Subgroups — #13131668

Order 1664.

370 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	#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	#11719066	C₂	1	2	yes	maximal, complemented
8	#11719064	C₂	1	2	yes	maximal, complemented
9	#11719062	C₂	1	2	yes	maximal, complemented
10	#11719051	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×#6118	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	1	yes	characteristic
15	C₂×#6118	C₂²	1	1	yes	characteristic
16	C₁₃×#124	C₄	1	1	yes	characteristic
17	#88200	C₄	1	1	yes	characteristic
18	Q₈×Dic₁₃	C₂²	1	1	yes	characteristic
19	#88192	C₄	1	1	yes	characteristic
20	D₄×Dic₁₃	C₂²	1	1	yes	characteristic
21	#88137	C₄	1	2	yes	—
22	#88135	C₂²	1	2	yes	—
23	#88133	C₄	1	2	yes	—
24	#88131	C₂²	1	2	yes	—
25	C₄²×D₁₃	C₂²	1	1	yes	characteristic
26	#88124	C₄	1	1	yes	characteristic
27	C₂²⋊#1057	C₂²	1	2	yes	—
28	C₂²⋊#1057	C₂²	1	2	yes	—
29	#88112	C₂²	1	2	yes	—
30	#88111	C₂²	1	2	yes	—
31	#88110	C₂²	1	1	yes	characteristic
32	#88110	C₂²	1	1	yes	characteristic
33	#88110	C₂²	1	2	yes	—
34	C₄⋊#1057	C₂²	1	1	yes	characteristic
35	C₄⋊#1057	C₂²	1	1	yes	characteristic
36	C₄⋊#1057	C₂²	1	2	yes	—
37	C₄×#1057	C₂²	1	1	yes	characteristic
38	C₄×#1057	C₂²	1	1	yes	characteristic
39	C₄×#1057	C₂²	1	2	yes	—
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	—	2	4	no	—
49	C₂×#1057	—	2	4	no	—
50	C₂×#1057	C₂³	1	2	yes	—
51	C₂×#1057	C₂³	1	1	yes	characteristic
52	C₂×#1057	C₂³	1	2	yes	—
53	C₂×#1057	C₂³	1	1	yes	characteristic
54	C₂×#1057	C₂³	1	1	yes	characteristic
55	C₂×#1057	C₂³	1	1	yes	characteristic
56	#6118	—	2	2	no	—
57	#6118	—	2	2	no	—
58	C₁₃×C₂²⋊C₄	C₂×C₄	1	2	yes	—
59	C₁₃×C₂²⋊C₄	C₂×C₄	1	2	yes	—
60	C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
61	C₂²⋊Dic₁₃	C₂×C₄	1	2	yes	—
62	C₂²⋊Dic₁₃	C₂×C₄	1	2	yes	—
63	#6103	C₂×C₄	1	2	yes	—
64	C₄⋊Dic₁₃	C₂×C₄	1	1	yes	characteristic
65	C₄⋊Dic₁₃	C₂×C₄	1	1	yes	characteristic
66	C₄⋊Dic₁₃	C₂×C₄	1	2	yes	—
67	#6101	C₂×C₄	1	2	yes	—
68	#6101	C₂×C₄	1	2	yes	—
69	#6101	C₂×C₄	1	2	yes	—
70	C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
71	C₄×Dic₁₃	C₂³	1	1	yes	characteristic
72	C₄×Dic₁₃	C₂³	1	1	yes	characteristic
73	C₄×Dic₁₃	C₂³	1	2	yes	—
74	#2969	—	13	1	no	maximal, Sylow
75	C₂³×C₁₃	C₂²×C₄	1	2	yes	—
76	C₂²×D₁₃	C₂²×C₄	1	1	yes	characteristic
