Subgroups — #15863159

Order 3328.

818 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718585	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×#6099	C₂	1	1	yes	maximal, characteristic
3	#13114976	C₂	1	4	yes	maximal, complemented
4	#13114973	C₂	1	4	yes	maximal, complemented
5	#13114562	C₂	1	4	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	C₂²×#6099	C₂²	1	2	yes	—
10	C₂²×#6099	C₂²	1	2	yes	—
11	C₁₃×#423	—	2	2	no	—
12	C₁₃×#423	C₂²	1	2	yes	—
13	C₁₃×#423	C₂²	1	2	yes	—
14	C₂×C₁₃×C₄⋊C₈	—	2	2	no	—
15	#11718585	—	2	2	no	—
16	#11718585	C₂²	1	2	yes	—
17	#11718585	C₂²	1	2	yes	—
18	C₂×C₄⋊#1055	—	2	2	no	—
19	C₄×#6099	C₄	1	2	yes	—
20	C₄×#6099	C₄	1	2	yes	—
21	C₄×#6099	C₄	1	2	yes	—
22	C₄×#6099	C₄	1	2	yes	—
23	C₂×#88043	C₂²	1	2	yes	—
24	C₂×C₄×#1055	C₂²	1	2	yes	—
25	#11718150	C₂²	1	16	yes	—
26	C₂³×C₄×C₁₃	Q₈	1	2	yes	—
27	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
28	C₂×C₁₃×M₁₆	—	2	2	no	—
29	C₂×C₁₃×M₁₆	—	2	8	no	—
30	C₂×C₁₃×M₁₆	C₂³	1	4	yes	—
31	C₂²×C₈×C₁₃	—	2	2	no	—
32	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄²×C₁₃	D₄	1	2	yes	—
37	C₂×C₄²×C₁₃	D₄	1	2	yes	—
38	C₂×C₄²×C₁₃	Q₈	1	2	yes	—
39	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
40	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
41	C₂×#6099	—	2	4	no	—
42	C₂×#6099	—	2	4	no	—
43	C₂×#6099	—	2	2	no	—
44	C₂×#6099	—	2	8	no	—
45	C₂×#6099	C₂×C₄	1	4	yes	—
46	C₂×#6099	C₂×C₄	1	4	yes	—
47	C₂×#6099	C₂³	1	4	yes	—
48	C₂²×#1055	—	2	2	no	—
49	C₂²×#1055	C₂³	1	4	yes	—
50	C₁₃×C₄⋊C₈	—	2	4	no	—
51	C₁₃×C₄⋊C₈	—	2	4	no	—
52	C₄⋊#1055	—	2	4	no	—
53	C₄⋊#1055	—	2	4	no	—
54	#88043	C₂×C₄	1	4	yes	—
55	#88043	C₂×C₄	1	4	yes	—
56	C₄×#1055	C₂×C₄	1	4	yes	—
57	C₄×#1055	C₂×C₄	1	4	yes	—
58	#20769	—	13	1	no	maximal, Sylow
59	C₂⁴×C₁₃	C₂×Q₈	1	1	yes	characteristic
60	C₂²×C₄×C₁₃	—	4	4	no	—
61	C₂²×C₄×C₁₃	C₄⋊C₄	1	2	yes	—
62	C₂²×C₄×C₁₃	C₄⋊C₄	1	2	yes	—
63	C₂²×C₄×C₁₃	C₄⋊C₄	1	2	yes	—
64	C₂²×C₄×C₁₃	C₄⋊C₄	1	2	yes	—
65	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₁₃	C₂×Q₈	1	2	yes	—
70	C₂²×C₄×C₁₃	C₂×Q₈	1	2	yes	—
71	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
72	C₂²×C₄×C₁₃	#41	1	4	yes	—
73	C₂²×C₄×C₁₃	#41	1	4	yes	—
74	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
75	C₁₃×M₁₆	—	2	8	no	—
76	C₁₃×M₁₆	—	4	4	no	—
77	C₂×C₈×C₁₃	—	2	4	no	—
78	C₂×C₈×C₁₃	—	4	4	no	—
79	C₄²×C₁₃	C₄⋊C₄	1	2	yes	—
80	C₄²×C₁₃	C₄⋊C₄	1	2	yes	—
81	C₄²×C₁₃	C₄⋊C₄	1	2	yes	—
82	C₄²×C₁₃	C₄⋊C₄	1	2	yes	—
83	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
