Subgroups — #16780919

Order 3328.

438 conjugacy classes of subgroups.

The subgroup lattice has 254 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13130622	C₂	1	2	yes	maximal
2	C₂×#11718254	C₂	1	2	yes	maximal
3	#13129666	C₂	1	2	yes	maximal
4	C₄×C₈×Dic₁₃	C₂	1	1	yes	maximal, characteristic
5	C₂×#88110	—	2	2	no	—
6	C₂×C₄⋊#1057	C₂²	1	4	yes	—
7	C₂×C₄×C₈×C₁₃	C₄	1	1	yes	characteristic
8	C₂×C₈×Dic₁₃	C₄	1	1	yes	characteristic
9	C₂×C₈×Dic₁₃	C₄	1	1	yes	characteristic
10	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
11	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
12	C₄²×Dic₁₃	C₂²	1	1	yes	characteristic
13	C₂×C₄×#1055	C₄	1	1	yes	characteristic
14	#11718290	—	2	2	no	—
15	#11718254	C₄	1	8	yes	complemented
16	C₂²×#1057	—	2	4	no	—
17	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄×Dic₁₃	D₄	1	1	yes	characteristic
22	C₂×C₄×Dic₁₃	D₄	1	1	yes	characteristic
23	C₂×C₄×Dic₁₃	Q₈	1	1	yes	characteristic
24	C₂×C₄×Dic₁₃	Q₈	1	1	yes	characteristic
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	#88110	—	4	4	no	—
29	C₄⋊#1057	—	2	4	no	—
30	C₄⋊#1057	C₂×C₄	1	8	yes	—
31	C₄×C₈×C₁₃	—	2	2	no	—
32	C₈×Dic₁₃	—	2	2	no	—
33	C₈×Dic₁₃	—	2	2	no	—
34	C₈×Dic₁₃	—	2	2	no	—
35	C₈×Dic₁₃	C₂×C₄	1	4	yes	—
36	C₄×#1055	—	2	2	no	—
37	#8274	—	13	1	no	maximal, Sylow
38	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
39	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
40	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
41	C₂²×Dic₁₃	C₄⋊C₄	1	1	yes	characteristic
42	C₂²×Dic₁₃	C₄⋊C₄	1	1	yes	characteristic
43	C₂²×Dic₁₃	C₂×D₄	1	1	yes	characteristic
44	C₂²×Dic₁₃	C₂×Q₈	1	1	yes	characteristic
45	C₂×#1057	—	2	8	no	—
46	C₂×#1057	—	4	4	no	—
47	C₂×#1057	—	4	4	no	—
48	C₂×C₈×C₁₃	—	2	1	no	—
49	C₂×C₈×C₁₃	—	2	1	no	—
50	C₂×C₈×C₁₃	C₄²	1	2	yes	—
51	C₂×C₈×C₁₃	M₁₆	1	2	yes	—
52	C₂×C₈×C₁₃	M₁₆	1	2	yes	—
53	C₂×C₈×C₁₃	M₁₆	1	2	yes	—
54	C₄²×C₁₃	—	2	1	no	—
55	C₄²×C₁₃	—	2	1	no	—
56	C₄×Dic₁₃	—	2	2	no	—
57	C₄×Dic₁₃	—	2	1	no	—
58	C₄×Dic₁₃	—	2	1	no	—
59	C₄×Dic₁₃	—	2	2	no	—
60	C₄×Dic₁₃	—	2	2	no	—
61	C₄×Dic₁₃	D₈	1	2	yes	—
62	C₄×Dic₁₃	Q₁₆	1	2	yes	—
63	C₄×Dic₁₃	C₂²×C₄	1	2	yes	—
64	C₄×Dic₁₃	#41	1	2	yes	—
65	C₂×#1055	—	2	1	no	—
66	C₂×#1055	—	2	1	no	—
67	C₂×#1055	—	2	1	no	—
68	C₂×#1055	—	2	1	no	—
69	C₂×#1055	C₄²	1	2	yes	—
70	C₂×#1055	M₁₆	1	2	yes	—
71	#1767	—	13	2	no	—
72	C₄²×C₈	—	13	1	no	—
73	C₂×#335	—	13	2	no	—
74	#1277	—	13	2	no	—
75	C₂³×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
76	C₂×C₄×C₁₃	—	2	1	no	—
77	C₂×C₄×C₁₃	—	2	1	no	—
78	C₂×C₄×C₁₃	—	2	1	no	—
79	C₂×C₄×C₁₃	—	2	1	no	—
80	C₂×C₄×C₁₃	#107	1	2	yes	—
81	C₂×C₄×C₁₃	#108	1	2	yes	—
82	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
83	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
84	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
