Subgroups — #16732608

Order 3328.

1762 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13128662	C₂	1	2	yes	maximal
2	#13125388	C₂	1	6	yes	maximal, complemented
3	#13125375	C₂	1	6	yes	maximal, complemented
4	#13125367	C₂	1	6	yes	maximal, complemented
5	#13125350	C₂	1	3	yes	maximal, complemented
6	C₂×#11718344	C₂	1	3	yes	maximal, complemented
7	#13123436	C₂	1	2	yes	maximal
8	#13123431	C₂	1	2	yes	maximal
9	C₄×C₁₃×#131	C₂	1	1	yes	maximal, characteristic
10	C₂×C₁₃×#131	C₂²	1	1	yes	characteristic
11	C₂×C₁₃×#131	C₂²	1	1	yes	characteristic
12	C₄×C₁₃×#41	C₂²	1	1	yes	characteristic
13	#11719532	C₂²	1	4	yes	—
14	#11719274	—	2	4	no	—
15	#11719244	—	2	12	no	—
16	#11719224	C₂²	1	12	yes	—
17	#11719175	C₂²	1	6	yes	—
18	C₂×#88124	C₂²	1	6	yes	—
19	C₁₃×#405	C₂²	1	3	yes	—
20	C₁₃×#405	C₂²	1	3	yes	—
21	C₄×C₁₃×M₁₆	C₂²	1	3	yes	—
22	C₂×C₄×C₈×C₁₃	C₂²	1	3	yes	—
23	#11718782	C₂²	1	12	yes	—
24	#11718782	C₂²	1	12	yes	—
25	#11718767	C₂²	1	6	yes	—
26	#11718767	C₂²	1	6	yes	—
27	C₂×#88058	C₂²	1	6	yes	—
28	C₂×#88057	C₂²	1	6	yes	—
29	#11718363	C₂²	1	24	yes	—
30	#11718344	C₂²	1	6	yes	—
31	#11718344	C₂²	1	6	yes	—
32	#11718344	C₂²	1	3	yes	—
33	#11718344	C₂²	1	6	yes	—
34	#11718344	C₂²	1	6	yes	—
35	#11718344	C₂²	1	1	yes	characteristic
36	#11718343	C₂²	1	6	yes	—
37	#11718341	C₂²	1	6	yes	—
38	C₂×C₁₃×#41	C₂³	1	1	yes	characteristic
39	C₁₃×#131	Q₈	1	8	yes	complemented
40	C₁₃×#131	Q₈	1	8	yes	complemented
41	C₂×C₁₃×M₁₆	C₂³	1	3	yes	—
42	C₂×C₁₃×M₁₆	C₂³	1	3	yes	—
43	C₂²×C₈×C₁₃	C₂³	1	3	yes	—
44	C₂²×C₈×C₁₃	C₂³	1	3	yes	—
45	C₄×Q₈×C₁₃	D₄	1	2	yes	—
46	C₄×D₄×C₁₃	D₄	1	6	yes	—
47	C₁₃×#117	C₂³	1	3	yes	—
48	C₂×C₄²×C₁₃	C₂³	1	3	yes	—
49	Q₈×Dic₁₃	—	2	4	no	—
50	D₄×Dic₁₃	—	2	12	no	—
51	#88180	C₂³	1	12	yes	—
52	C₂×C₄⋊Dic₁₃	C₂³	1	12	yes	—
53	#88178	—	2	12	no	—
54	C₂×#6101	—	2	12	no	—
55	#88144	—	2	12	no	—
56	#88142	—	2	12	no	—
57	#88132	—	2	24	no	—
58	#88124	—	2	12	no	—
59	#88124	C₂³	1	6	yes	—
60	#88124	C₂³	1	12	yes	—
61	#88124	C₂³	1	2	yes	—
62	#88123	C₂³	1	12	yes	—
63	C₄×Dic₂₆	—	2	12	no	—
64	C₁₃×#97	C₂³	1	6	yes	—
65	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
66	C₄×C₈×C₁₃	C₂³	1	3	yes	—
67	C₄×C₈×C₁₃	C₂³	1	3	yes	—
68	C₄×C₈×C₁₃	C₂³	1	3	yes	—
69	#88058	C₂³	1	6	yes	—
70	#88058	C₂³	1	12	yes	—
