Subgroups — #15833451

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	#13123771	C₂	1	2	yes	maximal
2	#13113733	C₂	1	4	yes	maximal, complemented
3	Q₁₆×#1055	C₂	1	4	yes	maximal, complemented
4	#13113595	C₂	1	1	yes	maximal, characteristic
5	#13113291	C₂	1	2	yes	maximal
6	C₂×C₈×#1055	C₂	1	1	yes	maximal, characteristic
7	C₁₃×#3170	C₂	1	1	yes	maximal, characteristic
8	C₁₃×#533	C₂²	1	2	yes	—
9	C₁₃×#500	C₄	1	1	yes	characteristic
10	C₁₃×#494	C₄	1	1	yes	characteristic
11	C₁₃×#487	C₄	1	2	yes	—
12	C₁₃×#467	C₄	1	4	yes	—
13	C₄×C₁₃×Q₁₆	C₂²	1	4	yes	—
14	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
15	#11718685	—	2	2	no	—
16	#11718685	C₂²	1	4	yes	—
17	Q₈×#1055	—	2	2	no	—
18	Q₈×#1055	C₂²	1	4	yes	—
19	#11718660	—	2	2	no	—
20	#11718658	—	2	2	no	—
21	#11718585	C₂²	1	2	yes	—
22	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
23	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
24	Q₈⋊#1055	C₂²	1	8	yes	—
25	#11718050	C₂²	1	4	yes	—
26	C₈×#1055	C₂²	1	2	yes	—
27	C₈×#1055	C₂²	1	2	yes	—
28	C₂×C₁₃×Q₁₆	C₂×C₄	1	4	yes	—
29	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
30	C₁₃×C₄⋊Q₈	C₂×C₄	1	2	yes	—
31	C₁₃×C₄⋊Q₈	C₂×C₄	1	2	yes	—
32	C₁₃×#125	—	2	2	no	—
33	C₁₃×C₂²⋊Q₈	—	2	2	no	—
34	C₁₃×C₂²⋊Q₈	C₂×C₄	1	4	yes	—
35	C₄×Q₈×C₁₃	—	2	2	no	—
36	C₄×Q₈×C₁₃	C₂³	1	4	yes	—
37	C₁₃×#117	—	2	2	no	—
38	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
39	C₂×#6099	—	2	2	no	—
40	C₂²×#1055	—	2	1	no	—
41	C₂²×#1055	D₄	1	1	yes	characteristic
42	C₂²×#1055	D₄	1	1	yes	characteristic
43	C₁₃×#107	C₂×C₄	1	4	yes	—
44	C₁₃×#103	C₂×C₄	1	8	yes	—
45	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
46	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
47	C₄×C₈×C₁₃	C₂³	1	2	yes	—
48	C₄⋊#1055	—	2	2	no	—
49	C₄⋊#1055	—	2	4	no	—
50	C₄⋊#1055	—	2	4	no	—
51	C₄⋊#1055	C₂³	1	4	yes	—
52	#88043	—	2	4	no	—
53	C₄×#1055	—	2	1	no	—
54	C₄×#1055	—	2	4	no	—
55	C₄×#1055	D₄	1	1	yes	characteristic
56	C₄×#1055	D₄	1	1	yes	characteristic
57	C₄×#1055	C₂³	1	2	yes	—
58	C₄×#1055	C₂³	1	1	yes	characteristic
59	C₄×#1055	C₂³	1	1	yes	characteristic
60	#17958	—	13	1	no	maximal, Sylow
61	C₂×Q₈×C₁₃	—	2	2	no	—
62	C₂×Q₈×C₁₃	C₂²×C₄	1	4	yes	—
63	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
66	C₁₃×Q₁₆	—	2	8	no	—
67	C₂×C₈×C₁₃	—	2	4	no	—
68	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
69	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
70	C₁₃×C₄⋊C₄	—	2	2	no	—
71	C₁₃×C₄⋊C₄	—	2	2	no	—
72	C₁₃×C₄⋊C₄	—	2	2	no	—
73	C₁₃×C₄⋊C₄	—	2	4	no	—
74	C₁₃×C₄⋊C₄	—	2	4	no	—
75	C₁₃×C₄⋊C₄	C₂²×C₄	1	4	yes	—
76	C₁₃×C₂²⋊C₄	—	2	4	no	—
77	C₄²×C₁₃	—	2	4	no	—
78	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
79	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
80	C₄²×C₁₃	#41	1	1	yes	characteristic
81	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
