Subgroups — #16069032

Order 3328.

2502 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11719175	C₂	1	2	yes	maximal
2	#13123234	C₂	1	2	yes	maximal, complemented
3	#13123214	C₂	1	2	yes	maximal, complemented
4	#13123199	C₂	1	4	yes	maximal, complemented
5	#13123188	C₂	1	8	yes	maximal
6	C₄×#88123	C₂	1	8	yes	maximal, complemented
7	#13117243	C₂	1	4	yes	maximal
8	C₂×C₄³×C₁₃	C₂	1	1	yes	maximal, characteristic
9	C₂²×C₄²×C₁₃	C₂²	1	4	yes	—
10	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
11	C₂²×C₄²×C₁₃	C₂²	1	2	yes	—
12	C₂×#88180	C₂²	1	4	yes	—
13	C₂×#88178	C₂²	1	8	yes	—
14	#11719175	—	2	16	no	—
15	C₂×#88124	C₂²	1	4	yes	—
16	C₂×#88123	C₂²	1	8	yes	—
17	C₂×#88123	C₂²	1	4	yes	—
18	C₂×C₄×Dic₂₆	C₂²	1	8	yes	—
19	C₄³×C₁₃	C₂²	1	2	yes	—
20	C₄³×C₁₃	C₂²	1	1	yes	characteristic
21	C₄³×C₁₃	C₂²	1	1	yes	characteristic
22	C₄³×C₁₃	C₂²	1	4	yes	—
23	#11718873	C₂²	1	16	yes	—
24	#11718597	C₂²	1	8	yes	—
25	#11718596	C₂²	1	8	yes	—
26	#11718596	C₂²	1	8	yes	—
27	C₄×C₄⋊Dic₁₃	C₂²	1	16	yes	—
28	#11718590	C₂²	1	16	yes	—
29	#11718589	C₂²	1	32	yes	—
30	C₂³×C₄×C₁₃	C₂³	1	2	yes	—
31	C₂³×C₄×C₁₃	C₂³	1	4	yes	—
32	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
33	C₂²×Dic₂₆	C₂³	1	8	yes	—
34	C₂×C₄²×C₁₃	—	2	16	no	—
35	C₂×C₄²×C₁₃	D₄	1	8	yes	—
36	C₂×C₄²×C₁₃	Q₈	1	16	yes	complemented
37	C₂×C₄²×C₁₃	C₂³	1	8	yes	—
38	C₂×C₄²×C₁₃	C₂³	1	4	yes	—
39	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
40	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
41	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
42	C₂×C₄²×C₁₃	C₂³	1	4	yes	—
43	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
44	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
45	C₂×C₄²×C₁₃	C₂³	1	4	yes	—
46	C₂×C₂²⋊Dic₁₃	C₂³	1	8	yes	—
47	#88180	—	2	16	no	—
48	C₂×C₄⋊Dic₁₃	C₂³	1	8	yes	—
49	C₂×C₄⋊Dic₁₃	C₂³	1	8	yes	—
50	C₂×C₄⋊Dic₁₃	C₂³	1	32	yes	—
51	#88178	—	2	32	no	—
52	C₂×#6101	C₂³	1	16	yes	—
53	C₂×C₄×Dic₁₃	C₂³	1	8	yes	—
54	#88124	—	2	16	no	—
55	#88123	—	2	16	no	—
56	#88123	—	2	32	no	—
57	C₄×Dic₂₆	—	2	32	no	—
58	#88071	C₂³	1	32	yes	—
59	#36645	—	13	1	no	maximal, Sylow
60	C₂⁴×C₁₃	C₂⁴	1	1	yes	characteristic
61	C₂²×C₄×C₁₃	—	2	8	no	—
62	C₂²×C₄×C₁₃	—	2	16	no	—
