Subgroups — #16781853

Order 3328.

598 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13130690	C₂	1	2	yes	maximal
2	C₂×#11718254	C₂	1	2	yes	maximal, complemented
3	#13129666	C₂	1	2	yes	maximal, complemented
4	C₂²×C₈×Dic₁₃	C₂	1	1	yes	maximal, characteristic
5	C₂³×C₈×C₁₃	C₄	1	1	yes	characteristic
6	C₂²×C₄×Dic₁₃	C₂²	1	1	yes	characteristic
7	C₂³×#1055	C₄	1	1	yes	characteristic
8	C₂×C₂²⋊#1057	—	2	2	no	—
9	C₂×C₄⋊#1057	C₂²	1	4	yes	—
10	C₂×C₈×Dic₁₃	—	2	4	no	—
11	C₂×C₈×Dic₁₃	C₄	1	1	yes	characteristic
12	C₂×C₈×Dic₁₃	C₄	1	1	yes	characteristic
13	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
14	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
15	#11718290	—	2	2	no	—
16	#11718254	—	2	4	no	—
17	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
18	C₂³×Dic₁₃	D₄	1	1	yes	characteristic
19	C₂³×Dic₁₃	Q₈	1	1	yes	characteristic
20	C₂²×#1057	—	2	4	no	—
21	C₂²×C₈×C₁₃	—	2	2	no	—
22	C₂²×C₈×C₁₃	—	2	2	no	—
23	C₂²×C₈×C₁₃	C₈	1	4	yes	complemented
24	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄×Dic₁₃	—	2	2	no	—
27	C₂×C₄×Dic₁₃	—	2	2	no	—
28	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄×Dic₁₃	D₄	1	1	yes	characteristic
30	C₂×C₄×Dic₁₃	Q₈	1	1	yes	characteristic
31	C₂×C₄×Dic₁₃	C₂³	1	1	yes	characteristic
32	C₂²×#1055	—	2	2	no	—
33	C₂²×#1055	—	2	2	no	—
34	C₂²×#1055	—	2	2	no	—
35	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
36	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
37	C₂²⋊#1057	—	4	4	no	—
38	C₄⋊#1057	—	2	4	no	—
39	C₄⋊#1057	—	2	4	no	—
40	C₈×Dic₁₃	—	2	2	no	—
41	C₈×Dic₁₃	—	2	2	no	—
42	C₈×Dic₁₃	—	2	2	no	—
43	C₈×Dic₁₃	D₄	1	4	yes	—
44	#8596	—	13	1	no	maximal, Sylow
45	C₂⁴×C₁₃	C₄⋊C₄	1	1	yes	characteristic
46	C₂²×C₄×C₁₃	—	2	1	no	—
47	C₂²×C₄×C₁₃	—	2	1	no	—
48	C₂²×C₄×C₁₃	—	2	1	no	—
49	C₂²×C₄×C₁₃	—	2	1	no	—
50	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
51	C₂²×C₄×C₁₃	C₂×C₈	1	2	yes	—
52	C₂²×C₄×C₁₃	M₁₆	1	2	yes	—
53	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
54	C₂²×Dic₁₃	—	2	2	no	—
55	C₂²×Dic₁₃	—	2	2	no	—
56	C₂²×Dic₁₃	—	2	2	no	—
57	C₂²×Dic₁₃	—	2	2	no	—
58	C₂²×Dic₁₃	—	2	2	no	—
