Subgroups — #16796347

Order 3328.

638 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13131637	C₂	1	2	yes	maximal
2	#13130475	C₂	1	2	yes	maximal, complemented
3	#13130448	C₂	1	4	yes	maximal, complemented
4	#13130424	C₂	1	2	yes	maximal, complemented
5	#13130393	C₂	1	1	yes	maximal, characteristic
6	#13130166	C₂	1	2	yes	maximal
7	C₈×#6122	C₂	1	1	yes	maximal, characteristic
8	C₂×#11718534	C₂	1	1	yes	maximal, characteristic
9	C₂²×Dic₅₂	C₄	1	1	yes	characteristic
10	C₂×#88141	C₂²	1	2	yes	—
11	#11719129	—	2	2	no	—
12	#11719129	C₂²	1	2	yes	—
13	#11719125	C₂²	1	1	yes	characteristic
14	C₄×#6122	—	2	2	no	—
15	C₄×#6122	C₂²	1	1	yes	characteristic
16	#11719080	C₂²	1	4	yes	—
17	#11719078	—	2	4	no	—
18	#11719074	C₂²	1	4	yes	—
19	C₂×C₁₃×#107	C₄	1	1	yes	characteristic
20	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
21	C₂×#88050	C₄	1	2	yes	—
22	#11718534	C₄	1	2	yes	—
23	#11718534	C₄	1	1	yes	characteristic
24	#11718534	C₄	1	1	yes	characteristic
25	#11718534	C₂²	1	1	yes	characteristic
26	#11718534	C₂²	1	2	yes	—
27	#11718534	C₂²	1	1	yes	characteristic
28	#11718256	C₂²	1	2	yes	—
29	#11718254	C₂²	1	2	yes	—
30	#1057⋊C₈	C₂²	1	2	yes	—
31	C₈×#1057	C₂²	1	2	yes	—
32	#11718244	C₂²	1	4	yes	—
33	#11718244	C₂²	1	4	yes	—
34	C₂²×Dic₂₆	C₂×C₄	1	2	yes	—
35	C₂×#6122	—	2	2	no	—
36	C₂×#6122	D₄	1	1	yes	characteristic
37	C₂×#6122	D₄	1	1	yes	characteristic
38	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
39	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
40	C₂×#6101	—	2	2	no	—
41	C₂×C₄×Dic₁₃	—	2	2	no	—
42	C₂×C₄×Dic₁₃	C₂³	1	1	yes	characteristic
43	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
44	C₂×Dic₅₂	—	2	4	no	—
45	C₂×Dic₅₂	C₂×C₄	1	2	yes	—
46	C₂×Dic₅₂	C₂×C₄	1	1	yes	characteristic
47	C₂×Dic₅₂	C₂×C₄	1	1	yes	characteristic
48	#88141	—	2	2	no	—
49	#88141	—	2	2	no	—
50	#88141	C₂×C₄	1	2	yes	—
51	#88141	C₂×C₄	1	2	yes	—
52	#88141	C₂³	1	2	yes	—
53	#88141	C₂³	1	2	yes	—
54	C₂²⋊#1057	—	2	1	no	—
55	#88112	—	2	4	no	—
56	#88112	—	2	2	no	—
57	#88112	—	2	1	no	—
58	#88110	—	2	4	no	—
59	#88110	D₄	1	2	yes	—
60	C₄⋊#1057	—	2	4	no	—
61	C₄⋊#1057	C₂³	1	4	yes	—
62	C₄×#1057	—	2	4	no	—
63	C₄×#1057	C₂³	1	2	yes	—
64	C₄×#1057	C₂³	1	2	yes	—
65	C₁₃×#107	C₂×C₄	1	2	yes	—
66	C₁₃×#107	C₂×C₄	1	1	yes	characteristic
67	C₁₃×#107	C₂×C₄	1	1	yes	characteristic
68	C₈×Dic₁₃	C₂×C₄	1	1	yes	characteristic
69	C₈×Dic₁₃	C₂×C₄	1	1	yes	characteristic
70	C₈×Dic₁₃	C₂³	1	1	yes	characteristic
71	C₈×Dic₁₃	C₂³	1	1	yes	characteristic
72	#88050	C₂×C₄	1	2	yes	—
73	#88050	C₂×C₄	1	2	yes	—
74	#88050	C₂×C₄	1	2	yes	—
75	#88050	C₂×C₄	1	2	yes	—
