Subgroups — #16018350

Order 3328.

918 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718854	C₂	1	4	yes	maximal, complemented
2	C₂×#11718356	C₂	1	1	yes	maximal, characteristic, complemented
3	#13117103	C₂	1	4	yes	maximal, complemented
4	#13117093	C₂	1	2	yes	maximal, complemented
5	#13117085	C₂	1	2	yes	maximal, complemented
6	C₄×#88052	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₃×#3147	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₁₃×#135	—	2	1	no	—
9	C₂×C₁₃×#135	C₂²	1	2	yes	—
10	C₁₃×#532	—	2	1	no	—
11	C₂²×D₁₀₄	—	2	1	no	—
12	C₂²×D₁₀₄	C₂²	1	2	yes	—
13	C₂×#88128	—	2	1	no	—
14	C₁₃×#495	—	2	1	no	—
15	C₁₃×#495	C₂²	1	2	yes	complemented
16	C₁₃×#494	C₂²	1	1	yes	characteristic, complemented
17	C₁₃×#493	C₂²	1	1	yes	characteristic, complemented
18	C₁₃×#492	—	2	1	no	—
19	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
20	#11718854	—	2	16	no	—
21	C₂×#88068	C₂²	1	8	yes	complemented
22	C₂×#88066	C₂²	1	8	yes	complemented
23	C₂×#88052	C₂²	1	2	yes	—
24	#11718356	—	2	1	no	—
25	#11718356	—	2	1	no	—
26	#11718356	C₂²	1	2	yes	—
27	#11718356	C₂²	1	2	yes	—
28	#11718053	C₂²	1	2	yes	complemented
29	C₄×#6090	C₂²	1	2	yes	complemented
30	C₂×C₁₃×#41	—	2	2	no	—
31	C₂²×D₅₂	—	2	2	no	—
32	C₂²×D₅₂	—	4	2	no	—
33	C₁₃×#135	—	2	8	no	—
34	C₁₃×#135	—	4	4	no	—
35	C₂×C₁₃×Q₁₆	—	2	1	no	—
36	C₂×C₁₃×Q₁₆	C₂³	1	2	yes	—
37	C₂×C₁₃×SD₁₆	—	2	2	no	—
38	C₂×C₁₃×SD₁₆	—	2	2	no	—
39	C₂×C₁₃×D₈	—	2	1	no	—
40	C₂×C₁₃×D₈	C₂³	1	2	yes	—
41	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
42	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
43	C₁₃×C₄⋊Q₈	—	2	1	no	—
44	C₁₃×#127	—	2	1	no	—
45	C₁₃×#124	—	2	2	no	—
46	C₁₃×#121	—	4	2	no	—
47	C₄×D₄×C₁₃	—	4	2	no	—
48	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
49	C₂×D₁₀₄	—	2	8	no	—
50	C₂×D₁₀₄	—	2	2	no	—
51	C₂×D₁₀₄	—	2	2	no	—
52	C₂×D₁₀₄	—	2	2	no	—
53	C₂×D₁₀₄	—	4	4	no	—
54	C₂×D₁₀₄	C₂³	1	4	yes	—
55	#88128	—	2	2	no	—
56	#88128	—	2	2	no	—
57	#88128	—	4	2	no	—
58	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
59	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
60	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
61	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
62	#88068	—	2	16	no	—
63	#88066	—	2	16	no	—
64	#88052	D₄	1	8	yes	complemented
65	C₂×#6090	C₂³	1	4	yes	—
66	#33717	—	13	1	no	maximal, Sylow
67	C₂³×D₁₃	—	4	2	no	—
68	C₁₃×#41	—	4	4	no	—
69	C₁₃×#41	—	4	4	no	—
70	C₂×Q₈×C₁₃	—	2	2	no	—
71	C₂×D₄×C₁₃	—	2	2	no	—
72	C₂×D₄×C₁₃	—	4	2	no	—
73	C₂×D₄×C₁₃	—	4	2	no	—
74	C₂²×C₄×C₁₃	—	4	2	no	—
75	C₂²×C₄×C₁₃	D₈	1	2	yes	—
76	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
