×M₁₆	Q₈	1	8	yes	—
31	C₂²×C₈×C₁₃	C₂³	1	4	yes	—
32	C₂×C₄²×C₁₃	C₂×C₄	1	4	yes	—
33	C₂×C₄²×C₁₃	D₄	1	2	yes	—
34	C₂×C₄²×C₁₃	Q₈	1	4	yes	—
35	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
36	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
37	C₂×C₂²⋊Dic₁₃	—	2	2	no	—
38	#88180	—	2	8	no	—
39	C₂×C₄⋊Dic₁₃	—	2	8	no	—
40	C₂×C₄⋊Dic₁₃	—	2	2	no	—
41	C₂×C₄⋊Dic₁₃	—	2	4	no	—
42	C₂×C₄⋊Dic₁₃	C₂³	1	4	yes	—
43	C₂×C₄⋊Dic₁₃	C₂³	1	4	yes	—
44	C₂×C₄×Dic₁₃	—	2	2	no	—
45	C₁₃×#97	C₂×C₄	1	8	yes	complemented
46	C₄×C₈×C₁₃	C₂×C₄	1	8	yes	complemented
47	#88071	—	2	4	no	—
48	#88071	—	2	4	no	—
49	#88071	—	2	4	no	—
50	#88058	—	2	16	no	—
51	#88057	—	2	16	no	—
52	#20801	—	13	1	no	maximal, Sylow
53	C₂⁴×C₁₃	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄×C₁₃	—	4	4	no	—
55	C₂²×C₄×C₁₃	C₄⋊C₄	1	8	yes	—
56	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
57	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
58	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
59	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₁₃	C₂×D₄	1	4	yes	—
61	C₂²×C₄×C₁₃	C₂×Q₈	1	4	yes	—
62	C₂²×C₄×C₁₃	C₂×Q₈	1	2	yes	—
63	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
64	C₂²×Dic₁₃	—	2	4	no	—
65	C₂²×Dic₁₃	—	2	4	no	—
66	C₂²×Dic₁₃	—	4	8	no	—
67	C₂²×Dic₁₃	—	4	4	no	—
68	C₁₃×M₁₆	C₄⋊C₄	1	32	yes	complemented
69	C₂×C₈×C₁₃	C₂²×C₄	1	8	yes	—
70	C₂×C₈×C₁₃	C₂×D₄	1	8	yes	—
71	C₂×C₈×C₁₃	C₂×Q₈	1	8	yes	—
72	C₄²×C₁₃	C₄⋊C₄	1	8	yes	—
73	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
74	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
75	C₄²×C₁₃	C₂×Q₈	1	2	yes	—
76	C₄²×C₁₃	C₂×Q₈	1	2	yes	—
77	C₂²⋊Dic₁₃	—	4	4	no	—
78	C₄⋊Dic₁₃	—	2	8	no	—
79	C₄⋊Dic₁₃	—	2	4	no	—
80	C₄⋊Dic₁₃	—	2	4	no	—
81	C₄⋊Dic₁₃	—	4	8	no	—
82	C₄⋊Dic₁₃	—	4	4	no	—
83	C₄×Dic₁₃	—	4	4	no	—
84	C₂×#428	—	13	4	no	—
85	C₂×C₄×M₁₆	D₁₃	1	1	yes	characteristic
86	#2285	—	13	1	no	—
87	C₂×#382	—	13	1	no	—
88	#1847	—	13	4	no	—
89	#1846	—	13	4	no	—
90	C₂³×C₁₃	—	2	4	no	—
91	C₂³×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
92	C₂³×C₁₃	#127	1	2	yes	—
93	C₂³×C₁₃	C₄⋊Q₈	1	2	yes	—
94	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
95	C₂×C₄×C₁₃	—	2	4	no	—
96	C₂×C₄×C₁₃	—	2	4	no	—
97	C₂×C₄×C₁₃	—	4	8	no	—
98	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	4	yes	—
99	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	4	yes	—
100	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	8	yes	—
101	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
102	C₂×C₄×C₁₃	#127	1	2	yes	—
103	C₂×C₄×C₁₃	C₄⋊Q₈	1	8	yes	—
104	C₂×C₄×C₁₃	C₄⋊Q₈	1	2	yes	—
105	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
106	C₂×C₄×C₁₃	C₂²×D₄	1	2	yes	—
