Subgroups — #13131631

Order 1664.

356 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11719327	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	C₄×#6122	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊#1057	C₂	1	1	yes	maximal, characteristic
6	#11719080	C₂	1	2	yes	maximal, complemented
7	#11719080	C₂	1	1	yes	maximal, characteristic, complemented
8	#11719078	C₂	1	2	yes	maximal, complemented
9	#11719074	C₂	1	1	yes	maximal, characteristic, complemented
10	#11719067	C₂	1	2	yes	maximal
11	#11719062	C₂	1	2	yes	maximal
12	C₂×#6122	C₂²	1	1	yes	characteristic
13	C₂×#6122	C₂²	1	1	yes	characteristic
14	C₂²×#1057	C₂²	1	2	yes	—
15	C₁₃×C₂²⋊Q₈	C₄	1	1	yes	characteristic
16	Q₈×Dic₁₃	C₂²	1	1	yes	characteristic
17	#88198	C₄	1	1	yes	characteristic
18	#88178	C₄	1	1	yes	characteristic
19	C₂×C₄×Dic₁₃	C₂²	1	1	yes	characteristic
20	#88143	C₄	1	2	yes	—
21	#88142	C₄	1	1	yes	characteristic
22	#88141	C₂²	1	1	yes	characteristic
23	#88141	C₂²	1	2	yes	—
24	#88132	C₄	1	2	yes	—
25	#88131	C₂²	1	2	yes	—
26	#88112	C₂²	1	2	yes	—
27	#88112	C₂²	1	2	yes	—
28	#88110	C₂²	1	2	yes	—
29	#88110	C₂²	1	2	yes	—
30	#88110	C₂²	1	1	yes	characteristic
31	#88110	C₂²	1	1	yes	characteristic
32	C₄⋊#1057	C₂²	1	2	yes	—
33	C₄⋊#1057	C₂²	1	2	yes	—
34	C₄⋊#1057	C₂²	1	1	yes	characteristic
35	C₄⋊#1057	C₂²	1	2	yes	—
36	C₄⋊#1057	C₂²	1	1	yes	characteristic
37	C₄×#1057	C₂²	1	2	yes	—
38	C₄×#1057	C₂²	1	2	yes	—
39	C₄×#1057	C₂²	1	1	yes	characteristic
40	C₄×#1057	C₂²	1	1	yes	characteristic
41	C₂×Q₈×C₁₃	C₂×C₄	1	1	yes	characteristic
42	C₂²×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
43	C₂²×Dic₁₃	C₂×C₄	1	1	yes	characteristic
44	C₂²×Dic₁₃	C₂³	1	1	yes	characteristic
45	C₂×Dic₂₆	C₂×C₄	1	1	yes	characteristic
46	C₂×Dic₂₆	C₂×C₄	1	2	yes	—
47	#6122	—	2	2	no	—
48	#6122	Q₈	1	4	yes	complemented
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	1	yes	characteristic
53	C₂×#1057	C₂³	1	2	yes	—
54	C₂×#1057	C₂³	1	1	yes	characteristic
55	C₂×#1057	C₂³	1	1	yes	characteristic
56	C₁₃×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
57	C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
58	C₁₃×C₂²⋊C₄	C₂×C₄	1	2	yes	—
59	C₂²⋊Dic₁₃	C₂×C₄	1	2	yes	—
60	C₄⋊Dic₁₃	C₂×C₄	1	2	yes	—
61	C₄⋊Dic₁₃	C₂×C₄	1	1	yes	characteristic
62	#6101	C₂×C₄	1	2	yes	—
63	#6101	C₂×C₄	1	2	yes	—
64	#6101	C₂×C₄	1	2	yes	—
65	#6101	C₂×C₄	1	2	yes	—
66	#6101	C₂×C₄	1	2	yes	—
67	C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
68	C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