59	C₂²×Dic₁₃	C₄⋊C₄	1	1	yes	characteristic
60	C₂²×Dic₁₃	C₄⋊C₄	1	1	yes	characteristic
61	C₂²×Dic₁₃	D₈	1	2	yes	—
62	C₂²×Dic₁₃	Q₁₆	1	2	yes	—
63	C₂²×Dic₁₃	C₂×D₄	1	1	yes	characteristic
64	C₂²×Dic₁₃	C₂×Q₈	1	1	yes	characteristic
65	C₂×#1057	—	4	4	no	—
66	C₂×#1057	—	4	4	no	—
67	C₂×#1057	—	4	4	no	—
68	C₂×C₈×C₁₃	—	2	1	no	—
69	C₂×C₈×C₁₃	—	2	1	no	—
70	C₂×C₈×C₁₃	—	2	8	no	—
71	C₂×C₈×C₁₃	C₂²⋊C₄	1	2	yes	—
72	C₂×C₈×C₁₃	C₂×C₈	1	2	yes	—
73	C₂×C₈×C₁₃	C₂×C₈	1	2	yes	—
74	C₂×C₈×C₁₃	M₁₆	1	2	yes	—
75	C₄×Dic₁₃	—	2	1	no	—
76	C₄×Dic₁₃	—	2	1	no	—
77	C₄×Dic₁₃	—	2	2	no	—
78	C₄×Dic₁₃	—	2	2	no	—
79	C₄×Dic₁₃	C₂×D₄	1	2	yes	—
80	C₄×Dic₁₃	#41	1	2	yes	—
81	C₂×#1055	—	2	8	no	—
82	C₂×#1055	—	2	1	no	—
83	C₂×#1055	—	2	1	no	—
84	C₂×#1055	—	2	1	no	—
85	C₂×#1055	—	2	1	no	—
86	C₂×#1055	C₂²⋊C₄	1	2	yes	—
87	C₂×#1055	M₁₆	1	2	yes	—
88	C₂²×C₄×C₈	—	13	1	no	—
89	#1797	—	13	2	no	—
90	C₂×#335	—	13	2	no	—
91	#1277	—	13	2	no	—
92	C₂³×C₁₃	—	2	1	no	—
93	C₂³×C₁₃	—	2	1	no	—
94	C₂³×C₁₃	—	2	1	no	—
95	C₂³×C₁₃	—	2	1	no	—
96	C₂³×C₁₃	C₄⋊C₈	1	2	yes	—
97	C₂³×C₁₃	#107	1	2	yes	—
98	C₂³×C₁₃	#108	1	2	yes	—
99	C₂³×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
100	C₂×C₄×C₁₃	—	2	1	no	—
101	C₂×C₄×C₁₃	—	2	1	no	—
102	C₂×C₄×C₁₃	—	2	2	no	—
103	C₂×C₄×C₁₃	—	2	2	no	—
104	C₂×C₄×C₁₃	—	2	2	no	—
105	C₂×C₄×C₁₃	—	2	2	no	—
106	C₂×C₄×C₁₃	C₄⋊C₈	1	2	yes	—
107	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
108	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
109	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
110	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
111	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
112	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
113	C₂×Dic₁₃	—	2	8	no	—
114	C₂×Dic₁₃	—	2	1	no	—
115	C₂×Dic₁₃	—	2	1	no	—
116	C₂×Dic₁₃	—	2	1	no	—
117	C₂×Dic₁₃	—	2	1	no	—
118	C₂×Dic₁₃	—	2	1	no	—
119	C₂×Dic₁₃	—	2	1	no	—
120	C₂×Dic₁₃	—	2	4	no	—
121	C₂×Dic₁₃	—	2	4	no	—
122	C₂×Dic₁₃	#107	1	2	yes	—
123	C₂×Dic₁₃	#108	1	2	yes	—
124	C₂×Dic₁₃	#121	1	2	yes	—
125	C₂×Dic₁₃	C₂²⋊Q₈	1	2	yes	—
126	C₂×Dic₁₃	C₂×D₈	1	1	yes	characteristic
127	C₂×Dic₁₃	C₂×Q₁₆	1	1	yes	characteristic
