Subgroups — #15867704

Order 3328.

954 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718586	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×#6099	C₂	1	1	yes	maximal, characteristic
3	#13115560	C₂	1	4	yes	maximal, complemented
4	#13115021	C₂	1	4	yes	maximal
5	C₄×C₄⋊#1055	C₂	1	4	yes	maximal, complemented
6	C₄×C₁₃×#117	C₂	1	1	yes	maximal, characteristic
7	C₂×C₁₃×#117	C₄	1	2	yes	—
8	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
9	C₂²×#6099	C₂²	1	2	yes	—
10	C₄×C₁₃×C₄⋊C₄	C₄	1	2	yes	—
11	C₄×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
12	C₄×C₁₃×C₂²⋊C₄	C₄	1	2	yes	—
13	C₄×C₁₃×C₂²⋊C₄	C₂²	1	2	yes	—
14	C₁₃×#376	C₄	1	1	yes	characteristic
15	C₁₃×#376	C₄	1	1	yes	characteristic
16	C₄³×C₁₃	C₂²	1	2	yes	—
17	C₂×C₂²⋊#1055	C₂²	1	2	yes	—
18	#11718586	—	2	4	no	—
19	C₂×C₄⋊#1055	C₂²	1	8	yes	—
20	C₂×C₄⋊#1055	C₂²	1	2	yes	—
21	C₄×#6099	—	2	4	no	—
22	C₂×#88043	C₂²	1	2	yes	—
23	C₂×C₄×#1055	C₂²	1	2	yes	—
24	C₂×C₄×#1055	C₂²	1	2	yes	—
25	#11718116	C₂²	1	8	yes	—
26	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
27	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
28	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
29	C₁₃×#117	—	2	8	no	—
30	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	4	yes	—
31	C₂×C₁₃×C₂²⋊C₄	C₂×C₄	1	4	yes	—
32	C₂×C₄²×C₁₃	C₂×C₄	1	4	yes	—
33	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
34	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
37	C₂×C₄²×C₁₃	D₄	1	4	yes	—
38	C₂×C₄²×C₁₃	Q₈	1	4	yes	—
39	C₂×C₄²×C₁₃	C₂³	1	4	yes	—
40	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
41	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
42	C₂×#6099	—	2	4	no	—
43	C₂×#6099	—	2	8	no	—
44	C₂²×#1055	C₂³	1	4	yes	—
45	C₂²×#1055	C₂³	1	4	yes	—
46	C₁₃×#95	C₂×C₄	1	2	yes	—
47	C₁₃×#95	C₂×C₄	1	2	yes	—
48	C₁₃×#95	C₂×C₄	1	2	yes	—
49	C₁₃×#95	C₂×C₄	1	2	yes	—
50	C₂²⋊#1055	—	2	4	no	—
51	C₄⋊#1055	—	2	4	no	—
52	C₄⋊#1055	—	2	16	no	—
53	#88043	—	2	4	no	—
54	C₄×#1055	—	2	4	no	—
55	C₄×#1055	—	2	4	no	—
56	#21350	—	13	1	no	maximal, Sylow
57	C₂⁴×C₁₃	C₂²×C₄	1	1	yes	characteristic
58	C₂²×C₄×C₁₃	—	2	4	no	—
59	C₂²×C₄×C₁₃	—	2	4	no	—
60	C₂²×C₄×C₁₃	—	2	4	no	—
61	C₂²×C₄×C₁₃	C₄⋊C₄	1	4	yes	—
62	C₂²×C₄×C₁₃	C₄⋊C₄	1	4	yes	—
63	C₂²×C₄×C₁₃	C₂²×C₄	1	4	yes	—
64	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₁₃	C₂²×C₄	1	4	yes	—
68	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
