Subgroups — #16795093

Order 3328.

702 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13131481	C₂	1	2	yes	maximal
2	#13130406	C₂	1	2	yes	maximal, complemented
3	#13130405	C₂	1	4	yes	maximal, complemented
4	#13130394	C₂	1	1	yes	maximal, characteristic, complemented
5	#13130393	C₂	1	2	yes	maximal, complemented
6	D₁₃⋊#335	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#11718254	C₂	1	2	yes	maximal, complemented
8	C₈×#6129	C₂	1	1	yes	maximal, characteristic
9	C₂×#88238	—	2	2	no	—
10	C₄×#6129	C₂²	1	1	yes	characteristic
11	#11719129	C₂²	1	4	yes	—
12	C₂×C₄⋊#1057	C₂²	1	4	yes	—
13	#11719079	—	2	2	no	—
14	#11719067	—	2	4	no	—
15	#11719052	C₂²	1	2	yes	—
16	D₁₃⋊(C₄⋊C₈)	C₂²	1	2	yes	—
17	C₈×D₄×C₁₃	C₄	1	1	yes	characteristic
18	C₈×C₂²⋊D₁₃	C₄	1	2	yes	—
19	C₂×C₈×Dic₁₃	C₂²	1	2	yes	—
20	D₄×#1055	C₄	1	1	yes	characteristic
21	#11718494	C₂²	1	1	yes	characteristic
22	#11718491	C₄	1	1	yes	characteristic
23	#11718386	C₄	1	2	yes	—
24	#11718379	C₂²	1	2	yes	—
25	C₄×C₈×D₁₃	C₂²	1	1	yes	characteristic
26	C₈×Dic₂₆	C₄	1	1	yes	characteristic
27	#11718256	C₂²	1	2	yes	—
28	#11718256	C₂²	1	4	yes	—
29	#11718254	C₂²	1	1	yes	characteristic
30	#11718254	C₂²	1	2	yes	—
31	#11718254	C₂²	1	4	yes	—
32	#11718254	C₂²	1	1	yes	characteristic
33	#11718254	C₂²	1	2	yes	—
34	C₂×#6129	D₄	1	1	yes	characteristic
35	C₂×#6122	—	2	4	no	—
36	C₂²×#1057	—	2	4	no	—
37	#88238	—	4	8	no	—
38	C₂×#6118	—	2	2	no	—
39	C₂×D₁₃⋊C₈	—	2	2	no	—
40	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
41	C₄×D₄×C₁₃	C₂×C₄	1	1	yes	characteristic
42	D₄×Dic₁₃	Q₈	1	1	yes	characteristic
43	C₄×C₂²⋊D₁₃	C₂×C₄	1	2	yes	—
44	C₂×C₄×Dic₁₃	C₂³	1	2	yes	—
45	C₂²×#1055	C₂×C₄	1	2	yes	—
46	C₂×C₈×D₁₃	C₂×C₄	1	1	yes	characteristic
47	C₂×C₈×D₁₃	C₂×C₄	1	1	yes	characteristic
48	#88146	D₄	1	1	yes	characteristic
49	#88141	Q₈	1	1	yes	characteristic
50	#88135	Q₈	1	2	yes	—
51	#88131	D₄	1	2	yes	—
52	C₄²×D₁₃	C₂³	1	1	yes	characteristic
53	C₄×Dic₂₆	C₂×C₄	1	1	yes	characteristic
54	#88112	—	2	2	no	—
55	#88112	—	2	4	no	—
56	C₄⋊#1057	—	2	2	no	—
57	C₄⋊#1057	—	2	4	no	—
58	C₄⋊#1057	C₂³	1	2	yes	—
59	C₄⋊#1057	C₂³	1	4	yes	—
60	C₄⋊#1057	C₂³	1	2	yes	—
61	C₁₃×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
62	C₁₃×C₂²⋊C₈	C₂×C₄	1	2	yes	—