63	C₂²×C₄×C₁₃	C₂×D₄	1	8	yes	—
64	C₂²×C₄×C₁₃	C₂×D₄	1	4	yes	—
65	C₂²×C₄×C₁₃	C₂×Q₈	1	16	yes	—
66	C₂²×C₄×C₁₃	C₂×Q₈	1	8	yes	—
67	C₂²×C₄×C₁₃	C₂⁴	1	2	yes	—
68	C₂²×C₄×C₁₃	C₂⁴	1	4	yes	—
69	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
70	C₂²×C₄×C₁₃	C₂⁴	1	2	yes	—
71	C₂²×C₄×C₁₃	C₂⁴	1	4	yes	—
72	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
73	C₂²×Dic₁₃	C₂⁴	1	16	yes	—
74	C₂×Dic₂₆	—	2	16	no	—
75	C₂×Dic₂₆	—	2	32	no	—
76	C₄²×C₁₃	—	2	8	no	—
77	C₄²×C₁₃	—	2	16	no	—
78	C₄²×C₁₃	C₂×D₄	1	8	yes	—
79	C₄²×C₁₃	C₂×D₄	1	4	yes	—
80	C₄²×C₁₃	C₂×D₄	1	4	yes	—
81	C₄²×C₁₃	C₂×Q₈	1	8	yes	—
82	C₄²×C₁₃	C₂×Q₈	1	16	yes	—
83	C₄²×C₁₃	C₂×Q₈	1	8	yes	—
84	C₄²×C₁₃	#41	1	16	yes	—
85	C₂²⋊Dic₁₃	—	2	16	no	—
86	C₄⋊Dic₁₃	—	2	16	no	—
87	C₄⋊Dic₁₃	—	2	16	no	—
88	C₄⋊Dic₁₃	—	2	64	no	—
89	#6101	—	2	32	no	—
90	C₄×Dic₁₃	—	2	16	no	—
91	C₂×#533	—	13	2	no	—
92	#2868	—	13	2	no	—
93	#2862	—	13	2	no	—
94	#2859	—	13	4	no	—
95	#2857	—	13	8	no	—
96	C₄×C₄⋊Q₈	—	13	8	no	—
97	#2285	—	13	4	no	—
98	C₂×C₄³	D₁₃	1	1	yes	characteristic
99	C₂³×C₁₃	—	2	4	no	—
100	C₂³×C₁₃	C₂²×D₄	1	2	yes	—
101	C₂³×C₁₃	C₂²×Q₈	1	4	yes	—
102	C₂³×C₁₃	C₂⁵	1	1	yes	characteristic
103	C₂×C₄×C₁₃	—	2	4	no	—
104	C₂×C₄×C₁₃	—	2	8	no	—
105	C₂×C₄×C₁₃	—	2	16	no	—
106	C₂×C₄×C₁₃	—	2	32	no	—
107	C₂×C₄×C₁₃	C₄⋊Q₈	1	16	yes	complemented
108	C₂×C₄×C₁₃	C₂²×D₄	1	4	yes	—
109	C₂×C₄×C₁₃	C₂²×D₄	1	2	yes	—
110	C₂×C₄×C₁₃	C₂²×D₄	1	2	yes	—
111	C₂×C₄×C₁₃	C₂²×D₄	1	4	yes	—
112	C₂×C₄×C₁₃	C₂²×Q₈	1	8	yes	—
113	C₂×C₄×C₁₃	C₂²×Q₈	1	4	yes	—
114	C₂×C₄×C₁₃	C₂²×Q₈	1	4	yes	—
115	C₂×C₄×C₁₃	C₂²×Q₈	1	8	yes	—
116	C₂×C₄×C₁₃	C₂×#41	1	8	yes	—
117	C₂×C₄×C₁₃	C₂×#41	1	16	yes	—
118	C₂×Dic₁₃	—	2	16	no	—
119	C₂×Dic₁₃	—	2	32	no	—
120	Dic₂₆	—	4	32	no	—
121	#533	—	26	16	no	—
122	C₂×C₄⋊Q₈	—	13	8	no	—
123	C₂×C₄⋊Q₈	—	13	4	no	—
124	C₂×#125	—	13	4	no	—
125	C₂×C₂²⋊Q₈	—	13	8	no	—
126	C₂×C₄×Q₈	—	13	8	no	—
127	C₂×#117	—	13	4	no	—
128	C₂²×C₄²	C₂×D₁₃	1	4	yes	—
129	C₂²×C₄²	C₂×D₁₃	1	2	yes	—
130	C₂²×C₄²	C₂×D₁₃	1	1	yes	characteristic
131	#391	—	13	16	no	—
132	#389	—	13	32	no	—
133	#384	—	13	8	no	—
134	#384	—	13	8	no	—
135	#382	—	13	8	no	—
136	#380	—	13	16	no	—
137	C₄×C₄⋊C₄	—	13	16	no	—
138	C₄³	C₂×D₁₃	1	4	yes	—
