Subgroups — #13131593

Order 1664.

506 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11719631	C₂	1	1	yes	maximal, characteristic, complemented
2	#11719623	C₂	1	1	yes	maximal, characteristic, complemented
3	D₁₃×AGL(1,ℤ/8)	C₂	1	1	yes	maximal, characteristic
4	#11719165	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#88115	C₂	1	1	yes	maximal, characteristic, complemented
6	#11719106	C₂	1	2	yes	maximal, complemented
7	SD₁₆×#261	C₂	1	2	yes	maximal, complemented
8	#11719102	C₂	1	2	yes	maximal, complemented
9	D₈×#261	C₂	1	2	yes	maximal, complemented
10	#11719099	C₂	1	1	yes	maximal, characteristic, complemented
11	#11719098	C₂	1	1	yes	maximal, characteristic, complemented
12	#41×D₁₃	C₂²	1	1	yes	characteristic
13	C₂×D₄×D₁₃	C₂²	1	1	yes	characteristic
14	C₂×C₂²⋊#261	—	2	2	no	—
15	Q₈×#261	C₂²	1	2	yes	complemented
16	D₄×#261	—	2	2	no	—
17	D₄×#261	—	2	1	no	—
18	D₄×#261	—	2	1	no	—
19	D₄×#261	C₂²	1	2	yes	—
20	D₄×#261	C₂²	1	2	yes	—
21	D₄×#261	C₂²	1	2	yes	complemented
22	#88237	—	2	1	no	—
23	#88237	C₂²	1	1	yes	characteristic
24	#88237	C₂²	1	1	yes	characteristic
25	C₂×C₄⋊#261	—	2	1	no	—
26	C₂×C₄⋊#261	C₂²	1	1	yes	characteristic
27	C₂×D₁₃⋊C₈	C₂²	1	1	yes	characteristic
28	C₁₃×AGL(1,ℤ/8)	C₄	1	1	yes	characteristic, complemented
29	#88203	C₄	1	1	yes	characteristic, complemented
30	#88186	C₄	1	1	yes	characteristic, complemented
31	#88168	C₄	1	2	yes	complemented
32	SD₁₆×D₁₃	C₂²	1	2	yes	—
33	#88165	C₄	1	2	yes	complemented
34	D₈×D₁₃	C₂²	1	2	yes	—
35	#88162	C₄	1	1	yes	characteristic, complemented
36	M₁₆×D₁₃	C₂²	1	1	yes	characteristic
37	#88117	C₂²	1	1	yes	characteristic, complemented
38	#88117	C₂²	1	1	yes	characteristic, complemented
39	#88115	C₂²	1	2	yes	—
40	#88115	C₂²	1	2	yes	—
41	#88115	C₂²	1	1	yes	characteristic, complemented
42	#88115	C₂²	1	1	yes	characteristic, complemented
43	#88102	C₂²	1	2	yes	complemented
44	#88101	C₂²	1	2	yes	complemented
45	#88100	C₂²	1	2	yes	complemented
46	C₈×#261	C₂²	1	2	yes	complemented
47	C₂³×D₁₃	—	2	1	no	—
48	C₂²×#261	—	2	1	no	—
49	C₂²×#261	—	2	2	no	—
50	C₂²×#261	—	2	2	no	—
51	C₂²×#261	—	2	2	no	—
