Subgroups — #16807427

Order 3328.

484 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13131253	C₂	1	2	yes	maximal, complemented
2	#13131221	C₂	1	2	yes	maximal, complemented
3	#13131219	C₂	1	2	yes	maximal, complemented
4	#13131217	C₂	1	2	yes	maximal, complemented
5	#13131204	C₂	1	2	yes	maximal, complemented
6	#13131197	C₂	1	1	yes	maximal, characteristic, complemented
7	#13131172	C₂	1	2	yes	maximal, complemented
8	#13131168	C₂	1	1	yes	maximal, characteristic, complemented
9	#13123805	C₂	1	1	yes	maximal, characteristic
10	#11719184	C₄	1	1	yes	characteristic
11	C₂×#88106	C₂²	1	1	yes	characteristic
12	C₂×#88106	C₂²	1	1	yes	characteristic
13	C₂×#88098	C₂²	1	2	yes	—
14	C₂×D₁₃⋊C₁₆	C₂²	1	2	yes	—
15	C₁₃×#432	C₄	1	1	yes	characteristic
16	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
17	#11718499	C₄	1	1	yes	characteristic
18	#11718495	C₂²	1	1	yes	characteristic
19	#11718492	C₄	1	1	yes	characteristic
20	#11718491	C₂²	1	1	yes	characteristic
21	#11718391	C₄	1	2	yes	—
22	#11718386	C₂²	1	2	yes	—
23	#11718370	C₄	1	2	yes	—
24	#11718369	C₂²	1	2	yes	—
25	C₂²⋊#6092	C₂²	1	2	yes	—
26	#11718260	C₂²	1	2	yes	—
27	#11718260	C₂²	1	2	yes	—
28	#11718259	C₂²	1	2	yes	—
29	#11718259	C₂²	1	2	yes	—
30	#11718258	C₂²	1	2	yes	—
31	#11718257	C₂²	1	2	yes	—
32	#11718231	C₂²	1	2	yes	—
33	C₄⋊#6092	C₂²	1	2	yes	—
34	C₄⋊#6092	C₂²	1	2	yes	—
35	C₄⋊#6092	C₂²	1	1	yes	characteristic
36	C₄⋊#6092	C₂²	1	1	yes	characteristic
37	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
38	#88184	C₈	1	1	yes	characteristic
39	#88183	C₈	1	2	yes	—
40	C₄×C₂²⋊D₁₃	C₂×C₄	1	2	yes	—
41	#88178	C₈	1	1	yes	characteristic
42	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
43	C₂²×#1055	C₂³	1	1	yes	characteristic
44	C₂×#6094	C₂×C₄	1	2	yes	—
45	C₂×C₈×D₁₃	C₂³	1	2	yes	—
46	#88130	C₈	1	2	yes	—
47	#88129	C₈	1	1	yes	characteristic
48	C₄×D₅₂	C₂×C₄	1	1	yes	characteristic
49	#88126	C₂×C₄	1	2	yes	—
50	#88124	C₈	1	1	yes	characteristic
51	C₄×Dic₂₆	C₂×C₄	1	1	yes	characteristic
52	#88106	—	4	1	no	—
53	#88106	—	4	1	no	—
54	C₂×#6092	C₂³	1	2	yes	—
55	C₂×#6092	C₂³	1	2	yes	—
56	C₂×#6092	C₂³	1	2	yes	—
57	C₂×#6092	C₂³	1	2	yes	—
58	#88098	—	4	2	no	—
59	D₁₃⋊C₁₆	—	4	2	no	—
60	C₁₃×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
61	C₁₃×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
62	C₁₃×C₂²⋊C₈	C₂×C₄	1	2	yes	—
63	C₁₃×#97	C₂×C₄	1	2	yes	—
64	#88060	C₂×C₄	1	2	yes	—
65	#88060	C₂×C₄	1	2	yes	—
66	#88055	C₂×C₄	1	2	yes	—
67	#88054	C₂×C₄	1	2	yes	—
68	#88054	C₂×C₄	1	2	yes	—
69	C₈×Dic₁₃	C₂³	1	2	yes	—
70	C₄×#1055	C₂×C₄	1	1	yes	characteristic
71	C₄×#1055	C₂×C₄	1	1	yes	characteristic
72	C₄×#1055	C₂³	1	1	yes	characteristic
