Subgroups — #15866194

Order 3328.

1038 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718780	C₂	1	2	yes	maximal
2	C₂²×C₄×#1055	C₂	1	1	yes	maximal, characteristic
3	#13117243	C₂	1	1	yes	maximal, characteristic
4	#13115552	C₂	1	8	yes	maximal, complemented
5	#13114534	C₂	1	2	yes	maximal
6	C₂×C₁₃×#423	C₂	1	1	yes	maximal, characteristic
7	C₂²×C₁₃×M₁₆	C₂²	1	2	yes	—
8	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
9	C₂×#88180	C₂²	1	2	yes	—
10	C₂³×#1055	C₂²	1	1	yes	characteristic
11	C₂³×#1055	C₂²	1	1	yes	characteristic
12	C₁₃×#423	—	2	4	no	—
13	C₂×C₁₃×C₄⋊C₈	C₂²	1	4	yes	—
14	#11718873	C₄	1	4	yes	—
15	#11718780	—	2	8	no	—
16	C₂×#88054	C₂²	1	8	yes	—
17	#11718597	C₄	1	2	yes	—
18	#11718596	C₄	1	2	yes	—
19	C₄×C₄⋊Dic₁₃	C₂²	1	4	yes	—
20	C₂×C₄×#1055	—	2	4	no	—
21	C₂×C₄×#1055	C₂²	1	4	yes	—
22	#11718142	C₂²	1	8	yes	—
23	C₂³×C₄×C₁₃	C₂×C₄	1	2	yes	—
24	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
25	C₂×C₁₃×M₁₆	—	2	4	no	—
26	C₂×C₁₃×M₁₆	—	2	4	no	—
27	C₂×C₁₃×M₁₆	—	2	4	no	—
28	C₂²×C₈×C₁₃	C₂³	1	4	yes	—
29	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄²×C₁₃	D₄	1	4	yes	—
32	C₂×C₄²×C₁₃	Q₈	1	4	yes	—
33	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
34	C₂×C₂²⋊Dic₁₃	C₂×C₄	1	4	yes	—
35	#88180	—	2	4	no	—
36	#88180	—	2	4	no	—
37	C₂×C₄⋊Dic₁₃	C₂×C₄	1	4	yes	—
38	C₂×C₄⋊Dic₁₃	C₂×C₄	1	8	yes	—
39	C₂×C₄×Dic₁₃	C₂³	1	4	yes	—
40	C₂²×#1055	—	2	2	no	—
41	C₂²×#1055	—	2	2	no	—
42	C₂²×#1055	—	2	2	no	—
43	C₂²×#1055	—	2	2	no	—
44	C₂²×#1055	D₄	1	4	yes	—
45	C₂²×#1055	Q₈	1	4	yes	—
46	C₂²×#1055	C₂³	1	2	yes	—
47	C₂²×#1055	C₂³	1	2	yes	—
48	C₁₃×C₄⋊C₈	—	2	8	no	—
49	#88071	C₂×C₄	1	8	yes	—
50	#88054	—	2	16	no	—
51	C₄×#1055	—	2	8	no	—
52	#21143	—	13	1	no	maximal, Sylow
53	C₂⁴×C₁₃	C₂²×C₄	1	1	yes	characteristic
54	C₂²×C₄×C₁₃	—	2	8	no	—
55	C₂²×C₄×C₁₃	C₄⋊C₄	1	8	yes	—
56	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
57	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
58	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
59	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
60	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
61	C₂²×C₄×C₁₃	C₂×Q₈	1	2	yes	—
62	C₂²×C₄×C₁₃	C₂×Q₈	1	2	yes	—
63	C₂²×C₄×C₁₃	#41	1	2	yes	—
64	C₂²×C₄×C₁₃	#41	1	2	yes	—
65	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
66	C₂²×Dic₁₃	C₂²×C₄	1	8	yes	—
