Subgroups — #13128921

Order 1664.

1104 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×C₄×Q₈×C₁₃	C₂	1	1	yes	maximal, characteristic, complemented
2	#11719522	C₂	1	2	yes	maximal, complemented
3	#11719290	C₂	1	3	yes	maximal, complemented
4	#11719287	C₂	1	6	yes	maximal, complemented
5	#11719284	C₂	1	3	yes	maximal, complemented
6	#11719282	C₂	1	1	yes	maximal, characteristic, complemented
7	#11719279	C₂	1	2	yes	maximal, complemented
8	Q₈×Dic₂₆	C₂	1	1	yes	maximal, characteristic, complemented
9	#11719212	C₂	1	6	yes	maximal, complemented
10	#11719184	C₂	1	3	yes	maximal, complemented
11	#11719182	C₂	1	3	yes	maximal, complemented
12	C₂²×Q₈×C₁₃	C₂²	1	1	yes	characteristic
13	C₂×#6131	C₂²	1	2	yes	—
14	C₂×#6127	C₂²	1	6	yes	—
15	C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic
16	C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic
17	C₄×Q₈×C₁₃	C₂²	1	3	yes	—
18	C₄×Q₈×C₁₃	C₂²	1	3	yes	—
19	C₂×C₁₃×C₄⋊C₄	C₂²	1	3	yes	—
20	C₂×C₄²×C₁₃	C₂²	1	3	yes	—
21	#88201	C₂²	1	6	yes	—
22	#88200	C₂²	1	6	yes	—
23	Q₈×Dic₁₃	C₂²	1	2	yes	—
24	#88184	C₂²	1	3	yes	—
25	#88184	C₂²	1	6	yes	—
26	#88183	C₂²	1	6	yes	—
27	C₄×C₂²⋊D₁₃	C₂²	1	6	yes	—
28	#88178	C₂²	1	1	yes	characteristic
29	#88150	C₂²	1	12	yes	—
30	#88149	C₂²	1	6	yes	—
31	#88148	C₂²	1	12	yes	—
32	#88147	C₂²	1	6	yes	—
33	C₄⋊C₄×D₁₃	C₂²	1	6	yes	—
34	#88142	C₂²	1	6	yes	—
35	#88130	C₂²	1	6	yes	—
36	#88129	C₂²	1	6	yes	—
37	#88129	C₂²	1	3	yes	—
38	#88128	C₂²	1	3	yes	—
39	C₄×D₅₂	C₂²	1	6	yes	—
40	C₄×D₅₂	C₂²	1	6	yes	—
41	C₄×D₅₂	C₂²	1	3	yes	—
42	#88126	C₂²	1	6	yes	—
43	#88124	C₂²	1	3	yes	—
44	#88123	C₂²	1	3	yes	—
45	C₄×Dic₂₆	C₂²	1	3	yes	—
46	C₂×Q₈×C₁₃	—	2	4	no	—
47	C₂×Q₈×C₁₃	C₂³	1	3	yes	—
48	C₂×Q₈×C₁₃	C₂³	1	1	yes	characteristic
49	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
50	C₂²×C₄×C₁₃	C₂³	1	3	yes	—
51	C₂²×C₄×C₁₃	C₂³	1	3	yes	—
52	C₂×C₂²⋊D₁₃	C₂³	1	6	yes	—
53	#6131	—	2	8	no	—
54	#6127	—	4	12	no	—
55	C₂×D₅₂	C₂³	1	3	yes	—
56	C₂×D₅₂	C₂³	1	6	yes	—
57	C₂×D₅₂	C₂³	1	6	yes	—
58	C₂×C₄×D₁₃	C₂³	1	6	yes	—
59	C₂×C₄×D₁₃	C₂³	1	12	yes	—