52	C₁₃×#41	C₂×C₄	1	1	yes	characteristic, complemented
53	C₂×D₄×C₁₃	C₂×C₄	1	1	yes	characteristic, complemented
54	C₂×C₂²⋊D₁₃	—	2	1	no	—
55	#6131	C₂×C₄	1	1	yes	characteristic, complemented
56	Q₈×D₁₃	C₂³	1	1	yes	characteristic
57	#6129	—	2	1	no	—
58	#6129	C₂×C₄	1	1	yes	characteristic, complemented
59	D₄×D₁₃	—	2	1	no	—
60	D₄×D₁₃	—	2	1	no	—
61	D₄×D₁₃	—	2	1	no	—
62	D₄×D₁₃	C₂×C₄	1	2	yes	complemented
63	D₄×D₁₃	C₂³	1	2	yes	—
64	D₄×D₁₃	C₂³	1	1	yes	characteristic
65	#6127	—	2	1	no	—
66	#6127	C₂×C₄	1	1	yes	characteristic, complemented
67	C₂×D₅₂	C₂×C₄	1	1	yes	characteristic, complemented
68	C₂×C₄×D₁₃	—	2	1	no	—
69	C₂×C₄×D₁₃	C₂³	1	1	yes	characteristic
70	C₂²⋊#261	—	2	1	no	—
71	C₂²⋊#261	—	2	2	no	—
72	C₂²⋊#261	—	2	2	no	—
73	C₂²⋊#261	—	2	2	no	—
74	C₂²⋊#261	D₄	1	2	yes	—
75	C₂×#1057	—	2	1	no	—
76	C₄⋊#261	—	2	2	no	—
77	C₄⋊#261	—	2	1	no	—
78	C₄⋊#261	D₄	1	1	yes	characteristic
79	C₄⋊#261	D₄	1	1	yes	characteristic
80	C₄⋊#261	C₂³	1	1	yes	characteristic
81	C₄⋊#261	C₂³	1	2	yes	—
82	C₄⋊#261	C₂³	1	1	yes	characteristic
83	C₄×#261	—	2	2	no	—
84	C₄×#261	—	2	1	no	—
85	C₄×#261	C₂³	1	2	yes	—
86	D₁₃⋊C₈	—	2	1	no	—
87	D₁₃⋊C₈	C₂³	1	1	yes	characteristic
88	D₁₃⋊C₈	C₂³	1	1	yes	characteristic
89	C₁₃×SD₁₆	C₂×C₄	1	2	yes	complemented
90	C₁₃×D₈	C₂×C₄	1	2	yes	complemented
91	C₁₃×M₁₆	C₂×C₄	1	1	yes	characteristic, complemented
92	#6106	C₂×C₄	1	2	yes	complemented
93	#6105	C₂×C₄	1	2	yes	complemented
94	#6104	C₂×C₄	1	2	yes	complemented
95	#6104	C₂×C₄	1	2	yes	complemented
96	#6099	C₂×C₄	1	1	yes	characteristic, complemented
97	D₁₀₄	C₂×C₄	1	2	yes	complemented
98	#6095	C₂×C₄	1	2	yes	complemented
99	#6094	C₂×C₄	1	2	yes	complemented
100	C₈×D₁₃	C₂³	1	2	yes	—
101	#2946	—	13	1	no	maximal, Sylow
102	C₂³×C₁₃	—	2	1	no	—
103	C₂²×D₁₃	—	2	1	no	—
104	C₂²×D₁₃	—	2	1	no	—
105	C₂²×D₁₃	—	2	2	no	—
106	C₂²×D₁₃	—	2	2	no	—
107	C₂²×D₁₃	—	8	1	no	—
108	C₂²×D₁₃	—	8	1	no	—
109	C₂²×D₁₃	C₂×D₄	1	1	yes	characteristic
110	C₂×#261	—	2	2	no	—
111	C₂×#261	—	2	1	no	—
112	C₂×#261	—	2	1	no	—
113	C₂×#261	—	8	2	no	—
114	C₂×#261	—	8	2	no	—
115	C₂×#261	—	8	2	no	—
116	C₂×#261	—	8	1	no	—
117	C₂×#261	C₂×D₄	1	2	yes	—
