Subgroups — #16795824

Order 3328.

614 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13131625	C₂	1	2	yes	maximal
2	#13130476	C₂	1	4	yes	maximal, complemented
3	Q₁₆×#1057	C₂	1	4	yes	maximal, complemented
4	#13130394	C₂	1	1	yes	maximal, characteristic
5	#13130185	C₂	1	2	yes	maximal
6	C₈×D₁₃⋊C₈	C₂	1	1	yes	maximal, characteristic
7	#13125420	C₂	1	1	yes	maximal, characteristic
8	#11719433	C₂²	1	2	yes	—
9	#11719158	C₂²	1	4	yes	—
10	Q₈×#1057	C₂²	1	4	yes	—
11	#11719080	—	2	2	no	—
12	#11719079	—	2	2	no	—
13	#11719074	—	2	2	no	—
14	#11719072	—	2	2	no	—
15	#11719052	C₂²	1	2	yes	—
16	C₄×D₁₃⋊C₈	C₂²	1	1	yes	characteristic
17	C₄×D₁₃⋊C₈	C₂²	1	1	yes	characteristic
18	C₁₃×#494	C₄	1	1	yes	characteristic
19	#11718849	C₄	1	4	yes	—
20	Q₁₆×Dic₁₃	C₂²	1	4	yes	—
21	#11718742	C₄	1	2	yes	—
22	C₄×C₈×D₁₃	C₂²	1	1	yes	characteristic
23	#11718343	C₄	1	1	yes	characteristic
24	Q₈⋊#1057	C₂²	1	8	yes	—
25	#11718254	C₂²	1	4	yes	—
26	C₈×#1057	C₂²	1	2	yes	—
27	C₈×#1057	C₂²	1	2	yes	—
28	C₂×#6118	—	2	2	no	—
29	C₂×D₁₃⋊C₈	—	2	1	no	—
30	C₂×D₁₃⋊C₈	D₄	1	1	yes	characteristic
31	C₂×D₁₃⋊C₈	D₄	1	1	yes	characteristic
32	C₂×C₁₃×Q₁₆	C₂×C₄	1	4	yes	—
33	C₁₃×C₄⋊Q₈	C₂×C₄	1	2	yes	—
34	#88200	C₂×C₄	1	4	yes	—
35	Q₈×Dic₁₃	C₂³	1	4	yes	—
36	C₂×C₈×D₁₃	C₂×C₄	1	2	yes	—
37	#88151	—	2	2	no	—
38	#88146	—	2	2	no	—
39	#88144	—	2	2	no	—
40	#88141	—	2	2	no	—
41	C₄²×D₁₃	C₂³	1	1	yes	characteristic
42	#88123	C₂×C₄	1	2	yes	—
43	#88112	—	2	4	no	—
44	#88110	—	2	4	no	—
45	C₄⋊#1057	—	2	4	no	—
46	C₄⋊#1057	—	2	2	no	—
47	C₄⋊#1057	C₂³	1	4	yes	—
48	C₄×#1057	—	2	4	no	—
49	C₄×#1057	—	2	1	no	—
50	C₄×#1057	D₄	1	1	yes	characteristic
51	C₄×#1057	D₄	1	1	yes	characteristic
52	C₄×#1057	C₂³	1	2	yes	—
53	C₄×#1057	C₂³	1	1	yes	characteristic
54	C₄×#1057	C₂³	1	1	yes	characteristic
55	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
56	#88075	C₂×C₄	1	8	yes	—
57	#88058	C₂×C₄	1	4	yes	—
58	C₈×Dic₁₃	C₂³	1	2	yes	—
59	C₄×#1055	C₂×C₄	1	1	yes	characteristic
60	#17958	—	13	1	no	maximal, Sylow
61	C₂×Q₈×C₁₃	C₂²×C₄	1	4	yes	—