107	C₂×C₄×C₁₃	C₂²×Q₈	1	2	yes	—
108	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
109	C₂×Dic₁₃	—	4	4	no	—
110	C₂×Dic₁₃	—	4	4	no	—
111	C₂×Dic₁₃	—	4	8	no	—
112	C₂×Dic₁₃	—	4	8	no	—
113	C₂×Dic₁₃	—	8	8	no	—
114	C₈×C₁₃	C₂×C₄⋊C₄	1	16	yes	—
115	C₂²×M₁₆	C₂×D₁₃	1	2	yes	—
116	C₂×#117	—	13	2	no	—
117	C₂²×C₄⋊C₄	—	13	2	no	—
118	C₂²×C₄²	C₂×D₁₃	1	1	yes	characteristic
119	#428	—	26	16	no	—
120	C₂×#107	—	13	8	no	—
121	C₂×#106	—	13	8	no	—
122	C₄×M₁₆	Dic₁₃	1	8	yes	complemented
123	C₂×#97	C₂×D₁₃	1	2	yes	—
124	C₂×C₄×C₈	C₂×D₁₃	1	2	yes	—
125	#384	—	13	2	no	—
126	#382	—	13	2	no	—
127	#382	—	13	2	no	—
128	#382	—	13	2	no	—
129	#382	—	26	2	no	—
130	#380	—	26	2	no	—
131	C₄×C₄⋊C₄	—	26	2	no	—
132	C₂×#95	—	26	2	no	—
133	C₂²×C₁₃	—	2	4	no	—
134	C₂²×C₁₃	—	2	4	no	—
135	C₂²×C₁₃	—	2	4	no	—
136	C₂²×C₁₃	#384	1	4	yes	—
137	C₂²×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
138	C₂²×C₁₃	C₂×#127	1	1	yes	characteristic
139	C₂²×C₁₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
140	C₂²×C₁₃	#575	1	4	yes	—
141	C₄×C₁₃	—	2	4	no	—
142	C₄×C₁₃	—	2	4	no	—
143	C₄×C₁₃	#384	1	4	yes	—
144	C₄×C₁₃	C₂²×C₄⋊C₄	1	2	yes	—
145	C₄×C₁₃	C₂×C₄⋊Q₈	1	2	yes	—
146	Dic₁₃	—	8	4	no	—
147	Dic₁₃	—	8	4	no	—
148	C₂³×C₄	—	26	2	no	—
149	C₂³×C₄	D₅₂	1	2	yes	—
150	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
151	C₂×M₁₆	Dic₂₆	1	8	yes	—
152	C₂×M₁₆	D₅₂	1	8	yes	—
153	C₂×M₁₆	C₂×Dic₁₃	1	8	yes	—
154	C₂²×C₈	C₂²×D₁₃	1	4	yes	—
155	#117	—	26	8	no	—
156	C₂×C₄⋊C₄	—	13	4	no	—
157	C₂×C₄⋊C₄	—	13	4	no	—
158	C₂×C₄⋊C₄	—	26	8	no	—
159	C₂×C₄⋊C₄	—	26	2	no	—
160	C₂×C₄⋊C₄	—	26	4	no	—
161	C₂×C₂²⋊C₄	—	26	2	no	—
162	C₂×C₄²	—	26	2	no	—
163	C₂×C₄²	Dic₂₆	1	4	yes	—
164	C₂×C₄²	D₅₂	1	2	yes	—
165	C₂×C₄²	C₂×Dic₁₃	1	4	yes	—
166	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
167	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
168	#107	—	26	16	no	—
169	#106	—	26	16	no	—
170	#97	C₂×Dic₁₃	1	8	yes	complemented
171	C₄×C₈	C₂×Dic₁₃	1	8	yes	complemented
172	#95	—	26	4	no	—
173	#95	—	26	4	no	—
174	#95	—	26	4	no	—
175	C₂×C₁₃	—	2	4	no	—
176	C₂×C₁₃	C₂×#384	1	1	yes	characteristic
177	C₂×C₁₃	#2911	1	2	yes	—
178	C₂×C₁₃	#3153	1	2	yes	—
179	C₂×C₁₃	#3166	1	2	yes	—
180	C₂⁴	C₂×D₅₂	1	1	yes	characteristic
181	C₂²×C₄	—	4	4	no	—
182	C₂²×C₄	—	26	4	no	—
183	C₂²×C₄	—	26	4	no	—
184	C₂²×C₄	—	52	8	no	—
185	C₂²×C₄	—	52	4	no	—
186	C₂²×C₄	C₄⋊Dic₁₃	1	8	yes	—
187	C₂²×C₄	C₂×Dic₂₆	1	2	yes	—
188	C₂²×C₄	C₂×Dic₂₆	1	4	yes	—
189	C₂²×C₄	C₂×D₅₂	1	2	yes	—
190	C₂²×C₄	C₂×D₅₂	1	2	yes	—
191	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
192	C₂²×C₄	C₂×D₅₂	1	4	yes	—
193	C₂²×C₄	C₂²×Dic₁₃	1	2	yes	—
194	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