118	C₂×#261	C₂×D₄	1	2	yes	—
119	Q₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
120	D₄×C₁₃	—	2	1	no	—
121	D₄×C₁₃	—	2	1	no	—
122	D₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
123	D₄×C₁₃	C₂²×C₄	1	2	yes	—
124	C₂×C₄×C₁₃	—	2	1	no	—
125	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
126	C₂²⋊D₁₃	—	2	1	no	—
127	C₂²⋊D₁₃	—	2	1	no	—
128	C₂²⋊D₁₃	—	2	2	no	—
129	C₂²⋊D₁₃	—	2	1	no	—
130	C₂²⋊D₁₃	—	2	2	no	—
131	C₂×Dic₁₃	—	2	1	no	—
132	C₂×Dic₁₃	#41	1	1	yes	characteristic
133	D₅₂	—	2	1	no	—
134	D₅₂	—	2	1	no	—
135	D₅₂	C₂²×C₄	1	2	yes	—
136	D₅₂	C₂²×C₄	1	1	yes	characteristic
137	C₄×D₁₃	—	2	1	no	—
138	C₄×D₁₃	—	2	1	no	—
139	C₄×D₁₃	—	2	1	no	—
140	C₄×D₁₃	C₂²×C₄	1	1	yes	characteristic
141	C₄×D₁₃	C₂×D₄	1	1	yes	characteristic
142	C₄×D₁₃	#41	1	1	yes	characteristic
143	C₄×D₁₃	C₂⁴	1	1	yes	characteristic
144	Dic₂₆	—	2	1	no	—
145	Dic₂₆	C₂²×C₄	1	1	yes	characteristic
146	#1057	—	2	1	no	—
147	#1057	—	2	1	no	—
148	C₈×C₁₃	C₂²×C₄	1	2	yes	—
149	#1055	C₂²×C₄	1	2	yes	—
150	C₂×AGL(1,ℤ/8)	—	13	1	no	—
151	#520	—	13	1	no	—
152	#518	—	13	1	no	—
153	#442	—	13	2	no	—
154	#440	—	13	2	no	—
155	C₄×SD₁₆	—	13	2	no	—
156	C₄×D₈	—	13	2	no	—
157	#428	—	13	1	no	—
158	#416	—	13	1	no	—
159	C₂×#102	—	13	1	no	—
160	#405	—	13	1	no	—
161	C₂²×C₁₃	—	2	1	no	—
162	C₂²×C₁₃	—	2	2	no	—
163	C₂²×C₁₃	—	8	1	no	—
164	C₂²×C₁₃	C₄×D₄	1	1	yes	characteristic
165	C₂×D₁₃	—	2	1	no	—
166	C₂×D₁₃	—	2	1	no	—
167	C₂×D₁₃	—	2	1	no	—
168	C₂×D₁₃	—	2	2	no	—
169	C₂×D₁₃	—	4	1	no	—
170	C₂×D₁₃	—	4	2	no	—
171	C₂×D₁₃	—	4	2	no	—
172	C₂×D₁₃	—	4	1	no	—
173	C₂×D₁₃	—	8	2	no	—
174	C₂×D₁₃	—	8	1	no	—
175	C₂×D₁₃	C₄×D₄	1	1	yes	characteristic
176	C₂×D₁₃	C₂²×D₄	1	1	yes	characteristic
177	#261	—	2	2	no	—
178	#261	—	2	2	no	—
179	#261	—	4	2	no	—
180	#261	—	4	1	no	—
181	#261	—	4	1	no	—
182	C₄×C₁₃	—	2	1	no	—
183	C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
184	C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
185	Dic₁₃	—	2	1	no	—
186	Dic₁₃	C₄×D₄	1	1	yes	characteristic
187	Dic₁₃	C₂×#41	1	1	yes	characteristic
188	C₂×#41	—	13	1	no	—
189	C₂²×D₄	—	13	1	no	—
190	AGL(1,ℤ/8)	—	13	1	no	—
191	AGL(1,ℤ/8)	—	13	1	no	—