71	C₂²×C₄×C₁₃	C₂×Q₈	1	2	yes	—
72	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
73	C₁₃×C₄⋊C₄	—	2	8	no	—
74	C₁₃×C₂²⋊C₄	—	2	8	no	—
75	C₄²×C₁₃	—	2	4	no	—
76	C₄²×C₁₃	—	2	8	no	—
77	C₄²×C₁₃	—	2	8	no	—
78	C₄²×C₁₃	C₄⋊C₄	1	4	yes	—
79	C₄²×C₁₃	C₄⋊C₄	1	4	yes	—
80	C₄²×C₁₃	M₁₆	1	8	yes	complemented
81	C₄²×C₁₃	C₂×D₄	1	2	yes	—
82	C₄²×C₁₃	C₂×D₄	1	2	yes	—
83	C₄²×C₁₃	C₂×Q₈	1	2	yes	—
84	C₄²×C₁₃	C₂×Q₈	1	2	yes	—
85	#6099	—	2	16	no	—
86	C₂×#1055	—	2	4	no	—
87	C₂×#1055	—	2	4	no	—
88	C₂×#1055	—	2	4	no	—
89	C₂×#1055	—	2	8	no	—
90	C₂×#1055	#41	1	8	yes	—
91	C₂×#431	—	13	1	no	—
92	C₂×C₄×M₁₆	—	13	1	no	—
93	C₄×#117	D₁₃	1	1	yes	characteristic
94	#1940	—	13	4	no	—
95	#1851	—	13	4	no	—
96	C₄×C₄⋊C₈	—	13	4	no	—
97	C₂³×C₁₃	—	2	2	no	—
98	C₂³×C₁₃	—	2	2	no	—
99	C₂³×C₁₃	—	2	2	no	—
100	C₂³×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
101	C₂³×C₁₃	C₂³×C₄	1	1	yes	characteristic
102	C₂×C₄×C₁₃	—	2	4	no	—
103	C₂×C₄×C₁₃	—	2	2	no	—
104	C₂×C₄×C₁₃	—	2	2	no	—
105	C₂×C₄×C₁₃	—	2	4	no	—
106	C₂×C₄×C₁₃	—	2	2	no	—
107	C₂×C₄×C₁₃	—	2	4	no	—
108	C₂×C₄×C₁₃	—	2	4	no	—
109	C₂×C₄×C₁₃	—	2	8	no	—
110	C₂×C₄×C₁₃	—	2	8	no	—
111	C₂×C₄×C₁₃	—	2	4	no	—
112	C₂×C₄×C₁₃	—	2	4	no	—
113	C₂×C₄×C₁₃	—	2	4	no	—
114	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
115	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
116	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
117	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
118	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
119	C₂×C₄×C₁₃	C₂×M₁₆	1	8	yes	—
120	C₂×C₄×C₁₃	C₂×M₁₆	1	4	yes	—
121	C₂×C₄×C₁₃	#131	1	4	yes	—
122	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
123	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
124	C₂×C₄×C₁₃	C₂×#41	1	4	yes	—
125	#1055	—	2	8	no	—
126	#1055	—	4	4	no	—
127	C₂²×M₁₆	—	13	2	no	—
128	C₂×#117	Dic₁₃	1	2	yes	—
129	C₂²×C₄²	C₂×D₁₃	1	1	yes	characteristic
130	#431	—	26	4	no	—
131	C₂×C₄⋊C₈	—	13	8	no	—
132	C₂×C₄⋊C₈	—	13	2	no	—
133	C₂×C₂²⋊C₈	—	13	2	no	—
134	C₄×M₁₆	—	26	4	no	—
135	C₂×#97	—	13	2	no	—
136	C₂×C₄×C₈	—	13	2	no	—
137	C₂×C₄×C₈	—	13	2	no	—
138	C₄×C₄⋊C₄	Dic₁₃	1	2	yes	—
139	C₄×C₄⋊C₄	C₂×D₁₃	1	2	yes	—
140	C₄×C₂²⋊C₄	Dic₁₃	1	2	yes	—
141	C₄×C₂²⋊C₄	C₂×D₁₃	1	2	yes	—
142	#376	Dic₁₃	1	1	yes	characteristic
143	#376	Dic₁₃	1	1	yes	characteristic
144	C₄³	C₂×D₁₃	1	2	yes	—
145	#336	—	13	8	no	—
146	C₂²×C₁₃	—	2	2	no	—
147	C₂²×C₁₃	—	2	2	no	—
148	C₂²×C₁₃	—	2	2	no	—
