Subgroups — #16807308

Order 3328.

552 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13131251	C₂	1	2	yes	maximal, complemented
2	#13131219	C₂	1	4	yes	maximal, complemented
3	#13131215	C₂	1	2	yes	maximal, complemented
4	#13131200	C₂	1	1	yes	maximal, characteristic, complemented
5	#13131200	C₂	1	2	yes	maximal, complemented
6	C₄×#88098	C₂	1	1	yes	maximal, characteristic, complemented
7	#13131037	C₂	1	2	yes	maximal, complemented
8	C₈×#6131	C₂	1	1	yes	maximal, characteristic
9	C₄×#6131	C₄	1	1	yes	characteristic
10	C₂×#88098	C₂²	1	2	yes	—
11	C₂×D₁₃⋊C₁₆	C₂²	1	4	yes	—
12	C₈×Q₈×C₁₃	C₄	1	1	yes	characteristic
13	Q₈×#1055	C₂²	1	1	yes	characteristic
14	#11718495	C₂²	1	1	yes	characteristic
15	#11718495	C₂²	1	2	yes	—
16	#11718494	C₄	1	1	yes	characteristic
17	#11718494	C₄	1	2	yes	—
18	C₈×D₅₂	C₄	1	2	yes	—
19	C₈×D₅₂	C₄	1	1	yes	characteristic
20	C₄×C₈×D₁₃	C₂²	1	2	yes	—
21	C₄×C₈×D₁₃	C₂²	1	1	yes	characteristic
22	#11718259	C₂²	1	4	yes	—
23	#11718259	C₂²	1	2	yes	—
24	#11718258	C₂²	1	1	yes	characteristic
25	#11718258	C₂²	1	2	yes	—
26	#11718258	C₂²	1	1	yes	characteristic
27	#11718231	C₂²	1	4	yes	—
28	C₄⋊#6092	C₂²	1	2	yes	—
29	C₄⋊#6092	C₂²	1	4	yes	—
30	C₄×#6092	C₂²	1	2	yes	—
31	C₂×#6131	C₈	1	1	yes	characteristic
32	C₄×Q₈×C₁₃	C₂×C₄	1	1	yes	characteristic
33	Q₈×Dic₁₃	C₈	1	1	yes	characteristic
34	C₂×C₈×D₁₃	C₂×C₄	1	2	yes	—
35	C₂×C₈×D₁₃	C₂×C₄	1	1	yes	characteristic
36	C₂×C₈×D₁₃	C₂³	1	2	yes	—
37	C₂×C₈×D₁₃	C₂³	1	1	yes	characteristic
38	#88147	C₈	1	2	yes	—
39	#88147	C₈	1	1	yes	characteristic
40	#88146	C₈	1	2	yes	—
41	#88146	C₈	1	1	yes	characteristic
42	C₄×D₅₂	C₂×C₄	1	2	yes	—
43	C₄×D₅₂	C₂×C₄	1	1	yes	characteristic
44	C₄²×D₁₃	C₂×C₄	1	2	yes	—
45	C₄²×D₁₃	C₂×C₄	1	1	yes	characteristic
46	C₂×#6092	C₂³	1	2	yes	—
47	C₂×#6092	C₂³	1	4	yes	—
48	C₂×#6092	C₂³	1	2	yes	—
49	#88098	—	2	4	no	—
50	D₁₃⋊C₁₆	—	2	8	no	—
51	C₁₃×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
52	C₁₃×C₄⋊C₈	C₂×C₄	1	2	yes	—
53	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
54	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
55	#88060	C₂×C₄	1	2	yes	—
56	#88060	C₂×C₄	1	2	yes	—
57	#88060	C₂×C₄	1	2	yes	—
58	C₈×Dic₁₃	C₂×C₄	1	2	yes	—
59	C₈×Dic₁₃	C₂×C₄	1	1	yes	characteristic
60	C₈×Dic₁₃	C₂³	1	1	yes	characteristic
61	C₄⋊#1055	C₂×C₄	1	2	yes	—
62	C₄⋊#1055	C₂×C₄	1	1	yes	characteristic
63	C₄×#1055	C₂³	1	1	yes	characteristic
64	C₄×#1055	C₂³	1	2	yes	—
65	#33484	—	13	1	no	maximal, Sylow
66	C₂×Q₈×C₁₃	C₂×C₈	1	1	yes	characteristic
67	#6131	—	2	4	no	—