77	Q₈×C₁₃	—	2	2	no	—
78	D₄×C₁₃	—	2	2	no	—
79	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₄×C₁₃	C₂²×C₄	1	2	yes	—
82	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
83	C₂²⋊D₁₃	—	4	2	no	—
84	C₂×Dic₁₃	—	2	2	no	—
85	C₂×Dic₁₃	—	2	2	no	—
86	C₂×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
87	C₂×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
88	C₂×Dic₁₃	C₂²×C₄	1	2	yes	—
89	C₂×Dic₁₃	C₂²×C₄	1	2	yes	—
90	C₂×Dic₁₃	C₂⁴	1	1	yes	characteristic
91	C₄×D₁₃	—	2	2	no	—
92	C₄×D₁₃	—	2	1	no	—
93	C₄×D₁₃	—	2	1	no	—
94	C₄×D₁₃	—	4	1	no	—
95	#1057	—	2	1	no	—
96	#1057	—	2	1	no	—
97	#1057	—	2	2	no	—
98	#1057	—	2	2	no	—
99	#1057	#41	1	2	yes	—
100	#1057	#41	1	2	yes	—
101	#529	—	13	1	no	—
102	#446	—	13	1	no	—
103	C₈×Q₈	—	13	1	no	—
104	#436	—	13	2	no	—
105	#436	—	13	1	no	—
106	#435	—	13	2	no	—
107	C₈×D₄	—	13	1	no	—
108	#433	—	13	2	no	—
109	#431	—	13	1	no	—
110	#405	—	13	2	no	—
111	C₄×M₁₆	—	13	1	no	—
112	C₂²×C₁₃	—	2	2	no	—
113	C₂²×C₁₃	—	2	2	no	—
114	C₂²×C₁₃	—	2	2	no	—
115	C₂²×C₁₃	C₂³×C₄	1	1	yes	characteristic
116	C₂×D₁₃	—	2	1	no	—
117	C₂×D₁₃	—	2	1	no	—
118	C₂×D₁₃	—	2	1	no	—
119	C₄×C₁₃	—	2	1	no	—
120	C₄×C₁₃	—	2	2	no	—
121	C₄×C₁₃	—	2	1	no	—
122	C₄×C₁₃	#131	1	2	yes	—
123	Dic₁₃	—	2	2	no	—
124	Dic₁₃	—	2	1	no	—
125	Dic₁₃	—	2	1	no	—
126	Dic₁₃	—	2	2	no	—
127	Dic₁₃	C₂×#41	1	1	yes	characteristic
128	Dic₁₃	C₂×#41	1	1	yes	characteristic
129	C₂×M₁₆	—	13	1	no	—
130	C₂×M₁₆	—	13	1	no	—
131	C₂×M₁₆	—	13	2	no	—
132	C₂²×C₈	—	13	2	no	—
133	#126	—	13	2	no	—
134	#125	—	13	1	no	—
135	#124	#261	1	1	yes	characteristic
136	#123	—	13	2	no	—
137	C₂²⋊Q₈	—	13	1	no	—
138	#121	—	13	1	no	—
139	C₄×Q₈	—	13	1	no	—
140	C₄×D₄	—	13	1	no	—
141	C₄×D₄	—	13	2	no	—
142	#117	—	13	2	no	—
143	C₂×C₄²	—	13	1	no	—
144	C₄⋊C₈	—	13	1	no	—
145	C₄⋊C₈	—	13	1	no	—
146	C₄⋊C₈	—	13	2	no	—
147	C₄⋊C₈	—	13	2	no	—
148	C₂²⋊C₈	—	13	2	no	—
149	C₂²⋊C₈	—	13	2	no	—
150	C₂²⋊C₈	—	13	2	no	—
151	#97	—	13	1	no	—
152	#97	—	13	1	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	C₄×C₈	—	13	2	no	—
158	C₂×C₁₃	—	4	2	no	—
159	C₂×C₁₃	C₄×#41	1	1	yes	characteristic
160	C₂×C₁₃	C₂×#131	1	1	yes	characteristic
161	C₂×C₁₃	#568	1	1	yes	characteristic
162	D₁₃	—	4	1	no	—
163	C₂×Q₈	C₂×#261	1	1	yes	characteristic
164	C₂×D₄	—	13	2	no	—
165	C₂×D₄	C₂×#261	1	1	yes	characteristic