76	M₁₆×Q₁₆	—	13	1	no	maximal, Sylow
77	C₂²×C₄×C₁₃	—	2	2	no	—
78	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
79	C₂²×Dic₁₃	—	2	2	no	—
80	C₂²×Dic₁₃	C₂×D₄	1	1	yes	characteristic
81	C₂²×Dic₁₃	#41	1	1	yes	characteristic
82	C₂×Dic₂₆	—	2	4	no	—
83	C₂×Dic₂₆	—	2	2	no	—
84	C₂×Dic₂₆	—	4	2	no	—
85	C₂×Dic₂₆	C₂²×C₄	1	2	yes	—
86	C₂×Dic₂₆	C₂²×C₄	1	2	yes	—
87	#6122	—	2	2	no	—
88	#6122	—	4	4	no	—
89	#6122	Q₁₆	1	4	yes	complemented
90	C₂×#1057	—	2	4	no	—
91	C₂×#1057	—	2	2	no	—
92	C₂×#1057	C₂×D₄	1	2	yes	—
93	C₂×#1057	C₂×D₄	1	2	yes	—
94	C₂×C₈×C₁₃	—	2	2	no	—
95	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
96	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
97	C₁₃×C₄⋊C₄	—	2	2	no	—
98	C₁₃×C₄⋊C₄	C₂²×C₄	1	2	yes	—
99	C₁₃×C₄⋊C₄	C₂²×C₄	1	2	yes	—
100	#6101	—	2	2	no	—
101	#6101	—	2	2	no	—
102	#6101	—	2	2	no	—
103	#6101	—	2	2	no	—
104	C₄×Dic₁₃	—	2	2	no	—
105	C₄×Dic₁₃	—	2	2	no	—
106	C₄×Dic₁₃	—	2	2	no	—
107	C₄×Dic₁₃	—	2	2	no	—
108	C₄×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
109	C₄×Dic₁₃	C₂×D₄	1	1	yes	characteristic
110	C₄×Dic₁₃	#41	1	1	yes	characteristic
111	C₄×Dic₁₃	C₂⁴	1	1	yes	characteristic
112	C₂×#1055	—	4	1	no	—
113	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
114	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
115	Dic₅₂	—	2	4	no	—
116	Dic₅₂	—	2	2	no	—
117	Dic₅₂	M₁₆	1	4	yes	complemented
118	Q₈×M₁₆	—	13	2	no	—
119	C₂×C₄×Q₁₆	—	13	1	no	—
120	#1591	—	13	2	no	—
121	#1584	—	13	4	no	—
122	C₈×Q₁₆	—	13	2	no	—
123	#1567	—	13	1	no	—
124	#1487	—	13	2	no	—
125	C₈×M₁₆	—	13	1	no	—
126	C₂³×C₁₃	C₄×D₄	1	1	yes	characteristic
127	C₂×C₄×C₁₃	—	2	1	no	—
128	C₂×C₄×C₁₃	—	2	1	no	—
129	C₂×C₄×C₁₃	—	2	2	no	—
130	C₂×C₄×C₁₃	—	2	2	no	—
131	C₂×C₄×C₁₃	—	8	2	no	—
132	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
133	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
134	C₂×Dic₁₃	—	2	2	no	—
135	C₂×Dic₁₃	—	2	2	no	—
136	C₂×Dic₁₃	—	4	1	no	—
137	C₂×Dic₁₃	—	4	4	no	—
138	C₂×Dic₁₃	C₄×D₄	1	1	yes	characteristic
139	C₂×Dic₁₃	C₄×D₄	1	1	yes	characteristic
140	C₂×Dic₁₃	C₂×Q₁₆	1	2	yes	—
141	C₂×Dic₁₃	#135	1	2	yes	—
142	C₂×Dic₁₃	C₂²×D₄	1	1	yes	characteristic
143	C₂×Dic₁₃	C₂×#41	1	1	yes	characteristic
144	Dic₂₆	—	2	2	no	—
145	Dic₂₆	—	2	2	no	—
146	Dic₂₆	—	2	2	no	—
147	Dic₂₆	—	4	4	no	—
148	Dic₂₆	C₂×M₁₆	1	4	yes	—
149	#1057	—	2	2	no	—
150	#1057	—	4	2	no	—
151	#1057	—	4	4	no	—
152	#1057	C₂×Q₁₆	1	4	yes	complemented
153	C₈×C₁₃	—	2	1	no	—
154	C₈×C₁₃	C₂×M₁₆	1	2	yes	—
155	#1055	—	2	1	no	—
156	#1055	—	2	1	no	—
157	C₂²×Q₁₆	—	13	1	no	—
158	C₂×C₄×Q₈	—	13	2	no	—
159	#446	—	13	4	no	—
160	#446	—	26	4	no	—
161	C₈×Q₈	—	13	4	no	—