84	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
85	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
86	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
87	C₄²×C₁₃	C₂×D₄	1	2	yes	—
88	C₄²×C₁₃	C₂×D₄	1	2	yes	—
89	#6099	—	2	8	no	—
90	#6099	—	2	8	no	—
91	#6099	—	2	8	no	—
92	#6099	—	4	4	no	—
93	C₂×#1055	—	2	4	no	—
94	C₂×#1055	—	2	4	no	—
95	C₂×#1055	—	2	4	no	—
96	C₂×#1055	—	4	4	no	—
97	C₂×#1055	C₂²×C₄	1	8	yes	—
98	C₂×#423	—	13	1	no	—
99	C₂×#423	D₁₃	1	1	yes	characteristic
100	C₂×C₄×M₁₆	—	13	1	no	—
101	#1838	—	13	4	no	—
102	#1837	—	13	4	no	—
103	#1738	—	13	4	no	—
104	C₂³×C₁₃	—	4	2	no	—
105	C₂³×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
106	C₂³×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
107	C₂³×C₁₃	#125	1	4	yes	—
108	C₂³×C₁₃	C₂²×Q₈	1	1	yes	characteristic
109	C₂×C₄×C₁₃	—	2	4	no	—
110	C₂×C₄×C₁₃	—	2	4	no	—
111	C₂×C₄×C₁₃	—	2	4	no	—
112	C₂×C₄×C₁₃	—	4	2	no	—
113	C₂×C₄×C₁₃	—	4	2	no	—
114	C₂×C₄×C₁₃	—	4	2	no	—
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	1	yes	characteristic
119	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
120	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
121	C₂×C₄×C₁₃	#117	1	4	yes	—
122	C₂×C₄×C₁₃	#117	1	4	yes	—
123	C₂×C₄×C₁₃	#124	1	2	yes	—
124	C₂×C₄×C₁₃	#124	1	2	yes	—
125	C₂×C₄×C₁₃	#124	1	2	yes	—
126	C₂×C₄×C₁₃	#124	1	2	yes	—
127	C₂×C₄×C₁₃	#125	1	4	yes	—
128	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
129	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
130	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
131	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
132	C₈×C₁₃	—	4	4	no	—
133	#1055	—	2	8	no	—
134	#1055	—	4	4	no	—
135	C₂²×M₁₆	—	13	2	no	—
136	C₂²×M₁₆	—	13	2	no	—
137	C₂²×M₁₆	C₂×D₁₃	1	2	yes	—
138	C₂²×C₄²	C₂×D₁₃	1	1	yes	characteristic
139	#423	—	2	2	no	—
140	#423	—	13	2	no	—
141	#423	—	13	2	no	—
142	#423	—	26	2	no	—
143	#423	C₂×D₁₃	1	2	yes	—
144	#423	C₂×D₁₃	1	2	yes	—
145	C₂×C₄⋊C₈	—	2	2	no	—
146	C₂×C₄⋊C₈	—	26	2	no	—
147	C₄×M₁₆	—	13	2	no	—
148	C₄×M₁₆	—	13	2	no	—
149	C₄×M₁₆	—	13	2	no	—
150	C₄×M₁₆	—	13	2	no	—
151	C₂×#97	—	13	2	no	—
152	C₂×C₄×C₈	—	13	2	no	—
153	#341	—	13	16	no	—
154	C₂²×C₁₃	—	2	2	no	—
155	C₂²×C₁₃	—	2	2	no	—
156	C₂²×C₁₃	—	2	2	no	—
157	C₂²×C₁₃	—	4	4	no	—
158	C₂²×C₁₃	#382	1	2	yes	—
159	C₂²×C₁₃	#382	1	2	yes	—
160	C₂²×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
161	C₂²×C₁₃	C₂×#125	1	2	yes	—
162	C₄×C₁₃	—	2	2	no	—
