Subgroups — #16782362

Order 3328.

378 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	#13130769	C₂	1	1	yes	maximal, characteristic
2	#13130753	C₂	1	1	yes	maximal, characteristic
3	#13129666	C₂	1	2	yes	maximal
4	#13129666	C₂	1	2	yes	maximal
5	#97×Dic₁₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#88112	—	2	2	no	—
7	C₂×C₄⋊#1057	C₂²	1	2	yes	—
8	C₂×C₄⋊#1057	C₂²	1	2	yes	—
9	C₂×C₁₃×#97	C₄	1	1	yes	characteristic
10	C₂×#88055	C₄	1	1	yes	characteristic
11	C₂×#88055	C₄	1	1	yes	characteristic
12	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
13	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
14	C₄²×Dic₁₃	C₂²	1	1	yes	characteristic
15	C₂×#88043	C₄	1	1	yes	characteristic
16	#11718290	—	2	2	no	—
17	C₂²×#1057	—	2	2	no	—
18	C₂²×#1057	—	2	2	no	—
19	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄×Dic₁₃	D₄	1	2	yes	—
24	C₂×C₄×Dic₁₃	Q₈	1	2	yes	—
25	C₂×C₄×Dic₁₃	C₂³	1	1	yes	characteristic
26	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
27	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
28	#88112	—	4	4	no	—
29	C₄⋊#1057	—	2	2	no	—
30	C₄⋊#1057	—	2	2	no	—
31	C₄⋊#1057	—	2	2	no	—
32	C₄⋊#1057	—	2	2	no	—
33	C₁₃×#97	—	2	2	no	—
34	#88055	—	2	2	no	—
35	#88055	C₈	1	4	yes	complemented
36	C₈×Dic₁₃	—	2	1	no	—
37	C₈×Dic₁₃	—	2	1	no	—
38	C₈×Dic₁₃	—	2	1	no	—
39	C₈×Dic₁₃	—	2	1	no	—
40	#88043	—	2	2	no	—
41	#8786	—	13	1	no	maximal, Sylow
42	C₂²×C₄×C₁₃	C₄⋊C₄	1	2	yes	—
43	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
44	C₂²×Dic₁₃	C₄⋊C₄	1	2	yes	—
45	C₂²×Dic₁₃	C₂×D₄	1	1	yes	characteristic
46	C₂²×Dic₁₃	C₂×Q₈	1	1	yes	characteristic
47	C₂×#1057	—	4	2	no	—
48	C₂×#1057	—	4	2	no	—
49	C₂×#1057	—	4	4	no	—
50	C₂×#1057	—	4	2	no	—
51	C₂×#1057	—	4	2	no	—
52	C₂×C₈×C₁₃	—	2	1	no	—
53	C₂×C₈×C₁₃	—	2	1	no	—
54	C₂×C₈×C₁₃	—	2	1	no	—
55	C₂×C₈×C₁₃	—	2	1	no	—
56	C₂×C₈×C₁₃	C₂×C₈	1	2	yes	—
57	C₂×C₈×C₁₃	M₁₆	1	2	yes	—
58	C₄²×C₁₃	—	2	1	no	—
59	C₄²×C₁₃	—	2	1	no	—
60	C₄×Dic₁₃	—	2	2	no	—
61	C₄×Dic₁₃	—	2	2	no	—
62	C₄×Dic₁₃	—	4	2	no	—
63	C₄×Dic₁₃	C₂×C₈	1	2	yes	—
64	C₄×Dic₁₃	M₁₆	1	2	yes	—
65	C₄×Dic₁₃	#41	1	1	yes	characteristic
66	C₄×Dic₁₃	#41	1	1	yes	characteristic
67	C₄×Dic₁₃	#41	1	1	yes	characteristic
68	C₄×Dic₁₃	#41	1	1	yes	characteristic
69	C₂×#1055	—	2	1	no	—
70	C₂×#1055	—	2	1	no	—
71	C₂×#1055	—	2	1	no	—
72	C₂×#1055	—	2	1	no	—
73	C₂×#1055	C₂×C₈	1	2	yes	—
74	C₂×#1055	M₁₆	1	2	yes	—
75	#1842	—	13	1	no	—
76	#1831	—	13	1	no	—
77	C₄×#97	—	13	1	no	—
78	#1277	—	13	2	no	—
79	#1277	—	13	2	no	—
80	C₂³×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
81	C₂×C₄×C₁₃	—	2	2	no	—
82	C₂×C₄×C₁₃	—	2	2	no	—