73	C₄×#1055	C₂³	1	1	yes	characteristic
74	#33509	—	13	1	no	maximal, Sylow
75	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
76	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
77	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₂²⋊D₁₃	C₂×C₈	1	2	yes	—
79	C₂×D₅₂	C₂×C₈	1	1	yes	characteristic
80	C₂×C₄×D₁₃	C₂²×C₄	1	2	yes	—
81	C₂×Dic₂₆	C₂×C₈	1	1	yes	characteristic
82	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
83	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
84	C₄²×C₁₃	C₂×C₈	1	1	yes	characteristic
85	C₄²×C₁₃	C₂×C₈	1	1	yes	characteristic
86	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
87	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
88	C₂²⋊Dic₁₃	C₂×C₈	1	2	yes	—
89	#6103	C₂×C₈	1	2	yes	—
90	#6103	C₂×C₈	1	2	yes	—
91	#6103	C₂×C₈	1	2	yes	—
92	#6103	C₂×C₈	1	2	yes	—
93	C₄⋊Dic₁₃	C₂×C₈	1	1	yes	characteristic
94	C₄⋊Dic₁₃	C₂×C₈	1	1	yes	characteristic
95	#6101	C₂×C₈	1	2	yes	—
96	#6101	C₂×C₈	1	2	yes	—
97	#6101	C₂×C₈	1	2	yes	—
98	#6101	C₂×C₈	1	2	yes	—
99	C₄×Dic₁₃	C₂²×C₄	1	2	yes	—
100	C₂×#1055	—	2	1	no	—
101	C₂×#1055	—	2	1	no	—
102	C₂×#1055	—	4	1	no	—
103	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
104	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
105	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
106	C₂×#1055	C₂⁴	1	1	yes	characteristic
107	#6094	—	4	2	no	—
108	C₈×D₁₃	—	2	2	no	—
109	C₈×D₁₃	—	2	2	no	—
110	#6092	—	2	2	no	—
111	#6092	—	2	2	no	—
112	#6092	—	2	2	no	—
113	#6092	—	2	2	no	—
114	#2966	—	13	1	no	—
115	#2183	—	13	2	no	—
116	#2169	—	13	2	no	—
117	#2168	—	13	2	no	—
118	#2168	—	13	2	no	—
119	#2164	—	13	2	no	—
120	#2163	—	13	1	no	—
121	#2151	—	13	2	no	—
122	#2150	—	13	1	no	—
123	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
124	C₂²×D₁₃	C₂²×C₈	1	2	yes	—
125	C₂×C₄×C₁₃	—	2	1	no	—
126	C₂×C₄×C₁₃	—	2	1	no	—
127	C₂×C₄×C₁₃	—	4	1	no	—
128	C₂×C₄×C₁₃	—	4	1	no	—
129	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
130	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
131	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
132	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
133	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
134	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
135	C₂²⋊D₁₃	—	4	2	no	—
136	C₂×Dic₁₃	C₂²×C₈	1	2	yes	—
137	C₂×Dic₁₃	C₂²×C₈	1	2	yes	—
138	C₂×Dic₁₃	C₂²×C₈	1	2	yes	—
139	D₅₂	—	4	1	no	—
140	C₄×D₁₃	—	2	2	no	—
141	C₄×D₁₃	—	2	2	no	—
142	Dic₂₆	—	4	1	no	—
143	C₈×C₁₃	—	2	2	no	—
144	C₈×C₁₃	—	2	2	no	—
145	#1055	—	2	1	no	—
146	#1055	—	2	1	no	—
147	#1055	—	2	1	no	—
148	#1055	H(5,𝔽₂)	1	1	yes	characteristic
149	#1055	2₋¹⁺⁴	1	1	yes	characteristic
