Subgroups — #15777987

Order 3328.

494 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	C₂×M₁₆×Dic₁₃	C₂	1	1	yes	maximal, characteristic
2	C₂×#11718145	C₂	1	1	yes	maximal, characteristic
3	C₂×#11718104	C₂	1	2	yes	maximal, complemented
4	#13112917	C₂	1	2	yes	maximal, complemented
5	C₂×C₁₃×#349	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₁₃×M₁₆	C₂²	1	1	yes	characteristic
7	C₂²×C₄×Dic₁₃	C₄	1	1	yes	characteristic
8	C₂²×#6099	C₄	1	1	yes	characteristic
9	C₂×C₁₃×M₃₂	C₂²	1	2	yes	—
10	M₁₆×Dic₁₃	—	2	4	no	—
11	C₂×#88055	C₄	1	1	yes	characteristic
12	C₂×#88055	C₂²	1	1	yes	characteristic
13	C₂×C₈×Dic₁₃	C₄	1	1	yes	characteristic
14	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
15	C₂×#88052	C₂²	1	2	yes	—
16	C₁₃×#349	C₄	1	4	yes	complemented
17	#11718145	C₄	1	4	yes	complemented
18	#11718104	—	2	4	no	—
19	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
20	C₂³×Dic₁₃	—	2	1	no	—
21	C₂×C₁₃×M₁₆	—	2	2	no	—
22	C₂×C₁₃×M₁₆	—	2	2	no	—
23	C₂×C₁₃×M₁₆	C₂×C₄	1	2	yes	—
24	C₂×C₁₃×M₁₆	C₂×C₄	1	2	yes	—
25	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
27	C₂×C₄×Dic₁₃	—	2	1	no	—
28	C₂×C₄×Dic₁₃	—	2	2	no	—
29	C₂×C₄×Dic₁₃	—	2	2	no	—
30	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
32	C₂×#6099	—	2	2	no	—
33	C₂×#6099	—	2	2	no	—
34	C₂×#6099	—	2	2	no	—
35	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
36	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
37	C₁₃×M₃₂	—	2	2	no	—
38	C₁₃×M₃₂	C₂×C₄	1	4	yes	complemented
39	C₂×C₁₃×C₁₆	—	2	2	no	—
40	#88055	—	2	2	no	—
41	#88055	D₄	1	4	yes	—
42	C₈×Dic₁₃	—	2	2	no	—
43	C₈×Dic₁₃	—	2	2	no	—
44	#88052	—	2	2	no	—
45	#88052	C₂×C₄	1	4	yes	complemented
46	C₂×#6090	—	2	2	no	—
47	#12607	—	13	1	no	maximal, Sylow
48	C₂⁴×C₁₃	M₁₆	1	1	yes	characteristic
49	C₂²×C₄×C₁₃	—	2	1	no	—
50	C₂²×C₄×C₁₃	—	2	1	no	—
51	C₂²×C₄×C₁₃	—	2	1	no	—
52	C₂²×C₄×C₁₃	—	2	1	no	—
53	C₂²×C₄×C₁₃	C₄²	1	2	yes	—
54	C₂²×C₄×C₁₃	M₁₆	1	1	yes	characteristic
55	C₂²×C₄×C₁₃	M₁₆	1	2	yes	—
56	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
57	C₂²×Dic₁₃	—	2	1	no	—
58	C₂²×Dic₁₃	—	2	4	no	—
59	C₂²×Dic₁₃	—	2	1	no	—
60	C₂²×Dic₁₃	—	4	4	no	—
61	C₁₃×M₁₆	—	2	4	no	—
62	C₁₃×M₁₆	—	2	4	no	—
63	C₂×C₈×C₁₃	—	2	1	no	—
64	C₂×C₈×C₁₃	—	2	1	no	—
65	C₂×C₈×C₁₃	C₄²	1	2	yes	—
66	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
67	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
68	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
69	C₂×C₈×C₁₃	#41	1	2	yes	—
70	C₄×Dic₁₃	—	2	4	no	—
71	C₄×Dic₁₃	—	2	1	no	—
72	C₄×Dic₁₃	—	2	1	no	—
73	C₄×Dic₁₃	C₂²⋊C₄	1	2	yes	—
74	C₄×Dic₁₃	M₁₆	1	2	yes	—
75	#6099	—	2	8	no	—
76	C₂×#1055	—	2	1	no	—
77	C₂×#1055	—	2	1	no	—
78	C₂×#1055	—	2	1	no	—
79	C₂×#1055	—	2	1	no	—
80	C₂×#1055	C₂²⋊C₄	1	2	yes	—
81	C₂×#1055	M₁₆	1	2	yes	—
82	C₁₃×C₁₆	—	2	4	no	—
83	#6090	—	2	4	no	—
84	C₂×C₄×M₁₆	—	13	1	no	—
85	C₂×#349	—	13	1	no	—
86	C₂×#349	D₁₃	1	1	yes	characteristic
87	C₂×#347	—	13	2	no	—