69	C₄×Dic₁₃	C₂³	1	1	yes	characteristic
70	C₄×Dic₁₃	C₂³	1	2	yes	—
71	C₄×Dic₁₃	C₂³	1	2	yes	—
72	C₄×Dic₁₃	C₂³	1	1	yes	characteristic
73	#2962	—	13	1	no	maximal, Sylow
74	C₂³×C₁₃	C₂²×C₄	1	1	yes	characteristic
75	Q₈×C₁₃	—	2	2	no	—
76	C₂×C₄×C₁₃	—	2	2	no	—
77	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₄×C₁₃	C₂²×C₄	1	2	yes	—
79	C₂×C₄×C₁₃	C₂²×C₄	1	2	yes	—
80	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
81	C₂×Dic₁₃	—	4	1	no	—
82	C₂×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
83	C₂×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
84	C₂×Dic₁₃	C₂²×C₄	1	2	yes	—
85	C₂×Dic₁₃	C₂²×C₄	1	2	yes	—
86	C₂×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
87	C₂×Dic₁₃	C₂×Q₈	1	2	yes	—
88	C₂×Dic₁₃	#41	1	2	yes	—
89	C₂×Dic₁₃	C₂⁴	1	1	yes	characteristic
90	Dic₂₆	—	2	2	no	—
91	Dic₂₆	—	4	2	no	—
92	#1057	—	2	2	no	—
93	#1057	—	2	2	no	—
94	#1057	—	2	1	no	—
95	#1057	—	2	1	no	—
96	#1057	C₂×Q₈	1	2	yes	—
97	#1057	C₂×Q₈	1	2	yes	—
98	#533	—	13	1	no	—
99	#446	—	13	2	no	—
100	#446	—	13	1	no	—
101	#446	—	13	2	no	—
102	#446	—	13	1	no	—
103	C₈×Q₈	—	13	1	no	—
104	C₈×Q₈	—	13	1	no	—
105	#424	—	13	2	no	—
106	C₂×C₄⋊C₈	—	13	1	no	—
107	#405	—	13	2	no	—
108	C₄×M₁₆	—	13	1	no	—
109	C₂²×C₁₃	—	2	1	no	—
110	C₂²×C₁₃	—	2	1	no	—
111	C₂²×C₁₃	C₄×Q₈	1	2	yes	—
112	C₂²×C₁₃	C₂³×C₄	1	1	yes	characteristic
113	C₄×C₁₃	—	2	1	no	—
114	C₄×C₁₃	—	2	2	no	—
115	C₄×C₁₃	—	2	2	no	—
116	C₄×C₁₃	#131	1	2	yes	—
117	Dic₁₃	—	2	2	no	—
118	Dic₁₃	—	2	1	no	—
119	Dic₁₃	—	2	2	no	—
120	Dic₁₃	—	2	1	no	—
121	Dic₁₃	C₄×Q₈	1	2	yes	—
122	Dic₁₃	C₂²×Q₈	1	1	yes	characteristic
123	Dic₁₃	C₂×#41	1	1	yes	characteristic
124	C₂×M₁₆	—	13	1	no	—
125	C₂×M₁₆	—	13	1	no	—
126	C₂²×C₈	—	13	2	no	—
127	C₄⋊Q₈	—	13	1	no	—
128	C₄⋊Q₈	—	13	1	no	—
129	#125	—	13	2	no	—
130	C₂²⋊Q₈	—	13	1	no	—
131	C₂²⋊Q₈	—	13	2	no	—
132	C₂²⋊Q₈	#261	1	1	yes	characteristic
133	C₄×Q₈	—	13	1	no	—
134	C₄×Q₈	—	13	1	no	—
135	C₄×Q₈	—	13	2	no	—
136	#117	—	13	2	no	—
137	C₂×C₄²	—	13	1	no	—
138	C₄⋊C₈	—	13	1	no	—
139	C₄⋊C₈	—	13	1	no	—
140	C₄⋊C₈	—	13	2	no	—
141	C₄⋊C₈	—	13	2	no	—
142	C₄⋊C₈	—	13	2	no	—
143	C₄⋊C₈	—	13	2	no	—
144	C₄⋊C₈	—	13	2	no	—
145	#97	—	13	1	no	—
146	#97	—	13	1	no	—
147	#97	—	13	2	no	—
148	#97	—	13	2	no	—
149	C₄×C₈	—	13	1	no	—
150	C₄×C₈	—	13	1	no	—
151	C₄×C₈	—	13	2	no	—
152	C₄×C₈	—	13	2	no	—
153	C₂×C₁₃	—	2	2	no	—
154	C₂×C₁₃	C₂×C₄×Q₈	1	1	yes	characteristic
155	C₂×C₁₃	C₂×#131	1	1	yes	characteristic
156	C₂×C₁₃	#568	1	1	yes	characteristic
157	C₂×Q₈	—	13	1	no	—
158	C₂×Q₈	—	13	2	no	—
159	C₂×Q₈	C₂×#261	1	1	yes	characteristic