68	C₂×D₅₂	C₂×C₈	1	2	yes	—
69	C₂×D₅₂	C₂×C₈	1	1	yes	characteristic
70	C₂×C₄×D₁₃	C₂×C₈	1	2	yes	—
71	C₂×C₄×D₁₃	C₂×C₈	1	1	yes	characteristic
72	C₂×C₄×D₁₃	C₂×C₈	1	1	yes	characteristic
73	C₂×C₄×D₁₃	C₂×C₈	1	2	yes	—
74	C₂×C₄×D₁₃	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₄×D₁₃	C₂²×C₄	1	2	yes	—
76	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
77	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
79	C₁₃×C₄⋊C₄	C₂×C₈	1	2	yes	—
80	C₁₃×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
81	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
82	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
83	#6103	C₂×C₈	1	2	yes	—
84	#6103	C₂×C₈	1	2	yes	—
85	#6103	C₂×C₈	1	2	yes	—
86	C₄⋊Dic₁₃	C₂×C₈	1	2	yes	—
87	C₄⋊Dic₁₃	C₂×C₈	1	1	yes	characteristic
88	C₄×Dic₁₃	C₂×C₈	1	2	yes	—
89	C₄×Dic₁₃	C₂×C₈	1	1	yes	characteristic
90	C₄×Dic₁₃	C₂×C₈	1	2	yes	—
91	C₄×Dic₁₃	C₂×C₈	1	1	yes	characteristic
92	C₄×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
93	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
94	C₂×#1055	C₂²×C₄	1	2	yes	—
95	C₂×#1055	C₂⁴	1	1	yes	characteristic
96	C₈×D₁₃	—	2	4	no	—
97	C₈×D₁₃	—	2	2	no	—
98	C₈×D₁₃	—	2	4	no	—
99	C₈×D₁₃	—	2	2	no	—
100	#6092	—	2	4	no	—
101	#6092	—	2	2	no	—
102	#6092	#41	1	4	yes	complemented
103	C₈×#41	—	13	1	no	—
104	#2183	—	13	2	no	—
105	#2168	—	13	4	no	—
106	D₄×C₁₆	—	13	2	no	—
107	#2163	—	13	1	no	—
108	#2163	—	13	2	no	—
109	C₄×M₃₂	—	13	1	no	—
110	C₂×#346	—	13	2	no	—
111	C₂²×D₁₃	C₂²×C₈	1	1	yes	characteristic
112	C₂²×D₁₃	C₂²×C₈	1	2	yes	—
113	Q₈×C₁₃	—	2	2	no	—
114	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
115	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
116	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
117	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
118	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
119	C₂×Dic₁₃	C₂²×C₈	1	1	yes	characteristic
120	C₂×Dic₁₃	C₂²×C₈	1	1	yes	characteristic
121	C₂×Dic₁₃	C₂²×C₈	1	2	yes	—
122	C₂×Dic₁₃	C₂²×C₈	1	1	yes	characteristic
123	D₅₂	—	2	4	no	—
124	D₅₂	M₃₂	1	4	yes	complemented
125	C₄×D₁₃	—	2	4	no	—
126	C₄×D₁₃	—	2	2	no	—
127	C₄×D₁₃	—	2	1	no	—
128	C₄×D₁₃	—	2	2	no	—
129	C₄×D₁₃	—	2	1	no	—
130	C₄×D₁₃	—	2	2	no	—
131	C₄×D₁₃	—	4	2	no	—
132	C₄×D₁₃	—	4	1	no	—
133	C₈×C₁₃	—	2	2	no	—
134	C₈×C₁₃	—	2	1	no	—
135	C₈×C₁₃	#131	1	2	yes	—
136	#1055	—	2	1	no	—
137	#1055	—	2	2	no	—
138	#1055	C₂×#41	1	2	yes	—
139	C₄×#41	—	13	1	no	—
140	C₂×M₃₂	—	13	2	no	—
141	C₂²×C₁₆	—	13	4	no	—