63	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
64	C₂²⋊#1055	C₂×C₄	1	2	yes	—
65	#88060	C₂×C₄	1	2	yes	—
66	#88054	C₂×C₄	1	2	yes	—
67	#88054	C₂×C₄	1	2	yes	—
68	C₈×Dic₁₃	C₂×C₄	1	2	yes	—
69	C₈×Dic₁₃	C₂×C₄	1	2	yes	—
70	C₈×Dic₁₃	C₂³	1	2	yes	—
71	C₈×Dic₁₃	C₂³	1	1	yes	characteristic
72	C₈×Dic₁₃	C₂³	1	1	yes	characteristic
73	C₄⋊#1055	C₂×C₄	1	1	yes	characteristic
74	C₄×#1055	C₂×C₄	1	1	yes	characteristic
75	#17828	—	13	1	no	maximal, Sylow
76	C₂×D₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
77	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
78	C₂×C₂²⋊D₁₃	C₄⋊C₄	1	2	yes	—
79	C₂²×Dic₁₃	C₂×D₄	1	2	yes	—
80	C₂²×Dic₁₃	C₂×Q₈	1	2	yes	—
81	#6129	D₈	1	4	yes	—
82	#6129	Q₁₆	1	4	yes	—
83	C₂×C₄×D₁₃	C₂²×C₄	1	1	yes	characteristic
84	C₂×C₄×D₁₃	C₂×D₄	1	1	yes	characteristic
85	C₂×C₄×D₁₃	C₂×Q₈	1	1	yes	characteristic
86	C₂×Dic₂₆	C₄⋊C₄	1	1	yes	characteristic
87	#6122	—	4	8	no	—
88	C₂×#1057	—	2	2	no	—
89	C₂×#1057	—	2	4	no	—
90	C₂×#1057	—	2	2	no	—
91	C₂×#1057	—	4	8	no	—
92	#6118	—	4	4	no	—
93	D₁₃⋊C₈	—	4	4	no	—
94	C₂×C₈×C₁₃	—	2	4	no	—
95	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
96	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
97	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
98	C₁₃×C₄⋊C₄	C₄⋊C₄	1	1	yes	characteristic
99	C₁₃×C₂²⋊C₄	C₄⋊C₄	1	2	yes	—
100	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
101	C₂²⋊Dic₁₃	C₄⋊C₄	1	2	yes	—
102	#6103	C₄⋊C₄	1	2	yes	—
103	C₄⋊Dic₁₃	C₄⋊C₄	1	1	yes	characteristic
104	#6101	C₄⋊C₄	1	2	yes	—
105	#6101	C₄⋊C₄	1	2	yes	—
106	C₄×Dic₁₃	C₂²×C₄	1	2	yes	—
107	C₄×Dic₁₃	C₂×D₄	1	1	yes	characteristic
108	C₄×Dic₁₃	C₂×D₄	1	2	yes	—
109	C₄×Dic₁₃	C₂×Q₈	1	1	yes	characteristic
110	C₄×Dic₁₃	C₂×Q₈	1	2	yes	—
111	C₄×Dic₁₃	C₂⁴	1	1	yes	characteristic
112	C₂×#1055	—	4	2	no	—
113	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
114	C₂×#1055	C₂²×C₄	1	2	yes	—
115	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
116	C₈×D₁₃	—	2	2	no	—
117	C₈×D₁₃	—	4	1	no	—
118	C₈×#41	—	13	1	no	—
119	#2906	—	13	2	no	—
120	#1571	—	13	4	no	—
121	#1571	—	13	2	no	—
122	#1567	—	13	2	no	—
123	#1567	—	13	1	no	—
124	C₂×#335	—	13	2	no	—
125	C₂×#335	—	13	1	no	—
126	C₂³×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
127	C₂²×D₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