60	C₂×Dic₂₆	C₂³	1	1	yes	characteristic
61	C₂×Dic₂₆	C₂³	1	6	yes	—
62	C₁₃×C₄⋊C₄	C₂³	1	3	yes	—
63	C₁₃×C₄⋊C₄	C₂³	1	3	yes	—
64	C₁₃×C₄⋊C₄	C₂³	1	6	yes	—
65	C₄²×C₁₃	C₂³	1	3	yes	—
66	C₄²×C₁₃	C₂³	1	3	yes	—
67	C₄²×C₁₃	C₂³	1	3	yes	—
68	C₄²×C₁₃	C₂³	1	3	yes	—
69	C₂²⋊Dic₁₃	C₂³	1	2	yes	—
70	#6103	C₂³	1	6	yes	—
71	#6103	C₂³	1	6	yes	—
72	#6103	C₂³	1	12	yes	—
73	#6103	C₂³	1	6	yes	—
74	#6103	C₂³	1	6	yes	—
75	C₄⋊Dic₁₃	C₂³	1	3	yes	—
76	C₄⋊Dic₁₃	C₂³	1	6	yes	—
77	C₄⋊Dic₁₃	C₂³	1	1	yes	characteristic
78	#6101	C₂³	1	6	yes	—
79	#6101	C₂³	1	6	yes	—
80	#6101	C₂³	1	6	yes	—
81	#6101	C₂³	1	2	yes	—
82	C₄×Dic₁₃	C₂³	1	6	yes	—
83	#3522	—	13	1	no	maximal, Sylow
84	C₂³×C₁₃	C₂⁴	1	1	yes	characteristic
85	C₂²×D₁₃	C₂⁴	1	6	yes	—
86	Q₈×C₁₃	—	2	6	no	—
87	Q₈×C₁₃	#41	1	4	yes	complemented
88	C₂×C₄×C₁₃	—	2	6	no	—
89	C₂×C₄×C₁₃	—	2	6	no	—
90	C₂×C₄×C₁₃	—	2	2	no	—
91	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
92	C₂×C₄×C₁₃	C₂⁴	1	3	yes	—
93	C₂×C₄×C₁₃	C₂⁴	1	3	yes	—
94	C₂×C₄×C₁₃	C₂⁴	1	3	yes	—
95	C₂×C₄×C₁₃	C₂⁴	1	3	yes	—
96	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
97	C₂²⋊D₁₃	—	2	12	no	—
98	C₂×Dic₁₃	C₂⁴	1	2	yes	—
99	C₂×Dic₁₃	C₂⁴	1	6	yes	—
100	C₂×Dic₁₃	C₂⁴	1	2	yes	—
101	D₅₂	—	2	12	no	—
102	D₅₂	—	2	6	no	—
103	D₅₂	—	4	6	no	—
104	C₄×D₁₃	—	2	12	no	—
105	C₄×D₁₃	—	4	12	no	—
106	Dic₂₆	—	2	2	no	—
107	Dic₂₆	—	4	6	no	—
108	#559	—	13	3	no	—
109	Q₈²	—	13	1	no	—
110	#554	—	13	3	no	—
111	#550	—	13	1	no	—
112	#548	—	13	2	no	—
113	#547	—	13	6	no	—
114	#542	—	13	6	no	—
115	#532	—	13	3	no	—
116	#529	—	13	3	no	—
117	#520	—	13	2	no	—
118	C₂×C₄×Q₈	D₁₃	1	1	yes	characteristic, complemented
119	C₂²×C₁₃	—	2	1	no	—
120	C₂²×C₁₃	—	2	1	no	—
121	C₂²×C₁₃	H(5,𝔽₂)	1	2	yes	—
122	C₂²×C₁₃	C₂⁵	1	1	yes	characteristic
123	C₂×D₁₃	—	2	6	no	—
124	C₂×D₁₃	—	2	6	no	—
125	C₂×D₁₃	—	2	6	no	—
126	C₄×C₁₃	—	2	3	no	—
127	C₄×C₁₃	—	2	3	no	—
128	C₄×C₁₃	—	2	3	no	—
129	C₄×C₁₃	—	2	1	no	—
130	C₄×C₁₃	C₂×#41	1	6	yes	—
131	C₄×C₁₃	2₋¹⁺⁴	1	2	yes	—
132	Dic₁₃	—	2	2	no	—
133	Dic₁₃	—	2	6	no	—
134	Dic₁₃	—	2	2	no	—
135	C₂×#41	—	13	2	no	—
136	C₂×#41	—	13	6	no	—