150	#529	—	13	1	no	—
151	C₂×M₃₂	—	13	1	no	—
152	C₂×M₃₂	—	13	1	no	—
153	C₂×M₃₂	—	13	2	no	—
154	C₂²×C₁₆	—	13	2	no	—
155	#446	—	13	1	no	—
156	C₈×Q₈	—	13	1	no	—
157	#436	—	13	1	no	—
158	#435	—	13	2	no	—
159	C₈×D₄	—	13	1	no	—
160	C₈×D₄	—	13	2	no	—
161	#433	—	13	2	no	—
162	#432	#261	1	1	yes	characteristic
163	#405	—	13	2	no	—
164	C₂×C₄×C₈	—	13	1	no	—
165	C₄⋊C₁₆	—	13	2	no	—
166	C₄⋊C₁₆	—	13	2	no	—
167	C₄⋊C₁₆	—	13	1	no	—
168	C₄⋊C₁₆	—	13	1	no	—
169	C₄⋊C₁₆	—	13	2	no	—
170	C₄⋊C₁₆	—	13	2	no	—
171	C₂²⋊C₁₆	—	13	2	no	—
172	C₂²⋊C₁₆	—	13	2	no	—
173	C₂²⋊C₁₆	—	13	2	no	—
174	#346	—	13	2	no	—
175	#346	—	13	2	no	—
176	C₄×C₁₆	—	13	2	no	—
177	C₂²×C₁₃	—	2	1	no	—
178	C₂²×C₁₃	—	2	1	no	—
179	C₂²×C₁₃	—	2	1	no	—
180	C₂²×C₁₃	C₂³×C₈	1	1	yes	characteristic
181	C₂×D₁₃	—	2	2	no	—
182	C₂×D₁₃	—	2	2	no	—
183	C₂×D₁₃	—	2	2	no	—
184	C₄×C₁₃	—	2	1	no	—
185	C₄×C₁₃	—	2	1	no	—
186	C₄×C₁₃	—	2	1	no	—
187	C₄×C₁₃	—	2	1	no	—
188	C₄×C₁₃	—	2	1	no	—
189	C₄×C₁₃	#520	1	1	yes	characteristic
190	C₄×C₁₃	#568	1	1	yes	characteristic
191	Dic₁₃	—	2	2	no	—
192	Dic₁₃	—	2	2	no	—
193	Dic₁₃	—	2	2	no	—
194	C₂×M₁₆	—	13	2	no	—
195	C₂²×C₈	—	13	1	no	—
196	C₂²×C₈	—	13	1	no	—
197	C₂²×C₈	—	13	2	no	—
198	#126	—	13	2	no	—
199	#125	—	13	1	no	—
200	#124	—	13	1	no	—
201	#123	—	13	2	no	—
202	C₂²⋊Q₈	—	13	1	no	—
203	#121	—	13	1	no	—
204	C₄×Q₈	—	13	1	no	—
205	C₄×D₄	—	13	2	no	—
206	C₄×D₄	—	13	1	no	—
207	#117	—	13	2	no	—
208	C₂×C₄²	C₂×#261	1	1	yes	characteristic
209	M₃₂	—	52	2	no	—
210	M₃₂	—	52	1	no	—
211	M₃₂	—	52	1	no	—
212	C₂×C₁₆	—	13	2	no	—
213	C₂×C₁₆	—	13	2	no	—
214	C₂×C₁₆	—	13	2	no	—
215	C₂×C₁₆	—	13	2	no	—
216	C₂×C₁₆	—	52	2	no	—
217	C₄⋊C₈	—	13	2	no	—
218	C₄⋊C₈	—	13	2	no	—
219	C₄⋊C₈	C₂×#261	1	1	yes	characteristic
220	C₄⋊C₈	C₂×#261	1	1	yes	characteristic
221	C₂²⋊C₈	—	13	2	no	—
222	C₂²⋊C₈	—	13	2	no	—
223	C₂²⋊C₈	C₂×#261	1	2	yes	—
224	#97	—	13	2	no	—
225	#97	C₂×#261	1	2	yes	—
226	C₄×C₈	—	13	2	no	—
227	C₄×C₈	—	13	1	no	—
228	C₄×C₈	—	13	1	no	—
229	C₄×C₈	—	13	1	no	—
230	C₄×C₈	—	13	1	no	—
231	C₂×C₁₃	—	4	1	no	—
232	C₂×C₁₃	#2988	1	1	yes	characteristic
233	C₂×C₁₃	#3407	1	1	yes	characteristic
234	C₂×C₁₃	#3407	1	1	yes	characteristic
235	D₁₃	—	4	2	no	—
236	C₂×Q₈	—	13	1	no	—
237	C₂×D₄	—	13	2	no	—
238	C₂×D₄	—	13	1	no	—
239	C₂²×C₄	—	13	2	no	—
240	C₂²×C₄	D₁₃⋊C₈	1	1	yes	characteristic
241	C₂²×C₄	D₁₃⋊C₈	1	1	yes	characteristic
242	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
243	M₁₆	—	52	2	no	—
244	C₂×C₈	—	13	1	no	—
245	C₂×C₈	—	13	1	no	—
246	C₂×C₈	—	13	1	no	—
247	C₂×C₈	—	13	1	no	—
248	C₂×C₈	—	26	2	no	—
249	C₂×C₈	—	26	2	no	—
250	C₂×C₈	—	26	1	no	—
251	C₂×C₈	—	26	1	no	—
252	C₂×C₈	—	52	1	no	—
253	C₂×C₈	C₂²×#261	1	2	yes	—
254	C₂×C₈	C₂²×#261	1	2	yes	—