128	D₄×C₁₃	#107	1	4	yes	—
129	C₂×C₄×C₁₃	—	2	2	no	—
130	C₂×C₄×C₁₃	—	2	2	no	—
131	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
132	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
133	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
134	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
135	C₂²⋊D₁₃	#107	1	8	yes	—
136	C₂×Dic₁₃	—	4	2	no	—
137	C₂×Dic₁₃	C₂×C₄⋊C₄	1	2	yes	—
138	C₂×Dic₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
139	C₂×Dic₁₃	C₂×C₄⋊C₄	1	2	yes	—
140	C₂×Dic₁₃	C₂×D₈	1	4	yes	—
141	C₂×Dic₁₃	C₂×Q₁₆	1	4	yes	—
142	C₂×Dic₁₃	C₂²×D₄	1	1	yes	characteristic
143	C₂×Dic₁₃	C₂²×Q₈	1	1	yes	characteristic
144	C₄×D₁₃	—	2	1	no	—
145	C₄×D₁₃	—	2	1	no	—
146	C₄×D₁₃	—	4	1	no	—
147	C₄×D₁₃	C₂×D₈	1	2	yes	—
148	C₄×D₁₃	C₂×Q₁₆	1	2	yes	—
149	Dic₂₆	#107	1	4	yes	—
150	#1057	—	4	2	no	—
151	#1057	—	4	2	no	—
152	#1057	—	4	4	no	—
153	C₈×C₁₃	—	2	2	no	—
154	C₈×C₁₃	—	2	1	no	—
155	C₈×C₁₃	#131	1	2	yes	complemented
156	#1055	—	2	1	no	—
157	#1055	—	2	1	no	—
158	#1055	—	2	2	no	—
159	C₂×#131	—	26	2	no	—
160	C₄×#41	—	13	1	no	—
161	C₈×Q₈	—	13	1	no	—
162	C₈×Q₈	—	13	1	no	—
163	C₈×D₄	—	13	2	no	—
164	C₈×D₄	—	13	1	no	—
165	C₈×D₄	—	13	2	no	—
166	C₈×D₄	#261	1	1	yes	characteristic
167	#431	—	13	1	no	—
168	#431	—	13	2	no	—
169	#424	—	26	4	no	—
170	#424	—	26	2	no	—
171	#423	—	13	2	no	—
172	#423	—	13	4	no	—
173	C₂×C₄⋊C₈	—	13	2	no	—
174	C₂×C₄⋊C₈	—	13	4	no	—
175	C₂×C₄×C₈	—	13	2	no	—
176	C₂×C₄×C₈	—	13	1	no	—
177	#335	—	13	1	no	—
178	#335	—	13	2	no	—
179	#335	—	13	1	no	—
180	#335	—	13	4	no	—
181	#335	—	13	2	no	—
182	#335	—	13	2	no	—
183	#335	—	13	4	no	—
184	C₂²×C₁₃	—	2	2	no	—
185	C₂²×C₁₃	—	2	2	no	—
186	C₂²×C₁₃	C₂×#107	1	4	yes	—
187	C₂²×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
188	C₂×D₁₃	—	2	1	no	—
189	C₂×D₁₃	—	2	1	no	—
190	C₂×D₁₃	C₂×#107	1	2	yes	—
191	C₄×C₁₃	—	2	2	no	—
192	C₄×C₁₃	—	2	1	no	—
193	C₄×C₁₃	C₂×#107	1	2	yes	—
194	C₄×C₁₃	C₂×#131	1	1	yes	characteristic
195	C₄×C₁₃	#568	1	1	yes	characteristic
196	Dic₁₃	—	2	1	no	—
197	Dic₁₃	—	2	1	no	—
198	Dic₁₃	—	2	2	no	—
199	Dic₁₃	C₂×#107	1	4	yes	—
200	Dic₁₃	C₂²×D₈	1	1	yes	characteristic
201	Dic₁₃	C₂²×Q₁₆	1	1	yes	characteristic
202	C₂×#41	—	13	1	no	—
203	#131	—	52	8	no	—
204	C₂×M₁₆	—	26	2	no	—
