Subgroups — #13128749

Order 1664.

1104 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	#11719522	C₂	1	4	yes	maximal, complemented
3	#11719439	C₂	1	2	yes	maximal, complemented
4	#11719438	C₂	1	4	yes	maximal, complemented
5	#11719437	C₂	1	2	yes	maximal, complemented
6	#11719435	C₂	1	2	yes	maximal, complemented
7	#11719433	C₂	1	2	yes	maximal, complemented
8	#11719431	C₂	1	2	yes	maximal, complemented
9	#11719430	C₂	1	2	yes	maximal, complemented
10	#11719219	C₂	1	4	yes	maximal, complemented
11	#11719215	C₂	1	2	yes	maximal, complemented
12	#11719209	C₂	1	2	yes	maximal, complemented
13	#11719182	C₂	1	1	yes	maximal, characteristic, complemented
14	C₄×#6127	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂²×Q₈×C₁₃	C₂²	1	2	yes	—
16	C₂×#6131	C₂²	1	4	yes	—
17	C₂×#6127	C₂²	1	1	yes	characteristic
18	C₂×#6127	C₂²	1	2	yes	—
19	C₁₃×C₄⋊Q₈	C₂²	1	2	yes	—
20	C₁₃×C₄⋊Q₈	C₂²	1	2	yes	—
21	C₁₃×C₄⋊Q₈	C₂²	1	2	yes	—
22	C₁₃×C₄⋊Q₈	C₂²	1	2	yes	—
23	C₂×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
24	C₂×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
25	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
26	#88201	C₂²	1	4	yes	—
27	#88201	C₂²	1	8	yes	—
28	#88200	C₂²	1	4	yes	—
29	#88200	C₂²	1	8	yes	—
30	Q₈×Dic₁₃	C₂²	1	4	yes	—
31	#88184	C₂²	1	2	yes	—
32	#88184	C₂²	1	2	yes	—
33	#88184	C₂²	1	8	yes	—
34	C₄×C₂²⋊D₁₃	C₂²	1	2	yes	—
35	C₄×C₂²⋊D₁₃	C₂²	1	2	yes	—
36	C₄×C₂²⋊D₁₃	C₂²	1	8	yes	—
37	#88180	C₂²	1	1	yes	characteristic
38	#88152	C₂²	1	8	yes	—
39	#88151	C₂²	1	4	yes	—
40	#88151	C₂²	1	4	yes	—
41	#88149	C₂²	1	4	yes	—
42	#88149	C₂²	1	4	yes	—
43	#88149	C₂²	1	4	yes	—
44	#88148	C₂²	1	8	yes	—
45	#88147	C₂²	1	4	yes	—
46	#88147	C₂²	1	4	yes	—
47	#88146	C₂²	1	4	yes	—
48	#88146	C₂²	1	4	yes	—
49	C₄⋊C₄×D₁₃	C₂²	1	4	yes	—
50	#88144	C₂²	1	4	yes	—
51	#88141	C₂²	1	4	yes	—
52	#88129	C₂²	1	2	yes	—
53	#88128	C₂²	1	1	yes	characteristic
54	C₄×D₅₂	C₂²	1	1	yes	characteristic
55	C₄×D₅₂	C₂²	1	2	yes	—
56	C₄×D₅₂	C₂²	1	1	yes	characteristic
57	C₄×D₅₂	C₂²	1	2	yes	—
58	#88126	C₂²	1	2	yes	—
59	C₄²×D₁₃	C₂²	1	2	yes	—
60	#88123	C₂²	1	1	yes	characteristic
61	C₄×Dic₂₆	C₂²	1	1	yes	characteristic
62	C₄×Dic₂₆	C₂²	1	1	yes	characteristic
63	C₂×Q₈×C₁₃	—	4	4	no	—
64	C₂×Q₈×C₁₃	C₂³	1	4	yes	—