162	C₄×Q₁₆	—	13	1	no	—
163	C₄×Q₁₆	—	13	2	no	—
164	C₄×Q₁₆	—	13	1	no	—
165	C₄×Q₁₆	—	13	1	no	—
166	C₄×Q₁₆	—	13	2	no	—
167	C₄×Q₁₆	—	13	1	no	—
168	#431	—	13	1	no	—
169	C₂×#107	#261	1	1	yes	characteristic
170	#423	—	13	2	no	—
171	#423	—	26	2	no	—
172	C₂×#103	—	13	2	no	—
173	C₄×M₁₆	—	13	1	no	—
174	C₄×M₁₆	—	26	2	no	—
175	C₂×C₄×C₈	—	13	1	no	—
176	#335	—	13	2	no	—
177	#335	—	13	2	no	—
178	Q₈⋊C₈	—	13	4	no	—
179	Q₈⋊C₈	—	13	4	no	—
180	#322	—	13	2	no	—
181	C₈²	—	13	2	no	—
182	C₂²×C₁₃	—	2	1	no	—
183	C₂²×C₁₃	—	2	1	no	—
184	C₂²×C₁₃	C₄×Q₁₆	1	2	yes	—
185	C₂²×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
186	C₄×C₁₃	—	2	1	no	—
187	C₄×C₁₃	—	4	2	no	—
188	C₄×C₁₃	—	4	2	no	—
189	C₄×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
190	C₄×C₁₃	#568	1	1	yes	characteristic
191	Dic₁₃	—	2	4	no	—
192	Dic₁₃	—	2	1	no	—
193	Dic₁₃	—	2	1	no	—
194	Dic₁₃	—	4	2	no	—
195	Dic₁₃	C₄×Q₁₆	1	2	yes	—
196	Dic₁₃	C₂²×Q₁₆	1	1	yes	characteristic
197	Dic₁₃	C₂×#135	1	1	yes	characteristic
198	C₂²×Q₈	—	13	2	no	—
199	C₂×Q₁₆	—	13	1	no	—
200	C₂×Q₁₆	—	13	2	no	—
201	C₂×Q₁₆	—	13	1	no	—
202	C₂×Q₁₆	—	26	4	no	—
203	C₂×M₁₆	—	13	1	no	—
204	C₂×M₁₆	—	13	1	no	—
205	C₂×M₁₆	—	26	2	no	—
206	C₂²×C₈	—	13	1	no	—
207	C₂²×C₈	C₂×#261	1	1	yes	characteristic
208	C₄×Q₈	—	13	2	no	—
209	C₄×Q₈	—	13	2	no	—
210	C₄×Q₈	—	13	2	no	—
211	C₄×Q₈	—	13	2	no	—
212	C₄×Q₈	—	26	2	no	—
213	C₄×Q₈	—	26	2	no	—
214	C₂×C₄⋊C₄	—	26	2	no	—
215	C₂×C₄⋊C₄	C₂×#261	1	2	yes	—
216	C₂×C₄²	—	13	1	no	—
217	C₂×C₄²	—	26	2	no	—
218	#107	C₂×#261	1	2	yes	—
219	#107	C₂×#261	1	1	yes	characteristic
220	#107	C₂×#261	1	1	yes	characteristic
221	C₄⋊C₈	—	13	4	no	—
222	C₄⋊C₈	—	26	2	no	—
223	C₄⋊C₈	—	26	1	no	—
224	C₄⋊C₈	—	26	4	no	—
225	C₄⋊C₈	—	26	4	no	—
226	#103	—	13	2	no	—
227	#103	—	13	2	no	—
228	#103	—	13	2	no	—
229	#103	—	13	2	no	—
230	C₂²⋊C₈	—	26	1	no	—
231	#97	—	13	2	no	—
232	#97	—	26	4	no	—
233	C₄×C₈	—	13	2	no	—
234	C₄×C₈	—	13	1	no	—
235	C₄×C₈	—	13	1	no	—
236	C₄×C₈	—	13	1	no	—
237	C₄×C₈	—	13	1	no	—
238	C₄×C₈	—	13	2	no	—
239	C₄×C₈	—	26	4	no	—
240	C₂×C₁₃	—	2	2	no	—
241	C₂×C₁₃	D₄×M₁₆	1	1	yes	characteristic
242	C₂×C₁₃	C₂×C₄×Q₁₆	1	1	yes	characteristic
243	C₂×C₁₃	#2957	1	1	yes	characteristic
244	C₂×Q₈	—	13	2	no	—
245	C₂×Q₈	—	13	2	no	—
246	C₂×Q₈	—	26	4	no	—
247	C₂×Q₈	—	26	2	no	—
248	C₂×Q₈	—	52	2	no	—
249	C₂²×C₄	—	2	2	no	—
250	C₂²×C₄	—	13	1	no	—
251	C₂²×C₄	—	13	1	no	—
252	C₂²×C₄	—	26	2	no	—
253	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
254	Q₁₆	—	13	4	no	—
255	Q₁₆	—	26	4	no	—
256	Q₁₆	—	26	2	no	—
257	M₁₆	—	13	4	no	—
258	M₁₆	—	26	2	no	—
259	M₁₆	—	52	4	no	—
260	C₂×C₈	—	2	2	no	—
261	C₂×C₈	—	13	2	no	—