71	#88058	C₂³	1	2	yes	—
72	#88058	C₂³	1	12	yes	—
73	#88057	C₂³	1	6	yes	—
74	#88057	C₂³	1	12	yes	—
75	#88057	C₂³	1	2	yes	—
76	#88057	C₂³	1	12	yes	—
77	#60666	—	13	1	no	maximal, Sylow
78	C₁₃×#41	C₂×Q₈	1	4	yes	—
79	C₁₃×#41	C₂×Q₈	1	4	yes	—
80	C₂×Q₈×C₁₃	C₂×D₄	1	1	yes	characteristic
81	C₂×D₄×C₁₃	C₂×D₄	1	3	yes	—
82	C₂²×C₄×C₁₃	C₂×D₄	1	6	yes	—
83	C₂²×C₄×C₁₃	C₂⁴	1	3	yes	—
84	C₂²×Dic₁₃	—	2	12	no	—
85	C₂×Dic₂₆	—	2	12	no	—
86	C₁₃×M₁₆	C₂×Q₈	1	12	yes	—
87	C₁₃×M₁₆	C₂×Q₈	1	12	yes	—
88	C₂×C₈×C₁₃	C₂×Q₈	1	12	yes	—
89	C₂×C₈×C₁₃	C₂×Q₈	1	12	yes	—
90	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
91	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
92	C₂×C₈×C₁₃	C₂⁴	1	3	yes	—
93	C₂×C₈×C₁₃	C₂⁴	1	3	yes	—
94	C₁₃×C₄⋊C₄	C₂×D₄	1	6	yes	—
95	C₁₃×C₂²⋊C₄	C₂×D₄	1	6	yes	—
96	C₄²×C₁₃	C₂×D₄	1	6	yes	—
97	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
98	C₄²×C₁₃	C₂⁴	1	3	yes	—
99	C₂²⋊Dic₁₃	—	2	12	no	—
100	C₄⋊Dic₁₃	—	2	12	no	—
101	C₄⋊Dic₁₃	—	2	12	no	—
102	C₄⋊Dic₁₃	C₂⁴	1	4	yes	—
103	C₄⋊Dic₁₃	C₂⁴	1	12	yes	—
104	#6101	—	2	4	no	—
105	#6101	—	2	12	no	—
106	#6101	—	2	24	no	—
107	C₄×Dic₁₃	—	2	12	no	—
108	#3504	—	13	2	no	—
109	#3166	—	13	6	no	—
110	#3164	—	13	6	no	—
111	#3163	—	13	6	no	—
112	#3160	—	13	3	no	—
113	C₂×#499	—	13	3	no	—
114	#2913	—	13	2	no	—
115	#2912	—	13	2	no	—
116	C₄×#131	D₁₃	1	1	yes	characteristic
117	C₂³×C₁₃	C₂²×D₄	1	3	yes	—
118	Q₈×C₁₃	C₄⋊Q₈	1	4	yes	—
119	D₄×C₁₃	C₄⋊Q₈	1	12	yes	—
120	C₂×C₄×C₁₃	—	4	6	no	—
121	C₂×C₄×C₁₃	C₂²×D₄	1	2	yes	—
122	C₂×C₄×C₁₃	C₂²×D₄	1	6	yes	—
123	C₂×C₄×C₁₃	C₂²×D₄	1	3	yes	—
124	C₂×C₄×C₁₃	C₂²×Q₈	1	6	yes	—
125	C₂×C₄×C₁₃	C₂²×Q₈	1	6	yes	—
126	C₂×C₄×C₁₃	C₂⁵	1	1	yes	characteristic
127	C₂×Dic₁₃	—	2	4	no	—
128	C₂×Dic₁₃	—	2	12	no	—
129	C₂×Dic₁₃	—	8	12	no	—
130	Dic₂₆	—	8	12	no	—
131	C₈×C₁₃	C₂²×Q₈	1	2	yes	—
132	C₈×C₁₃	C₂²×Q₈	1	2	yes	—
133	C₈×C₁₃	C₂²×Q₈	1	6	yes	—
134	C₈×C₁₃	C₂²×Q₈	1	6	yes	—
135	C₂×#131	C₂×D₁₃	1	1	yes	characteristic
136	C₂×#131	C₂×D₁₃	1	1	yes	characteristic
137	#557	—	26	4	no	—
138	#554	—	26	12	no	—
139	#544	—	13	12	no	—
140	#533	—	13	6	no	—
141	C₂×#125	—	13	6	no	—
142	#520	—	13	4	no	—
143	C₄×#41	C₂×D₁₃	1	1	yes	characteristic
144	#501	—	13	24	no	—
145	#500	—	13	6	no	—
146	#499	—	13	3	no	—
147	#499	—	13	6	no	—
148	#499	—	13	6	no	—