65	C₂×Q₈×C₁₃	C₂³	1	4	yes	—
66	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
67	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
68	C₂²×C₄×C₁₃	C₂³	1	4	yes	—
69	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
70	C₂×C₂²⋊D₁₃	C₂³	1	2	yes	—
71	C₂×C₂²⋊D₁₃	C₂³	1	4	yes	—
72	#6131	—	4	8	no	—
73	#6127	—	2	4	no	—
74	#6127	—	2	8	no	—
75	C₂×D₅₂	C₂³	1	1	yes	characteristic
76	C₂×D₅₂	C₂³	1	2	yes	—
77	C₂×D₅₂	C₂³	1	2	yes	—
78	C₂×D₅₂	C₂³	1	2	yes	—
79	C₂×D₅₂	C₂³	1	8	yes	—
80	C₂×C₄×D₁₃	C₂³	1	2	yes	—
81	C₂×C₄×D₁₃	C₂³	1	2	yes	—
82	C₂×C₄×D₁₃	C₂³	1	2	yes	—
83	C₂×C₄×D₁₃	C₂³	1	4	yes	—
84	C₂×C₄×D₁₃	C₂³	1	8	yes	—
85	C₂×Dic₂₆	C₂³	1	1	yes	characteristic
86	C₂×Dic₂₆	C₂³	1	2	yes	—
87	C₁₃×C₄⋊C₄	C₂³	1	4	yes	—
88	C₁₃×C₄⋊C₄	C₂³	1	4	yes	—
89	C₁₃×C₄⋊C₄	C₂³	1	4	yes	—
90	C₁₃×C₄⋊C₄	C₂³	1	4	yes	—
91	C₄²×C₁₃	C₂³	1	1	yes	characteristic
92	C₄²×C₁₃	C₂³	1	1	yes	characteristic
93	C₄²×C₁₃	C₂³	1	1	yes	characteristic
94	C₄²×C₁₃	C₂³	1	1	yes	characteristic
95	C₂²⋊Dic₁₃	C₂³	1	2	yes	—
96	#6103	C₂³	1	2	yes	—
97	#6103	C₂³	1	2	yes	—
98	#6103	C₂³	1	4	yes	—
99	#6103	C₂³	1	4	yes	—
100	#6103	C₂³	1	8	yes	—
101	#6103	C₂³	1	8	yes	—
102	#6103	C₂³	1	8	yes	—
103	C₄⋊Dic₁₃	C₂³	1	1	yes	characteristic
104	C₄⋊Dic₁₃	C₂³	1	1	yes	characteristic
105	C₄⋊Dic₁₃	C₂³	1	2	yes	—
106	C₄⋊Dic₁₃	C₂³	1	2	yes	—
107	C₄⋊Dic₁₃	C₂³	1	8	yes	—
108	#6101	C₂³	1	2	yes	—
109	#6101	C₂³	1	2	yes	—
110	#6101	C₂³	1	8	yes	—
111	C₄×Dic₁₃	C₂³	1	2	yes	—
112	C₄×Dic₁₃	C₂³	1	2	yes	—
113	C₄×Dic₁₃	C₂³	1	2	yes	—
114	C₄×Dic₁₃	C₂³	1	8	yes	—
115	#3511	—	13	1	no	maximal, Sylow
116	C₂³×C₁₃	C₂⁴	1	1	yes	characteristic
117	C₂²×D₁₃	C₂⁴	1	2	yes	—
118	C₂²×D₁₃	C₂⁴	1	4	yes	—
119	Q₈×C₁₃	—	2	8	no	—
120	Q₈×C₁₃	—	4	4	no	—
121	C₂×C₄×C₁₃	—	2	2	no	—
122	C₂×C₄×C₁₃	—	2	2	no	—
123	C₂×C₄×C₁₃	—	4	4	no	—
124	C₂×C₄×C₁₃	#41	1	4	yes	complemented
125	C₂×C₄×C₁₃	C₂⁴	1	8	yes	—
126	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
127	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
128	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
129	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
130	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
131	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
132	C₂²⋊D₁₃	—	2	4	no	—
