Subgroups — #16015530

Order 3328.

806 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718363	C₂	1	2	yes	maximal
2	C₂×#11718344	C₂	1	1	yes	maximal, characteristic
3	C₂×#11718341	C₂	1	1	yes	maximal, characteristic
4	#13120955	C₂	1	2	yes	maximal
5	#13115004	C₂	1	4	yes	maximal
6	#13115000	C₂	1	2	yes	maximal
7	#13114993	C₂	1	2	yes	maximal
8	C₁₃×#1726	C₂	1	1	yes	maximal, characteristic
9	C₂×#88124	C₂²	1	2	yes	—
10	C₂×#88123	C₂²	1	2	yes	—
11	C₂×C₁₃×#97	C₂²	1	1	yes	characteristic
12	C₂×C₁₃×#97	C₂²	1	2	yes	—
13	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈×C₁₃	C₂²	1	2	yes	—
15	C₁₃×#376	C₂²	1	1	yes	characteristic
16	C₂×#88058	C₂²	1	4	yes	—
17	C₂×#88058	C₂²	1	2	yes	—
18	C₂×#88057	C₂²	1	1	yes	characteristic
19	C₂×#88057	C₂²	1	2	yes	—
20	C₂×#88057	C₂²	1	4	yes	—
21	C₂×#88057	C₂²	1	1	yes	characteristic
22	C₂×#88057	C₂²	1	2	yes	—
23	#11718597	C₂²	1	4	yes	—
24	#11718596	C₂²	1	4	yes	—
25	#11718363	—	2	8	no	—
26	#11718344	—	2	4	no	—
27	#11718341	—	2	4	no	—
28	#11718322	—	2	4	no	—
29	#11718317	—	2	4	no	—
30	C₂²×Dic₂₆	—	2	2	no	—
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	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
35	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
36	C₂×C₄⋊Dic₁₃	—	2	2	no	—
37	C₂×C₄⋊Dic₁₃	—	2	8	no	—
38	C₂×C₄⋊Dic₁₃	C₂³	1	4	yes	—
39	C₂×C₄⋊Dic₁₃	C₂³	1	2	yes	—
40	C₂×C₄⋊Dic₁₃	C₂³	1	2	yes	—
41	C₂×#6101	—	2	2	no	—
42	C₂×#6101	—	2	2	no	—
43	#88124	—	2	4	no	—
44	#88124	—	2	4	no	—
45	#88123	—	2	4	no	—
46	#88123	—	2	4	no	—
47	C₁₃×#97	D₄	1	4	yes	—
48	C₁₃×#97	Q₈	1	8	yes	complemented
49	C₄×C₈×C₁₃	D₄	1	4	yes	—
50	C₄×C₈×C₁₃	Q₈	1	8	yes	complemented
51	C₁₃×#95	D₄	1	4	yes	—
52	#88071	—	2	8	no	—
53	#88058	—	2	4	no	—
54	#88058	—	2	8	no	—
55	#88057	—	2	4	no	—
56	#88057	—	2	4	no	—
57	#88057	—	2	2	no	—
58	#88057	—	2	2	no	—
59	#88057	—	2	8	no	—
60	#33318	—	13	1	no	maximal, Sylow
61	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
62	C₂²×C₄×C₁₃	C₂×D₄	1	4	yes	—
63	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
64	C₂²×Dic₁₃	—	2	2	no	—
65	C₂²×Dic₁₃	—	2	4	no	—
66	C₂²×Dic₁₃	—	2	2	no	—
67	C₂×Dic₂₆	—	4	4	no	—
68	C₂×Dic₂₆	—	8	4	no	—
69	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
70	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
71	C₂×C₈×C₁₃	C₂×D₄	1	4	yes	—
72	C₂×C₈×C₁₃	C₂×Q₈	1	4	yes	—
73	C₂×C₈×C₁₃	C₂×Q₈	1	4	yes	—
74	C₂×C₈×C₁₃	C₂×Q₈	1	4	yes	—
75	C₂×C₈×C₁₃	C₂×Q₈	1	4	yes	—
76	C₄²×C₁₃	C₂×D₄	1	2	yes	—
77	C₄²×C₁₃	C₂×D₄	1	2	yes	—
78	C₄²×C₁₃	C₂×Q₈	1	4	yes	—
79	C₄²×C₁₃	C₂×Q₈	1	4	yes	—
80	C₄⋊Dic₁₃	—	2	4	no	—
81	C₄⋊Dic₁₃	—	2	4	no	—
82	C₄⋊Dic₁₃	—	2	2	no	—
83	C₄⋊Dic₁₃	—	2	2	no	—
84	C₄⋊Dic₁₃	—	2	2	no	—