192	AGL(1,ℤ/8)	—	13	2	no	—
193	AGL(1,ℤ/8)	—	13	2	no	—
194	AGL(1,ℤ/8)	—	13	1	no	—
195	AGL(1,ℤ/8)	#261	1	1	yes	characteristic, complemented
196	C₂×SD₁₆	—	13	2	no	—
197	C₂×D₈	—	13	2	no	—
198	C₂×M₁₆	—	13	1	no	—
199	C₂²×C₈	—	13	1	no	—
200	C₄×Q₈	—	13	2	no	—
201	C₄×D₄	—	13	2	no	—
202	C₄×D₄	—	13	2	no	—
203	C₄×D₄	—	13	2	no	—
204	C₄×D₄	—	26	2	no	—
205	C₄×D₄	—	26	1	no	—
206	C₄×D₄	—	26	1	no	—
207	#117	—	13	1	no	—
208	#117	—	13	1	no	—
209	#117	—	26	1	no	—
210	C₂×C₄⋊C₄	—	13	1	no	—
211	C₂×C₄⋊C₄	—	26	1	no	—
212	C₂×C₂²⋊C₄	—	26	2	no	—
213	#107	—	13	2	no	—
214	#106	—	13	2	no	—
215	#103	—	13	1	no	—
216	#103	—	13	1	no	—
217	#102	—	13	2	no	—
218	#102	—	13	2	no	—
219	#102	—	13	1	no	—
220	#102	—	13	1	no	—
221	#97	—	13	2	no	—
222	C₄×C₈	—	13	2	no	—
223	C₂×C₁₃	—	2	1	no	—
224	C₂×C₁₃	—	4	2	no	—
225	C₂×C₁₃	—	4	1	no	—
226	C₂×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
227	D₁₃	—	2	1	no	—
228	D₁₃	—	4	1	no	—
229	D₁₃	—	4	2	no	—
230	D₁₃	#576	1	1	yes	characteristic
231	D₁₃	#577	1	1	yes	characteristic
232	C₂⁴	—	26	1	no	—
233	#41	—	13	1	no	—
234	#41	—	13	1	no	—
235	#41	—	13	1	no	—
236	#41	—	26	1	no	—
237	#41	—	26	1	no	—
238	#41	C₂×#261	1	1	yes	characteristic, complemented
239	C₂×Q₈	—	13	1	no	—
240	C₂×D₄	—	13	1	no	—
241	C₂×D₄	—	13	2	no	—
242	C₂×D₄	—	13	2	no	—
243	C₂×D₄	—	13	1	no	—
244	C₂×D₄	—	26	1	no	—
245	C₂×D₄	—	26	1	no	—
246	C₂×D₄	—	26	1	no	—
247	C₂×D₄	—	26	1	no	—
248	C₂×D₄	C₂×#261	1	1	yes	characteristic, complemented
249	C₂²×C₄	—	13	1	no	—
250	C₂²×C₄	—	26	1	no	—
251	C₂²×C₄	—	26	2	no	—
252	C₂²×C₄	—	26	2	no	—
253	C₂²×C₄	—	26	2	no	—
254	C₂²×C₄	—	26	1	no	—
255	SD₁₆	—	13	2	no	—
256	SD₁₆	—	13	2	no	—
257	SD₁₆	—	13	2	no	—
258	SD₁₆	C₂×#261	1	2	yes	complemented
259	D₈	—	13	2	no	—
260	D₈	—	13	2	no	—
261	D₈	—	13	2	no	—
262	D₈	C₂×#261	1	2	yes	complemented
263	M₁₆	—	13	1	no	—
264	M₁₆	—	13	2	no	—
265	M₁₆	C₂×#261	1	1	yes	characteristic, complemented
266	C₂×C₈	—	13	2	no	—
267	C₂×C₈	—	13	1	no	—
268	C₂×C₈	—	13	1	no	—
269	C₂×C₈	—	26	1	no	—
270	C₂×C₈	—	26	1	no	—
271	C₄⋊C₄	—	13	1	no	—
272	C₄⋊C₄	—	13	1	no	—
273	C₄⋊C₄	—	13	2	no	—
274	C₄⋊C₄	—	13	1	no	—
275	C₄⋊C₄	—	13	1	no	—