133	C₂²⋊D₁₃	—	2	8	no	—
134	C₂×Dic₁₃	C₂⁴	1	2	yes	—
135	C₂×Dic₁₃	C₂⁴	1	2	yes	—
136	C₂×Dic₁₃	C₂⁴	1	2	yes	—
137	C₂×Dic₁₃	C₂⁴	1	4	yes	—
138	D₅₂	—	2	4	no	—
139	D₅₂	—	2	2	no	—
140	D₅₂	—	2	4	no	—
141	D₅₂	—	2	4	no	—
142	D₅₂	—	4	8	no	—
143	C₄×D₁₃	—	2	4	no	—
144	C₄×D₁₃	—	2	8	no	—
145	C₄×D₁₃	—	2	4	no	—
146	C₄×D₁₃	—	2	4	no	—
147	C₄×D₁₃	—	4	8	no	—
148	Dic₂₆	—	2	4	no	—
149	Dic₂₆	—	2	2	no	—
150	#556	—	13	2	no	—
151	#555	—	13	2	no	—
152	#554	—	13	2	no	—
153	#553	—	13	4	no	—
154	#548	—	13	4	no	—
155	#547	—	13	2	no	—
156	#542	—	13	4	no	—
157	#537	—	13	2	no	—
158	#533	—	13	2	no	—
159	#532	—	13	2	no	—
160	#532	—	13	1	no	—
161	C₂×C₄⋊Q₈	D₁₃	1	1	yes	characteristic, complemented
162	#520	—	13	2	no	—
163	C₄×#41	—	13	1	no	—
164	C₂²×C₁₃	—	2	1	no	—
165	C₂²×C₁₃	—	2	1	no	—
166	C₂²×C₁₃	C₂×#41	1	2	yes	—
167	C₂²×C₁₃	C₂⁵	1	1	yes	characteristic
168	C₂×D₁₃	—	2	2	no	—
169	C₂×D₁₃	—	2	4	no	—
170	C₂×D₁₃	—	2	2	no	—
171	C₂×D₁₃	—	2	4	no	—
172	C₂×D₁₃	—	2	2	no	—
173	C₂×D₁₃	—	2	4	no	—
174	C₄×C₁₃	—	2	8	no	—
175	C₄×C₁₃	—	2	1	no	—
176	C₄×C₁₃	—	2	1	no	—
177	C₄×C₁₃	C₂×#41	1	2	yes	—
178	C₄×C₁₃	C₂×#41	1	2	yes	—
179	C₄×C₁₃	H(5,𝔽₂)	1	2	yes	—
180	C₄×C₁₃	2₋¹⁺⁴	1	2	yes	—
181	Dic₁₃	—	2	4	no	—
182	Dic₁₃	—	2	2	no	—
183	Dic₁₃	—	2	2	no	—
184	Dic₁₃	—	2	2	no	—
185	C₂×#41	—	13	4	no	—
186	C₂×#41	—	13	1	no	—
187	C₂×#41	—	13	2	no	—
188	C₂²×Q₈	C₂×D₁₃	1	2	yes	—
189	C₄⋊Q₈	—	13	1	no	—
190	C₄⋊Q₈	C₂×D₁₃	1	2	yes	—
191	C₄⋊Q₈	C₂×D₁₃	1	2	yes	—
192	C₄⋊Q₈	C₂×D₁₃	1	2	yes	—
193	C₄⋊Q₈	C₂×D₁₃	1	2	yes	—
194	#127	—	13	1	no	—
195	#126	—	13	8	no	—
196	#125	—	13	4	no	—
197	#124	—	13	4	no	—
198	#124	—	13	8	no	—
199	#124	—	13	2	no	—
200	#123	—	13	8	no	—
201	C₂²⋊Q₈	—	13	4	no	—
202	C₂²⋊Q₈	—	13	8	no	—
203	C₂²⋊Q₈	—	13	4	no	—
204	C₂²⋊Q₈	—	13	4	no	—
205	#121	—	13	2	no	—
206	#121	—	13	2	no	—
207	#121	—	13	8	no	—
208	#121	—	13	4	no	—
209	#121	—	13	4	no	—
210	#121	—	13	4	no	—
211	C₄×Q₈	—	13	4	no	—
212	C₄×Q₈	—	13	4	no	—
213	C₄×Q₈	—	13	1	no	—
214	C₄×Q₈	—	13	1	no	—
215	C₄×D₄	—	13	2	no	—
216	C₄×D₄	—	13	2	no	—
217	C₄×D₄	—	13	8	no	—
218	C₄×D₄	—	13	4	no	—
219	C₄×D₄	—	13	4	no	—
220	C₄×D₄	—	13	1	no	—
221	C₄×D₄	—	13	1	no	—
222	C₄×D₄	—	13	2	no	—
223	C₄×D₄	—	13	2	no	—
224	#117	—	13	1	no	—
225	#117	—	13	4	no	—
226	#117	—	13	4	no	—
227	#117	—	13	2	no	—
