Subgroups — #13131779

Order 1664.

354 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11719385	C₂	1	1	yes	maximal, characteristic
2	#11719158	C₂	1	1	yes	maximal, characteristic, complemented
3	Q₈×#1057	C₂	1	1	yes	maximal, characteristic, complemented
4	#11719145	C₂	1	1	yes	maximal, characteristic, complemented
5	D₄×#1057	C₂	1	1	yes	maximal, characteristic, complemented
6	#11719068	C₂	1	2	yes	maximal, complemented
7	#11719067	C₂	1	2	yes	maximal, complemented
8	#11719066	C₂	1	2	yes	maximal, complemented
9	#11719064	C₂	1	2	yes	maximal, complemented
10	#11719052	C₂	1	1	yes	maximal, characteristic, complemented
11	#11719049	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#6122	C₂²	1	2	yes	—
13	C₂²×#1057	C₂²	1	2	yes	—
14	C₂×#6118	C₂²	1	2	yes	—
15	C₁₃×#124	C₄	1	1	yes	characteristic
16	#88200	C₄	1	1	yes	characteristic
17	Q₈×Dic₁₃	C₂²	1	1	yes	characteristic
18	#88192	C₄	1	1	yes	characteristic
19	D₄×Dic₁₃	C₂²	1	1	yes	characteristic
20	#88137	C₄	1	2	yes	—
21	#88135	C₂²	1	2	yes	—
22	#88133	C₄	1	2	yes	—
23	#88131	C₂²	1	2	yes	—
24	C₄²×D₁₃	C₂²	1	1	yes	characteristic
25	#88124	C₄	1	1	yes	characteristic
26	C₂²⋊#1057	C₂²	1	2	yes	—
27	C₂²⋊#1057	C₂²	1	2	yes	—
28	#88112	C₂²	1	1	yes	characteristic
29	#88112	C₂²	1	1	yes	characteristic
30	#88112	C₂²	1	2	yes	—
31	#88111	C₂²	1	1	yes	characteristic
32	#88111	C₂²	1	1	yes	characteristic
33	#88110	C₂²	1	2	yes	—
34	C₄⋊#1057	C₂²	1	2	yes	—
35	C₄⋊#1057	C₂²	1	1	yes	characteristic
36	C₄⋊#1057	C₂²	1	1	yes	characteristic
37	C₄⋊#1057	C₂²	1	2	yes	—
38	C₄×#1057	C₂²	1	1	yes	characteristic
39	C₄×#1057	C₂²	1	1	yes	characteristic
40	C₄×#1057	C₂²	1	2	yes	—
41	C₂×Q₈×C₁₃	C₂×C₄	1	1	yes	characteristic
42	C₂×D₄×C₁₃	C₂×C₄	1	1	yes	characteristic
43	C₂×C₂²⋊D₁₃	C₂×C₄	1	2	yes	—
44	C₂²×Dic₁₃	C₂³	1	2	yes	—
45	C₂×C₄×D₁₃	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄×D₁₃	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄×D₁₃	C₂³	1	1	yes	characteristic
48	#6122	—	4	2	no	—
49	C₂×#1057	—	4	2	no	—
50	C₂×#1057	C₂³	1	1	yes	characteristic
51	C₂×#1057	C₂³	1	2	yes	—
52	C₂×#1057	C₂³	1	1	yes	characteristic
53	C₂×#1057	C₂³	1	1	yes	characteristic
54	C₂×#1057	C₂³	1	1	yes	characteristic
55	C₂×#1057	C₂³	1	2	yes	—
56	#6118	—	4	2	no	—
57	C₁₃×C₂²⋊C₄	C₂×C₄	1	2	yes	—
58	C₁₃×C₂²⋊C₄	C₂×C₄	1	2	yes	—
59	C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
60	C₂²⋊Dic₁₃	C₂×C₄	1	2	yes	—
61	C₂²⋊Dic₁₃	C₂×C₄	1	2	yes	—
62	#6103	C₂×C₄	1	2	yes	—
63	C₄⋊Dic₁₃	C₂×C₄	1	1	yes	characteristic
64	C₄⋊Dic₁₃	C₂×C₄	1	1	yes	characteristic
65	C₄⋊Dic₁₃	C₂×C₄	1	2	yes	—
66	#6101	C₂×C₄	1	2	yes	—
67	#6101	C₂×C₄	1	2	yes	—
68	#6101	C₂×C₄	1	2	yes	—
69	C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