88	#1376	—	13	2	no	—
89	C₂³×C₁₃	—	2	1	no	—
90	C₂³×C₁₃	—	2	1	no	—
91	C₂³×C₁₃	—	4	2	no	—
92	C₂³×C₁₃	#97	1	2	yes	—
93	C₂³×C₁₃	C₂×M₁₆	1	1	yes	characteristic
94	C₂×C₄×C₁₃	—	2	1	no	—
95	C₂×C₄×C₁₃	—	2	1	no	—
96	C₂×C₄×C₁₃	—	2	2	no	—
97	C₂×C₄×C₁₃	—	2	2	no	—
98	C₂×C₄×C₁₃	—	2	2	no	—
99	C₂×C₄×C₁₃	—	2	2	no	—
100	C₂×C₄×C₁₃	#97	1	2	yes	—
101	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
102	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
103	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
104	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
105	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
106	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
107	C₂×Dic₁₃	—	2	4	no	—
108	C₂×Dic₁₃	—	2	2	no	—
109	C₂×Dic₁₃	—	2	2	no	—
110	C₂×Dic₁₃	—	4	8	no	—
111	C₂×Dic₁₃	—	4	2	no	—
112	C₈×C₁₃	—	2	2	no	—
113	C₈×C₁₃	C₄×D₄	1	2	yes	—
114	C₈×C₁₃	C₄×D₄	1	2	yes	—
115	#1055	—	2	2	no	—
116	#1055	—	2	2	no	—
117	C₂²×M₁₆	—	13	1	no	—
118	C₂²×M₁₆	C₂×D₁₃	1	1	yes	characteristic
119	C₂²×C₄²	—	13	1	no	—
120	C₂×M₃₂	—	13	2	no	—
121	C₂×M₃₂	C₂×D₁₃	1	2	yes	—
122	C₄×M₁₆	—	26	4	no	—
123	C₂×#97	—	13	1	no	—
124	C₂×#97	—	13	1	no	—
125	C₂×C₄×C₈	—	13	1	no	—
126	C₂×C₄×C₈	—	13	1	no	—
127	#349	—	13	4	no	—
128	#349	Dic₁₃	1	4	yes	complemented
129	#347	—	26	4	no	—
130	C₂²×C₁₃	—	2	2	no	—
131	C₂²×C₁₃	—	2	2	no	—
132	C₂²×C₁₃	—	4	2	no	—
133	C₂²×C₁₃	—	4	2	no	—
134	C₂²×C₁₃	—	4	2	no	—
135	C₂²×C₁₃	C₂×#97	1	1	yes	characteristic
136	C₂²×C₁₃	#407	1	1	yes	characteristic
137	C₂²×C₁₃	#432	1	1	yes	characteristic
138	C₄×C₁₃	—	2	2	no	—
139	C₄×C₁₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
140	C₄×C₁₃	C₄×M₁₆	1	1	yes	characteristic
141	C₄×C₁₃	#407	1	1	yes	characteristic
142	C₄×C₁₃	#432	1	1	yes	characteristic
143	Dic₁₃	—	2	4	no	—
144	Dic₁₃	—	4	2	no	—
145	C₂³×C₄	—	26	1	no	—
146	C₂³×C₄	C₄×D₁₃	1	1	yes	characteristic
147	C₂×M₁₆	—	2	2	no	—
148	C₂×M₁₆	—	2	2	no	—
149	C₂×M₁₆	—	26	2	no	—
150	C₂×M₁₆	—	26	2	no	—
151	C₂×M₁₆	—	26	2	no	—
152	C₂×M₁₆	C₄×D₁₃	1	2	yes	—
153	C₂×M₁₆	C₂×Dic₁₃	1	2	yes	—
154	C₂²×C₈	—	13	1	no	—
155	C₂²×C₈	—	13	1	no	—
156	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
157	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
158	C₂×C₄²	—	13	1	no	—
159	C₂×C₄²	—	13	1	no	—
160	C₂×C₄²	—	26	2	no	—
161	C₂×C₄²	—	26	2	no	—
162	C₂×C₄²	—	26	1	no	—
163	M₃₂	—	2	2	no	—
164	M₃₂	—	13	4	no	—
165	M₃₂	—	26	2	no	—
166	M₃₂	C₂×Dic₁₃	1	4	yes	complemented
167	C₂×C₁₆	—	2	2	no	—
168	C₂×C₁₆	—	26	2	no	—
169	#97	—	13	4	no	—
170	#97	—	26	2	no	—
171	C₄×C₈	—	26	2	no	—
172	C₄×C₈	—	26	2	no	—
173	C₂×C₁₃	—	2	1	no	—
174	C₂×C₁₃	—	2	1	no	—
175	C₂×C₁₃	—	4	2	no	—
176	C₂×C₁₃	#1749	1	1	yes	characteristic
177	C₂×C₁₃	#2165	1	2	yes	—
178	C₂⁴	#6094	1	1	yes	characteristic
179	C₂²×C₄	—	2	1	no	—
180	C₂²×C₄	—	2	1	no	—
181	C₂²×C₄	—	2	1	no	—
182	C₂²×C₄	—	2	1	no	—
183	C₂²×C₄	—	26	1	no	—
184	C₂²×C₄	—	26	4	no	—
185	C₂²×C₄	—	26	1	no	—
186	C₂²×C₄	—	52	4	no	—