137	C₂²×Q₈	C₂×D₁₃	1	1	yes	characteristic
138	C₄⋊Q₈	—	13	6	no	—
139	C₄⋊Q₈	—	13	3	no	—
140	#127	—	13	3	no	—
141	#126	—	13	6	no	—
142	#125	—	13	3	no	—
143	#124	—	13	6	no	—
144	#124	—	13	3	no	—
145	#124	—	13	6	no	—
146	#123	—	13	6	no	—
147	#123	—	13	12	no	—
148	C₂²⋊Q₈	—	13	6	no	—
149	C₂²⋊Q₈	—	13	1	no	—
150	C₂²⋊Q₈	—	13	12	no	—
151	#121	—	13	3	no	—
152	#121	—	13	6	no	—
153	#121	—	13	6	no	—
154	C₄×Q₈	—	13	2	no	—
155	C₄×Q₈	—	13	3	no	—
156	C₄×Q₈	C₂×D₁₃	1	1	yes	characteristic
157	C₄×Q₈	C₂×D₁₃	1	3	yes	—
158	C₄×Q₈	C₂×D₁₃	1	1	yes	characteristic
159	C₄×Q₈	C₂×D₁₃	1	3	yes	—
160	C₄×D₄	—	13	6	no	—
161	C₄×D₄	—	13	6	no	—
162	C₄×D₄	—	13	6	no	—
163	C₄×D₄	—	13	3	no	—
164	C₄×D₄	—	13	6	no	—
165	#117	—	13	6	no	—
166	C₂×C₄⋊C₄	—	13	6	no	—
167	C₂×C₄⋊C₄	C₂×D₁₃	1	3	yes	—
168	C₂×C₄²	C₂×D₁₃	1	3	yes	—
169	C₂×C₁₃	—	2	2	no	—
170	C₂×C₁₃	C₂²×#41	1	1	yes	characteristic
171	C₂×C₁₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
172	C₂×C₁₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
173	D₁₃	—	4	6	no	—
174	#41	—	26	8	no	—
175	#41	—	52	12	no	—
176	C₂×Q₈	—	2	4	no	—
177	C₂×Q₈	—	13	1	no	—
178	C₂×Q₈	—	13	6	no	—
179	C₂×Q₈	C₂²×D₁₃	1	3	yes	—
180	C₂×Q₈	C₂²×D₁₃	1	1	yes	characteristic
181	C₂×D₄	—	13	6	no	—
182	C₂×D₄	—	13	3	no	—
183	C₂×D₄	—	13	6	no	—
184	C₂×D₄	—	13	6	no	—
185	C₂²×C₄	—	13	6	no	—
186	C₂²×C₄	—	13	12	no	—
187	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
188	C₂²×C₄	C₂²×D₁₃	1	3	yes	—
189	C₂²×C₄	C₂²×D₁₃	1	3	yes	—
190	C₄⋊C₄	—	13	6	no	—
191	C₄⋊C₄	—	13	6	no	—
192	C₄⋊C₄	—	13	2	no	—
193	C₄⋊C₄	—	13	1	no	—
194	C₄⋊C₄	—	13	3	no	—
195	C₄⋊C₄	—	13	6	no	—
196	C₄⋊C₄	—	13	6	no	—
197	C₄⋊C₄	C₂²×D₁₃	1	3	yes	—
198	C₄⋊C₄	C₂²×D₁₃	1	3	yes	—
199	C₄⋊C₄	C₂²×D₁₃	1	6	yes	—
200	C₂²⋊C₄	—	13	2	no	—
201	C₂²⋊C₄	—	13	6	no	—
202	C₂²⋊C₄	—	13	6	no	—
203	C₂²⋊C₄	—	13	6	no	—
204	C₂²⋊C₄	—	13	12	no	—
205	C₂²⋊C₄	—	13	6	no	—
206	C₄²	—	13	6	no	—
207	C₄²	C₂²×D₁₃	1	3	yes	—
208	C₄²	C₂²×D₁₃	1	3	yes	—
209	C₄²	C₂²×D₁₃	1	3	yes	—
210	C₄²	C₂²×D₁₃	1	3	yes	—
211	C₁₃	#3522	1	1	yes	characteristic, complemented, Sylow
212	C₂³	—	13	6	no	—