276	C₄⋊C₄	—	26	1	no	—
277	C₄⋊C₄	—	26	2	no	—
278	C₂²⋊C₄	—	13	2	no	—
279	C₂²⋊C₄	—	26	2	no	—
280	C₂²⋊C₄	—	26	2	no	—
281	C₂²⋊C₄	—	26	2	no	—
282	C₂²⋊C₄	—	26	1	no	—
283	C₄²	—	13	2	no	—
284	C₄²	—	26	1	no	—
285	C₄²	—	26	2	no	—
286	C₁₃	#2946	1	1	yes	characteristic, complemented, Sylow
287	C₂³	—	2	1	no	—
288	C₂³	—	13	1	no	—
289	C₂³	—	26	1	no	—
290	C₂³	—	26	2	no	—
291	C₂³	—	26	2	no	—
292	C₂³	—	26	1	no	—
293	C₂³	—	104	1	no	—
294	C₂³	—	104	1	no	—
295	Q₈	—	13	1	no	—
296	Q₈	—	26	1	no	—
297	Q₈	C₂²×#261	1	1	yes	characteristic
298	D₄	—	2	1	no	—
299	D₄	—	2	1	no	—
300	D₄	—	13	1	no	—
301	D₄	—	13	2	no	—
302	D₄	—	26	1	no	—
303	D₄	—	26	2	no	—
304	D₄	—	26	2	no	—
305	D₄	—	26	1	no	—
306	D₄	—	26	1	no	—
307	D₄	—	26	1	no	—
308	D₄	—	26	1	no	—
309	D₄	C₂²×#261	1	1	yes	characteristic
310	D₄	C₂²×#261	1	2	yes	—
311	C₂×C₄	—	2	1	no	—
312	C₂×C₄	—	13	1	no	—
313	C₂×C₄	—	13	1	no	—
314	C₂×C₄	—	13	1	no	—
315	C₂×C₄	—	13	1	no	—
316	C₂×C₄	—	13	1	no	—
317	C₂×C₄	—	13	2	no	—
318	C₂×C₄	—	13	2	no	—
319	C₂×C₄	—	26	1	no	—
320	C₂×C₄	—	26	1	no	—
321	C₂×C₄	—	26	1	no	—
322	C₂×C₄	—	26	1	no	—
323	C₂×C₄	—	26	1	no	—
324	C₂×C₄	—	26	1	no	—
325	C₂×C₄	—	26	2	no	—
326	C₂×C₄	—	104	2	no	—
327	C₂×C₄	—	104	2	no	—
328	C₂×C₄	—	104	2	no	—
329	C₂×C₄	—	104	1	no	—
330	C₂×C₄	C₂²×#261	1	1	yes	characteristic
331	C₈	—	13	2	no	—
332	C₈	—	26	1	no	—
333	C₈	—	26	1	no	—
334	C₈	C₂²×#261	1	2	yes	—
335	C₂²	—	2	1	no	—
336	C₂²	—	2	2	no	—
337	C₂²	—	8	1	no	—
338	C₂²	—	13	1	no	—
339	C₂²	—	13	1	no	—
340	C₂²	—	26	1	no	—
341	C₂²	—	26	1	no	—
342	C₂²	—	26	1	no	—
343	C₂²	—	26	2	no	—
344	C₂²	—	52	1	no	—
345	C₂²	—	52	2	no	—
346	C₂²	—	52	2	no	—
347	C₂²	—	52	1	no	—
348	C₂²	—	104	1	no	—
349	C₂²	—	104	2	no	—
350	C₂²	D₄×#261	1	1	yes	characteristic
351	C₄	—	2	1	no	—
352	C₄	—	13	1	no	—
353	C₄	—	13	1	no	—
354	C₄	—	26	2	no	—
355	C₄	—	26	2	no	—
356	C₄	—	26	1	no	—
357	C₄	—	52	1	no	—
358	C₄	—	52	2	no	—
359	C₄	—	52	1	no	—
360	C₄	D₄×#261	1	1	yes	characteristic
361	C₄	C₂³×#261	1	1	yes	characteristic
362	C₂	—	2	1	no	—
363	C₂	—	4	2	no	—
364	C₂	—	4	1	no	—
365	C₂	—	13	1	no	—
366	C₂	—	13	1	no	—
367	C₂	—	26	1	no	—
368	C₂	—	52	2	no	—