85	C₄⋊Dic₁₃	—	2	2	no	—
86	C₄⋊Dic₁₃	—	4	4	no	—
87	C₄⋊Dic₁₃	—	8	8	no	—
88	#6101	—	4	4	no	—
89	#6101	—	4	4	no	—
90	C₂×#501	—	13	2	no	—
91	C₂×#499	—	13	1	no	—
92	C₂×#498	—	13	1	no	—
93	#2660	—	13	2	no	—
94	#1847	—	13	4	no	—
95	#1846	—	13	2	no	—
96	#1844	—	13	2	no	—
97	#1726	D₁₃	1	1	yes	characteristic
98	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
99	C₂×C₄×C₁₃	—	2	8	no	—
100	C₂×C₄×C₁₃	—	2	4	no	—
101	C₂×C₄×C₁₃	#127	1	4	yes	—
102	C₂×C₄×C₁₃	C₄⋊Q₈	1	8	yes	—
103	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
104	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
105	C₂×C₄×C₁₃	C₂²×Q₈	1	2	yes	—
106	C₂×C₄×C₁₃	C₂²×Q₈	1	2	yes	—
107	C₂×Dic₁₃	—	4	2	no	—
108	C₂×Dic₁₃	—	4	2	no	—
109	C₂×Dic₁₃	—	4	4	no	—
110	C₂×Dic₁₃	—	4	4	no	—
111	C₂×Dic₁₃	—	4	8	no	—
112	C₂×Dic₁₃	—	4	4	no	—
113	Dic₂₆	—	8	8	no	—
114	C₈×C₁₃	C₄⋊Q₈	1	4	yes	—
115	C₈×C₁₃	C₄⋊Q₈	1	4	yes	—
116	C₈×C₁₃	C₄⋊Q₈	1	8	yes	—
117	C₂×C₄⋊Q₈	—	13	2	no	—
118	C₂×#125	—	13	2	no	—
119	#501	—	26	8	no	—
120	#499	—	26	4	no	—
121	#498	—	26	4	no	—
122	C₂×#107	—	13	4	no	—
123	C₂×#107	—	13	2	no	—
124	C₂×#106	—	13	1	no	—
125	C₂×#106	—	13	2	no	—
126	C₂×#106	—	13	4	no	—
127	C₂×#106	—	13	1	no	—
128	C₂×#106	—	13	2	no	—
129	C₂×#97	C₂×D₁₃	1	2	yes	—
130	C₂×#97	C₂×D₁₃	1	1	yes	characteristic
131	C₂×C₄×C₈	C₂×D₁₃	1	2	yes	—
132	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
133	#398	—	26	4	no	—
134	#395	—	26	4	no	—
135	#384	—	13	4	no	—
136	#382	—	13	4	no	—
137	#376	C₂×D₁₃	1	1	yes	characteristic
138	C₂²×C₁₃	C₂×#127	1	1	yes	characteristic
139	C₂²×C₁₃	C₂×C₄⋊Q₈	1	2	yes	—
140	C₂²×C₁₃	#577	1	4	yes	—
141	C₄×C₁₃	—	2	4	no	—
142	C₄×C₁₃	—	2	8	no	—
143	C₄×C₁₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
144	C₄×C₁₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
145	C₄×C₁₃	C₂×C₄⋊Q₈	1	2	yes	—
146	Dic₁₃	—	8	4	no	—
147	Dic₁₃	—	8	2	no	—
148	Dic₁₃	—	8	2	no	—
149	C₂²×Q₈	—	26	2	no	—
150	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
151	C₂²×C₈	C₂²×D₁₃	1	2	yes	—
152	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
153	C₄⋊Q₈	—	26	4	no	—
154	C₄⋊Q₈	—	26	4	no	—
155	#125	—	26	4	no	—
156	#125	—	26	4	no	—
157	C₂×C₄⋊C₄	—	13	2	no	—
158	C₂×C₄⋊C₄	—	13	2	no	—
159	C₂×C₄⋊C₄	—	13	4	no	—
160	C₂×C₄⋊C₄	—	26	2	no	—
161	C₂×C₄⋊C₄	—	26	2	no	—
162	C₂×C₄⋊C₄	—	26	2	no	—
163	C₂×C₄⋊C₄	—	26	8	no	—
164	C₂×C₄²	C₂²×D₁₃	1	2	yes	—
165	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
166	#107	—	26	4	no	—
167	#107	—	26	8	no	—
168	#106	—	26	4	no	—
169	#106	—	26	4	no	—
170	#106	—	26	2	no	—
171	#106	—	26	2	no	—
172	#106	—	26	8	no	—
173	#97	Dic₂₆	1	8	yes	complemented
174	#97	D₅₂	1	4	yes	—