255	C₄⋊C₄	—	13	2	no	—
256	C₄⋊C₄	—	13	2	no	—
257	C₄⋊C₄	—	13	1	no	—
258	C₄⋊C₄	—	13	2	no	—
259	C₄⋊C₄	—	13	1	no	—
260	C₄⋊C₄	—	13	2	no	—
261	C₂²⋊C₄	—	13	2	no	—
262	C₂²⋊C₄	—	13	2	no	—
263	C₂²⋊C₄	—	13	2	no	—
264	C₂²⋊C₄	—	13	2	no	—
265	C₂²⋊C₄	—	13	2	no	—
266	C₄²	—	13	2	no	—
267	C₄²	D₁₃⋊C₈	1	1	yes	characteristic
268	C₄²	D₁₃⋊C₈	1	1	yes	characteristic
269	C₄²	C₂²×#261	1	1	yes	characteristic
270	C₄²	C₂²×#261	1	1	yes	characteristic
271	C₁₆	—	26	2	no	—
272	C₁₆	—	26	2	no	—
273	C₁₆	—	26	2	no	—
274	C₁₆	—	26	2	no	—
275	C₁₃	#33509	1	1	yes	characteristic, complemented, Sylow
276	C₂³	—	13	2	no	—
277	C₂³	C₂×D₁₃⋊C₈	1	1	yes	characteristic
278	Q₈	—	52	1	no	—
279	D₄	—	52	1	no	—
280	D₄	—	52	2	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	4	1	no	—
284	C₂×C₄	—	4	1	no	—
285	C₂×C₄	—	13	2	no	—
286	C₂×C₄	—	13	2	no	—
287	C₂×C₄	—	13	2	no	—
288	C₂×C₄	—	26	2	no	—
289	C₂×C₄	—	26	2	no	—
290	C₂×C₄	C₂×D₁₃⋊C₈	1	1	yes	characteristic
291	C₂×C₄	C₂×D₁₃⋊C₈	1	1	yes	characteristic
292	C₂×C₄	C₂×D₁₃⋊C₈	1	1	yes	characteristic
293	C₂×C₄	C₂×D₁₃⋊C₈	1	1	yes	characteristic
294	C₂×C₄	C₂×D₁₃⋊C₈	1	1	yes	characteristic
295	C₂×C₄	C₂³×#261	1	1	yes	characteristic
296	C₈	—	2	2	no	—
297	C₈	—	2	2	no	—
298	C₈	—	13	1	no	—
299	C₈	—	13	1	no	—
300	C₈	—	26	1	no	—
301	C₈	—	26	1	no	—
302	C₈	—	26	1	no	—
303	C₂²	—	2	1	no	—
304	C₂²	—	2	1	no	—
305	C₂²	—	2	1	no	—
306	C₂²	—	26	2	no	—
307	C₂²	—	26	2	no	—
308	C₂²	—	26	2	no	—
309	C₂²	C₂²×D₁₃⋊C₈	1	1	yes	characteristic
310	C₄	—	2	1	no	—
311	C₄	—	2	1	no	—
312	C₄	—	2	1	no	—
313	C₄	—	2	1	no	—
314	C₄	—	2	1	no	—
315	C₄	—	26	2	no	—
316	C₄	—	26	2	no	—
317	C₄	—	26	2	no	—
318	C₄	#11719629	1	1	yes	characteristic
319	C₄	#11719631	1	1	yes	characteristic
320	C₂	—	4	1	no	—
321	C₂	—	52	2	no	—
322	C₂	#13131753	1	1	yes	characteristic
323	C₂	#13131805	1	1	yes	characteristic
324	C₂	#13131805	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×#261
commutator, socle	C₂²×C₁₃	C₂³×C₈
Fitting	C₁₃×#432	C₄
prime radical	#88184	C₈
cosocle	C₂×#1055	C₂⁴
hypercenter	#432	#261
hypocenter	C₁₃	#33509
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ #432

Complements

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

N	H ≅ G/N	type
1. #13131253	C₂	semidirect
2. #13131221	C₂	semidirect
3. #13131219	C₂	semidirect
4. #13131217	C₂	semidirect
5. #13131204	C₂	semidirect
6. #13131197	C₂	semidirect
7. #13131172	C₂	semidirect
8. #13131168	C₂	semidirect
275. C₁₃	#33509	semidirect

Maximal subgroups

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

subgroup	index
#13131253	2
#13131221	2
#13131219	2
#13131217	2
#13131204	2
#13131197	2
#13131172	2
#13131168	2
#13123805	2
#33509	13