77	C₂×D₅₂	—	2	4	no	—
78	C₂×D₅₂	—	4	4	no	—
79	C₂×D₅₂	—	4	4	no	—
80	C₂×D₅₂	—	4	4	no	—
81	C₂×D₅₂	—	4	4	no	—
82	C₂×D₅₂	—	4	2	no	—
83	C₂×D₅₂	—	8	8	no	—
84	C₁₃×Q₁₆	—	2	4	no	—
85	C₁₃×Q₁₆	—	4	2	no	—
86	C₁₃×SD₁₆	—	4	4	no	—
87	C₁₃×SD₁₆	—	4	4	no	—
88	C₁₃×D₈	—	2	4	no	—
89	C₁₃×D₈	—	4	2	no	—
90	C₂×C₈×C₁₃	D₈	1	4	yes	—
91	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
92	C₂×C₈×C₁₃	C₂×D₄	1	4	yes	—
93	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
94	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
95	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
96	C₁₃×C₄⋊C₄	—	4	2	no	—
97	C₁₃×C₂²⋊C₄	—	4	4	no	—
98	C₄²×C₁₃	D₈	1	2	yes	—
99	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
100	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
101	D₁₀₄	—	2	8	no	—
102	D₁₀₄	—	4	4	no	—
103	D₁₀₄	—	4	4	no	—
104	D₁₀₄	—	4	4	no	—
105	#6090	C₂×D₄	1	8	yes	complemented
106	C₂×#509	—	13	4	no	—
107	#3147	D₁₃	1	1	yes	characteristic, complemented
108	C₂×#493	—	13	1	no	—
109	#2250	—	13	4	no	—
110	#2248	—	13	2	no	—
111	#2246	—	13	2	no	—
112	C₄×M₃₂	—	13	1	no	—
113	C₂³×C₁₃	—	4	2	no	—
114	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
115	C₂²×D₁₃	—	4	4	no	—
116	C₂²×D₁₃	—	8	8	no	—
117	C₂²×D₁₃	—	8	4	no	—
118	Q₈×C₁₃	—	4	2	no	—
119	Q₈×C₁₃	—	4	2	no	—
120	D₄×C₁₃	—	4	2	no	—
121	D₄×C₁₃	—	4	2	no	—
122	D₄×C₁₃	—	8	4	no	—
123	D₄×C₁₃	—	8	4	no	—
124	C₂×C₄×C₁₃	—	2	4	no	—
125	C₂×C₄×C₁₃	—	4	2	no	—
126	C₂×C₄×C₁₃	—	8	4	no	—
127	C₂×C₄×C₁₃	#127	1	2	yes	—
128	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
129	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
130	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
131	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
132	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
133	D₅₂	—	4	4	no	—
134	D₅₂	—	4	4	no	—
135	D₅₂	—	8	8	no	—
136	D₅₂	—	8	4	no	—
137	D₅₂	—	8	8	no	—
138	C₈×C₁₃	#127	1	2	yes	—
139	C₈×C₁₃	C₂×D₈	1	2	yes	—
140	C₈×C₁₃	C₂×D₈	1	2	yes	—
141	C₈×C₁₃	C₂²×D₄	1	2	yes	—
142	C₂×#135	—	2	1	no	—
143	C₂×#135	C₂×D₁₃	1	2	yes	—
144	C₂²×D₈	—	13	2	no	—
145	C₂²×D₈	—	26	1	no	—
146	#532	—	2	1	no	—
147	C₂×#127	—	26	1	no	—
148	#509	—	26	16	no	—
149	C₂×SD₃₂	—	13	8	no	—
150	C₂×D₁₆	—	13	8	no	—
151	C₂×M₃₂	—	13	2	no	—
152	#495	—	2	1	no	—
153	#495	C₂×D₁₃	1	2	yes	complemented
154	#494	C₂×D₁₃	1	1	yes	characteristic, complemented
155	#493	—	13	2	no	—
156	#493	—	13	2	no	—
157	#493	—	26	1	no	—
158	#493	—	26	1	no	—
159	#493	C₂×D₁₃	1	1	yes	characteristic, complemented
160	#492	—	2	1	no	—
161	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
162	#346	—	13	2	no	—
163	C₄×C₁₆	—	13	2	no	—