142	C₈×Q₈	—	13	1	no	—
143	C₈×Q₈	#261	1	1	yes	characteristic
144	C₈×D₄	—	13	1	no	—
145	C₈×D₄	—	13	2	no	—
146	C₈×D₄	—	13	2	no	—
147	C₈×D₄	—	13	1	no	—
148	#431	—	13	1	no	—
149	#431	—	13	2	no	—
150	C₂×C₄×C₈	—	13	2	no	—
151	C₂×C₄×C₈	—	13	1	no	—
152	C₄⋊C₁₆	—	13	2	no	—
153	C₄⋊C₁₆	—	13	4	no	—
154	C₂²⋊C₁₆	—	13	4	no	—
155	C₂²⋊C₁₆	—	13	2	no	—
156	#346	—	13	1	no	—
157	#346	—	13	4	no	—
158	#346	—	13	2	no	—
159	#346	—	13	1	no	—
160	C₄×C₁₆	—	13	2	no	—
161	C₂²×C₁₃	C₂³×C₈	1	1	yes	characteristic
162	C₂×D₁₃	—	2	2	no	—
163	C₂×D₁₃	—	2	2	no	—
164	C₂×D₁₃	—	2	1	no	—
165	C₂×D₁₃	—	2	1	no	—
166	C₂×D₁₃	—	2	1	no	—
167	C₂×D₁₃	C₂×M₃₂	1	4	yes	—
168	C₄×C₁₃	—	2	2	no	—
169	C₄×C₁₃	—	2	2	no	—
170	C₄×C₁₃	—	2	1	no	—
171	C₄×C₁₃	C₂×M₃₂	1	2	yes	—
172	C₄×C₁₃	C₄×#41	1	1	yes	characteristic
173	C₄×C₁₃	C₂×#131	1	1	yes	characteristic
174	Dic₁₃	—	2	1	no	—
175	Dic₁₃	—	2	1	no	—
176	Dic₁₃	—	2	2	no	—
177	Dic₁₃	—	2	1	no	—
178	C₂×#41	—	13	1	no	—
179	C₂²×C₈	—	13	2	no	—
180	C₂²×C₈	—	13	1	no	—
181	C₂²×C₈	—	13	2	no	—
182	C₂²×C₈	—	13	1	no	—
183	C₄×Q₈	—	13	1	no	—
184	C₄×Q₈	C₂×#261	1	1	yes	characteristic
185	C₄×D₄	—	13	1	no	—
186	C₄×D₄	—	13	2	no	—
187	C₄×D₄	—	13	1	no	—
188	C₄×D₄	—	13	2	no	—
189	#117	—	13	2	no	—
190	#117	—	13	1	no	—
191	C₂×C₄²	—	13	2	no	—
192	C₂×C₄²	—	13	1	no	—
193	M₃₂	—	26	4	no	—
194	C₂×C₁₆	—	13	2	no	—
195	C₂×C₁₆	—	13	2	no	—
196	C₂×C₁₆	—	13	4	no	—
197	C₂×C₁₆	—	26	8	no	—
198	C₄⋊C₈	—	13	2	no	—
199	C₄⋊C₈	—	13	1	no	—
200	C₄⋊C₈	C₂×#261	1	1	yes	characteristic
201	C₄⋊C₈	C₂×#261	1	2	yes	—
202	C₂²⋊C₈	—	13	2	no	—
203	C₂²⋊C₈	—	13	2	no	—
204	C₂²⋊C₈	—	13	2	no	—
205	C₄×C₈	—	13	1	no	—
206	C₄×C₈	—	13	2	no	—
207	C₄×C₈	—	13	1	no	—
208	C₄×C₈	—	13	1	no	—
209	C₄×C₈	—	13	2	no	—
210	C₄×C₈	C₂×#261	1	2	yes	—
211	C₄×C₈	C₂×#261	1	1	yes	characteristic
212	C₂×C₁₃	C₈×#41	1	1	yes	characteristic
213	C₂×C₁₃	C₂²×M₃₂	1	1	yes	characteristic
214	C₂×C₁₃	#3407	1	1	yes	characteristic
215	D₁₃	—	2	4	no	—
216	D₁₃	—	4	1	no	—
217	#41	—	26	4	no	—
218	C₂×Q₈	D₁₃⋊C₈	1	1	yes	characteristic
219	C₂×D₄	—	13	1	no	—
220	C₂×D₄	—	13	2	no	—
221	C₂²×C₄	—	13	2	no	—
222	C₂²×C₄	—	13	1	no	—
223	C₂²×C₄	—	13	2	no	—
224	C₂²×C₄	—	13	1	no	—
225	C₂²×C₄	—	13	2	no	—
226	C₂²×C₄	—	13	1	no	—
227	C₂×C₈	—	13	1	no	—
228	C₂×C₈	—	13	1	no	—
229	C₂×C₈	—	13	2	no	—
230	C₂×C₈	—	26	4	no	—
231	C₂×C₈	—	26	2	no	—
232	C₂×C₈	—	26	4	no	—
233	C₂×C₈	—	26	2	no	—
234	C₂×C₈	C₂²×#261	1	2	yes	—
235	C₂×C₈	C₂²×#261	1	1	yes	characteristic
236	C₂×C₈	C₂²×#261	1	1	yes	characteristic
237	C₄⋊C₄	—	13	2	no	—
238	C₄⋊C₄	—	13	1	no	—