149	#499	—	13	6	no	—
150	#499	—	13	1	no	—
151	#499	—	13	6	no	—
152	#498	—	13	6	no	—
153	#428	—	13	12	no	—
154	#428	—	13	12	no	—
155	#427	—	13	6	no	—
156	#427	—	13	6	no	—
157	C₂×#107	—	13	6	no	—
158	C₂×#106	—	13	6	no	—
159	#405	C₂×D₁₃	1	3	yes	—
160	#405	C₂×D₁₃	1	3	yes	—
161	C₄×M₁₆	C₂×D₁₃	1	3	yes	—
162	C₂×C₄×C₈	C₂×D₁₃	1	3	yes	—
163	C₂²×C₁₃	—	2	6	no	—
164	C₂²×C₁₃	C₂×C₄⋊Q₈	1	6	yes	—
165	C₂²×C₁₃	C₂³×D₄	1	1	yes	characteristic
166	C₄×C₁₃	—	2	6	no	—
167	C₄×C₁₃	—	2	2	no	—
168	C₄×C₁₃	C₂×C₄⋊Q₈	1	6	yes	—
169	C₄×C₁₃	C₂³×Q₈	1	1	yes	characteristic
170	C₄×C₁₃	C₂³×Q₈	1	1	yes	characteristic
171	Dic₁₃	—	4	4	no	—
172	Dic₁₃	—	4	12	no	—
173	C₂×#41	C₂²×D₁₃	1	1	yes	characteristic
174	#131	Dic₂₆	1	8	yes	complemented
175	#131	Dic₂₆	1	8	yes	complemented
176	C₂×M₁₆	C₂²×D₁₃	1	3	yes	—
177	C₂×M₁₆	C₂²×D₁₃	1	3	yes	—
178	C₂²×C₈	C₂²×D₁₃	1	3	yes	—
179	C₂²×C₈	C₂²×D₁₃	1	3	yes	—
180	C₄⋊Q₈	—	13	12	no	—
181	C₄⋊Q₈	—	26	12	no	—
182	#125	—	13	6	no	—
183	#125	—	13	12	no	—
184	#125	—	13	2	no	—
185	#125	—	26	12	no	—
186	#125	—	26	12	no	—
187	C₂²⋊Q₈	—	26	12	no	—
188	C₂²⋊Q₈	—	26	24	no	—
189	C₄×Q₈	—	26	4	no	—
190	C₄×Q₈	—	26	12	no	—
191	C₄×Q₈	D₅₂	1	2	yes	—
192	C₄×D₄	—	26	12	no	—
193	C₄×D₄	D₅₂	1	6	yes	—
194	#117	—	13	12	no	—
195	#117	C₂²×D₁₃	1	3	yes	—
196	C₂×C₄⋊C₄	—	13	12	no	—
197	C₂×C₄⋊C₄	—	26	12	no	—
198	C₂×C₄²	C₂²×D₁₃	1	3	yes	—
199	#107	—	13	6	no	—
200	#107	—	13	12	no	—
201	#107	—	13	2	no	—
202	#107	—	13	12	no	—
203	#106	—	13	6	no	—
204	#106	—	13	12	no	—
205	#106	—	13	2	no	—
206	#106	—	13	12	no	—
207	#97	C₂²×D₁₃	1	6	yes	—
208	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
209	C₄×C₈	C₂²×D₁₃	1	3	yes	—
210	C₄×C₈	C₂²×D₁₃	1	3	yes	—
211	C₄×C₈	C₂²×D₁₃	1	3	yes	—
212	C₂×C₁₃	—	2	6	no	—
213	C₂×C₁₃	C₂²×C₄⋊Q₈	1	1	yes	characteristic
214	C₂×C₁₃	#3584	1	2	yes	—
215	#41	C₂×Dic₂₆	1	4	yes	—
216	#41	C₂×Dic₂₆	1	4	yes	—
217	C₂×Q₈	—	26	12	no	—
218	C₂×Q₈	C₂×D₅₂	1	1	yes	characteristic
219	C₂×D₄	C₂×D₅₂	1	3	yes	—
220	C₂²×C₄	—	26	12	no	—
221	C₂²×C₄	C₂×D₅₂	1	6	yes	—
222	C₂²×C₄	C₂³×D₁₃	1	3	yes	—
223	M₁₆	C₂×Dic₂₆	1	12	yes	—
224	M₁₆	C₂×Dic₂₆	1	12	yes	—
225	C₂×C₈	C₂×Dic₂₆	1	12	yes	—
226	C₂×C₈	C₂×Dic₂₆	1	12	yes	—
227	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
228	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