213	C₂³	C₂³×D₁₃	1	1	yes	characteristic
214	Q₈	—	2	6	no	—
215	Q₈	—	26	2	no	—
216	Q₈	—	52	6	no	—
217	Q₈	#6127	1	4	yes	complemented
218	D₄	—	26	12	no	—
219	D₄	—	26	12	no	—
220	D₄	—	26	6	no	—
221	D₄	—	52	6	no	—
222	C₂×C₄	—	2	6	no	—
223	C₂×C₄	—	2	6	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	13	2	no	—
226	C₂×C₄	—	13	6	no	—
227	C₂×C₄	—	13	2	no	—
228	C₂×C₄	—	26	12	no	—
229	C₂×C₄	—	52	12	no	—
230	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
231	C₂×C₄	C₂³×D₁₃	1	3	yes	—
232	C₂×C₄	C₂³×D₁₃	1	3	yes	—
233	C₂×C₄	C₂³×D₁₃	1	3	yes	—
234	C₂×C₄	C₂³×D₁₃	1	3	yes	—
235	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
236	C₂²	—	2	1	no	—
237	C₂²	—	2	1	no	—
238	C₂²	—	26	6	no	—
239	C₂²	—	26	6	no	—
240	C₂²	—	26	6	no	—
241	C₂²	#88256	1	2	yes	—
242	C₂²	C₂⁴×D₁₃	1	1	yes	characteristic
243	C₄	—	2	3	no	—
244	C₄	—	2	1	no	—
245	C₄	—	2	3	no	—
246	C₄	—	2	3	no	—
247	C₄	—	26	2	no	—
248	C₄	—	26	2	no	—
249	C₄	—	26	6	no	—
250	C₄	C₂×#6127	1	6	yes	—
251	C₄	#88254	1	2	yes	—
252	C₂	—	2	2	no	—
253	C₂	—	52	6	no	—
254	C₂	C₂²×#6127	1	1	yes	characteristic
255	C₂	C₂×#88254	1	1	yes	characteristic
256	C₂	C₂×#88256	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₁₃
commutator, socle, cosocle	C₂²×C₁₃	C₂⁵
Fitting	C₂×C₄×Q₈×C₁₃	C₂
hypercenter	C₂×C₄×Q₈	D₁₃
hypocenter	C₁₃	#3522
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄×Q₈

Complements

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

N	H ≅ G/N	type
1. C₂×C₄×Q₈×C₁₃	C₂	semidirect
2. #11719522	C₂	semidirect
3. #11719290	C₂	semidirect
4. #11719287	C₂	semidirect
5. #11719284	C₂	semidirect
6. #11719282	C₂	semidirect
7. #11719279	C₂	semidirect
8. Q₈×Dic₂₆	C₂	semidirect
9. #11719212	C₂	semidirect
10. #11719184	C₂	semidirect
11. #11719182	C₂	semidirect
87. Q₈×C₁₃	#41	semidirect
118. C₂×C₄×Q₈	D₁₃	semidirect
211. C₁₃	#3522	semidirect
217. Q₈	#6127	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×Q₈×C₁₃	2
#11719522	2
#11719290	2
#11719287	2
#11719284	2
#11719282	2
#11719279	2
Q₈×Dic₂₆	2
#11719212	2
#11719184	2
#11719182	2
#3522	13