82	#6099	—	4	4	no	—
83	C₂×#1055	—	2	4	no	—
84	C₂×#1055	—	2	2	no	—
85	C₂×#1055	—	2	2	no	—
86	C₂×#1055	—	2	4	no	—
87	C₂×#1055	Q₁₆	1	4	yes	complemented
88	C₂×#1055	C₂×D₄	1	1	yes	characteristic
89	C₂×#1055	C₂×D₄	1	1	yes	characteristic
90	C₂×#1055	C₂×D₄	1	1	yes	characteristic
91	C₂×#1055	C₂×D₄	1	1	yes	characteristic
92	#3170	D₁₃	1	1	yes	characteristic
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	C₂³×C₁₃	C₄×D₄	1	1	yes	characteristic
100	Q₈×C₁₃	—	2	8	no	—
101	Q₈×C₁₃	—	4	4	no	—
102	C₂×C₄×C₁₃	—	2	2	no	—
103	C₂×C₄×C₁₃	—	2	2	no	—
104	C₂×C₄×C₁₃	—	2	4	no	—
105	C₂×C₄×C₁₃	—	2	4	no	—
106	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
107	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
108	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
109	C₂×C₄×C₁₃	C₂×Q₁₆	1	2	yes	—
110	C₂×C₄×C₁₃	#135	1	2	yes	—
111	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
112	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
113	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
114	C₈×C₁₃	—	2	2	no	—
115	C₈×C₁₃	#131	1	4	yes	—
116	#1055	—	2	4	no	—
117	#1055	—	2	1	no	—
118	#1055	—	2	1	no	—
119	#1055	—	4	4	no	—
120	#1055	C₂×Q₁₆	1	2	yes	—
121	#1055	C₂×Q₁₆	1	2	yes	—
122	#533	C₂×D₁₃	1	2	yes	—
123	#500	Dic₁₃	1	1	yes	characteristic
124	#494	Dic₁₃	1	1	yes	characteristic
125	#487	Dic₁₃	1	2	yes	—
126	#467	Dic₁₃	1	4	yes	—
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	C₂×D₁₃	1	4	yes	—
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	C₂×D₁₃	1	1	yes	characteristic
138	#335	—	13	4	no	—
139	Q₈⋊C₈	—	13	8	no	—
140	C₈²	—	13	2	no	—
141	C₈²	—	13	2	no	—
142	C₂²×C₁₃	—	2	1	no	—
143	C₂²×C₁₃	—	2	1	no	—
144	C₂²×C₁₃	C₄×Q₁₆	1	2	yes	—
145	C₂²×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
146	C₄×C₁₃	—	2	1	no	—
147	C₄×C₁₃	—	2	1	no	—
148	C₄×C₁₃	—	4	4	no	—
149	C₄×C₁₃	—	4	4	no	—
150	C₄×C₁₃	C₄×Q₁₆	1	2	yes	—
151	C₄×C₁₃	#443	1	2	yes	—
152	C₄×C₁₃	C₂×#131	1	2	yes	—
153	C₄×C₁₃	C₂²×Q₁₆	1	1	yes	characteristic
154	C₄×C₁₃	C₂×#135	1	1	yes	characteristic
155	C₂×Q₁₆	C₂×Dic₁₃	1	4	yes	—
156	C₂×M₁₆	—	26	2	no	—
157	C₂²×C₈	—	13	1	no	—
158	C₂²×C₈	—	13	1	no	—
159	C₂²×C₈	—	26	1	no	—
160	C₂²×C₈	C₂×Dic₁₃	1	2	yes	—
161	C₄⋊Q₈	C₂×Dic₁₃	1	2	yes	—
162	C₄⋊Q₈	C₂×Dic₁₃	1	2	yes	—
163	#125	—	2	2	no	—
164	C₂²⋊Q₈	—	2	2	no	—
165	C₂²⋊Q₈	C₂×Dic₁₃	1	4	yes	—
166	C₄×Q₈	—	2	2	no	—
167	C₄×Q₈	C₂²×D₁₃	1	4	yes	—
168	#117	—	2	2	no	—
169	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
170	#107	C₂×Dic₁₃	1	4	yes	—
171	C₄⋊C₈	—	13	4	no	—
172	C₄⋊C₈	—	26	4	no	—
173	C₄⋊C₈	—	26	4	no	—
174	C₄⋊C₈	—	26	2	no	—
175	#103	C₂×Dic₁₃	1	8	yes	—
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	—	26	4	no	—
183	C₄×C₈	—	26	1	no	—
184	C₄×C₈	C₂×Dic₁₃	1	1	yes	characteristic
185	C₄×C₈	C₂×Dic₁₃	1	1	yes	characteristic
186	C₄×C₈	C₂²×D₁₃	1	2	yes	—
187	C₂×C₁₃	—	2	2	no	—