187	C₂²×C₄	#6094	1	2	yes	—
188	C₂²×C₄	#6094	1	1	yes	characteristic
189	C₂²×C₄	C₄×Dic₁₃	1	2	yes	—
190	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
191	M₁₆	—	2	4	no	—
192	M₁₆	—	2	4	no	—
193	M₁₆	—	26	8	no	—
194	C₂×C₈	—	2	1	no	—
195	C₂×C₈	—	2	1	no	—
196	C₂×C₈	—	13	2	no	—
197	C₂×C₈	—	13	2	no	—
198	C₂×C₈	—	26	1	no	—
199	C₂×C₈	—	26	1	no	—
200	C₂×C₈	—	26	1	no	—
201	C₂×C₈	—	26	1	no	—
202	C₂×C₈	C₄×Dic₁₃	1	2	yes	—
203	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
204	C₂×C₈	D₄×D₁₃	1	2	yes	—
205	C₂×C₈	#6129	1	2	yes	—
206	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
207	C₄²	—	13	2	no	—
208	C₄²	—	13	2	no	—
209	C₄²	—	26	4	no	—
210	C₄²	—	26	1	no	—
211	C₄²	—	26	1	no	—
212	C₁₆	—	2	4	no	—
213	C₁₆	—	26	4	no	—
214	C₁₃	#12607	1	1	yes	characteristic, complemented, Sylow
215	C₂³	—	2	1	no	—
216	C₂³	—	2	1	no	—
217	C₂³	—	4	2	no	—
218	C₂³	#88055	1	2	yes	—
219	C₂³	C₂×#6094	1	1	yes	characteristic
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	26	4	no	—
227	C₂×C₄	—	26	2	no	—
228	C₂×C₄	—	26	2	no	—
229	C₂×C₄	—	52	8	no	—
230	C₂×C₄	—	52	2	no	—
231	C₂×C₄	#88055	1	2	yes	—
232	C₂×C₄	#88131	1	1	yes	characteristic
233	C₂×C₄	C₂²⋊C₄×D₁₃	1	1	yes	characteristic
234	C₂×C₄	C₂×#6094	1	1	yes	characteristic
235	C₂×C₄	M₁₆×D₁₃	1	1	yes	characteristic
236	C₂×C₄	M₁₆×D₁₃	1	1	yes	characteristic
237	C₂×C₄	C₂×C₄×Dic₁₃	1	1	yes	characteristic
238	C₈	—	2	2	no	—
239	C₈	—	26	2	no	—
240	C₈	—	26	2	no	—
241	C₈	#88135	1	2	yes	—
242	C₈	D₄×Dic₁₃	1	2	yes	—
243	C₂²	—	2	2	no	—
244	C₂²	—	2	2	no	—
245	C₂²	—	4	2	no	—
246	C₂²	—	4	2	no	—
247	C₂²	—	4	2	no	—
248	C₂²	#11718380	1	1	yes	characteristic
249	C₂²	#11718387	1	1	yes	characteristic
250	C₂²	C₂×#88055	1	1	yes	characteristic
251	C₄	—	2	2	no	—
252	C₄	—	26	4	no	—
253	C₄	—	52	2	no	—
254	C₄	#11718380	1	1	yes	characteristic
255	C₄	#11718387	1	1	yes	characteristic
256	C₄	C₂²⋊C₄×Dic₁₃	1	1	yes	characteristic
257	C₄	M₁₆×Dic₁₃	1	1	yes	characteristic
258	C₂	—	2	1	no	—
259	C₂	—	2	1	no	—
260	C₂	—	4	2	no	—
261	C₂	#13114600	1	1	yes	characteristic
262	C₂	#13116645	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#88131
commutator	C₂×C₄×C₁₃	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₁₃
Fitting	C₂×C₁₃×#349	C₂
socle	C₂²×C₁₃	#432
prime radical	C₂⁴×C₁₃	M₁₆
cosocle	C₂²×C₈×C₁₃	C₂³
hypercenter	C₂×#349	D₁₃
hypocenter	C₁₃	#12607
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂×#349

Complements

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

N	H ≅ G/N	type
3. C₂×#11718104	C₂	semidirect
4. #13112917	C₂	semidirect
16. C₁₃×#349	C₄	semidirect
17. #11718145	C₄	semidirect
38. C₁₃×M₃₂	C₂×C₄	semidirect
45. #88052	C₂×C₄	semidirect
128. #349	Dic₁₃	semidirect
166. M₃₂	C₂×Dic₁₃	semidirect
214. C₁₃	#12607	semidirect

Maximal subgroups

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

subgroup	index
C₂×M₁₆×Dic₁₃	2
C₂×#11718145	2
C₂×#11718104	2
#13112917	2
C₂×C₁₃×#349	2
#12607	13