164	C₂²×C₁₃	—	2	2	no	—
165	C₂²×C₁₃	—	8	2	no	—
166	C₂²×C₁₃	—	8	4	no	—
167	C₂²×C₁₃	#493	1	2	yes	—
168	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
169	C₂×D₁₃	—	8	4	no	—
170	C₂×D₁₃	—	8	8	no	—
171	C₂×D₁₃	—	16	8	no	—
172	C₄×C₁₃	—	2	2	no	—
173	C₄×C₁₃	—	8	2	no	—
174	C₄×C₁₃	#493	1	2	yes	—
175	C₄×C₁₃	#509	1	4	yes	complemented
176	C₄×C₁₃	C₂×#127	1	1	yes	characteristic
177	C₄×C₁₃	C₂²×D₈	1	1	yes	characteristic
178	C₂×#41	—	2	2	no	—
179	C₂²×D₄	—	26	2	no	—
180	C₂²×D₄	—	52	2	no	—
181	#135	—	2	8	no	—
182	#135	—	4	4	no	—
183	C₂×Q₁₆	—	2	1	no	—
184	C₂×Q₁₆	C₂²×D₁₃	1	2	yes	—
185	C₂×SD₁₆	—	2	2	no	—
186	C₂×SD₁₆	—	2	2	no	—
187	C₂×D₈	—	2	1	no	—
188	C₂×D₈	—	13	4	no	—
189	C₂×D₈	—	26	8	no	—
190	C₂×D₈	—	26	2	no	—
191	C₂×D₈	—	26	2	no	—
192	C₂×D₈	—	26	2	no	—
193	C₂×D₈	—	52	4	no	—
194	C₂×D₈	C₂²×D₁₃	1	2	yes	—
195	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
196	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
197	C₄⋊Q₈	—	2	1	no	—
198	#127	—	2	1	no	—
199	#127	—	26	2	no	—
200	#127	—	26	2	no	—
201	#127	—	52	2	no	—
202	#124	—	2	2	no	—
203	#121	—	4	2	no	—
204	C₄×D₄	—	4	2	no	—
205	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
206	SD₃₂	—	26	16	no	—
207	D₁₆	—	26	16	no	—
208	M₃₂	—	13	8	no	—
209	C₂×C₁₆	—	13	4	no	—
210	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
211	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
212	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
213	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
214	C₂×C₁₃	—	2	2	no	—
215	C₂×C₁₃	—	16	2	no	—
216	C₂×C₁₃	C₂×#493	1	1	yes	characteristic
217	C₂×C₁₃	C₂×#509	1	2	yes	—
218	D₁₃	—	16	4	no	—
219	C₂⁴	—	52	2	no	—
220	#41	—	4	4	no	—
221	#41	—	4	4	no	—
222	C₂×Q₈	—	2	2	no	—
223	C₂×D₄	—	2	2	no	—
224	C₂×D₄	—	4	2	no	—
225	C₂×D₄	—	4	2	no	—
226	C₂×D₄	—	26	4	no	—
227	C₂×D₄	—	52	4	no	—
228	C₂×D₄	—	52	4	no	—
229	C₂×D₄	—	52	4	no	—
230	C₂×D₄	—	52	2	no	—
231	C₂×D₄	—	52	4	no	—
232	C₂×D₄	—	104	8	no	—
233	C₂²×C₄	—	4	2	no	—
234	C₂²×C₄	#6104	1	2	yes	—
235	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
236	Q₁₆	—	2	4	no	—
237	Q₁₆	—	4	2	no	—
238	SD₁₆	—	4	4	no	—
239	SD₁₆	—	4	4	no	—
240	D₈	—	2	4	no	—
241	D₈	—	4	2	no	—
242	D₈	—	26	8	no	—
243	D₈	—	52	4	no	—
244	D₈	—	52	4	no	—
245	D₈	—	52	4	no	—
246	C₂×C₈	#6104	1	4	yes	—
247	C₂×C₈	D₄×D₁₃	1	4	yes	—
248	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
249	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
250	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
251	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
252	C₄⋊C₄	—	4	2	no	—
253	C₂²⋊C₄	—	4	4	no	—
254	C₄²	#6104	1	2	yes	—