188	C₂×C₁₃	#2931	1	1	yes	characteristic
189	C₂×C₁₃	C₂×C₄×Q₁₆	1	1	yes	characteristic
190	C₂×C₁₃	C₂×#443	1	1	yes	characteristic
191	C₂×Q₈	—	2	2	no	—
192	C₂×Q₈	C₂²×Dic₁₃	1	4	yes	—
193	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
194	C₂²×C₄	#6129	1	1	yes	characteristic
195	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
196	Q₁₆	—	2	8	no	—
197	M₁₆	—	52	4	no	—
198	C₂×C₈	—	2	4	no	—
199	C₂×C₈	—	13	4	no	—
200	C₂×C₈	—	13	1	no	—
201	C₂×C₈	—	13	1	no	—
202	C₂×C₈	—	13	1	no	—
203	C₂×C₈	—	13	1	no	—
204	C₂×C₈	—	26	4	no	—
205	C₂×C₈	—	26	2	no	—
206	C₂×C₈	—	26	2	no	—
207	C₂×C₈	—	26	4	no	—
208	C₂×C₈	C₂²×Dic₁₃	1	2	yes	—
209	C₂×C₈	C₂²×Dic₁₃	1	2	yes	—
210	C₄⋊C₄	—	2	2	no	—
211	C₄⋊C₄	—	2	2	no	—
212	C₄⋊C₄	—	2	2	no	—
213	C₄⋊C₄	—	2	4	no	—
214	C₄⋊C₄	—	2	4	no	—
215	C₄⋊C₄	C₂²×Dic₁₃	1	4	yes	—
216	C₂²⋊C₄	—	2	4	no	—
217	C₄²	—	2	4	no	—
218	C₄²	D₄×D₁₃	1	1	yes	characteristic
219	C₄²	#6129	1	1	yes	characteristic
220	C₄²	C₂²×Dic₁₃	1	1	yes	characteristic
221	C₄²	C₂³×D₁₃	1	1	yes	characteristic
222	C₁₃	#17958	1	1	yes	characteristic, complemented, Sylow
223	C₂³	D₄×Dic₁₃	1	1	yes	characteristic
224	Q₈	—	2	8	no	—
225	Q₈	—	4	4	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	4	no	—
229	C₂×C₄	—	2	4	no	—
230	C₂×C₄	Q₁₆×D₁₃	1	2	yes	—
231	C₂×C₄	#88173	1	2	yes	—
232	C₂×C₄	D₄×Dic₁₃	1	1	yes	characteristic
233	C₂×C₄	D₄×Dic₁₃	1	1	yes	characteristic
234	C₂×C₄	D₄×Dic₁₃	1	1	yes	characteristic
235	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
236	C₂×C₄	C₂×#6129	1	1	yes	characteristic
237	C₂×C₄	C₂³×Dic₁₃	1	1	yes	characteristic
238	C₈	—	2	2	no	—
239	C₈	—	13	2	no	—
240	C₈	—	13	2	no	—
241	C₈	—	26	4	no	—
242	C₈	—	26	1	no	—
243	C₈	—	26	1	no	—
244	C₈	—	52	4	no	—
245	C₈	#88202	1	4	yes	—
246	C₂²	—	2	1	no	—
247	C₂²	—	2	1	no	—
248	C₂²	Q₁₆×Dic₁₃	1	2	yes	—
249	C₂²	C₂×D₄×Dic₁₃	1	1	yes	characteristic
250	C₄	—	2	1	no	—
251	C₄	—	2	1	no	—
252	C₄	—	4	4	no	—
253	C₄	—	4	4	no	—
254	C₄	Q₁₆×Dic₁₃	1	2	yes	—
255	C₄	#11718857	1	2	yes	—
256	C₄	C₂×Q₁₆×D₁₃	1	1	yes	characteristic
257	C₄	C₂×#88173	1	1	yes	characteristic
258	C₄	C₂×#88202	1	2	yes	—
259	C₂	—	2	2	no	—
260	C₂	#13123548	1	1	yes	characteristic
261	C₂	C₂×Q₁₆×Dic₁₃	1	1	yes	characteristic
262	C₂	C₂×#11718857	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₁₃
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₄²	C₂³×D₁₃
Fitting	C₁₃×#3170	C₂
socle	C₂²×C₁₃	C₂×C₄×D₄
prime radical	C₂³×C₁₃	C₄×D₄
cosocle	C₄²×C₁₃	C₂⁴
hypercenter	#3170	D₁₃
hypocenter	C₁₃	#17958
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3170

Complements

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

N	H ≅ G/N	type
2. #13113733	C₂	semidirect
3. Q₁₆×#1055	C₂	semidirect
87. C₂×#1055	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
#13123771	2
#13113733	2
Q₁₆×#1055	2
#13113595	2
#13113291	2
C₂×C₈×#1055	2
C₁₃×#3170	2
#17958	13