369	C₂	—	52	1	no	—
370	C₂	C₂×D₄×#261	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×D₄×#261
commutator	C₄×C₁₃	C₂³×C₄
Frattini	C₄	C₂³×#261
Fitting	C₁₃×AGL(1,ℤ/8)	C₄
socle	C₂×C₁₃	C₂×C₄×D₄
prime radical	D₁₃×AGL(1,ℤ/8)	C₂
cosocle	C₄×D₁₃	C₂⁴
hypercenter	AGL(1,ℤ/8)	#261
hypocenter	C₁₃	#2946
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ AGL(1,ℤ/8)

Complements

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

N	H ≅ G/N	type
1. #11719631	C₂	semidirect
2. #11719623	C₂	semidirect
4. #11719165	C₂	semidirect
5. C₂×#88115	C₂	semidirect
6. #11719106	C₂	semidirect
7. SD₁₆×#261	C₂	semidirect
8. #11719102	C₂	semidirect
9. D₈×#261	C₂	semidirect
10. #11719099	C₂	semidirect
11. #11719098	C₂	semidirect
15. Q₈×#261	C₂²	semidirect
21. D₄×#261	C₂²	semidirect
28. C₁₃×AGL(1,ℤ/8)	C₄	semidirect
29. #88203	C₄	semidirect
30. #88186	C₄	semidirect
31. #88168	C₄	semidirect
33. #88165	C₄	semidirect
35. #88162	C₄	semidirect
37. #88117	C₂²	semidirect
38. #88117	C₂²	semidirect
41. #88115	C₂²	semidirect
42. #88115	C₂²	semidirect
43. #88102	C₂²	semidirect
44. #88101	C₂²	semidirect
45. #88100	C₂²	semidirect
46. C₈×#261	C₂²	semidirect
52. C₁₃×#41	C₂×C₄	semidirect
53. C₂×D₄×C₁₃	C₂×C₄	semidirect
55. #6131	C₂×C₄	semidirect
58. #6129	C₂×C₄	semidirect
62. D₄×D₁₃	C₂×C₄	semidirect
66. #6127	C₂×C₄	semidirect
67. C₂×D₅₂	C₂×C₄	semidirect
89. C₁₃×SD₁₆	C₂×C₄	semidirect
90. C₁₃×D₈	C₂×C₄	semidirect
91. C₁₃×M₁₆	C₂×C₄	semidirect
92. #6106	C₂×C₄	semidirect
93. #6105	C₂×C₄	semidirect
94. #6104	C₂×C₄	semidirect
95. #6104	C₂×C₄	semidirect
96. #6099	C₂×C₄	semidirect
97. D₁₀₄	C₂×C₄	semidirect
98. #6095	C₂×C₄	semidirect
99. #6094	C₂×C₄	semidirect
195. AGL(1,ℤ/8)	#261	semidirect
238. #41	C₂×#261	semidirect
248. C₂×D₄	C₂×#261	semidirect
258. SD₁₆	C₂×#261	semidirect
262. D₈	C₂×#261	semidirect
265. M₁₆	C₂×#261	semidirect
286. C₁₃	#2946	semidirect

Maximal subgroups

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

subgroup	index
#11719631	2
#11719623	2
D₁₃×AGL(1,ℤ/8)	2
#11719165	2
C₂×#88115	2
#11719106	2
SD₁₆×#261	2
#11719102	2
D₈×#261	2
#11719099	2
#11719098	2
#2946	13