205	C₂×M₁₆	—	26	4	no	—
206	C₂²×C₈	—	13	2	no	—
207	C₂²×C₈	—	13	1	no	—
208	C₂²×C₈	—	13	1	no	—
209	C₂²×C₈	—	26	2	no	—
210	C₂²×C₈	—	26	4	no	—
211	C₂²×C₈	C₂×#261	1	2	yes	—
212	C₄×Q₈	—	13	1	no	—
213	C₄×Q₈	—	13	1	no	—
214	C₄×D₄	—	13	1	no	—
215	C₄×D₄	—	13	2	no	—
216	C₄×D₄	—	13	2	no	—
217	C₄×D₄	C₂×#261	1	1	yes	characteristic
218	#117	—	13	1	no	—
219	#117	—	13	2	no	—
220	C₂×C₄²	—	13	2	no	—
221	C₂×C₄²	—	13	1	no	—
222	C₄⋊C₈	—	13	2	no	—
223	C₄⋊C₈	—	13	2	no	—
224	C₄⋊C₈	—	13	2	no	—
225	C₄⋊C₈	—	13	4	no	—
226	C₄⋊C₈	—	13	2	no	—
227	C₄⋊C₈	—	13	1	no	—
228	C₄⋊C₈	—	26	4	no	—
229	C₄⋊C₈	—	26	2	no	—
230	C₄⋊C₈	—	26	2	no	—
231	C₄⋊C₈	—	26	4	no	—
232	C₄⋊C₈	C₂×#261	1	1	yes	characteristic
233	C₂²⋊C₈	—	13	2	no	—
234	C₂²⋊C₈	—	13	2	no	—
235	C₂²⋊C₈	C₂×#261	1	2	yes	—
236	C₄×C₈	—	13	1	no	—
237	C₄×C₈	—	13	2	no	—
238	C₄×C₈	—	13	1	no	—
239	C₄×C₈	—	13	2	no	—
240	C₄×C₈	—	13	2	no	—
241	C₄×C₈	—	13	1	no	—
242	C₄×C₈	C₂×#261	1	1	yes	characteristic
243	C₂×C₁₃	—	2	4	no	—
244	C₂×C₁₃	#2906	1	1	yes	characteristic
245	C₂×C₁₃	C₂²×#107	1	1	yes	characteristic
246	C₂×C₁₃	#2916	1	1	yes	characteristic
247	D₁₃	—	2	2	no	—
248	#41	—	13	4	no	—
249	#41	—	13	4	no	—
250	C₂×Q₈	—	13	1	no	—
251	C₂×D₄	—	13	2	no	—
252	C₂×D₄	C₄⋊#261	1	1	yes	characteristic
253	C₂²×C₄	—	13	2	no	—
254	C₂²×C₄	—	13	1	no	—
255	C₂²×C₄	—	13	2	no	—
256	C₂²×C₄	—	13	1	no	—
257	C₂²×C₄	—	13	1	no	—
258	C₂²×C₄	C₂²×#261	1	2	yes	—
259	M₁₆	—	52	4	no	—
260	M₁₆	—	52	8	no	—
261	C₂×C₈	—	2	4	no	—
262	C₂×C₈	—	13	1	no	—
263	C₂×C₈	—	13	1	no	—
264	C₂×C₈	—	13	2	no	—
265	C₂×C₈	—	26	2	no	—
266	C₂×C₈	—	26	2	no	—
267	C₂×C₈	—	26	4	no	—
268	C₂×C₈	—	26	2	no	—
269	C₂×C₈	—	52	1	no	—
270	C₂×C₈	—	52	2	no	—
271	C₂×C₈	—	52	4	no	—
272	C₂×C₈	—	52	8	no	—
273	C₂×C₈	C₂²×#261	1	2	yes	—
274	C₂×C₈	C₂²×#261	1	1	yes	characteristic
275	C₂×C₈	C₂²×#261	1	1	yes	characteristic
276	C₄⋊C₄	—	13	2	no	—
277	C₄⋊C₄	—	13	2	no	—
278	C₄⋊C₄	—	13	1	no	—
279	C₄⋊C₄	C₄⋊#261	1	1	yes	characteristic
280	C₂²⋊C₄	—	13	2	no	—
281	C₂²⋊C₄	—	13	2	no	—
282	C₂²⋊C₄	C₄⋊#261	1	2	yes	—
283	C₄²	—	13	2	no	—
284	C₄²	—	13	1	no	—
285	C₄²	—	13	2	no	—
286	C₄²	—	13	1	no	—
287	C₄²	—	13	1	no	—
288	C₄²	—	13	2	no	—
289	C₄²	C₂²×#261	1	1	yes	characteristic
290	C₁₃	#17828	1	1	yes	characteristic, complemented, Sylow