85	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
86	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
87	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
88	C₂×Dic₁₃	—	2	1	no	—
89	C₂×Dic₁₃	—	2	1	no	—
90	C₂×Dic₁₃	—	2	1	no	—
91	C₂×Dic₁₃	—	2	1	no	—
92	C₂×Dic₁₃	—	2	1	no	—
93	C₂×Dic₁₃	—	2	1	no	—
94	C₂×Dic₁₃	#107	1	2	yes	—
95	C₂×Dic₁₃	#108	1	2	yes	—
96	C₂×Dic₁₃	C₄×D₄	1	2	yes	—
97	C₂×Dic₁₃	C₄×Q₈	1	2	yes	—
98	C₂×Dic₁₃	C₂×D₈	1	1	yes	characteristic
99	C₂×Dic₁₃	C₂×Q₁₆	1	1	yes	characteristic
100	C₂×Dic₁₃	#135	1	2	yes	—
101	#1057	—	4	8	no	—
102	C₈×C₁₃	—	2	2	no	—
103	C₈×C₁₃	#97	1	4	yes	complemented
104	#1055	—	2	2	no	—
105	#1055	—	2	2	no	—
106	C₂×C₄⋊C₈	—	13	4	no	—
107	C₂×#97	—	26	2	no	—
108	C₂×C₄×C₈	—	13	1	no	—
109	C₂×C₄×C₈	—	13	1	no	—
110	C₂×C₄×C₈	—	13	1	no	—
111	C₂×C₄×C₈	—	13	1	no	—
112	C₂×C₄×C₈	—	13	1	no	—
113	C₂×C₄×C₈	#261	1	1	yes	characteristic
114	C₄³	—	13	1	no	—
115	#336	—	26	2	no	—
116	#335	—	13	8	no	—
117	C₂²×C₁₃	C₄×C₄⋊C₄	1	1	yes	characteristic
118	C₂²×C₁₃	#423	1	1	yes	characteristic
119	C₂²×C₁₃	#423	1	1	yes	characteristic
120	C₂²×C₁₃	C₂×#107	1	1	yes	characteristic
121	C₂²×C₁₃	#427	1	1	yes	characteristic
122	C₂²×C₁₃	C₂×#108	1	1	yes	characteristic
123	C₂²×C₁₃	C₂×#108	1	1	yes	characteristic
124	C₄×C₁₃	—	2	2	no	—
125	C₄×C₁₃	—	2	2	no	—
126	C₄×C₁₃	C₂×#97	1	1	yes	characteristic
127	C₄×C₁₃	C₄×M₁₆	1	1	yes	characteristic
128	C₄×C₁₃	#432	1	1	yes	characteristic
129	C₄×C₁₃	#432	1	1	yes	characteristic
130	Dic₁₃	—	2	2	no	—
131	Dic₁₃	—	2	2	no	—
132	Dic₁₃	—	2	2	no	—
133	Dic₁₃	C₄×D₈	1	2	yes	—
134	Dic₁₃	C₄×Q₁₆	1	2	yes	—
135	C₂²×C₈	—	13	1	no	—
136	C₂²×C₈	—	13	1	no	—
137	C₂²×C₈	—	26	4	no	—
138	C₂²×C₈	C₂×#261	1	1	yes	characteristic
139	C₂²×C₈	C₂×#261	1	1	yes	characteristic
140	C₂×C₄²	—	13	1	no	—
141	C₂×C₄²	—	13	1	no	—
142	C₂×C₄²	—	13	1	no	—
143	C₂×C₄²	—	13	1	no	—
144	C₂×C₄²	—	13	1	no	—
145	C₂×C₄²	—	13	1	no	—
146	C₂×C₄²	C₂×#261	1	1	yes	characteristic
147	C₄⋊C₈	—	13	8	no	—
148	C₄⋊C₈	—	26	4	no	—
149	#97	—	52	4	no	—
150	C₄×C₈	—	2	2	no	—
151	C₄×C₈	—	13	4	no	—
152	C₄×C₈	—	26	2	no	—
153	C₄×C₈	—	26	2	no	—
154	C₄×C₈	—	26	2	no	—
155	C₄×C₈	—	26	2	no	—
156	C₂×C₁₃	#1567	1	1	yes	characteristic
157	C₂×C₁₃	#1567	1	1	yes	characteristic
158	C₂×C₁₃	#1568	1	1	yes	characteristic
159	C₂×C₁₃	#1568	1	1	yes	characteristic
160	C₂×C₁₃	#1767	1	1	yes	characteristic
161	C₂×C₁₃	C₄×#107	1	1	yes	characteristic
162	C₂×C₁₃	C₄×#108	1	1	yes	characteristic
163	C₂²×C₄	—	13	1	no	—
164	C₂²×C₄	—	13	1	no	—
165	C₂²×C₄	—	13	1	no	—
166	C₂²×C₄	—	13	1	no	—
167	C₂²×C₄	C₄⋊#261	1	1	yes	characteristic
168	C₂²×C₄	C₄⋊#261	1	1	yes	characteristic
169	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
170	C₂×C₈	—	2	1	no	—
171	C₂×C₈	—	2	1	no	—
172	C₂×C₈	—	13	2	no	—
173	C₂×C₈	—	13	2	no	—
