Subgroups — #16018297

Order 3328.

590 conjugacy classes of subgroups.

The subgroup lattice has 266 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718344	C₂	1	1	yes	maximal, characteristic, complemented
2	#13117081	C₂	1	2	yes	maximal, complemented
3	#13117077	C₂	1	2	yes	maximal, complemented
4	#13117067	C₂	1	4	yes	maximal, complemented
5	#13116548	C₂	1	4	yes	maximal, complemented
6	C₄×#88052	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₃×#3147	C₂	1	1	yes	maximal, characteristic
8	C₂×C₁₃×#135	—	2	1	no	—
9	C₂×C₁₃×#135	C₂²	1	2	yes	—
10	C₁₃×#532	—	2	1	no	—
11	C₂×#88124	—	2	1	no	—
12	C₁₃×#495	—	2	1	no	—
13	C₁₃×#495	C₂²	1	2	yes	—
14	C₁₃×#494	C₂²	1	1	yes	characteristic
15	C₁₃×#493	C₂²	1	1	yes	characteristic
16	C₁₃×#492	—	2	1	no	—
17	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
18	C₂×#88058	C₂²	1	2	yes	—
19	C₂×#88057	—	2	1	no	—
20	C₂×#88052	C₂²	1	2	yes	—
21	#11718344	—	2	1	no	—
22	#11718344	—	2	1	no	—
23	#11718344	C₂²	1	2	yes	—
24	#11718344	C₂²	1	2	yes	—
25	#11718156	C₂²	1	8	yes	—
26	#11718154	C₂²	1	8	yes	—
27	#11718053	C₂²	1	2	yes	—
28	C₄×#6090	C₂²	1	2	yes	—
29	C₂×C₁₃×#41	—	2	2	no	—
30	C₁₃×#135	—	2	8	no	—
31	C₁₃×#135	—	4	4	no	—
32	C₂×C₁₃×Q₁₆	—	2	1	no	—
33	C₂×C₁₃×Q₁₆	C₂³	1	2	yes	—
34	C₂×C₁₃×SD₁₆	—	2	2	no	—
35	C₂×C₁₃×SD₁₆	—	2	2	no	—
36	C₂×C₁₃×D₈	—	2	1	no	—
37	C₂×C₁₃×D₈	C₂³	1	2	yes	—
38	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
39	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
40	C₁₃×C₄⋊Q₈	—	2	1	no	—
41	C₁₃×#127	—	2	1	no	—
42	C₁₃×#124	—	2	2	no	—
43	C₁₃×#121	—	4	2	no	—
44	C₄×D₄×C₁₃	—	4	2	no	—
45	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
46	C₂×C₄⋊Dic₁₃	—	2	2	no	—
47	C₂×#6101	—	4	2	no	—
48	#88124	—	2	2	no	—
49	#88124	—	2	2	no	—
50	#88124	—	4	2	no	—
51	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
52	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
53	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
54	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
55	#88058	—	2	2	no	—
56	#88058	C₂³	1	4	yes	—
57	#88057	—	2	2	no	—
58	#88057	—	2	2	no	—
59	#88052	—	2	4	no	—
60	C₂×#6090	C₂³	1	4	yes	—
61	#33708	—	13	1	no	maximal, Sylow
62	C₁₃×#41	—	4	4	no	—
63	C₁₃×#41	—	4	4	no	—
64	C₂×Q₈×C₁₃	—	2	2	no	—
65	C₂×D₄×C₁₃	—	2	2	no	—
66	C₂×D₄×C₁₃	—	4	2	no	—
67	C₂×D₄×C₁₃	—	4	2	no	—
68	C₂²×C₄×C₁₃	—	4	2	no	—
69	C₂²×C₄×C₁₃	D₈	1	2	yes	—
70	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
71	C₂²×Dic₁₃	—	4	2	no	—
72	C₁₃×Q₁₆	—	2	4	no	—
73	C₁₃×Q₁₆	—	4	2	no	—
74	C₁₃×SD₁₆	—	4	4	no	—
75	C₁₃×SD₁₆	—	4	4	no	—
76	C₁₃×D₈	—	2	4	no	—
77	C₁₃×D₈	—	4	2	no	—
78	C₂×C₈×C₁₃	SD₁₆	1	4	yes	—
79	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
80	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
81	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
82	C₂×C₈×C₁₃	#41	1	4	yes	—
83	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
84	C₁₃×C₄⋊C₄	—	4	2	no	—
85	C₁₃×C₂²⋊C₄	—	4	4	no	—
86	C₄²×C₁₃	D₈	1	2	yes	—