128	C₂×Dic₁₃	#135	1	2	yes	—
129	#1057	—	8	4	no	—
130	C₈×C₁₃	—	2	2	no	—
131	C₈×C₁₃	C₂²⋊C₈	1	4	yes	complemented
132	#1055	—	2	2	no	—
133	#1055	—	2	2	no	—
134	C₂³×C₈	—	13	1	no	—
135	C₂³×C₈	#261	1	1	yes	characteristic
136	C₂²×C₄²	—	13	1	no	—
137	C₂×C₄⋊C₈	—	13	4	no	—
138	C₂×C₂²⋊C₈	—	26	2	no	—
139	C₂×C₄×C₈	—	13	1	no	—
140	C₂×C₄×C₈	—	13	1	no	—
141	C₂×C₄×C₈	—	13	1	no	—
142	C₂×C₄×C₈	—	13	1	no	—
143	C₂×C₄×C₈	—	26	4	no	—
144	#336	—	26	2	no	—
145	#335	—	26	4	no	—
146	C₂²×C₁₃	—	2	2	no	—
147	C₂²×C₁₃	—	2	2	no	—
148	C₂²×C₁₃	—	2	2	no	—
149	C₂²×C₁₃	—	2	2	no	—
150	C₂²×C₁₃	—	2	2	no	—
151	C₂²×C₁₃	—	2	2	no	—
152	C₂²×C₁₃	#335	1	4	yes	complemented
153	C₂²×C₁₃	#380	1	1	yes	characteristic
154	C₂²×C₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
155	C₂²×C₁₃	#423	1	1	yes	characteristic
156	C₂²×C₁₃	C₂×#107	1	1	yes	characteristic
157	C₂²×C₁₃	#427	1	1	yes	characteristic
158	C₂²×C₁₃	C₂×#108	1	1	yes	characteristic
159	C₂²×C₁₃	C₂×#108	1	1	yes	characteristic
160	C₄×C₁₃	—	2	2	no	—
161	C₄×C₁₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
162	C₄×C₁₃	#407	1	1	yes	characteristic
163	C₄×C₁₃	#431	1	1	yes	characteristic
164	C₄×C₁₃	#432	1	1	yes	characteristic
165	Dic₁₃	—	2	2	no	—
166	Dic₁₃	—	2	2	no	—
167	Dic₁₃	—	2	2	no	—
168	Dic₁₃	#466	1	2	yes	—
169	Dic₁₃	#467	1	2	yes	—
170	C₂³×C₄	—	13	1	no	—
171	C₂³×C₄	—	13	1	no	—
172	C₂³×C₄	C₂×#261	1	1	yes	characteristic
173	C₂²×C₈	—	2	2	no	—
174	C₂²×C₈	—	2	2	no	—
175	C₂²×C₈	—	13	1	no	—
176	C₂²×C₈	—	13	1	no	—
177	C₂²×C₈	—	26	2	no	—
178	C₂²×C₈	—	26	2	no	—
179	C₂²×C₈	—	26	2	no	—
180	C₂²×C₈	—	26	4	no	—
181	C₂²×C₈	#1057	1	4	yes	complemented
182	C₂²×C₈	C₂×#261	1	1	yes	characteristic
183	C₂²×C₈	C₂×#261	1	1	yes	characteristic
184	C₂×C₄²	—	13	1	no	—
185	C₂×C₄²	—	13	1	no	—
186	C₂×C₄²	—	13	1	no	—
187	C₂×C₄²	—	13	1	no	—
188	C₂×C₄²	—	26	2	no	—
189	C₂×C₄²	—	26	2	no	—
190	C₄⋊C₈	—	26	4	no	—
191	C₄⋊C₈	—	26	4	no	—
192	C₂²⋊C₈	—	52	4	no	—
193	C₄×C₈	—	13	4	no	—
194	C₄×C₈	—	26	2	no	—
195	C₄×C₈	—	26	2	no	—
196	C₄×C₈	—	26	2	no	—
197	C₂×C₁₃	—	2	4	no	—
198	C₂×C₁₃	C₂×#335	1	1	yes	characteristic
199	C₂×C₁₃	#1564	1	1	yes	characteristic