163	C₄×C₁₃	—	2	2	no	—
164	C₄×C₁₃	—	2	2	no	—
165	C₄×C₁₃	#382	1	2	yes	—
166	C₄×C₁₃	#382	1	2	yes	—
167	C₄×C₁₃	#430	1	4	yes	—
168	C₄×C₁₃	C₂×#117	1	2	yes	—
169	C₄×C₁₃	C₂×#124	1	1	yes	characteristic
170	C₄×C₁₃	C₂×#124	1	1	yes	characteristic
171	C₂³×C₄	Dic₂₆	1	2	yes	—
172	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
173	C₂×M₁₆	—	2	2	no	—
174	C₂×M₁₆	—	2	8	no	—
175	C₂×M₁₆	—	13	4	no	—
176	C₂×M₁₆	—	13	4	no	—
177	C₂×M₁₆	—	13	4	no	—
178	C₂×M₁₆	—	26	4	no	—
179	C₂×M₁₆	—	26	4	no	—
180	C₂×M₁₆	—	26	2	no	—
181	C₂×M₁₆	—	26	8	no	—
182	C₂×M₁₆	C₂²×D₁₃	1	4	yes	—
183	C₂²×C₈	—	2	2	no	—
184	C₂²×C₈	—	13	4	no	—
185	C₂²×C₈	—	26	2	no	—
186	C₂×C₄²	Dic₂₆	1	2	yes	—
187	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
188	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
189	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
190	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
191	C₂×C₄²	C₂²⋊D₁₃	1	2	yes	—
192	C₂×C₄²	C₂²⋊D₁₃	1	2	yes	—
193	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
194	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
195	C₄⋊C₈	—	2	4	no	—
196	C₄⋊C₈	—	2	4	no	—
197	C₄⋊C₈	—	26	4	no	—
198	C₄⋊C₈	—	26	4	no	—
199	#97	—	13	4	no	—
200	#97	—	13	4	no	—
201	C₄×C₈	—	13	4	no	—
202	C₄×C₈	—	13	4	no	—
203	C₂×C₁₃	—	2	2	no	—
204	C₂×C₁₃	—	2	2	no	—
205	C₂×C₁₃	—	2	2	no	—
206	C₂×C₁₃	C₂×#382	1	1	yes	characteristic
207	C₂×C₁₃	C₂×#430	1	2	yes	—
208	C₂⁴	C₂×Dic₂₆	1	1	yes	characteristic
209	C₂²×C₄	—	4	4	no	—
210	C₂²×C₄	#6101	1	2	yes	—
211	C₂²×C₄	#6101	1	2	yes	—
212	C₂²×C₄	#6101	1	2	yes	—
213	C₂²×C₄	#6101	1	2	yes	—
214	C₂²×C₄	C₂×Dic₂₆	1	2	yes	—
215	C₂²×C₄	C₂×Dic₂₆	1	2	yes	—
216	C₂²×C₄	C₂×Dic₂₆	1	1	yes	characteristic
217	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
218	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
219	C₂²×C₄	#6129	1	4	yes	—
220	C₂²×C₄	#6131	1	4	yes	—
221	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
222	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
223	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
224	M₁₆	—	2	8	no	—
225	M₁₆	—	4	4	no	—
226	M₁₆	—	26	8	no	—
227	M₁₆	—	26	8	no	—
228	M₁₆	—	26	8	no	—
229	M₁₆	—	52	4	no	—
230	C₂×C₈	—	2	4	no	—
231	C₂×C₈	—	4	4	no	—
232	C₂×C₈	—	13	8	no	—
233	C₂×C₈	—	26	4	no	—
234	C₂×C₈	—	26	4	no	—
235	C₂×C₈	—	26	4	no	—
236	C₂×C₈	—	52	4	no	—
237	C₄²	#6101	1	2	yes	—
238	C₄²	#6101	1	2	yes	—
239	C₄²	#6101	1	2	yes	—
240	C₄²	#6101	1	2	yes	—
241	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