62	C₂×C₄×D₁₃	C₂²×C₄	1	1	yes	characteristic
63	C₂×C₄×D₁₃	C₂×D₄	1	1	yes	characteristic
64	C₂×C₄×D₁₃	#41	1	1	yes	characteristic
65	C₂×Dic₂₆	—	2	2	no	—
66	C₂×#1057	—	2	2	no	—
67	C₂×#1057	—	2	4	no	—
68	C₂×#1057	C₂×D₄	1	1	yes	characteristic
69	C₂×#1057	C₂×D₄	1	1	yes	characteristic
70	C₂×#1057	C₂×D₄	1	1	yes	characteristic
71	C₂×#1057	C₂×D₄	1	1	yes	characteristic
72	#6118	—	4	4	no	—
73	D₁₃⋊C₈	—	2	4	no	—
74	D₁₃⋊C₈	—	2	2	no	—
75	D₁₃⋊C₈	Q₁₆	1	4	yes	complemented
76	C₁₃×Q₁₆	—	2	8	no	—
77	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
78	C₁₃×C₄⋊C₄	—	2	2	no	—
79	C₁₃×C₄⋊C₄	—	2	2	no	—
80	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
81	#6103	—	2	4	no	—
82	C₄⋊Dic₁₃	—	2	2	no	—
83	C₄⋊Dic₁₃	—	2	4	no	—
84	C₄⋊Dic₁₃	C₂²×C₄	1	4	yes	—
85	#6101	—	2	4	no	—
86	C₄×Dic₁₃	—	2	4	no	—
87	C₄×Dic₁₃	C₂×D₄	1	1	yes	characteristic
88	C₄×Dic₁₃	#41	1	1	yes	characteristic
89	C₄×Dic₁₃	C₂⁴	1	1	yes	characteristic
90	C₂×#1055	C₂²×C₄	1	2	yes	—
91	C₈×D₁₃	—	2	4	no	—
92	#3170	—	13	1	no	—
93	#2960	—	13	2	no	—
94	#1591	—	13	4	no	—
95	C₈×Q₁₆	—	13	4	no	—
96	#1567	—	13	1	no	—
97	#1493	—	13	2	no	—
98	C₂×C₈²	—	13	1	no	—
99	C₂²×D₁₃	C₄×D₄	1	1	yes	characteristic
100	Q₈×C₁₃	—	2	8	no	—
101	C₂×C₄×C₁₃	—	2	4	no	—
102	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
103	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
104	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
105	C₂×Dic₁₃	—	2	4	no	—
106	C₂×Dic₁₃	C₄×D₄	1	1	yes	characteristic
107	C₂×Dic₁₃	C₂²×D₄	1	1	yes	characteristic
108	C₂×Dic₁₃	C₂×#41	1	1	yes	characteristic
109	C₄×D₁₃	—	2	2	no	—
110	C₄×D₁₃	—	2	2	no	—
111	C₄×D₁₃	C₂×Q₁₆	1	2	yes	—
112	C₄×D₁₃	#135	1	2	yes	—
113	Dic₂₆	—	4	4	no	—
114	#1057	—	2	4	no	—
115	#1057	—	2	1	no	—
116	#1057	—	2	1	no	—
117	#1057	—	4	4	no	—
118	#1057	C₂×Q₁₆	1	2	yes	—
119	#1057	C₂×Q₁₆	1	2	yes	—
120	C₈×C₁₃	#131	1	4	yes	—
121	#1055	—	2	2	no	—
122	#533	—	13	2	no	—
123	#500	—	13	1	no	—
124	#494	#261	1	1	yes	characteristic
125	#487	—	13	2	no	—
126	#467	—	13	4	no	—
127	#446	—	13	4	no	—
128	#446	—	26	2	no	—