228	C₂×C₄⋊C₄	—	13	4	no	—
229	C₂×C₄⋊C₄	C₂×D₁₃	1	2	yes	—
230	C₂×C₄⋊C₄	C₂×D₁₃	1	2	yes	—
231	C₂×C₄²	—	13	2	no	—
232	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
233	C₂×C₁₃	—	2	2	no	—
234	C₂×C₁₃	C₂²×#41	1	1	yes	characteristic
235	C₂×C₁₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
236	C₂×C₁₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
237	D₁₃	—	4	2	no	—
238	D₁₃	—	4	4	no	—
239	#41	—	26	4	no	—
240	#41	—	26	8	no	—
241	#41	—	52	8	no	—
242	C₂×Q₈	—	4	4	no	—
243	C₂×Q₈	—	13	1	no	—
244	C₂×Q₈	—	13	2	no	—
245	C₂×Q₈	C₂²×D₁₃	1	4	yes	—
246	C₂×Q₈	C₂²×D₁₃	1	4	yes	—
247	C₂×D₄	—	13	2	no	—
248	C₂×D₄	—	13	4	no	—
249	C₂×D₄	—	13	2	no	—
250	C₂×D₄	—	13	1	no	—
251	C₂×D₄	—	13	2	no	—
252	C₂×D₄	—	13	2	no	—
253	C₂×D₄	—	13	8	no	—
254	C₂²×C₄	—	13	2	no	—
255	C₂²×C₄	—	13	2	no	—
256	C₂²×C₄	—	13	2	no	—
257	C₂²×C₄	—	13	4	no	—
258	C₂²×C₄	—	13	8	no	—
259	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
260	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
261	C₂²×C₄	C₂²×D₁₃	1	4	yes	—
262	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
263	C₄⋊C₄	—	13	1	no	—
264	C₄⋊C₄	—	13	8	no	—
265	C₄⋊C₄	—	13	1	no	—
266	C₄⋊C₄	—	13	2	no	—
267	C₄⋊C₄	—	13	2	no	—
268	C₄⋊C₄	—	13	2	no	—
269	C₄⋊C₄	—	13	2	no	—
270	C₄⋊C₄	—	13	8	no	—
271	C₄⋊C₄	C₂²×D₁₃	1	4	yes	—
272	C₄⋊C₄	C₂²×D₁₃	1	4	yes	—
273	C₄⋊C₄	C₂²×D₁₃	1	4	yes	—
274	C₄⋊C₄	C₂²×D₁₃	1	4	yes	—
275	C₂²⋊C₄	—	13	2	no	—
276	C₂²⋊C₄	—	13	8	no	—
277	C₂²⋊C₄	—	13	2	no	—
278	C₂²⋊C₄	—	13	2	no	—
279	C₂²⋊C₄	—	13	8	no	—
280	C₂²⋊C₄	—	13	8	no	—
281	C₂²⋊C₄	—	13	4	no	—
282	C₂²⋊C₄	—	13	4	no	—
283	C₄²	—	13	2	no	—
284	C₄²	—	13	2	no	—
285	C₄²	—	13	2	no	—
286	C₄²	—	13	8	no	—
287	C₄²	C₂²×D₁₃	1	1	yes	characteristic
288	C₄²	C₂²×D₁₃	1	1	yes	characteristic
289	C₄²	C₂²×D₁₃	1	1	yes	characteristic
290	C₄²	C₂²×D₁₃	1	1	yes	characteristic
291	C₁₃	#3511	1	1	yes	characteristic, complemented, Sylow
292	C₂³	—	13	2	no	—
293	C₂³	—	13	4	no	—
294	C₂³	C₂³×D₁₃	1	1	yes	characteristic
295	Q₈	—	2	8	no	—
296	Q₈	—	4	4	no	—
297	Q₈	—	26	2	no	—
298	Q₈	—	26	4	no	—
299	D₄	—	26	4	no	—
300	D₄	—	26	8	no	—
301	D₄	—	26	4	no	—
302	D₄	—	26	2	no	—
303	D₄	—	26	4	no	—
304	D₄	—	26	4	no	—
305	D₄	—	52	8	no	—
306	C₂×C₄	—	2	2	no	—
307	C₂×C₄	—	2	2	no	—