139	C₄³	C₂×D₁₃	1	2	yes	—
140	C₄³	C₂×D₁₃	1	1	yes	characteristic
141	C₄³	C₂×D₁₃	1	1	yes	characteristic
142	C₂²×C₁₃	—	2	4	no	—
143	C₂²×C₁₃	—	2	8	no	—
144	C₂²×C₁₃	C₂×C₄⋊Q₈	1	4	yes	—
145	C₂²×C₁₃	C₂³×D₄	1	1	yes	characteristic
146	C₂²×C₁₃	C₂³×Q₈	1	2	yes	—
147	C₂²×C₁₃	C₂²×#41	1	4	yes	—
148	C₄×C₁₃	—	2	4	no	—
149	C₄×C₁₃	—	2	8	no	—
150	C₄×C₁₃	C₂×C₄⋊Q₈	1	4	yes	—
151	C₄×C₁₃	C₂×C₄⋊Q₈	1	4	yes	—
152	C₄×C₁₃	#532	1	8	yes	—
153	C₄×C₁₃	#533	1	16	yes	complemented
154	Dic₁₃	—	4	16	no	—
155	C₂²×Q₈	—	13	8	no	—
156	C₂³×C₄	C₂²×D₁₃	1	4	yes	—
157	C₂³×C₄	C₂²×D₁₃	1	2	yes	—
158	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
159	C₄⋊Q₈	—	26	16	no	—
160	C₄⋊Q₈	—	26	32	no	—
161	#125	—	26	16	no	—
162	C₂²⋊Q₈	—	26	32	no	—
163	C₄×Q₈	—	26	32	no	—
164	#117	—	26	16	no	—
165	C₂×C₄⋊C₄	—	13	8	no	—
166	C₂×C₄⋊C₄	—	13	16	no	—
167	C₂×C₄⋊C₄	—	13	8	no	—
168	C₂×C₄⋊C₄	—	13	32	no	—
169	C₂×C₂²⋊C₄	—	13	8	no	—
170	C₂×C₄²	—	2	16	no	—
171	C₂×C₄²	—	13	8	no	—
172	C₂×C₄²	Dic₂₆	1	16	yes	complemented
173	C₂×C₄²	D₅₂	1	8	yes	—
174	C₂×C₄²	C₂²×D₁₃	1	8	yes	—
175	C₂×C₄²	C₂²×D₁₃	1	4	yes	—
176	C₂×C₄²	C₂²×D₁₃	1	2	yes	—
177	C₂×C₄²	C₂²×D₁₃	1	2	yes	—
178	C₂×C₄²	C₂²×D₁₃	1	4	yes	—
179	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
180	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
181	C₂×C₄²	C₂²×D₁₃	1	2	yes	—
182	C₂×C₄²	C₂²×D₁₃	1	4	yes	—
183	#95	—	13	32	no	—
184	C₂×C₁₃	—	2	4	no	—
185	C₂×C₁₃	C₂²×C₄⋊Q₈	1	1	yes	characteristic
186	C₂×C₁₃	C₂×#532	1	2	yes	—
187	C₂×C₁₃	C₂×#533	1	4	yes	—
188	C₂⁴	C₂³×D₁₃	1	1	yes	characteristic
189	C₂×Q₈	—	26	16	no	—
190	C₂×Q₈	—	26	32	no	—
191	C₂²×C₄	—	2	16	no	—
192	C₂²×C₄	—	2	8	no	—
193	C₂²×C₄	—	13	16	no	—
194	C₂²×C₄	C₂×Dic₂₆	1	16	yes	—
195	C₂²×C₄	C₂×Dic₂₆	1	8	yes	—
196	C₂²×C₄	C₂×D₅₂	1	8	yes	—
197	C₂²×C₄	C₂×D₅₂	1	4	yes	—
198	C₂²×C₄	C₂³×D₁₃	1	4	yes	—
199	C₂²×C₄	C₂³×D₁₃	1	2	yes	—
200	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
201	C₂²×C₄	C₂³×D₁₃	1	4	yes	—
202	C₂²×C₄	C₂³×D₁₃	1	2	yes	—
203	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
204	C₄⋊C₄	—	26	16	no	—
205	C₄⋊C₄	—	26	32	no	—
206	C₄⋊C₄	—	26	16	no	—
207	C₄⋊C₄	—	26	64	no	—
208	C₂²⋊C₄	—	26	16	no	—
209	C₄²	—	2	16	no	—
210	C₄²	—	2	8	no	—
211	C₄²	—	26	16	no	—