129	C₈×Q₈	—	13	4	no	—
130	C₈×Q₈	—	26	2	no	—
131	C₄×Q₁₆	—	13	4	no	—
132	#424	—	26	2	no	—
133	#423	—	13	2	no	—
134	#405	—	26	2	no	—
135	C₂×C₄×C₈	—	13	1	no	—
136	C₂×C₄×C₈	—	13	1	no	—
137	C₂×C₄×C₈	—	13	1	no	—
138	#335	—	13	4	no	—
139	Q₈⋊C₈	—	13	8	no	—
140	C₈²	—	13	2	no	—
141	C₈²	—	13	2	no	—
142	C₂²×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
143	C₂×D₁₃	—	2	1	no	—
144	C₂×D₁₃	—	2	1	no	—
145	C₂×D₁₃	C₄×Q₁₆	1	2	yes	—
146	C₄×C₁₃	—	4	4	no	—
147	C₄×C₁₃	C₄×Q₁₆	1	2	yes	—
148	C₄×C₁₃	#443	1	2	yes	—
149	C₄×C₁₃	C₂×#131	1	2	yes	—
150	Dic₁₃	—	2	1	no	—
151	Dic₁₃	—	2	1	no	—
152	Dic₁₃	—	4	4	no	—
153	Dic₁₃	C₂²×Q₁₆	1	1	yes	characteristic
154	Dic₁₃	C₂×#135	1	1	yes	characteristic
155	C₂×Q₁₆	C₂×#261	1	4	yes	—
156	C₂×M₁₆	—	26	2	no	—
157	C₂²×C₈	—	13	1	no	—
158	C₂²×C₈	—	13	1	no	—
159	C₂²×C₈	—	13	2	no	—
160	C₂²×C₈	—	26	1	no	—
161	C₄⋊Q₈	—	13	2	no	—
162	C₄⋊Q₈	C₂×#261	1	2	yes	—
163	#125	—	26	2	no	—
164	C₂²⋊Q₈	—	13	4	no	—
165	C₂²⋊Q₈	—	26	2	no	—
166	C₄×Q₈	—	13	4	no	—
167	C₄×Q₈	—	26	2	no	—
168	#117	—	26	2	no	—
169	C₂×C₄²	—	13	1	no	—
170	#107	—	13	4	no	—
171	C₄⋊C₈	—	13	4	no	—
172	C₄⋊C₈	—	26	2	no	—
173	C₄⋊C₈	—	26	4	no	—
174	C₄⋊C₈	—	26	4	no	—
175	#103	—	13	8	no	—
176	#97	—	26	4	no	—
177	C₄×C₈	—	13	1	no	—
178	C₄×C₈	—	13	1	no	—
179	C₄×C₈	—	13	1	no	—
180	C₄×C₈	—	13	1	no	—
181	C₄×C₈	—	13	2	no	—
182	C₄×C₈	—	13	2	no	—
183	C₄×C₈	—	13	1	no	—
184	C₄×C₈	—	26	1	no	—
185	C₄×C₈	—	26	4	no	—
186	C₄×C₈	C₂×#261	1	1	yes	characteristic
187	C₂×C₁₃	#2931	1	1	yes	characteristic
188	C₂×C₁₃	C₂×C₄×Q₁₆	1	1	yes	characteristic
189	C₂×C₁₃	C₂×#443	1	1	yes	characteristic
190	D₁₃	—	2	2	no	—
191	C₂×Q₈	—	26	2	no	—
192	C₂×Q₈	C₂²×#261	1	4	yes	—
193	C₂²×C₄	—	13	1	no	—
194	C₂²×C₄	—	13	1	no	—
195	C₂²×C₄	—	13	1	no	—
196	Q₁₆	—	2	8	no	—
197	M₁₆	—	52	4	no	—
198	C₂×C₈	—	13	1	no	—
199	C₂×C₈	—	13	1	no	—
200	C₂×C₈	—	13	4	no	—
201	C₂×C₈	—	13	1	no	—
202	C₂×C₈	—	13	1	no	—
203	C₂×C₈	—	13	2	no	—
204	C₂×C₈	—	26	2	no	—
205	C₂×C₈	—	26	2	no	—
206	C₂×C₈	—	26	4	no	—
207	C₂×C₈	—	26	4	no	—
208	C₂×C₈	—	26	4	no	—
209	C₂×C₈	C₂²×#261	1	2	yes	—