83	C₂×C₄×C₁₃	—	2	2	no	—
84	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
85	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
86	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
87	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
88	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
89	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
90	C₂×Dic₁₃	—	2	2	no	—
91	C₂×Dic₁₃	—	2	2	no	—
92	C₂×Dic₁₃	—	2	2	no	—
93	C₂×Dic₁₃	C₄⋊C₈	1	4	yes	—
94	C₂×Dic₁₃	#124	1	1	yes	characteristic
95	C₂×Dic₁₃	#124	1	1	yes	characteristic
96	C₂×Dic₁₃	#125	1	1	yes	characteristic
97	C₂×Dic₁₃	#125	1	1	yes	characteristic
98	C₂×Dic₁₃	AGL(1,ℤ/8)	1	2	yes	—
99	C₂×Dic₁₃	#137	1	2	yes	—
100	#1057	—	8	2	no	—
101	#1057	—	8	2	no	—
102	C₈×C₁₃	—	2	2	no	—
103	C₈×C₁₃	—	2	2	no	—
104	#1055	—	2	2	no	—
105	#1055	—	2	2	no	—
106	C₂×C₄⋊C₈	—	13	2	no	—
107	C₂×C₄⋊C₈	—	13	2	no	—
108	C₂×C₄⋊C₈	—	26	2	no	—
109	C₂×#97	—	13	1	no	—
110	C₂×#97	—	13	1	no	—
111	C₂×#97	—	13	1	no	—
112	C₂×#97	#261	1	1	yes	characteristic
113	C₂×C₄×C₈	—	13	1	no	—
114	C₂×C₄×C₈	—	13	1	no	—
115	C₄³	—	13	1	no	—
116	#336	—	26	2	no	—
117	C₂²×C₁₃	#382	1	1	yes	characteristic
118	C₂²×C₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
119	C₂²×C₁₃	#423	1	1	yes	characteristic
120	C₂²×C₁₃	#428	1	2	yes	—
121	C₂²×C₁₃	#430	1	2	yes	—
122	C₄×C₁₃	—	4	2	no	—
123	C₄×C₁₃	#431	1	1	yes	characteristic
124	C₄×C₁₃	#431	1	1	yes	characteristic
125	C₄×C₁₃	#432	1	1	yes	characteristic
126	C₄×C₁₃	#432	1	1	yes	characteristic
127	Dic₁₃	—	2	4	no	—
128	Dic₁₃	—	2	1	no	—
129	Dic₁₃	—	2	1	no	—
130	Dic₁₃	#489	1	2	yes	—
131	Dic₁₃	#490	1	1	yes	characteristic
132	Dic₁₃	#491	1	1	yes	characteristic
133	C₂²×C₈	—	13	1	no	—
134	C₂²×C₈	—	13	1	no	—
135	C₂²×C₈	—	26	2	no	—
136	C₂²×C₈	—	26	2	no	—
137	C₂²×C₈	C₂×#261	1	1	yes	characteristic
138	C₂²×C₈	C₂×#261	1	1	yes	characteristic
139	C₂×C₄²	—	13	1	no	—
140	C₂×C₄²	—	13	1	no	—
141	C₂×C₄²	—	13	2	no	—
142	C₂×C₄²	—	13	2	no	—
143	C₂×C₄²	C₂×#261	1	1	yes	characteristic
144	C₄⋊C₈	—	26	2	no	—
145	C₄⋊C₈	—	26	2	no	—
146	C₄⋊C₈	—	26	2	no	—
147	C₄⋊C₈	—	26	2	no	—
148	C₄⋊C₈	—	52	4	no	—
149	#97	—	2	2	no	—
150	#97	—	13	4	no	—
151	#97	—	26	2	no	—
152	#97	—	26	2	no	—
153	C₄×C₈	—	26	1	no	—
154	C₄×C₈	—	26	1	no	—
155	C₄×C₈	—	26	1	no	—
156	C₄×C₈	—	26	1	no	—
157	C₂×C₁₃	#1565	1	2	yes	—
158	C₂×C₁₃	#1569	1	2	yes	—
159	C₂×C₁₃	#1831	1	1	yes	characteristic
160	C₂×C₁₃	#1836	1	1	yes	characteristic
161	C₂×C₁₃	#1838	1	1	yes	characteristic
162	C₂²×C₄	—	13	1	no	—
163	C₂²×C₄	—	13	2	no	—
164	C₂²×C₄	—	13	1	no	—
165	C₂²×C₄	C₄⋊#261	1	2	yes	—
166	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