67	C₁₃×M₁₆	—	4	8	no	—
68	C₂×C₈×C₁₃	—	2	4	no	—
69	C₂×C₈×C₁₃	—	2	4	no	—
70	C₂×C₈×C₁₃	—	2	4	no	—
71	C₄²×C₁₃	C₄⋊C₄	1	4	yes	—
72	C₄²×C₁₃	C₄⋊C₄	1	4	yes	—
73	C₄²×C₁₃	C₂×D₄	1	4	yes	—
74	C₄²×C₁₃	C₂×Q₈	1	4	yes	—
75	C₂²⋊Dic₁₃	—	2	8	no	—
76	C₄⋊Dic₁₃	—	2	8	no	—
77	C₄⋊Dic₁₃	—	2	16	no	—
78	C₄×Dic₁₃	—	2	4	no	—
79	C₄×Dic₁₃	—	2	4	no	—
80	C₂×#1055	—	2	8	no	—
81	C₂×#1055	—	2	8	no	—
82	C₂×#1055	—	2	2	no	—
83	C₂×#1055	—	2	2	no	—
84	C₂×#1055	—	2	2	no	—
85	C₂×#1055	—	2	2	no	—
86	C₂×#1055	C₂×D₄	1	4	yes	—
87	C₂×#1055	C₂×Q₈	1	4	yes	—
88	C₂×#424	—	13	2	no	—
89	C₂×#423	D₁₃	1	1	yes	characteristic
90	C₂²×C₄×C₈	—	13	1	no	—
91	#2285	—	13	1	no	—
92	#1939	—	13	8	no	—
93	#1729	—	13	2	no	—
94	C₂³×C₁₃	—	2	4	no	—
95	C₂³×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
96	C₂³×C₁₃	C₄×D₄	1	2	yes	—
97	C₂³×C₁₃	C₄×Q₈	1	2	yes	—
98	C₂³×C₁₃	C₂³×C₄	1	1	yes	characteristic
99	C₂×C₄×C₁₃	—	2	4	no	—
100	C₂×C₄×C₁₃	—	2	16	no	—
101	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	4	yes	—
102	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
103	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	4	yes	—
104	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
105	C₂×C₄×C₁₃	C₄×Q₈	1	2	yes	—
106	C₂×C₄×C₁₃	#121	1	4	yes	—
107	C₂×C₄×C₁₃	C₂²⋊Q₈	1	4	yes	—
108	C₂×C₄×C₁₃	#125	1	4	yes	—
109	C₂×C₄×C₁₃	C₄⋊Q₈	1	4	yes	—
110	C₂×C₄×C₁₃	#131	1	8	yes	—
111	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
112	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
113	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
114	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
115	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
116	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
117	C₂×Dic₁₃	—	2	8	no	—
118	C₂×Dic₁₃	—	2	16	no	—
119	C₈×C₁₃	—	4	4	no	—
120	#1055	—	2	4	no	—
121	#1055	—	2	4	no	—
122	C₂²×M₁₆	C₂×D₁₃	1	2	yes	—
123	C₂³×C₈	—	13	1	no	—
124	C₂³×C₈	—	13	1	no	—
125	C₂×#117	—	13	2	no	—
126	C₂²×C₄²	C₂×D₁₃	1	1	yes	characteristic
127	#424	—	26	8	no	—
128	#423	—	2	4	no	—
129	C₂×C₄⋊C₈	—	13	8	no	—
130	C₂×C₄⋊C₈	C₂×D₁₃	1	4	yes	—
131	C₂×C₄×C₈	—	13	4	no	—
132	C₂×C₄×C₈	—	26	4	no	—
133	#384	—	13	2	no	—
134	#382	—	13	2	no	—
135	#380	—	13	4	no	—
136	C₄×C₄⋊C₄	—	13	4	no	—
137	#336	—	13	8	no	—
138	C₂²×C₁₃	—	2	4	no	—
139	C₂²×C₁₃	—	2	4	no	—
140	C₂²×C₁₃	—	2	4	no	—
141	C₂²×C₁₃	#389	1	4	yes	—