87	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
88	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
89	C₄⋊Dic₁₃	—	2	4	no	—
90	C₄⋊Dic₁₃	—	4	2	no	—
91	#6101	—	4	4	no	—
92	#6101	—	4	4	no	—
93	#6090	—	2	4	no	—
94	C₂×#499	—	13	1	no	—
95	#3147	D₁₃	1	1	yes	characteristic
96	#2245	—	13	2	no	—
97	#2244	—	13	2	no	—
98	#2242	—	13	4	no	—
99	#2139	—	13	4	no	—
100	C₄×M₃₂	—	13	1	no	—
101	C₂³×C₁₃	—	4	2	no	—
102	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
103	Q₈×C₁₃	—	4	2	no	—
104	Q₈×C₁₃	—	4	2	no	—
105	D₄×C₁₃	—	4	2	no	—
106	D₄×C₁₃	—	4	2	no	—
107	D₄×C₁₃	—	8	4	no	—
108	D₄×C₁₃	—	8	4	no	—
109	C₂×C₄×C₁₃	—	4	2	no	—
110	C₂×C₄×C₁₃	—	4	2	no	—
111	C₂×C₄×C₁₃	—	8	4	no	—
112	C₂×C₄×C₁₃	#124	1	2	yes	—
113	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
114	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
115	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
116	C₂×C₄×C₁₃	C₂×SD₁₆	1	2	yes	—
117	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
118	C₂×Dic₁₃	—	4	4	no	—
119	C₂×Dic₁₃	—	8	4	no	—
120	C₈×C₁₃	#124	1	2	yes	—
121	C₈×C₁₃	C₂×SD₁₆	1	2	yes	—
122	C₈×C₁₃	C₂×SD₁₆	1	2	yes	—
123	C₈×C₁₃	C₂×#41	1	2	yes	—
124	C₂×#135	—	2	1	no	—
125	C₂×#135	C₂×D₁₃	1	2	yes	—
126	#532	—	2	1	no	—
127	C₂×#125	—	26	1	no	—
128	C₂×M₃₂	—	13	2	no	—
129	#499	—	13	2	no	—
130	#499	—	13	2	no	—
131	#499	—	26	1	no	—
132	#499	—	26	1	no	—
133	#495	—	2	1	no	—
134	#495	C₂×D₁₃	1	2	yes	—
135	#494	C₂×D₁₃	1	1	yes	characteristic
136	#493	C₂×D₁₃	1	1	yes	characteristic
137	#492	—	2	1	no	—
138	C₂×#107	—	13	2	no	—
139	C₂×#106	—	26	1	no	—
140	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
141	#358	—	13	8	no	—
142	#357	—	13	8	no	—
143	#346	—	13	2	no	—
144	C₄×C₁₆	—	13	2	no	—
145	C₂²×C₁₃	—	2	2	no	—
146	C₂²×C₁₃	—	8	2	no	—
147	C₂²×C₁₃	—	8	4	no	—
148	C₂²×C₁₃	#486	1	2	yes	—
149	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
150	C₄×C₁₃	—	2	2	no	—
151	C₄×C₁₃	—	2	2	no	—
152	C₄×C₁₃	—	8	2	no	—
153	C₄×C₁₃	#486	1	2	yes	—
154	C₄×C₁₃	C₂×#124	1	1	yes	characteristic
155	C₄×C₁₃	C₂²×SD₁₆	1	1	yes	characteristic
156	Dic₁₃	—	8	4	no	—
157	C₂×#41	—	2	2	no	—
158	#135	—	2	8	no	—
159	#135	—	4	4	no	—
160	C₂×Q₁₆	—	2	1	no	—
161	C₂×Q₁₆	C₂²×D₁₃	1	2	yes	—
162	C₂×SD₁₆	—	2	2	no	—
163	C₂×SD₁₆	—	2	2	no	—
164	C₂×D₈	—	2	1	no	—
165	C₂×D₈	C₂²×D₁₃	1	2	yes	—
166	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
167	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
168	C₄⋊Q₈	—	2	1	no	—
169	#127	—	2	1	no	—
170	#125	—	26	2	no	—
171	#125	—	26	2	no	—
172	#125	—	52	2	no	—
173	#124	—	2	2	no	—
174	#121	—	4	2	no	—
175	C₄×D₄	—	4	2	no	—
176	C₂×C₄⋊C₄	—	26	2	no	—
177	C₂×C₄⋊C₄	—	52	2	no	—
178	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
179	M₃₂	—	26	4	no	—
180	C₂×C₁₆	—	13	4	no	—
181	#107	—	13	4	no	—
182	#107	—	26	2	no	—
183	#106	—	26	2	no	—
184	#106	—	26	2	no	—
185	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
186	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
187	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