212	C₄²	C₂×Dic₂₆	1	8	yes	—
213	C₄²	C₂×Dic₂₆	1	16	yes	—
214	C₄²	C₂×Dic₂₆	1	8	yes	—
215	C₄²	C₂×D₅₂	1	8	yes	—
216	C₄²	C₂×D₅₂	1	4	yes	—
217	C₄²	C₂×D₅₂	1	4	yes	—
218	C₄²	#6127	1	16	yes	—
219	C₁₃	#36645	1	1	yes	characteristic, complemented, Sylow
220	C₂³	—	2	4	no	—
221	C₂³	C₂²×Dic₂₆	1	4	yes	—
222	C₂³	C₂²×D₅₂	1	2	yes	—
223	C₂³	C₂⁴×D₁₃	1	1	yes	characteristic
224	Q₈	—	52	32	no	—
225	C₂×C₄	—	2	4	no	—
226	C₂×C₄	—	2	8	no	—
227	C₂×C₄	—	2	32	no	—
228	C₂×C₄	—	2	16	no	—
229	C₂×C₄	—	26	16	no	—
230	C₂×C₄	—	26	32	no	—
231	C₂×C₄	#88123	1	16	yes	complemented
232	C₂×C₄	C₂²×Dic₂₆	1	8	yes	—
233	C₂×C₄	C₂²×Dic₂₆	1	4	yes	—
234	C₂×C₄	C₂²×Dic₂₆	1	4	yes	—
235	C₂×C₄	C₂²×Dic₂₆	1	8	yes	—
236	C₂×C₄	C₂²×D₅₂	1	4	yes	—
237	C₂×C₄	C₂²×D₅₂	1	2	yes	—
238	C₂×C₄	C₂²×D₅₂	1	2	yes	—
239	C₂×C₄	C₂²×D₅₂	1	4	yes	—
240	C₂×C₄	C₂×#6127	1	16	yes	—
241	C₂×C₄	C₂×#6127	1	8	yes	—
242	C₂²	—	2	4	no	—
243	C₂²	—	2	8	no	—
244	C₂²	C₂×#88123	1	4	yes	—
245	C₂²	C₂³×Dic₂₆	1	2	yes	—
246	C₂²	C₂³×D₅₂	1	1	yes	characteristic
247	C₂²	C₂²×#6127	1	4	yes	—
248	C₄	—	2	4	no	—
249	C₄	—	2	8	no	—
250	C₄	—	52	16	no	—
251	C₄	C₂×#88123	1	4	yes	—
252	C₄	C₂×#88123	1	4	yes	—
253	C₄	#11719175	1	16	yes	complemented
254	C₄	#11719182	1	8	yes	—
255	C₂	—	2	4	no	—
256	C₂	C₂²×#88123	1	1	yes	characteristic
257	C₂	C₂×#11719182	1	2	yes	—
258	C₂	C₂×#11719175	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂³×D₁₃
commutator, socle, cosocle	C₂³×C₁₃	C₂⁵
Frattini	C₂³	C₂⁴×D₁₃
Fitting	C₂×C₄³×C₁₃	C₂
prime radical	C₂⁴×C₁₃	C₂⁴
hypercenter	C₂×C₄³	D₁₃
hypocenter	C₁₃	#36645
LCS	G ▷ C₂³×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄³

Complements

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

N	H ≅ G/N	type
2. #13123234	C₂	semidirect
3. #13123214	C₂	semidirect
4. #13123199	C₂	semidirect
6. C₄×#88123	C₂	semidirect
36. C₂×C₄²×C₁₃	Q₈	semidirect
107. C₂×C₄×C₁₃	C₄⋊Q₈	semidirect
153. C₄×C₁₃	#533	semidirect
172. C₂×C₄²	Dic₂₆	semidirect
219. C₁₃	#36645	semidirect
231. C₂×C₄	#88123	semidirect
253. C₄	#11719175	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11719175	2
#13123234	2
#13123214	2
#13123199	2
#13123188	2
C₄×#88123	2
#13117243	2
C₂×C₄³×C₁₃	2
#36645	13