239	C₄⋊C₄	D₁₃⋊C₈	1	2	yes	—
240	C₄⋊C₄	D₁₃⋊C₈	1	1	yes	characteristic
241	C₂²⋊C₄	—	13	2	no	—
242	C₂²⋊C₄	—	13	2	no	—
243	C₂²⋊C₄	—	13	2	no	—
244	C₄²	—	13	2	no	—
245	C₄²	—	13	1	no	—
246	C₄²	—	13	2	no	—
247	C₄²	—	13	1	no	—
248	C₄²	—	13	1	no	—
249	C₄²	C₂²×#261	1	2	yes	—
250	C₄²	C₂²×#261	1	1	yes	characteristic
251	C₁₆	—	13	4	no	—
252	C₁₆	—	26	4	no	—
253	C₁₆	—	26	2	no	—
254	C₁₃	#33484	1	1	yes	characteristic, complemented, Sylow
255	C₂³	—	13	2	no	—
256	C₂³	—	13	1	no	—
257	Q₈	—	2	2	no	—
258	D₄	—	13	4	no	—
259	D₄	—	26	4	no	—
260	C₂×C₄	—	13	1	no	—
261	C₂×C₄	—	13	1	no	—
262	C₂×C₄	—	13	2	no	—
263	C₂×C₄	—	13	1	no	—
264	C₂×C₄	—	26	4	no	—
265	C₂×C₄	—	26	2	no	—
266	C₂×C₄	—	26	2	no	—
267	C₂×C₄	—	26	1	no	—
268	C₂×C₄	—	26	2	no	—
269	C₂×C₄	—	26	1	no	—
270	C₂×C₄	—	52	2	no	—
271	C₂×C₄	—	52	1	no	—
272	C₂×C₄	C₂×D₁₃⋊C₈	1	2	yes	—
273	C₂×C₄	C₂×D₁₃⋊C₈	1	1	yes	characteristic
274	C₂×C₄	C₂×D₁₃⋊C₈	1	2	yes	—
275	C₂×C₄	C₂×D₁₃⋊C₈	1	1	yes	characteristic
276	C₂×C₄	C₂³×#261	1	1	yes	characteristic
277	C₈	—	2	2	no	—
278	C₈	—	2	1	no	—
279	C₈	—	13	2	no	—
280	C₈	—	26	1	no	—
281	C₈	—	26	2	no	—
282	C₈	#88240	1	2	yes	—
283	C₂²	—	13	4	no	—
284	C₂²	—	26	2	no	—
285	C₂²	—	26	2	no	—
286	C₂²	—	26	1	no	—
287	C₂²	—	26	1	no	—
288	C₂²	—	26	1	no	—
289	C₂²	C₂²×D₁₃⋊C₈	1	1	yes	characteristic
290	C₄	—	2	2	no	—
291	C₄	—	2	2	no	—
292	C₄	—	2	1	no	—
293	C₄	—	26	1	no	—
294	C₄	—	26	1	no	—
295	C₄	—	26	2	no	—
296	C₄	—	26	1	no	—
297	C₄	D₁₃⋊M₃₂	1	2	yes	—
298	C₄	C₂×#88240	1	1	yes	characteristic
299	C₄	#41×#261	1	1	yes	characteristic
300	C₂	—	26	4	no	—
301	C₂	—	52	1	no	—
302	C₂	#13131642	1	1	yes	characteristic
303	C₂	C₂×D₁₃⋊M₃₂	1	1	yes	characteristic
304	C₂	#13131804	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×#261
commutator, socle	C₂²×C₁₃	C₂³×C₈
Fitting	C₈×Q₈×C₁₃	C₄
prime radical	C₂×#6131	C₈
cosocle	C₂×#1055	C₂⁴
hypercenter	C₈×Q₈	#261
hypocenter	C₁₃	#33484
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₈×Q₈

Complements

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

N	H ≅ G/N	type
1. #13131251	C₂	semidirect
2. #13131219	C₂	semidirect
3. #13131215	C₂	semidirect
4. #13131200	C₂	semidirect
5. #13131200	C₂	semidirect
6. C₄×#88098	C₂	semidirect
7. #13131037	C₂	semidirect
102. #6092	#41	semidirect
124. D₅₂	M₃₂	semidirect
254. C₁₃	#33484	semidirect

Maximal subgroups

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

subgroup	index
#13131251	2
#13131219	2
#13131215	2
#13131200	2
C₄×#88098	2
#13131037	2
C₈×#6131	2
#33484	13