160	C₂²×C₄	—	13	1	no	—
161	C₂²×C₄	—	13	1	no	—
162	C₂²×C₄	C₂×#261	1	1	yes	characteristic
163	M₁₆	—	13	4	no	—
164	M₁₆	—	26	2	no	—
165	C₂×C₈	—	13	1	no	—
166	C₂×C₈	—	13	1	no	—
167	C₂×C₈	—	13	2	no	—
168	C₂×C₈	—	13	1	no	—
169	C₂×C₈	—	13	1	no	—
170	C₂×C₈	—	13	2	no	—
171	C₂×C₈	—	26	4	no	—
172	C₄⋊C₄	—	13	2	no	—
173	C₄⋊C₄	—	13	2	no	—
174	C₄⋊C₄	—	13	2	no	—
175	C₄⋊C₄	—	13	2	no	—
176	C₄⋊C₄	—	13	2	no	—
177	C₄⋊C₄	—	13	2	no	—
178	C₄⋊C₄	—	13	1	no	—
179	C₄⋊C₄	C₂×#261	1	2	yes	—
180	C₄⋊C₄	C₂×#261	1	1	yes	characteristic
181	C₂²⋊C₄	—	13	2	no	—
182	C₂²⋊C₄	C₂×#261	1	2	yes	—
183	C₄²	—	13	1	no	—
184	C₄²	—	13	2	no	—
185	C₄²	—	13	2	no	—
186	C₄²	—	13	1	no	—
187	C₄²	—	13	1	no	—
188	C₄²	—	13	1	no	—
189	C₁₃	#2962	1	1	yes	characteristic, complemented, Sylow
190	C₂³	C₂²×#261	1	1	yes	characteristic
191	Q₈	—	2	2	no	—
192	Q₈	—	26	2	no	—
193	Q₈	—	52	2	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	13	2	no	—
196	C₂×C₄	—	13	2	no	—
197	C₂×C₄	—	13	1	no	—
198	C₂×C₄	—	13	1	no	—
199	C₂×C₄	—	13	1	no	—
200	C₂×C₄	—	13	1	no	—
201	C₂×C₄	—	13	2	no	—
202	C₂×C₄	—	13	2	no	—
203	C₂×C₄	—	52	1	no	—
204	C₂×C₄	C₂²×#261	1	1	yes	characteristic
205	C₂×C₄	C₂²×#261	1	2	yes	—
206	C₂×C₄	C₂²×#261	1	2	yes	—
207	C₂×C₄	C₂²×#261	1	1	yes	characteristic
208	C₈	—	13	2	no	—
209	C₈	—	13	2	no	—
210	C₈	—	26	2	no	—
211	C₈	—	26	2	no	—
212	C₈	—	26	1	no	—
213	C₈	—	26	1	no	—
214	C₂²	—	2	1	no	—
215	C₂²	—	2	1	no	—
216	C₂²	Q₈×#261	1	2	yes	—
217	C₂²	C₂³×#261	1	1	yes	characteristic
218	C₄	—	2	1	no	—
219	C₄	—	2	2	no	—
220	C₄	—	2	2	no	—
221	C₄	—	13	2	no	—
222	C₄	—	13	1	no	—
223	C₄	—	13	1	no	—
224	C₄	—	26	2	no	—
225	C₄	—	26	1	no	—
226	C₄	—	26	1	no	—
227	C₄	—	26	2	no	—
228	C₄	#88238	1	2	yes	—
229	C₂	—	2	2	no	—
230	C₂	C₂×#88238	1	1	yes	characteristic
231	C₂	C₂×Q₈×#261	1	1	yes	characteristic
232	C₂	#11719629	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×#261
commutator, socle	C₂²×C₁₃	C₂³×C₄
Fitting	C₁₃×C₂²⋊Q₈	C₄
prime radical	C₂³×C₁₃	C₂²×C₄
cosocle	C₂×Dic₁₃	C₂⁴
hypercenter	C₂²⋊Q₈	#261
hypocenter	C₁₃	#2962
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂²⋊Q₈

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
6. #11719080	C₂	semidirect
7. #11719080	C₂	semidirect
8. #11719078	C₂	semidirect
9. #11719074	C₂	semidirect
48. #6122	Q₈	semidirect
189. C₁₃	#2962	semidirect

Maximal subgroups

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

subgroup	index
#11719327	2
#11719158	2
Q₈×#1057	2
C₄×#6122	2
C₂×C₄⋊#1057	2
#11719080	2
#11719078	2
#11719074	2
#11719067	2
#11719062	2
#2962	13