291	C₂³	—	13	1	no	—
292	C₂³	C₂×C₄⋊#261	1	2	yes	—
293	Q₈	—	13	4	no	—
294	D₄	—	13	8	no	—
295	D₄	#88102	1	4	yes	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	13	4	no	—
299	C₂×C₄	—	13	4	no	—
300	C₂×C₄	—	13	1	no	—
301	C₂×C₄	—	13	2	no	—
302	C₂×C₄	—	13	2	no	—
303	C₂×C₄	—	13	2	no	—
304	C₂×C₄	—	13	1	no	—
305	C₂×C₄	—	13	1	no	—
306	C₂×C₄	—	13	2	no	—
307	C₂×C₄	—	26	1	no	—
308	C₂×C₄	—	26	1	no	—
309	C₂×C₄	—	52	1	no	—
310	C₂×C₄	—	52	2	no	—
311	C₂×C₄	C₂×C₄⋊#261	1	2	yes	—
312	C₂×C₄	C₂×C₄⋊#261	1	1	yes	characteristic
313	C₂×C₄	C₂×C₄⋊#261	1	1	yes	characteristic
314	C₂×C₄	C₂³×#261	1	1	yes	characteristic
315	C₈	—	2	2	no	—
316	C₈	—	2	1	no	—
317	C₈	—	26	1	no	—
318	C₈	—	26	2	no	—
319	C₈	—	26	1	no	—
320	C₈	—	52	2	no	—
321	C₈	—	52	2	no	—
322	C₈	—	52	4	no	—
323	C₈	#88238	1	2	yes	complemented
324	C₂²	—	2	2	no	—
325	C₂²	—	2	2	no	—
326	C₂²	—	13	2	no	—
327	C₂²	—	26	1	no	—
328	C₂²	—	26	1	no	—
329	C₂²	C₂×#88102	1	4	yes	—
330	C₂²	C₂²×C₄⋊#261	1	1	yes	characteristic
331	C₄	—	2	2	no	—
332	C₄	—	2	1	no	—
333	C₄	—	13	1	no	—
334	C₄	—	13	1	no	—
335	C₄	—	13	4	no	—
336	C₄	—	26	1	no	—
337	C₄	—	26	2	no	—
338	C₄	—	26	1	no	—
339	C₄	C₂×#88102	1	2	yes	—
340	C₄	C₂×#88238	1	1	yes	characteristic
341	C₄	#11719629	1	1	yes	characteristic
342	C₂	—	2	4	no	—
343	C₂	—	26	2	no	—
344	C₂	#13131481	1	1	yes	characteristic
345	C₂	C₂²×#88102	1	1	yes	characteristic
346	C₂	#13131498	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×C₄⋊#261
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×#261
Fitting	C₈×D₄×C₁₃	C₄
socle	C₂²×C₁₃	C₂²×C₄⋊C₄
prime radical	C₂×#6129	D₄
cosocle	C₄×Dic₁₃	C₂⁴
hypercenter	C₈×D₄	#261
hypocenter	C₁₃	#17828
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ C₈×D₄

Complements

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

N	H ≅ G/N	type
2. #13130406	C₂	semidirect
3. #13130405	C₂	semidirect
4. #13130394	C₂	semidirect
5. #13130393	C₂	semidirect
6. D₁₃⋊#335	C₂	semidirect
7. C₂×#11718254	C₂	semidirect
155. C₈×C₁₃	#131	semidirect
290. C₁₃	#17828	semidirect
323. C₈	#88238	semidirect

Maximal subgroups

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

subgroup	index
#13131481	2
#13130406	2
#13130405	2
#13130394	2
#13130393	2
D₁₃⋊#335	2
C₂×#11718254	2
C₈×#6129	2
#17828	13