70	C₄×Dic₁₃	C₂³	1	1	yes	characteristic
71	C₄×Dic₁₃	C₂³	1	2	yes	—
72	C₄×Dic₁₃	C₂³	1	1	yes	characteristic
73	#2993	—	13	1	no	maximal, Sylow
74	C₂³×C₁₃	C₂²×C₄	1	2	yes	—
75	C₂²×D₁₃	C₂²×C₄	1	1	yes	characteristic
76	Q₈×C₁₃	—	2	2	no	—
77	D₄×C₁₃	—	2	2	no	—
78	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₄×C₁₃	C₂²×C₄	1	2	yes	—
81	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
82	C₂²⋊D₁₃	—	4	2	no	—
83	C₂×Dic₁₃	—	2	2	no	—
84	C₂×Dic₁₃	—	2	2	no	—
85	C₂×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
86	C₂×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
87	C₂×Dic₁₃	C₂²×C₄	1	2	yes	—
88	C₂×Dic₁₃	C₂²×C₄	1	2	yes	—
89	C₂×Dic₁₃	C₂⁴	1	1	yes	characteristic
90	C₄×D₁₃	—	2	2	no	—
91	C₄×D₁₃	—	2	1	no	—
92	C₄×D₁₃	—	2	1	no	—
93	C₄×D₁₃	—	4	1	no	—
94	#1057	—	2	2	no	—
95	#1057	—	2	2	no	—
96	#1057	—	2	1	no	—
97	#1057	—	2	1	no	—
98	#1057	—	2	1	no	—
99	#1057	—	2	1	no	—
100	#529	—	13	1	no	—
101	#446	—	13	1	no	—
102	C₈×Q₈	—	13	1	no	—
103	#436	—	13	2	no	—
104	#436	—	13	1	no	—
105	#435	—	13	2	no	—
106	C₈×D₄	—	13	1	no	—
107	#433	—	13	2	no	—
108	#431	—	13	1	no	—
109	#424	—	13	2	no	—
110	#423	—	13	1	no	—
111	C₂²×C₁₃	—	2	2	no	—
112	C₂²×C₁₃	—	2	2	no	—
113	C₂²×C₁₃	—	2	2	no	—
114	C₂²×C₁₃	C₂³×C₄	1	1	yes	characteristic
115	C₂×D₁₃	—	2	1	no	—
116	C₂×D₁₃	—	2	1	no	—
117	C₂×D₁₃	—	2	1	no	—
118	C₄×C₁₃	—	2	1	no	—
119	C₄×C₁₃	—	2	2	no	—
120	C₄×C₁₃	—	2	1	no	—
121	C₄×C₁₃	#131	1	2	yes	—
122	Dic₁₃	—	2	2	no	—
123	Dic₁₃	—	2	1	no	—
124	Dic₁₃	—	2	1	no	—
125	Dic₁₃	—	2	2	no	—
126	Dic₁₃	H(5,𝔽₂)	1	1	yes	characteristic
127	Dic₁₃	2₋¹⁺⁴	1	1	yes	characteristic
128	C₂×M₁₆	—	13	2	no	—
129	C₂×M₁₆	—	13	2	no	—
130	C₂²×C₈	—	13	2	no	—
131	#126	—	13	2	no	—
132	#125	—	13	1	no	—
133	#124	#261	1	1	yes	characteristic
134	#123	—	13	2	no	—
135	C₂²⋊Q₈	—	13	1	no	—
136	#121	—	13	1	no	—
137	C₄×Q₈	—	13	1	no	—
138	C₄×D₄	—	13	1	no	—
139	C₄×D₄	—	13	2	no	—
140	#117	—	13	2	no	—
141	C₂×C₄²	—	13	1	no	—
142	C₄⋊C₈	—	13	2	no	—
143	C₄⋊C₈	—	13	1	no	—
144	C₄⋊C₈	—	13	1	no	—
145	C₄⋊C₈	—	13	2	no	—
146	C₄⋊C₈	—	13	1	no	—
147	C₄⋊C₈	—	13	2	no	—
148	C₄⋊C₈	—	13	1	no	—
149	C₂²⋊C₈	—	13	1	no	—
150	C₂²⋊C₈	—	13	1	no	—
151	C₂²⋊C₈	—	13	2	no	—
152	C₂²⋊C₈	—	13	2	no	—
153	#97	—	13	2	no	—
154	C₄×C₈	—	13	1	no	—
155	C₄×C₈	—	13	1	no	—
156	C₄×C₈	—	13	2	no	—
157	C₂×C₁₃	—	4	2	no	—
158	C₂×C₁₃	#520	1	1	yes	characteristic
159	C₂×C₁₃	C₂×#131	1	1	yes	characteristic
160	C₂×C₁₃	#568	1	1	yes	characteristic
161	D₁₃	—	4	1	no	—
162	C₂×Q₈	C₂×#261	1	1	yes	characteristic
163	C₂×D₄	—	13	2	no	—
164	C₂×D₄	C₂×#261	1	1	yes	characteristic
165	C₂²×C₄	—	13	2	no	—
166	C₂²×C₄	—	13	1	no	—
167	C₂²×C₄	—	13	1	no	—
168	C₂²×C₄	—	13	1	no	—