210	C₄⋊C₄	—	2	2	no	—
211	C₄⋊C₄	—	2	2	no	—
212	C₄⋊C₄	—	13	4	no	—
213	C₄⋊C₄	—	26	4	no	—
214	C₄⋊C₄	—	26	4	no	—
215	C₄⋊C₄	—	26	2	no	—
216	C₂²⋊C₄	—	26	4	no	—
217	C₄²	—	13	1	no	—
218	C₄²	—	13	1	no	—
219	C₄²	—	13	1	no	—
220	C₄²	—	26	4	no	—
221	C₄²	C₂²×#261	1	1	yes	characteristic
222	C₁₃	#17958	1	1	yes	characteristic, complemented, Sylow
223	C₂³	—	13	1	no	—
224	Q₈	—	2	8	no	—
225	Q₈	—	52	4	no	—
226	C₂×C₄	—	2	4	no	—
227	C₂×C₄	—	13	1	no	—
228	C₂×C₄	—	13	2	no	—
229	C₂×C₄	—	13	2	no	—
230	C₂×C₄	—	13	1	no	—
231	C₂×C₄	—	13	1	no	—
232	C₂×C₄	—	26	2	no	—
233	C₂×C₄	—	26	2	no	—
234	C₂×C₄	—	26	4	no	—
235	C₂×C₄	D₄×#261	1	1	yes	characteristic
236	C₂×C₄	D₄×#261	1	1	yes	characteristic
237	C₂×C₄	C₂³×#261	1	1	yes	characteristic
238	C₈	—	13	2	no	—
239	C₈	—	13	2	no	—
240	C₈	—	26	1	no	—
241	C₈	—	26	1	no	—
242	C₈	—	26	4	no	—
243	C₈	—	26	2	no	—
244	C₈	—	52	4	no	—
245	C₈	#88238	1	4	yes	—
246	C₂²	—	13	2	no	—
247	C₂²	—	26	1	no	—
248	C₂²	—	26	1	no	—
249	C₂²	C₂×D₄×#261	1	1	yes	characteristic
250	C₄	—	4	4	no	—
251	C₄	—	13	1	no	—
252	C₄	—	13	1	no	—
253	C₄	—	26	1	no	—
254	C₄	—	26	1	no	—
255	C₄	—	52	4	no	—
256	C₄	Q₁₆×#261	1	2	yes	—
257	C₄	#11719111	1	2	yes	—
258	C₄	C₂×#88238	1	2	yes	—
259	C₂	—	26	2	no	—
260	C₂	#13131558	1	1	yes	characteristic
261	C₂	C₂×Q₁₆×#261	1	1	yes	characteristic
262	C₂	C₂×#11719111	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×#261
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×#261
Fitting	C₁₃×#494	C₄
socle	C₂²×C₁₃	C₂×C₄×D₄
prime radical	C₂²×D₁₃	C₄×D₄
cosocle	C₄×Dic₁₃	C₂⁴
hypercenter	#494	#261
hypocenter	C₁₃	#17958
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #494

Complements

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

N	H ≅ G/N	type
2. #13130476	C₂	semidirect
3. Q₁₆×#1057	C₂	semidirect
75. D₁₃⋊C₈	Q₁₆	semidirect
222. C₁₃	#17958	semidirect

Maximal subgroups

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

subgroup	index
#13131625	2
#13130476	2
Q₁₆×#1057	2
#13130394	2
#13130185	2
C₈×D₁₃⋊C₈	2
#13125420	2
#17958	13