166	C₂²×C₄	—	13	2	no	—
167	C₂²×C₄	—	13	1	no	—
168	C₂²×C₄	—	13	1	no	—
169	C₂²×C₄	—	13	1	no	—
170	M₁₆	—	26	2	no	—
171	M₁₆	—	26	2	no	—
172	M₁₆	—	26	4	no	—
173	C₂×C₈	—	13	1	no	—
174	C₂×C₈	—	13	1	no	—
175	C₂×C₈	—	13	2	no	—
176	C₂×C₈	—	13	1	no	—
177	C₂×C₈	—	13	1	no	—
178	C₂×C₈	—	13	2	no	—
179	C₂×C₈	—	26	4	no	—
180	C₄⋊C₄	—	13	2	no	—
181	C₄⋊C₄	—	13	2	no	—
182	C₄⋊C₄	—	13	1	no	—
183	C₄⋊C₄	—	13	2	no	—
184	C₄⋊C₄	—	13	1	no	—
185	C₄⋊C₄	—	13	2	no	—
186	C₂²⋊C₄	—	13	2	no	—
187	C₂²⋊C₄	—	13	2	no	—
188	C₂²⋊C₄	—	13	2	no	—
189	C₂²⋊C₄	C₂×#261	1	2	yes	—
190	C₂²⋊C₄	C₂×#261	1	2	yes	—
191	C₄²	—	13	1	no	—
192	C₄²	—	13	1	no	—
193	C₄²	—	13	2	no	—
194	C₄²	—	13	1	no	—
195	C₄²	C₂×#261	1	1	yes	characteristic
196	C₁₃	#2969	1	1	yes	characteristic, complemented, Sylow
197	C₂³	—	13	1	no	—
198	C₂³	C₂²×#261	1	2	yes	—
199	Q₈	—	2	2	no	—
200	D₄	—	2	2	no	—
201	D₄	—	52	2	no	—
202	C₂×C₄	—	13	1	no	—
203	C₂×C₄	—	13	1	no	—
204	C₂×C₄	—	13	1	no	—
205	C₂×C₄	—	13	2	no	—
206	C₂×C₄	—	13	2	no	—
207	C₂×C₄	—	26	1	no	—
208	C₂×C₄	—	26	1	no	—
209	C₂×C₄	—	26	2	no	—
210	C₂×C₄	—	26	2	no	—
211	C₂×C₄	—	26	2	no	—
212	C₂×C₄	—	52	1	no	—
213	C₂×C₄	C₂²×#261	1	1	yes	characteristic
214	C₂×C₄	C₂²×#261	1	1	yes	characteristic
215	C₂×C₄	C₂²×#261	1	2	yes	—
216	C₂×C₄	C₂²×#261	1	1	yes	characteristic
217	C₈	—	13	2	no	—
218	C₈	—	13	2	no	—
219	C₈	—	26	2	no	—
220	C₈	—	26	1	no	—
221	C₈	—	26	1	no	—
222	C₈	—	26	2	no	—
223	C₂²	—	2	2	no	—
224	C₂²	—	2	2	no	—
225	C₂²	—	2	2	no	—
226	C₂²	—	26	1	no	—
227	C₂²	—	26	1	no	—
228	C₂²	—	26	1	no	—
229	C₂²	C₂³×#261	1	1	yes	characteristic
230	C₄	—	2	2	no	—
231	C₄	—	2	1	no	—
232	C₄	—	2	1	no	—
233	C₄	—	13	1	no	—
234	C₄	—	13	1	no	—
235	C₄	—	26	1	no	—
236	C₄	—	26	1	no	—
237	C₄	—	26	2	no	—
238	C₄	—	26	2	no	—
239	C₄	#88238	1	2	yes	—
240	C₂	—	4	2	no	—
241	C₂	—	52	1	no	—
242	C₂	C₂×#88238	1	1	yes	characteristic
243	C₂	#11719629	1	1	yes	characteristic
244	C₂	#41×#261	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	#2969
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. #11719066	C₂	semidirect
8. #11719064	C₂	semidirect
9. #11719062	C₂	semidirect
10. #11719051	C₂	semidirect
11. C₄×#6118	C₂	semidirect
196. C₁₃	#2969	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
#11719066	2
#11719064	2
#11719062	2
#11719051	2
C₄×#6118	2
#2969	13