142	C₂²×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
143	C₂²×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
144	C₂²×C₁₃	C₂×C₄×Q₈	1	1	yes	characteristic
145	C₂²×C₁₃	C₂×#131	1	4	yes	—
146	C₄×C₁₃	—	2	4	no	—
147	C₄×C₁₃	#389	1	4	yes	—
148	C₄×C₁₃	#424	1	8	yes	complemented
149	C₄×C₁₃	C₂×#121	1	1	yes	characteristic
150	C₄×C₁₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
151	C₄×C₁₃	C₂×#125	1	1	yes	characteristic
152	C₄×C₁₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
153	Dic₁₃	—	4	8	no	—
154	C₂³×C₄	C₄×D₁₃	1	2	yes	—
155	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
156	C₂×M₁₆	—	2	4	no	—
157	C₂×M₁₆	—	2	4	no	—
158	C₂×M₁₆	—	2	4	no	—
159	C₂²×C₈	—	13	4	no	—
160	C₂²×C₈	—	13	4	no	—
161	C₂²×C₈	—	13	2	no	—
162	C₂²×C₈	—	13	2	no	—
163	C₂²×C₈	—	26	2	no	—
164	C₂²×C₈	—	26	2	no	—
165	C₂²×C₈	—	26	2	no	—
166	C₂²×C₈	—	26	2	no	—
167	C₂²×C₈	C₂²×D₁₃	1	4	yes	—
168	#117	—	26	4	no	—
169	#117	—	26	4	no	—
170	C₂×C₄⋊C₄	—	13	4	no	—
171	C₂×C₄⋊C₄	—	13	8	no	—
172	C₂×C₂²⋊C₄	—	13	4	no	—
173	C₂×C₄²	—	13	4	no	—
174	C₂×C₄²	Dic₂₆	1	4	yes	—
175	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
176	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
177	C₂×C₄²	C₂²⋊D₁₃	1	4	yes	—
178	C₂×C₄²	C₂²×D₁₃	1	2	yes	—
179	C₄⋊C₈	—	2	8	no	—
180	C₄⋊C₈	—	26	16	no	—
181	C₄×C₈	—	26	8	no	—
182	#95	—	13	8	no	—
183	C₂×C₁₃	—	2	4	no	—
184	C₂×C₁₃	C₂×#389	1	1	yes	characteristic
185	C₂×C₁₃	C₂×#424	1	2	yes	—
186	C₂×C₁₃	#2931	1	2	yes	—
187	C₂×C₁₃	#2960	1	2	yes	—
188	C₂⁴	C₂×C₄×D₁₃	1	1	yes	characteristic
189	C₂²×C₄	—	2	8	no	—
190	C₂²×C₄	—	13	8	no	—
191	C₂²×C₄	#6101	1	8	yes	—
192	C₂²×C₄	C₂×Dic₂₆	1	2	yes	—
193	C₂²×C₄	C₂×C₄×D₁₃	1	2	yes	—
194	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
195	C₂²×C₄	C₂×C₄×D₁₃	1	2	yes	—
196	C₂²×C₄	D₄×D₁₃	1	2	yes	—
197	C₂²×C₄	#6129	1	2	yes	—
198	C₂²×C₄	Q₈×D₁₃	1	2	yes	—
199	C₂²×C₄	#6131	1	2	yes	—
200	C₂²×C₄	C₂×C₂²⋊D₁₃	1	2	yes	—
201	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
202	M₁₆	—	4	8	no	—
203	C₂×C₈	—	2	4	no	—
204	C₂×C₈	—	2	4	no	—
205	C₂×C₈	—	2	4	no	—
206	C₂×C₈	—	13	4	no	—
207	C₂×C₈	—	13	4	no	—
208	C₂×C₈	—	26	8	no	—
209	C₂×C₈	—	26	8	no	—
210	C₂×C₈	—	26	2	no	—
211	C₂×C₈	—	26	2	no	—
212	C₂×C₈	—	26	2	no	—
213	C₂×C₈	—	26	2	no	—
214	C₄⋊C₄	—	26	8	no	—
215	C₄⋊C₄	—	26	16	no	—
216	C₂²⋊C₄	—	26	8	no	—
217	C₄²	—	26	4	no	—