174	C₂×C₈	—	26	1	no	—
175	C₂×C₈	—	26	1	no	—
176	C₂×C₈	—	26	1	no	—
177	C₂×C₈	—	26	1	no	—
178	C₂×C₈	—	26	8	no	—
179	C₂×C₈	—	52	4	no	—
180	C₂×C₈	—	52	4	no	—
181	C₂×C₈	#6118	1	2	yes	—
182	C₂×C₈	#6118	1	2	yes	—
183	C₂×C₈	C₄×#261	1	2	yes	—
184	C₂×C₈	#6122	1	2	yes	—
185	C₄²	—	2	1	no	—
186	C₄²	—	2	1	no	—
187	C₄²	—	13	2	no	—
188	C₄²	—	13	2	no	—
189	C₄²	—	13	2	no	—
190	C₄²	—	13	2	no	—
191	C₄²	—	26	2	no	—
192	C₄²	—	26	2	no	—
193	C₄²	—	26	2	no	—
194	C₄²	—	26	1	no	—
195	C₄²	—	26	1	no	—
196	C₁₃	#8274	1	1	yes	characteristic, complemented, Sylow
197	C₂³	C₂×C₄⋊#261	1	1	yes	characteristic
198	C₂×C₄	—	2	1	no	—
199	C₂×C₄	—	2	1	no	—
200	C₂×C₄	—	2	1	no	—
201	C₂×C₄	—	2	1	no	—
202	C₂×C₄	—	13	2	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	—	13	2	no	—
208	C₂×C₄	—	13	2	no	—
209	C₂×C₄	—	26	1	no	—
210	C₂×C₄	—	26	1	no	—
211	C₂×C₄	—	26	1	no	—
212	C₂×C₄	—	26	1	no	—
213	C₂×C₄	—	26	1	no	—
214	C₂×C₄	—	26	1	no	—
215	C₂×C₄	#88102	1	2	yes	—
216	C₂×C₄	#88104	1	2	yes	—
217	C₂×C₄	C₂×#6118	1	1	yes	characteristic
218	C₂×C₄	C₂×#6118	1	1	yes	characteristic
219	C₂×C₄	D₁₃⋊M₁₆	1	1	yes	characteristic
220	C₂×C₄	C₂×C₄×#261	1	1	yes	characteristic
221	C₂×C₄	#88237	1	1	yes	characteristic
222	C₂×C₄	C₂×#6122	1	1	yes	characteristic
223	C₈	—	2	2	no	—
224	C₈	—	26	2	no	—
225	C₈	—	26	2	no	—
226	C₈	—	52	8	no	—
227	C₈	#88110	1	4	yes	complemented
228	C₂²	#11719052	1	1	yes	characteristic
229	C₂²	C₄×C₄⋊#261	1	1	yes	characteristic
230	C₂²	C₂×#88102	1	1	yes	characteristic
231	C₂²	#11719092	1	1	yes	characteristic
232	C₂²	C₂×#88104	1	1	yes	characteristic
233	C₂²	D₁₃⋊#108	1	1	yes	characteristic
234	C₂²	#11719129	1	1	yes	characteristic
235	C₄	—	2	2	no	—
236	C₄	—	2	2	no	—
237	C₄	—	13	2	no	—
238	C₄	—	13	2	no	—
239	C₄	—	26	2	no	—
240	C₄	—	26	2	no	—
241	C₄	—	26	2	no	—
242	C₄	C₄×#6118	1	1	yes	characteristic
243	C₄	#11719054	1	1	yes	characteristic
244	C₄	C₂×#88110	1	1	yes	characteristic
245	C₄	#11719130	1	1	yes	characteristic
246	C₂	#13130393	1	1	yes	characteristic
247	C₂	#13130394	1	1	yes	characteristic
248	C₂	#13130397	1	1	yes	characteristic
249	C₂	#13130398	1	1	yes	characteristic
250	C₂	#13130622	1	1	yes	characteristic
251	C₂	C₄×#88102	1	1	yes	characteristic
252	C₂	C₄×#88104	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₄⋊#261
commutator	C₂×C₄×C₁₃	C₂×C₄²
Frattini	C₂²×C₄	C₂²×#261
Fitting	C₂×C₄×C₈×C₁₃	C₄
socle, prime radical	C₂³×C₁₃	C₂×C₄⋊C₄
cosocle	C₂×C₄×Dic₁₃	C₂³
hypercenter	C₂×C₄×C₈	#261
hypocenter	C₁₃	#8274
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
15. #11718254	C₄	semidirect
103. C₈×C₁₃	#97	semidirect
196. C₁₃	#8274	semidirect
227. C₈	#88110	semidirect

Maximal subgroups

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

subgroup	index
#13130622	2
C₂×#11718254	2
#13129666	2
C₄×C₈×Dic₁₃	2
#8274	13