167	C₂×C₈	—	2	1	no	—
168	C₂×C₈	—	2	1	no	—
169	C₂×C₈	—	2	1	no	—
170	C₂×C₈	—	2	1	no	—
171	C₂×C₈	—	13	2	no	—
172	C₂×C₈	—	13	2	no	—
173	C₂×C₈	—	26	1	no	—
174	C₂×C₈	—	26	1	no	—
175	C₂×C₈	—	26	1	no	—
176	C₂×C₈	—	26	1	no	—
177	C₂×C₈	—	52	2	no	—
178	C₂×C₈	—	52	2	no	—
179	C₂×C₈	—	52	4	no	—
180	C₂×C₈	—	52	2	no	—
181	C₂×C₈	—	52	2	no	—
182	C₂×C₈	D₁₃⋊C₈	1	2	yes	—
183	C₂×C₈	#6122	1	2	yes	—
184	C₄²	—	2	1	no	—
185	C₄²	—	2	1	no	—
186	C₄²	—	13	2	no	—
187	C₄²	—	13	2	no	—
188	C₄²	—	13	1	no	—
189	C₄²	—	13	1	no	—
190	C₄²	—	13	1	no	—
191	C₄²	—	13	1	no	—
192	C₄²	—	26	2	no	—
193	C₄²	—	26	2	no	—
194	C₄²	—	52	2	no	—
195	C₁₃	#8786	1	1	yes	characteristic, complemented, Sylow
196	C₂³	C₂×C₄⋊#261	1	1	yes	characteristic
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	13	1	no	—
201	C₂×C₄	—	13	1	no	—
202	C₂×C₄	—	13	2	no	—
203	C₂×C₄	—	13	2	no	—
204	C₂×C₄	—	13	4	no	—
205	C₂×C₄	—	13	1	no	—
206	C₂×C₄	—	13	1	no	—
207	C₂×C₄	—	26	2	no	—
208	C₂×C₄	—	26	2	no	—
209	C₂×C₄	—	26	2	no	—
210	C₂×C₄	C₂×D₁₃⋊C₈	1	1	yes	characteristic
211	C₂×C₄	D₁₃⋊M₁₆	1	1	yes	characteristic
212	C₂×C₄	D₁₃⋊M₁₆	1	1	yes	characteristic
213	C₂×C₄	#88237	1	1	yes	characteristic
214	C₂×C₄	#88237	1	1	yes	characteristic
215	C₂×C₄	C₂×#6122	1	1	yes	characteristic
216	C₈	—	2	2	no	—
217	C₈	—	2	2	no	—
218	C₈	—	26	2	no	—
219	C₈	—	26	2	no	—
220	C₈	—	104	2	no	—
221	C₈	—	104	2	no	—
222	C₂²	D₁₃⋊(C₄⋊C₈)	1	1	yes	characteristic
223	C₂²	#11719060	1	1	yes	characteristic
224	C₂²	#11719098	1	2	yes	—
225	C₂²	#11719100	1	2	yes	—
226	C₂²	#11719129	1	1	yes	characteristic
227	C₄	—	4	2	no	—
228	C₄	—	13	2	no	—
229	C₄	—	13	1	no	—
230	C₄	—	13	1	no	—
231	C₄	—	26	1	no	—
232	C₄	—	26	1	no	—
233	C₄	—	26	4	no	—
234	C₄	#11719049	1	1	yes	characteristic
235	C₄	#11719055	1	1	yes	characteristic
236	C₄	#11719125	1	1	yes	characteristic
237	C₄	#11719128	1	1	yes	characteristic
238	C₂	#13130388	1	2	yes	—
239	C₂	#13130400	1	2	yes	—
240	C₂	#13130752	1	1	yes	characteristic
241	C₂	#13130759	1	1	yes	characteristic
242	C₂	#13130762	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₄⋊#261
commutator	C₂×C₄×C₁₃	C₂²×C₈
Frattini	C₂²×C₄	C₂²×#261
Fitting	C₂×C₁₃×#97	C₄
socle, prime radical	C₂³×C₁₃	C₂×C₄⋊C₄
cosocle	C₂×C₄×Dic₁₃	C₂³
hypercenter	C₂×#97	#261
hypocenter	C₁₃	#8786
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#97

Complements

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

N	H ≅ G/N	type
35. #88055	C₈	semidirect
195. C₁₃	#8786	semidirect

Maximal subgroups

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

subgroup	index
#13130769	2
#13130753	2
#13129666	2
#97×Dic₁₃	2
#8786	13