188	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
189	C₂×C₁₃	—	2	2	no	—
190	C₂×C₁₃	—	16	2	no	—
191	C₂×C₁₃	C₂×#486	1	1	yes	characteristic
192	C₂×C₁₃	#3414	1	2	yes	—
193	#41	—	4	4	no	—
194	#41	—	4	4	no	—
195	C₂×Q₈	—	2	2	no	—
196	C₂×D₄	—	2	2	no	—
197	C₂×D₄	—	4	2	no	—
198	C₂×D₄	—	4	2	no	—
199	C₂²×C₄	—	4	2	no	—
200	C₂²×C₄	—	52	2	no	—
201	C₂²×C₄	#6104	1	2	yes	—
202	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
203	Q₁₆	—	2	4	no	—
204	Q₁₆	—	4	2	no	—
205	SD₁₆	—	4	4	no	—
206	SD₁₆	—	4	4	no	—
207	D₈	—	2	4	no	—
208	D₈	—	4	2	no	—
209	C₂×C₈	#6105	1	4	yes	—
210	C₂×C₈	#6129	1	4	yes	—
211	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
212	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
213	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
214	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
215	C₄⋊C₄	—	4	2	no	—
216	C₄⋊C₄	—	26	4	no	—
217	C₄⋊C₄	—	52	2	no	—
218	C₄⋊C₄	—	52	4	no	—
219	C₄⋊C₄	—	52	4	no	—
220	C₂²⋊C₄	—	4	4	no	—
221	C₄²	#6104	1	2	yes	—
222	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
223	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
224	C₁₆	—	26	4	no	—
225	C₁₃	#33708	1	1	yes	characteristic, complemented, Sylow
226	C₂³	—	4	2	no	—
227	C₂³	C₂×#6104	1	1	yes	characteristic
228	Q₈	—	4	2	no	—
229	Q₈	—	4	2	no	—
230	D₄	—	4	2	no	—
231	D₄	—	4	2	no	—
232	D₄	—	8	4	no	—
233	D₄	—	8	4	no	—
234	C₂×C₄	—	4	2	no	—
235	C₂×C₄	—	4	2	no	—
236	C₂×C₄	—	8	4	no	—
237	C₂×C₄	—	52	4	no	—
238	C₂×C₄	—	104	4	no	—
239	C₂×C₄	C₂×#6104	1	2	yes	—
240	C₂×C₄	C₂×#6104	1	2	yes	—
241	C₂×C₄	C₂×#6104	1	1	yes	characteristic
242	C₂×C₄	C₂×#6105	1	2	yes	—
243	C₂×C₄	#88190	1	2	yes	—
244	C₂×C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
245	C₈	C₂×#6105	1	2	yes	—
246	C₈	C₂×#6105	1	2	yes	—
247	C₈	#88190	1	2	yes	—
248	C₈	C₂×#6129	1	2	yes	—
249	C₂²	—	2	2	no	—
250	C₂²	—	8	2	no	—
251	C₂²	—	8	4	no	—
252	C₂²	#11718731	1	2	yes	—
253	C₂²	C₂²×#6104	1	1	yes	characteristic
254	C₄	—	2	2	no	—
255	C₄	—	2	2	no	—
256	C₄	—	8	2	no	—
257	C₄	—	104	4	no	—
258	C₄	#11718731	1	2	yes	—
259	C₄	C₂²×#6105	1	1	yes	characteristic
260	C₄	C₂×#88190	1	1	yes	characteristic
261	C₂	—	2	2	no	—
262	C₂	—	16	2	no	—
263	C₂	C₂×#11718731	1	1	yes	characteristic
264	C₂	#13127970	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂²×#6104
commutator, cosocle	C₂×C₈×C₁₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₃
Fitting, prime radical	C₁₃×#3147	C₂
socle	C₂²×C₁₃	C₂²×D₈
hypercenter	#3147	D₁₃
hypocenter	C₁₃	#33708
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #3147

Complements

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

N	H ≅ G/N	type
1. C₂×#11718344	C₂	semidirect
2. #13117081	C₂	semidirect
3. #13117077	C₂	semidirect
4. #13117067	C₂	semidirect
5. #13116548	C₂	semidirect
6. C₄×#88052	C₂	semidirect
225. C₁₃	#33708	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718344	2
#13117081	2
#13117077	2
#13117067	2
#13116548	2
C₄×#88052	2
C₁₃×#3147	2
#33708	13