218	C₄²	—	26	4	no	—
219	C₄²	#6101	1	4	yes	—
220	C₄²	#6101	1	4	yes	—
221	C₄²	C₂×Dic₂₆	1	4	yes	—
222	C₄²	C₂×C₂²⋊D₁₃	1	4	yes	—
223	C₁₃	#21143	1	1	yes	characteristic, complemented, Sylow
224	C₂³	—	2	4	no	—
225	C₂³	#88141	1	2	yes	—
226	C₂³	#88147	1	2	yes	—
227	C₂³	C₂×#6101	1	2	yes	—
228	C₂³	C₂²×C₄×D₁₃	1	1	yes	characteristic
229	C₂×C₄	—	2	4	no	—
230	C₂×C₄	—	2	16	no	—
231	C₂×C₄	—	26	16	no	—
232	C₂×C₄	—	26	8	no	—
233	C₂×C₄	#88141	1	2	yes	—
234	C₂×C₄	#88142	1	4	yes	—
235	C₂×C₄	#88144	1	4	yes	—
236	C₂×C₄	#88147	1	2	yes	—
237	C₂×C₄	#88161	1	8	yes	—
238	C₂×C₄	C₂×#6101	1	4	yes	—
239	C₂×C₄	C₂×#6101	1	4	yes	—
240	C₂×C₄	C₂×#6101	1	2	yes	—
241	C₂×C₄	#88192	1	4	yes	—
242	C₂×C₄	#88200	1	4	yes	—
243	C₂×C₄	C₂²×Dic₂₆	1	1	yes	characteristic
244	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
245	C₂×C₄	C₂×#6129	1	1	yes	characteristic
246	C₂×C₄	C₂×Q₈×D₁₃	1	1	yes	characteristic
247	C₂×C₄	C₂×#6131	1	1	yes	characteristic
248	C₂×C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
249	C₈	—	4	4	no	—
250	C₈	—	26	4	no	—
251	C₈	—	26	4	no	—
252	C₂²	—	2	4	no	—
253	C₂²	—	2	4	no	—
254	C₂²	—	2	4	no	—
255	C₂²	#11718634	1	4	yes	—
256	C₂²	C₂×#88141	1	1	yes	characteristic
257	C₂²	C₂×#88147	1	1	yes	characteristic
258	C₂²	C₂×#88161	1	4	yes	—
259	C₂²	C₂²×#6101	1	1	yes	characteristic
260	C₄	—	2	4	no	—
261	C₄	—	52	8	no	—
262	C₄	#11718634	1	4	yes	—
263	C₄	#11718780	1	8	yes	complemented
264	C₄	C₂×#88142	1	1	yes	characteristic
265	C₄	C₂×#88144	1	1	yes	characteristic
266	C₄	C₂×#88192	1	1	yes	characteristic
267	C₄	C₂×#88200	1	1	yes	characteristic
268	C₂	—	2	4	no	—
269	C₂	C₂×#11718634	1	1	yes	characteristic
270	C₂	C₂×#11718780	1	2	yes	—
271	C₂	#13123547	1	2	yes	—
272	C₂	#13123766	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₁₃
commutator, socle	C₂³×C₁₃	C₂³×C₄
Fitting	C₂×C₁₃×#423	C₂
prime radical	C₂⁴×C₁₃	C₂²×C₄
cosocle	C₂²×C₄×C₁₃	C₂⁴
hypercenter	C₂×#423	D₁₃
hypocenter	C₁₃	#21143
LCS	G ▷ C₂³×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#423

Complements

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

N	H ≅ G/N	type
4. #13115552	C₂	semidirect
148. C₄×C₁₃	#424	semidirect
223. C₁₃	#21143	semidirect
263. C₄	#11718780	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718780	2
C₂²×C₄×#1055	2
#13117243	2
#13115552	2
#13114534	2
C₂×C₁₃×#423	2
#21143	13