242	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
243	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
244	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
245	C₄²	C₂×C₂²⋊D₁₃	1	2	yes	—
246	C₄²	C₂×C₂²⋊D₁₃	1	2	yes	—
247	C₁₃	#20769	1	1	yes	characteristic, complemented, Sylow
248	C₂³	—	4	2	no	—
249	C₂³	#88144	1	4	yes	—
250	C₂³	C₂×#6101	1	1	yes	characteristic
251	C₂³	C₂×#6101	1	1	yes	characteristic
252	C₂³	C₂²×Dic₂₆	1	1	yes	characteristic
253	C₂×C₄	—	2	4	no	—
254	C₂×C₄	—	2	4	no	—
255	C₂×C₄	—	2	4	no	—
256	C₂×C₄	—	4	2	no	—
257	C₂×C₄	—	4	2	no	—
258	C₂×C₄	—	4	2	no	—
259	C₂×C₄	#88144	1	4	yes	—
260	C₂×C₄	#88146	1	4	yes	—
261	C₂×C₄	#88146	1	4	yes	—
262	C₂×C₄	C₂×#6101	1	2	yes	—
263	C₂×C₄	C₂×#6101	1	2	yes	—
264	C₂×C₄	C₂×#6101	1	2	yes	—
265	C₂×C₄	C₂×#6101	1	2	yes	—
266	C₂×C₄	C₂×#6101	1	1	yes	characteristic
267	C₂×C₄	C₂×#6101	1	1	yes	characteristic
268	C₂×C₄	#88190	1	2	yes	—
269	C₂×C₄	#88190	1	2	yes	—
270	C₂×C₄	#88201	1	2	yes	—
271	C₂×C₄	#88201	1	2	yes	—
272	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
273	C₂×C₄	C₂×#6129	1	2	yes	—
274	C₂×C₄	C₂×#6131	1	2	yes	—
275	C₂×C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
276	C₈	—	4	4	no	—
277	C₈	—	26	8	no	—
278	C₈	—	52	4	no	—
279	C₂²	—	2	2	no	—
280	C₂²	—	2	2	no	—
281	C₂²	—	2	2	no	—
282	C₂²	—	4	4	no	—
283	C₂²	#11718638	1	2	yes	—
284	C₂²	#11718638	1	2	yes	—
285	C₂²	C₂×#88144	1	2	yes	—
286	C₂²	C₂²×#6101	1	1	yes	characteristic
287	C₄	—	2	2	no	—
288	C₄	—	2	2	no	—
289	C₄	—	2	2	no	—
290	C₄	#11718638	1	2	yes	—
291	C₄	#11718638	1	2	yes	—
292	C₄	#11718785	1	4	yes	—
293	C₄	C₂×#88146	1	2	yes	—
294	C₄	C₂×#88190	1	1	yes	characteristic
295	C₄	C₂×#88201	1	1	yes	characteristic
296	C₂	—	2	2	no	—
297	C₂	—	2	2	no	—
298	C₂	—	2	2	no	—
299	C₂	C₂×#11718638	1	1	yes	characteristic
300	C₂	C₂×#11718785	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊D₁₃
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₁₃
Fitting	C₂×C₁₃×#423	C₂
socle	C₂²×C₁₃	C₂²×C₄⋊C₄
prime radical	C₂⁴×C₁₃	C₂×Q₈
cosocle	C₂²×C₄×C₁₃	C₂⁴
hypercenter	C₂×#423	D₁₃
hypocenter	C₁₃	#20769
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ C₂×#423

Complements

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

N	H ≅ G/N	type
3. #13114976	C₂	semidirect
4. #13114973	C₂	semidirect
247. C₁₃	#20769	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718585	2
C₂×C₄×#6099	2
#13114976	2
#13114973	2
#13114562	2
C₂×C₁₃×#423	2
#20769	13