200	C₂×C₁₃	#1567	1	1	yes	characteristic
201	C₂×C₁₃	#1568	1	1	yes	characteristic
202	C₂×C₁₃	#1797	1	1	yes	characteristic
203	C₂×C₁₃	#1808	1	1	yes	characteristic
204	C₂×C₁₃	#1810	1	1	yes	characteristic
205	C₂⁴	C₄⋊#261	1	1	yes	characteristic
206	C₂²×C₄	—	2	1	no	—
207	C₂²×C₄	—	2	1	no	—
208	C₂²×C₄	—	2	1	no	—
209	C₂²×C₄	—	2	1	no	—
210	C₂²×C₄	—	13	1	no	—
211	C₂²×C₄	—	13	2	no	—
212	C₂²×C₄	—	13	2	no	—
213	C₂²×C₄	—	13	1	no	—
214	C₂²×C₄	—	13	1	no	—
215	C₂²×C₄	—	13	1	no	—
216	C₂²×C₄	—	26	2	no	—
217	C₂²×C₄	—	26	2	no	—
218	C₂²×C₄	—	26	2	no	—
219	C₂²×C₄	—	26	2	no	—
220	C₂²×C₄	—	26	2	no	—
221	C₂²×C₄	#6118	1	2	yes	—
222	C₂²×C₄	C₄⋊#261	1	1	yes	characteristic
223	C₂²×C₄	C₂×#1057	1	2	yes	—
224	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
225	C₂×C₈	—	2	1	no	—
226	C₂×C₈	—	2	1	no	—
227	C₂×C₈	—	2	8	no	—
228	C₂×C₈	—	13	2	no	—
229	C₂×C₈	—	13	2	no	—
230	C₂×C₈	—	26	8	no	—
231	C₂×C₈	—	26	1	no	—
232	C₂×C₈	—	26	1	no	—
233	C₂×C₈	—	26	1	no	—
234	C₂×C₈	—	26	1	no	—
235	C₂×C₈	—	52	4	no	—
236	C₂×C₈	—	52	4	no	—
237	C₂×C₈	—	52	4	no	—
238	C₂×C₈	C₂×#1057	1	2	yes	—
239	C₂×C₈	C₂×#1057	1	2	yes	—
240	C₂×C₈	#6122	1	2	yes	—
241	C₂×C₈	C₂²⋊#261	1	2	yes	—
242	C₄²	—	13	2	no	—
243	C₄²	—	13	2	no	—
244	C₄²	—	26	2	no	—
245	C₄²	—	26	2	no	—
246	C₄²	—	26	1	no	—
247	C₄²	—	26	1	no	—
248	C₁₃	#8596	1	1	yes	characteristic, complemented, Sylow
249	C₂³	—	2	1	no	—
250	C₂³	—	2	1	no	—
251	C₂³	—	2	1	no	—
252	C₂³	—	2	1	no	—
253	C₂³	#88102	1	2	yes	—
254	C₂³	#88104	1	2	yes	—
255	C₂³	C₄⋊#1057	1	2	yes	—
256	C₂³	C₂×C₄⋊#261	1	1	yes	characteristic
257	C₂×C₄	—	2	1	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	13	1	no	—
264	C₂×C₄	—	13	1	no	—
265	C₂×C₄	—	13	2	no	—
266	C₂×C₄	—	13	2	no	—
267	C₂×C₄	—	13	2	no	—
268	C₂×C₄	—	13	2	no	—
269	C₂×C₄	—	13	2	no	—
270	C₂×C₄	—	26	4	no	—
271	C₂×C₄	—	26	4	no	—
272	C₂×C₄	—	26	8	no	—
273	C₂×C₄	—	26	1	no	—
274	C₂×C₄	—	26	1	no	—
275	C₂×C₄	—	26	1	no	—
276	C₂×C₄	—	26	1	no	—
277	C₂×C₄	—	26	1	no	—
278	C₂×C₄	—	26	1	no	—
279	C₂×C₄	C₄⋊#1057	1	2	yes	—
280	C₂×C₄	C₂×#6118	1	1	yes	characteristic
281	C₂×C₄	D₁₃⋊M₁₆	1	1	yes	characteristic
282	C₂×C₄	#88237	1	1	yes	characteristic