149	C₂²×C₁₃	—	4	4	no	—
150	C₂²×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
151	C₂²×C₁₃	C₄×#41	1	2	yes	—
152	C₂²×C₁₃	C₂²×M₁₆	1	2	yes	—
153	C₂²×C₁₃	C₂×#131	1	2	yes	—
154	C₄×C₁₃	—	2	2	no	—
155	C₄×C₁₃	—	2	2	no	—
156	C₄×C₁₃	—	2	8	no	—
157	C₄×C₁₃	—	2	2	no	—
158	C₄×C₁₃	—	4	4	no	—
159	C₄×C₁₃	#423	1	4	yes	—
160	C₄×C₁₃	#424	1	4	yes	—
161	C₄×C₁₃	#525	1	2	yes	—
162	C₄×C₁₃	#533	1	2	yes	—
163	C₂³×C₄	C₂×Dic₁₃	1	1	yes	characteristic
164	C₂³×C₄	C₂×Dic₁₃	1	1	yes	characteristic
165	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
166	C₂×M₁₆	—	26	8	no	—
167	C₂×M₁₆	—	26	4	no	—
168	C₂²×C₈	—	13	4	no	—
169	C₂²×C₈	—	13	4	no	—
170	#117	—	2	8	no	—
171	C₂×C₄⋊C₄	C₂×Dic₁₃	1	4	yes	—
172	C₂×C₂²⋊C₄	C₂×Dic₁₃	1	4	yes	—
173	C₂×C₄²	Dic₂₆	1	4	yes	—
174	C₂×C₄²	D₅₂	1	4	yes	—
175	C₂×C₄²	C₂×Dic₁₃	1	2	yes	—
176	C₂×C₄²	C₂×Dic₁₃	1	4	yes	—
177	C₂×C₄²	C₂×Dic₁₃	1	2	yes	—
178	C₂×C₄²	C₂×Dic₁₃	1	1	yes	characteristic
179	C₂×C₄²	C₂×Dic₁₃	1	1	yes	characteristic
180	C₂×C₄²	C₂²×D₁₃	1	4	yes	—
181	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
182	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
183	C₄⋊C₈	—	26	16	no	—
184	C₄⋊C₈	—	26	4	no	—
185	C₂²⋊C₈	—	26	4	no	—
186	#97	—	26	4	no	—
187	C₄×C₈	—	26	4	no	—
188	C₄×C₈	—	26	4	no	—
189	#95	C₂×Dic₁₃	1	2	yes	—
190	#95	C₂×Dic₁₃	1	2	yes	—
191	#95	C₂×Dic₁₃	1	2	yes	—
192	#95	C₂×Dic₁₃	1	2	yes	—
193	C₂×C₁₃	—	4	2	no	—
194	C₂×C₁₃	#2296	1	1	yes	characteristic
195	C₂×C₁₃	C₂×#423	1	1	yes	characteristic
196	C₂×C₁₃	C₂×#424	1	1	yes	characteristic
197	C₂×C₁₃	#2965	1	4	yes	—
198	C₂⁴	C₂²×Dic₁₃	1	1	yes	characteristic
199	C₂²×C₄	—	2	4	no	—
200	C₂²×C₄	—	2	4	no	—
201	C₂²×C₄	—	2	4	no	—
202	C₂²×C₄	C₄⋊Dic₁₃	1	4	yes	—
203	C₂²×C₄	C₄⋊Dic₁₃	1	4	yes	—
204	C₂²×C₄	C₂×Dic₂₆	1	2	yes	—
205	C₂²×C₄	C₂×D₅₂	1	2	yes	—
206	C₂²×C₄	C₂²×Dic₁₃	1	4	yes	—
207	C₂²×C₄	C₂²×Dic₁₃	1	4	yes	—
208	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
209	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
210	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
211	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
212	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
213	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
214	M₁₆	—	26	16	no	—
215	C₂×C₈	—	13	8	no	—
216	C₂×C₈	—	26	4	no	—
217	C₂×C₈	—	26	4	no	—
218	C₂×C₈	—	26	4	no	—
219	C₂×C₈	—	26	8	no	—
220	C₄⋊C₄	—	2	8	no	—
221	C₂²⋊C₄	—	2	8	no	—
222	C₄²	—	2	4	no	—
223	C₄²	—	2	8	no	—