308	C₂×C₄	—	4	4	no	—
309	C₂×C₄	—	13	2	no	—
310	C₂×C₄	—	13	2	no	—
311	C₂×C₄	—	13	4	no	—
312	C₂×C₄	—	13	2	no	—
313	C₂×C₄	—	26	4	no	—
314	C₂×C₄	—	26	8	no	—
315	C₂×C₄	—	26	4	no	—
316	C₂×C₄	—	26	4	no	—
317	C₂×C₄	—	52	8	no	—
318	C₂×C₄	#6131	1	4	yes	complemented
319	C₂×C₄	C₂³×D₁₃	1	8	yes	—
320	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
321	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
322	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
323	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
324	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
325	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
326	C₂²	—	2	1	no	—
327	C₂²	—	2	1	no	—
328	C₂²	—	26	2	no	—
329	C₂²	—	26	4	no	—
330	C₂²	—	26	2	no	—
331	C₂²	—	26	4	no	—
332	C₂²	—	26	2	no	—
333	C₂²	—	26	4	no	—
334	C₂²	C₂×#6131	1	2	yes	—
335	C₂²	C₂⁴×D₁₃	1	1	yes	characteristic
336	C₄	—	2	1	no	—
337	C₄	—	2	8	no	—
338	C₄	—	2	1	no	—
339	C₄	—	26	2	no	—
340	C₄	—	26	2	no	—
341	C₄	—	26	4	no	—
342	C₄	—	26	2	no	—
343	C₄	#88251	1	2	yes	—
344	C₄	C₂×#6131	1	2	yes	—
345	C₄	C₂×#6131	1	2	yes	—
346	C₄	#88254	1	2	yes	—
347	C₂	—	2	2	no	—
348	C₂	—	52	2	no	—
349	C₂	—	52	4	no	—
350	C₂	C₂×#88251	1	1	yes	characteristic
351	C₂	C₂²×#6131	1	1	yes	characteristic
352	C₂	C₂×#88254	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₁₃
commutator, socle, cosocle	C₂²×C₁₃	C₂⁵
Fitting	C₂×C₁₃×C₄⋊Q₈	C₂
hypercenter	C₂×C₄⋊Q₈	D₁₃
hypocenter	C₁₃	#3511
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
1. C₂×C₁₃×C₄⋊Q₈	C₂	semidirect
2. #11719522	C₂	semidirect
3. #11719439	C₂	semidirect
4. #11719438	C₂	semidirect
5. #11719437	C₂	semidirect
6. #11719435	C₂	semidirect
7. #11719433	C₂	semidirect
8. #11719431	C₂	semidirect
9. #11719430	C₂	semidirect
10. #11719219	C₂	semidirect
11. #11719215	C₂	semidirect
12. #11719209	C₂	semidirect
13. #11719182	C₂	semidirect
14. C₄×#6127	C₂	semidirect
124. C₂×C₄×C₁₃	#41	semidirect
161. C₂×C₄⋊Q₈	D₁₃	semidirect
291. C₁₃	#3511	semidirect
318. C₂×C₄	#6131	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₁₃×C₄⋊Q₈	2
#11719522	2
#11719439	2
#11719438	2
#11719437	2
#11719435	2
#11719433	2
#11719431	2
#11719430	2
#11719219	2
#11719215	2
#11719209	2
#11719182	2
C₄×#6127	2
#3511	13