229	C₂×C₈	C₂³×D₁₃	1	3	yes	—
230	C₂×C₈	C₂³×D₁₃	1	3	yes	—
231	C₄⋊C₄	—	13	12	no	—
232	C₄⋊C₄	—	13	4	no	—
233	C₄⋊C₄	—	26	12	no	—
234	C₄⋊C₄	—	26	12	no	—
235	C₄⋊C₄	—	26	24	no	—
236	C₄⋊C₄	—	26	4	no	—
237	C₄⋊C₄	—	26	12	no	—
238	C₄⋊C₄	C₂×D₅₂	1	6	yes	—
239	C₂²⋊C₄	—	26	12	no	—
240	C₂²⋊C₄	C₂×D₅₂	1	6	yes	—
241	C₄²	—	26	12	no	—
242	C₄²	C₂×D₅₂	1	6	yes	—
243	C₄²	C₂³×D₁₃	1	1	yes	characteristic
244	C₄²	C₂³×D₁₃	1	3	yes	—
245	C₁₃	#60666	1	1	yes	characteristic, complemented, Sylow
246	C₂³	C₂²×D₅₂	1	3	yes	—
247	Q₈	—	104	12	no	—
248	Q₈	#88123	1	4	yes	—
249	D₄	#88123	1	12	yes	—
250	C₂×C₄	—	4	6	no	—
251	C₂×C₄	—	26	4	no	—
252	C₂×C₄	—	26	12	no	—
253	C₂×C₄	—	104	12	no	—
254	C₂×C₄	C₂²×Dic₂₆	1	6	yes	—
255	C₂×C₄	C₂²×Dic₂₆	1	6	yes	—
256	C₂×C₄	C₂²×D₅₂	1	2	yes	—
257	C₂×C₄	C₂²×D₅₂	1	6	yes	—
258	C₂×C₄	C₂²×D₅₂	1	3	yes	—
259	C₂×C₄	C₂⁴×D₁₃	1	1	yes	characteristic
260	C₈	C₂²×Dic₂₆	1	2	yes	—
261	C₈	C₂²×Dic₂₆	1	2	yes	—
262	C₈	C₂²×Dic₂₆	1	6	yes	—
263	C₈	C₂²×Dic₂₆	1	6	yes	—
264	C₂²	—	2	6	no	—
265	C₂²	C₂×#88123	1	6	yes	—
266	C₂²	C₂³×D₅₂	1	1	yes	characteristic
267	C₄	—	2	2	no	—
268	C₄	—	2	6	no	—
269	C₄	—	52	12	no	—
270	C₄	—	52	4	no	—
271	C₄	C₂×#88123	1	6	yes	—
272	C₄	C₂³×Dic₂₆	1	1	yes	characteristic
273	C₄	C₂³×Dic₂₆	1	1	yes	characteristic
274	C₂	—	2	6	no	—
275	C₂	C₂²×#88123	1	1	yes	characteristic
276	C₂	#13129610	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂³×D₅₂
commutator, cosocle	C₂×C₄×C₁₃	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₁₃
Fitting	C₄×C₁₃×#131	C₂
socle	C₂²×C₁₃	C₂³×D₄
prime radical	C₂×C₁₃×#41	C₂³
hypercenter	C₄×#131	D₁₃
hypocenter	C₁₃	#60666
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ C₄×#131

Complements

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

N	H ≅ G/N	type
2. #13125388	C₂	semidirect
3. #13125375	C₂	semidirect
4. #13125367	C₂	semidirect
5. #13125350	C₂	semidirect
6. C₂×#11718344	C₂	semidirect
39. C₁₃×#131	Q₈	semidirect
40. C₁₃×#131	Q₈	semidirect
174. #131	Dic₂₆	semidirect
175. #131	Dic₂₆	semidirect
245. C₁₃	#60666	semidirect

Maximal subgroups

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

subgroup	index
#13128662	2
#13125388	2
#13125375	2
#13125367	2
#13125350	2
C₂×#11718344	2
#13123436	2
#13123431	2
C₄×C₁₃×#131	2
#60666	13