224	C₄²	—	2	8	no	—
225	C₄²	#6099	1	8	yes	complemented
226	C₄²	C₄⋊Dic₁₃	1	4	yes	—
227	C₄²	C₄⋊Dic₁₃	1	4	yes	—
228	C₄²	C₂×Dic₂₆	1	2	yes	—
229	C₄²	C₂×Dic₂₆	1	2	yes	—
230	C₄²	C₂×D₅₂	1	2	yes	—
231	C₄²	C₂×D₅₂	1	2	yes	—
232	C₁₃	#21350	1	1	yes	characteristic, complemented, Sylow
233	C₂³	—	2	2	no	—
234	C₂³	—	2	2	no	—
235	C₂³	—	2	2	no	—
236	C₂³	C₂×C₄⋊Dic₁₃	1	2	yes	—
237	C₂³	C₂³×Dic₁₃	1	1	yes	characteristic
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	4	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	4	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	2	8	no	—
244	C₂×C₄	—	2	4	no	—
245	C₂×C₄	—	2	4	no	—
246	C₂×C₄	—	2	8	no	—
247	C₂×C₄	—	2	4	no	—
248	C₂×C₄	—	2	4	no	—
249	C₂×C₄	—	2	4	no	—
250	C₂×C₄	C₂×#6099	1	8	yes	—
251	C₂×C₄	C₂×#6099	1	4	yes	—
252	C₂×C₄	C₂×C₄⋊Dic₁₃	1	2	yes	—
253	C₂×C₄	C₂×C₄⋊Dic₁₃	1	2	yes	—
254	C₂×C₄	C₂×C₄⋊Dic₁₃	1	2	yes	—
255	C₂×C₄	C₂×C₄⋊Dic₁₃	1	2	yes	—
256	C₂×C₄	C₂×C₄⋊Dic₁₃	1	2	yes	—
257	C₂×C₄	#88202	1	4	yes	—
258	C₂×C₄	C₂²×Dic₂₆	1	1	yes	characteristic
259	C₂×C₄	C₂²×D₅₂	1	1	yes	characteristic
260	C₂×C₄	#41×D₁₃	1	4	yes	—
261	C₈	—	26	8	no	—
262	C₈	—	52	4	no	—
263	C₂²	—	2	2	no	—
264	C₂²	—	2	2	no	—
265	C₂²	—	2	2	no	—
266	C₂²	—	4	4	no	—
267	C₂²	C₂²×#6099	1	2	yes	—
268	C₂²	C₂²×C₄⋊Dic₁₃	1	1	yes	characteristic
269	C₂²	C₂×#88202	1	2	yes	—
270	C₂²	#41×Dic₁₃	1	2	yes	—
271	C₄	—	2	2	no	—
272	C₄	—	2	2	no	—
273	C₄	—	2	2	no	—
274	C₄	—	2	8	no	—
275	C₄	—	4	4	no	—
276	C₄	#11718585	1	4	yes	—
277	C₄	#11718660	1	4	yes	—
278	C₄	#11719222	1	2	yes	—
279	C₄	#11719229	1	2	yes	—
280	C₂	—	4	2	no	—
281	C₂	#13117320	1	1	yes	characteristic
282	C₂	C₂×#11718585	1	1	yes	characteristic
283	C₂	C₂×#11718660	1	1	yes	characteristic
284	C₂	#13123803	1	4	yes	—

	subgroup	G/H
center	C₂×C₄²	C₂²×D₁₃
commutator, socle	C₂³×C₁₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₁₃
Fitting	C₄×C₁₃×#117	C₂
prime radical	C₂⁴×C₁₃	C₂²×C₄
cosocle	C₂²×C₄×C₁₃	C₂⁴
hypercenter	C₄×#117	D₁₃
hypocenter	C₁₃	#21350
LCS	G ▷ C₂³×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄² ◁ C₄×#117

Complements

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

N	H ≅ G/N	type
3. #13115560	C₂	semidirect
5. C₄×C₄⋊#1055	C₂	semidirect
80. C₄²×C₁₃	M₁₆	semidirect
225. C₄²	#6099	semidirect
232. C₁₃	#21350	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718586	2
C₂×C₄×#6099	2
#13115560	2
#13115021	2
C₄×C₄⋊#1055	2
C₄×C₁₃×#117	2
#21350	13