262	C₂×C₈	—	13	2	no	—
263	C₂×C₈	—	13	1	no	—
264	C₂×C₈	—	13	1	no	—
265	C₂×C₈	—	26	2	no	—
266	C₂×C₈	—	26	4	no	—
267	C₂×C₈	—	52	1	no	—
268	C₂×C₈	C₂²×#261	1	1	yes	characteristic
269	C₂×C₈	C₂²×#261	1	1	yes	characteristic
270	C₄⋊C₄	—	2	2	no	—
271	C₄⋊C₄	—	26	2	no	—
272	C₄⋊C₄	—	26	2	no	—
273	C₄⋊C₄	—	26	2	no	—
274	C₄⋊C₄	—	26	2	no	—
275	C₄⋊C₄	C₂²×#261	1	2	yes	—
276	C₄⋊C₄	C₂²×#261	1	2	yes	—
277	C₄²	—	13	1	no	—
278	C₄²	—	13	1	no	—
279	C₄²	—	13	1	no	—
280	C₄²	—	13	1	no	—
281	C₄²	—	26	2	no	—
282	C₄²	—	26	2	no	—
283	C₄²	—	26	2	no	—
284	C₄²	—	26	2	no	—
285	C₁₃	M₁₆×Q₁₆	1	1	yes	characteristic, complemented, Sylow
286	C₂³	D₄×#261	1	1	yes	characteristic
287	Q₈	—	13	4	no	—
288	Q₈	—	26	2	no	—
289	Q₈	—	26	2	no	—
290	Q₈	—	26	2	no	—
291	Q₈	—	52	4	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	8	2	no	—
297	C₂×C₄	—	13	2	no	—
298	C₂×C₄	—	13	2	no	—
299	C₂×C₄	—	13	1	no	—
300	C₂×C₄	—	13	1	no	—
301	C₂×C₄	—	13	1	no	—
302	C₂×C₄	—	13	1	no	—
303	C₂×C₄	—	26	2	no	—
304	C₂×C₄	—	26	2	no	—
305	C₂×C₄	—	52	1	no	—
306	C₂×C₄	—	52	4	no	—
307	C₂×C₄	D₄×#261	1	1	yes	characteristic
308	C₂×C₄	C₂³×#261	1	1	yes	characteristic
309	C₈	—	2	1	no	—
310	C₈	—	13	4	no	—
311	C₈	—	26	2	no	—
312	C₈	—	26	1	no	—
313	C₈	—	26	1	no	—
314	C₈	—	52	4	no	—
315	C₈	—	52	2	no	—
316	C₈	D₁₃⋊M₁₆	1	2	yes	—
317	C₂²	—	2	1	no	—
318	C₂²	—	2	1	no	—
319	C₂²	Q₁₆×#261	1	2	yes	—
320	C₂²	C₂×D₄×#261	1	1	yes	characteristic
321	C₄	—	2	1	no	—
322	C₄	—	4	2	no	—
323	C₄	—	4	2	no	—
324	C₄	—	13	1	no	—
325	C₄	—	13	1	no	—
326	C₄	—	13	2	no	—
327	C₄	—	26	4	no	—
328	C₄	—	26	1	no	—
329	C₄	—	26	1	no	—
330	C₄	—	52	2	no	—
331	C₄	C₂×D₁₃⋊M₁₆	1	1	yes	characteristic
332	C₄	#11719625	1	1	yes	characteristic
333	C₂	—	2	2	no	—
334	C₂	#13131576	1	1	yes	characteristic
335	C₂	C₂×Q₁₆×#261	1	1	yes	characteristic
336	C₂	#13131612	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×#261
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×#261
Fitting	C₂×C₁₃×#107	C₄
socle	C₂²×C₁₃	C₂×C₄×D₄
prime radical	C₂³×C₁₃	C₄×D₄
cosocle	C₄×Dic₁₃	C₂⁴
hypercenter	C₂×#107	#261
hypocenter	C₁₃	M₁₆×Q₁₆
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#107

Complements

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

N	H ≅ G/N	type
2. #13130475	C₂	semidirect
3. #13130448	C₂	semidirect
4. #13130424	C₂	semidirect
89. #6122	Q₁₆	semidirect
117. Dic₅₂	M₁₆	semidirect
152. #1057	C₂×Q₁₆	semidirect
285. C₁₃	M₁₆×Q₁₆	semidirect

Maximal subgroups

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

subgroup	index
#13131637	2
#13130475	2
#13130448	2
#13130424	2
#13130393	2
#13130166	2
C₈×#6122	2
C₂×#11718534	2
M₁₆×Q₁₆	13