169	M₁₆	—	52	2	no	—
170	M₁₆	—	52	2	no	—
171	C₂×C₈	—	13	1	no	—
172	C₂×C₈	—	13	1	no	—
173	C₂×C₈	—	13	1	no	—
174	C₂×C₈	—	13	1	no	—
175	C₂×C₈	—	13	2	no	—
176	C₂×C₈	—	13	2	no	—
177	C₂×C₈	—	52	2	no	—
178	C₄⋊C₄	—	13	2	no	—
179	C₄⋊C₄	—	13	2	no	—
180	C₄⋊C₄	—	13	1	no	—
181	C₄⋊C₄	—	13	2	no	—
182	C₄⋊C₄	—	13	1	no	—
183	C₄⋊C₄	—	13	2	no	—
184	C₂²⋊C₄	—	13	2	no	—
185	C₂²⋊C₄	—	13	2	no	—
186	C₂²⋊C₄	—	13	2	no	—
187	C₂²⋊C₄	C₂×#261	1	2	yes	—
188	C₂²⋊C₄	C₂×#261	1	2	yes	—
189	C₄²	—	13	1	no	—
190	C₄²	—	13	2	no	—
191	C₄²	—	13	1	no	—
192	C₄²	—	13	1	no	—
193	C₄²	C₂×#261	1	1	yes	characteristic
194	C₁₃	#2993	1	1	yes	characteristic, complemented, Sylow
195	C₂³	—	13	1	no	—
196	C₂³	C₂²×#261	1	2	yes	—
197	Q₈	—	2	2	no	—
198	D₄	—	2	2	no	—
199	D₄	—	52	2	no	—
200	C₂×C₄	—	13	1	no	—
201	C₂×C₄	—	13	1	no	—
202	C₂×C₄	—	13	1	no	—
203	C₂×C₄	—	13	2	no	—
204	C₂×C₄	—	13	2	no	—
205	C₂×C₄	—	26	1	no	—
206	C₂×C₄	—	26	1	no	—
207	C₂×C₄	—	26	2	no	—
208	C₂×C₄	—	26	2	no	—
209	C₂×C₄	—	26	2	no	—
210	C₂×C₄	—	52	1	no	—
211	C₂×C₄	C₂²×#261	1	1	yes	characteristic
212	C₂×C₄	C₂²×#261	1	1	yes	characteristic
213	C₂×C₄	C₂²×#261	1	2	yes	—
214	C₂×C₄	C₂²×#261	1	1	yes	characteristic
215	C₈	—	26	1	no	—
216	C₈	—	26	2	no	—
217	C₈	—	26	1	no	—
218	C₈	—	26	1	no	—
219	C₈	—	26	2	no	—
220	C₈	—	26	1	no	—
221	C₂²	—	2	2	no	—
222	C₂²	—	2	2	no	—
223	C₂²	—	2	2	no	—
224	C₂²	—	26	1	no	—
225	C₂²	—	26	1	no	—
226	C₂²	—	26	1	no	—
227	C₂²	C₂³×#261	1	1	yes	characteristic
228	C₄	—	2	2	no	—
229	C₄	—	2	1	no	—
230	C₄	—	2	1	no	—
231	C₄	—	13	1	no	—
232	C₄	—	13	1	no	—
233	C₄	—	26	1	no	—
234	C₄	—	26	1	no	—
235	C₄	—	26	2	no	—
236	C₄	—	26	2	no	—
237	C₄	#88238	1	2	yes	—
238	C₂	—	4	2	no	—
239	C₂	—	52	1	no	—
240	C₂	C₂×#88238	1	1	yes	characteristic
241	C₂	#11719629	1	1	yes	characteristic
242	C₂	#11719631	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×#261
commutator, socle	C₂²×C₁₃	C₂³×C₄
Fitting	C₁₃×#124	C₄
prime radical	#88192	C₄
cosocle	C₂×Dic₁₃	C₂⁴
hypercenter	#124	#261
hypocenter	C₁₃	#2993
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ #124

Complements

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

N	H ≅ G/N	type
2. #11719158	C₂	semidirect
3. Q₈×#1057	C₂	semidirect
4. #11719145	C₂	semidirect
5. D₄×#1057	C₂	semidirect
6. #11719068	C₂	semidirect
7. #11719067	C₂	semidirect
8. #11719066	C₂	semidirect
9. #11719064	C₂	semidirect
10. #11719052	C₂	semidirect
11. #11719049	C₂	semidirect
194. C₁₃	#2993	semidirect

Maximal subgroups

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

subgroup	index
#11719385	2
#11719158	2
Q₈×#1057	2
#11719145	2
D₄×#1057	2
#11719068	2
#11719067	2
#11719066	2
#11719064	2
#11719052	2
#11719049	2
#2993	13