283	C₂×C₄	C₂²×#1057	1	1	yes	characteristic
284	C₂×C₄	C₂×#6122	1	1	yes	characteristic
285	C₂×C₄	C₂×C₂²⋊#261	1	1	yes	characteristic
286	C₈	—	2	2	no	—
287	C₈	—	26	2	no	—
288	C₈	—	26	2	no	—
289	C₈	—	104	4	no	—
290	C₈	C₂²⋊#1057	1	4	yes	complemented
291	C₂²	—	2	2	no	—
292	C₂²	—	2	2	no	—
293	C₂²	—	2	2	no	—
294	C₂²	—	2	2	no	—
295	C₂²	—	2	2	no	—
296	C₂²	—	2	2	no	—
297	C₂²	#11718254	1	4	yes	complemented
298	C₂²	C₂×#88102	1	1	yes	characteristic
299	C₂²	#11719092	1	1	yes	characteristic
300	C₂²	C₂×#88104	1	1	yes	characteristic
301	C₂²	D₁₃⋊#108	1	1	yes	characteristic
302	C₂²	C₂×C₄⋊#1057	1	1	yes	characteristic
303	C₂²	#11719129	1	1	yes	characteristic
304	C₂²	#11719136	1	1	yes	characteristic
305	C₄	—	2	2	no	—
306	C₄	—	13	2	no	—
307	C₄	—	13	2	no	—
308	C₄	—	26	2	no	—
309	C₄	—	26	2	no	—
310	C₄	—	26	2	no	—
311	C₄	#11719119	1	1	yes	characteristic
312	C₄	#11719127	1	1	yes	characteristic
313	C₄	#11719130	1	1	yes	characteristic
314	C₄	C₂×C₂²⋊#1057	1	1	yes	characteristic
315	C₂	—	2	4	no	—
316	C₂	C₂×#11718254	1	1	yes	characteristic
317	C₂	#13130383	1	1	yes	characteristic
318	C₂	#13130393	1	1	yes	characteristic
319	C₂	#13130398	1	1	yes	characteristic
320	C₂	#13130690	1	1	yes	characteristic
321	C₂	#13130713	1	1	yes	characteristic
322	C₂	#13130716	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₄⋊#261
commutator	C₂×C₄×C₁₃	C₂²×C₈
Frattini	C₂²×C₄	C₂²×#261
Fitting	C₂³×C₈×C₁₃	C₄
socle	C₂³×C₁₃	C₂×C₄⋊C₄
prime radical	C₂⁴×C₁₃	C₄⋊C₄
cosocle	C₂×C₄×Dic₁₃	C₂³
hypercenter	C₂³×C₈	#261
hypocenter	C₁₃	#8596
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×C₈

Complements

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

N	H ≅ G/N	type
2. C₂×#11718254	C₂	semidirect
3. #13129666	C₂	semidirect
23. C₂²×C₈×C₁₃	C₈	semidirect
131. C₈×C₁₃	C₂²⋊C₈	semidirect
152. C₂²×C₁₃	#335	semidirect
181. C₂²×C₈	#1057	semidirect
248. C₁₃	#8596	semidirect
290. C₈	C₂²⋊#1057	semidirect
297. C₂²	#11718254	semidirect

Maximal subgroups

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

subgroup	index
#13130690	2
C₂×#11718254	2
#13129666	2
C₂²×C₈×Dic₁₃	2
#8596	13