255	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
256	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
257	C₁₆	—	13	8	no	—
258	C₁₃	#33717	1	1	yes	characteristic, complemented, Sylow
259	C₂³	—	4	2	no	—
260	C₂³	—	52	4	no	—
261	C₂³	—	104	8	no	—
262	C₂³	—	104	4	no	—
263	C₂³	C₂×#6104	1	1	yes	characteristic
264	Q₈	—	4	2	no	—
265	Q₈	—	4	2	no	—
266	D₄	—	4	2	no	—
267	D₄	—	4	2	no	—
268	D₄	—	8	4	no	—
269	D₄	—	8	4	no	—
270	D₄	—	52	4	no	—
271	D₄	—	52	4	no	—
272	D₄	—	104	8	no	—
273	D₄	—	104	4	no	—
274	D₄	—	104	8	no	—
275	C₂×C₄	—	2	4	no	—
276	C₂×C₄	—	4	2	no	—
277	C₂×C₄	—	8	4	no	—
278	C₂×C₄	C₂×#6104	1	2	yes	—
279	C₂×C₄	C₂×#6104	1	2	yes	—
280	C₂×C₄	C₂×#6104	1	1	yes	characteristic
281	C₂×C₄	C₂×#6104	1	2	yes	—
282	C₂×C₄	#88194	1	2	yes	—
283	C₂×C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
284	C₈	C₂×#6104	1	2	yes	—
285	C₈	C₂×#6104	1	2	yes	—
286	C₈	#88194	1	2	yes	—
287	C₈	C₂×D₄×D₁₃	1	2	yes	—
288	C₂²	—	2	2	no	—
289	C₂²	—	8	4	no	—
290	C₂²	—	8	2	no	—
291	C₂²	—	104	8	no	—
292	C₂²	—	104	4	no	—
293	C₂²	—	208	8	no	—
294	C₂²	#11718734	1	2	yes	—
295	C₂²	C₂²×#6104	1	1	yes	characteristic
296	C₄	—	2	2	no	—
297	C₄	—	8	2	no	—
298	C₄	#11718734	1	2	yes	—
299	C₄	#11718854	1	4	yes	complemented
300	C₄	C₂²×#6104	1	1	yes	characteristic
301	C₄	C₂×#88194	1	1	yes	characteristic
302	C₂	—	2	2	no	—
303	C₂	—	16	2	no	—
304	C₂	—	208	4	no	—
305	C₂	C₂×#11718734	1	1	yes	characteristic
306	C₂	C₂×#11718854	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂²×#6104
commutator, cosocle	C₂×C₈×C₁₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₁₃×#3147	C₂
socle	C₂²×C₁₃	C₂²×D₈
hypercenter	#3147	D₁₃
hypocenter	C₁₃	#33717
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #3147

Complements

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

N	H ≅ G/N	type
1. C₂×#11718854	C₂	semidirect
2. C₂×#11718356	C₂	semidirect
3. #13117103	C₂	semidirect
4. #13117093	C₂	semidirect
5. #13117085	C₂	semidirect
6. C₄×#88052	C₂	semidirect
7. C₁₃×#3147	C₂	semidirect
15. C₁₃×#495	C₂²	semidirect
16. C₁₃×#494	C₂²	semidirect
17. C₁₃×#493	C₂²	semidirect
21. C₂×#88068	C₂²	semidirect
22. C₂×#88066	C₂²	semidirect
28. #11718053	C₂²	semidirect
29. C₄×#6090	C₂²	semidirect
64. #88052	D₄	semidirect
105. #6090	C₂×D₄	semidirect
107. #3147	D₁₃	semidirect
153. #495	C₂×D₁₃	semidirect
154. #494	C₂×D₁₃	semidirect
159. #493	C₂×D₁₃	semidirect
175. C₄×C₁₃	#509	semidirect
258. C₁₃	#33717	semidirect
299. C₄	#11718854	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718854	2
C₂×#11718356	2
#13117103	2
#13117093	2
#13117085	2
C₄×#88052	2
C₁₃×#3147	2
#33717	13