175	C₄×C₈	Dic₂₆	1	8	yes	complemented
176	C₄×C₈	D₅₂	1	4	yes	—
177	#95	—	26	8	no	—
178	#95	D₅₂	1	4	yes	—
179	C₂×C₁₃	#2868	1	1	yes	characteristic
180	C₂×C₁₃	#3154	1	2	yes	—
181	C₂×C₁₃	#3163	1	4	yes	—
182	C₂×Q₈	—	52	4	no	—
183	C₂×Q₈	—	104	4	no	—
184	C₂²×C₄	—	26	2	no	—
185	C₂²×C₄	—	26	4	no	—
186	C₂²×C₄	—	26	2	no	—
187	C₂²×C₄	C₂×D₅₂	1	4	yes	—
188	C₂²×C₄	C₂×D₅₂	1	2	yes	—
189	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
190	C₂×C₈	C₂×Dic₂₆	1	4	yes	—
191	C₂×C₈	C₂×Dic₂₆	1	4	yes	—
192	C₂×C₈	C₂×Dic₂₆	1	4	yes	—
193	C₂×C₈	C₂×Dic₂₆	1	4	yes	—
194	C₂×C₈	C₂×D₅₂	1	2	yes	—
195	C₂×C₈	C₂×D₅₂	1	2	yes	—
196	C₂×C₈	C₂×D₅₂	1	4	yes	—
197	C₄⋊C₄	—	26	2	no	—
198	C₄⋊C₄	—	26	4	no	—
199	C₄⋊C₄	—	26	2	no	—
200	C₄⋊C₄	—	26	2	no	—
201	C₄⋊C₄	—	26	4	no	—
202	C₄⋊C₄	—	26	2	no	—
203	C₄⋊C₄	—	52	4	no	—
204	C₄⋊C₄	—	52	4	no	—
205	C₄⋊C₄	—	52	4	no	—
206	C₄⋊C₄	—	104	8	no	—
207	C₄²	C₂×Dic₂₆	1	4	yes	—
208	C₄²	C₂×Dic₂₆	1	4	yes	—
209	C₄²	C₂×D₅₂	1	2	yes	—
210	C₄²	C₂×D₅₂	1	2	yes	—
211	C₁₃	#33318	1	1	yes	characteristic, complemented, Sylow
212	C₂³	C₂²×D₅₂	1	1	yes	characteristic
213	Q₈	—	104	8	no	—
214	C₂×C₄	—	2	8	no	—
215	C₂×C₄	—	2	4	no	—
216	C₂×C₄	—	52	2	no	—
217	C₂×C₄	—	52	2	no	—
218	C₂×C₄	—	52	4	no	—
219	C₂×C₄	—	52	4	no	—
220	C₂×C₄	—	52	8	no	—
221	C₂×C₄	—	52	4	no	—
222	C₂×C₄	#88123	1	8	yes	—
223	C₂×C₄	#88128	1	4	yes	—
224	C₂×C₄	C₂²×Dic₂₆	1	2	yes	—
225	C₂×C₄	C₂²×Dic₂₆	1	2	yes	—
226	C₂×C₄	C₂²×D₅₂	1	1	yes	characteristic
227	C₂×C₄	C₂²×D₅₂	1	1	yes	characteristic
228	C₈	#88123	1	4	yes	—
229	C₈	#88123	1	8	yes	—
230	C₈	#88123	1	4	yes	—
231	C₂²	C₂×#88123	1	2	yes	—
232	C₂²	C₂×#88128	1	1	yes	characteristic
233	C₂²	#11719456	1	4	yes	—
234	C₄	—	2	4	no	—
235	C₄	—	2	8	no	—
236	C₄	—	104	2	no	—
237	C₄	—	104	2	no	—
238	C₄	—	104	4	no	—
239	C₄	C₂×#88123	1	1	yes	characteristic
240	C₄	C₂×#88123	1	2	yes	—
241	C₄	C₂×#88123	1	1	yes	characteristic
242	C₂	#13123234	1	1	yes	characteristic
243	C₂	#13125312	1	2	yes	—
244	C₂	#13125367	1	4	yes	—

	subgroup	G/H
center	C₂³	C₂²×D₅₂
commutator, cosocle	C₂²×C₄×C₁₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₁₃
Fitting	C₁₃×#1726	C₂
socle, prime radical	C₂³×C₁₃	C₂²×D₄
hypercenter	#1726	D₁₃
hypocenter	C₁₃	#33318
LCS	G ▷ C₂²×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ #376 ◁ #1726

Complements

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

N	H ≅ G/N	type
48. C₁₃×#97	Q₈	semidirect
50. C₄×C₈×C₁₃	Q₈	semidirect
173. #97	Dic₂₆	semidirect
175. C₄×C₈	Dic₂₆	semidirect
211. C₁₃	#33318	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718363	2
C₂×#11718344	2
C₂×#11718341	2
#13120955	2
#13115004	2
#13115000	2
#13114993	2
C₁₃×#1726	2
#33318	13