195	M₁₆	C₄⋊Dic₁₃	1	32	yes	complemented
196	C₂×C₈	C₂×Dic₂₆	1	8	yes	—
197	C₂×C₈	C₂×D₅₂	1	8	yes	—
198	C₂×C₈	C₂²×Dic₁₃	1	8	yes	—
199	C₄⋊C₄	—	26	4	no	—
200	C₄⋊C₄	—	26	8	no	—
201	C₄⋊C₄	—	26	4	no	—
202	C₄⋊C₄	—	52	8	no	—
203	C₄⋊C₄	—	52	4	no	—
204	C₂²⋊C₄	—	52	4	no	—
205	C₄²	—	52	4	no	—
206	C₄²	C₄⋊Dic₁₃	1	8	yes	—
207	C₄²	C₂×Dic₂₆	1	2	yes	—
208	C₄²	C₂×Dic₂₆	1	2	yes	—
209	C₄²	C₂²×Dic₁₃	1	2	yes	—
210	C₄²	C₂²×Dic₁₃	1	2	yes	—
211	C₁₃	#20801	1	1	yes	characteristic, complemented, Sylow
212	C₂³	—	2	4	no	—
213	C₂³	#88123	1	2	yes	—
214	C₂³	#88128	1	2	yes	—
215	C₂³	C₂×C₄⋊Dic₁₃	1	2	yes	—
216	C₂³	C₂²×D₅₂	1	1	yes	characteristic
217	C₂×C₄	—	2	4	no	—
218	C₂×C₄	—	2	4	no	—
219	C₂×C₄	—	4	8	no	—
220	C₂×C₄	—	52	4	no	—
221	C₂×C₄	—	52	4	no	—
222	C₂×C₄	—	52	8	no	—
223	C₂×C₄	—	52	8	no	—
224	C₂×C₄	—	104	8	no	—
225	C₂×C₄	#88123	1	8	yes	—
226	C₂×C₄	#88123	1	2	yes	—
227	C₂×C₄	#88128	1	2	yes	—
228	C₂×C₄	C₂×C₄⋊Dic₁₃	1	4	yes	—
229	C₂×C₄	C₂×C₄⋊Dic₁₃	1	4	yes	—
230	C₂×C₄	C₂×C₄⋊Dic₁₃	1	8	yes	—
231	C₂×C₄	C₂×C₄⋊Dic₁₃	1	2	yes	—
232	C₂×C₄	C₂²×Dic₂₆	1	1	yes	characteristic
233	C₂×C₄	C₂²×Dic₂₆	1	2	yes	—
234	C₂×C₄	C₂²×D₅₂	1	2	yes	—
235	C₂×C₄	C₂³×Dic₁₃	1	1	yes	characteristic
236	C₈	C₂×C₄⋊Dic₁₃	1	16	yes	—
237	C₂²	—	2	4	no	—
238	C₂²	—	2	4	no	—
239	C₂²	—	2	4	no	—
240	C₂²	#11718596	1	4	yes	—
241	C₂²	C₂×#88123	1	1	yes	characteristic
242	C₂²	C₂×#88128	1	1	yes	characteristic
243	C₂²	#11719453	1	4	yes	—
244	C₂²	C₂²×C₄⋊Dic₁₃	1	1	yes	characteristic
245	C₄	—	2	4	no	—
246	C₄	—	2	4	no	—
247	C₄	—	104	4	no	—
248	C₄	—	104	4	no	—
249	C₄	#11718596	1	4	yes	—
250	C₄	C₂×#88123	1	2	yes	—
251	C₄	C₂²×C₄⋊Dic₁₃	1	2	yes	—
252	C₂	—	2	4	no	—
253	C₂	C₂×#11718596	1	1	yes	characteristic
254	C₂	#13123426	1	2	yes	—
255	C₂	#13125302	1	2	yes	—
256	C₂	#13125388	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂²×D₅₂
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₁₃
Fitting	C₂×C₄×C₁₃×M₁₆	C₂
socle	C₂³×C₁₃	C₂²×D₄
prime radical	C₂⁴×C₁₃	C₂×D₄
cosocle	C₂²×C₄×C₁₃	C₂⁴
hypercenter	C₂×C₄×M₁₆	D₁₃
hypocenter	C₁₃	#20801
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂×C₄×M₁₆

Complements

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

N	H ≅ G/N	type
4. #13115004	C₂	semidirect
5. #13115000	C₂	semidirect
11. C₄×C₁₃×M₁₆	C₄	semidirect
45. C₁₃×#97	C₂×C₄	semidirect
46. C₄×C₈×C₁₃	C₂×C₄	semidirect
68. C₁₃×M₁₆	C₄⋊C₄	semidirect
122. C₄×M₁₆	Dic₁₃	semidirect
170. #97	C₂×Dic₁₃	semidirect
171. C₄×C₈	C₂×Dic₁₃	semidirect
195. M₁₆	C₄⋊Dic₁₃	semidirect
211. C₁₃	#20801	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718782	2
#13117243	2
C₂×#11718597	2
#13115004	2
#13115000	2
C₂×C₄×C₁₃×M₁₆	2
#20801	13