Subgroups — #13128688

Order 1664.

1140 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×D₄×C₁₃	C₂	1	1	yes	maximal, characteristic, complemented
2	#11719510	C₂	1	2	yes	maximal, complemented
3	#11719496	C₂	1	4	yes	maximal, complemented
4	#11719268	C₂	1	2	yes	maximal, complemented
5	#11719264	C₂	1	4	yes	maximal, complemented
6	#11719263	C₂	1	4	yes	maximal, complemented
7	#11719258	C₂	1	2	yes	maximal, complemented
8	#11719256	C₂	1	2	yes	maximal, complemented
9	#11719247	C₂	1	2	yes	maximal, complemented
10	#11719209	C₂	1	2	yes	maximal, complemented
11	#11719199	C₂	1	4	yes	maximal, complemented
12	#11719182	C₂	1	1	yes	maximal, characteristic, complemented
13	#11719175	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂²×D₄×C₁₃	C₂²	1	1	yes	characteristic
15	C₂³×C₄×C₁₃	C₂²	1	2	yes	—
16	C₂×#6127	C₂²	1	2	yes	—
17	C₂×#6127	C₂²	1	4	yes	complemented
18	C₄×D₄×C₁₃	C₂²	1	2	yes	—
19	C₄×D₄×C₁₃	C₂²	1	2	yes	—
20	C₄×D₄×C₁₃	C₂²	1	2	yes	—
21	C₄×D₄×C₁₃	C₂²	1	2	yes	—
22	C₂×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
23	C₂×C₁₃×C₂²⋊C₄	C₂²	1	2	yes	—
24	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
25	#88207	C₂²	1	8	yes	—
26	#88194	C₂²	1	4	yes	—
27	#88192	C₂²	1	4	yes	—
28	#88192	C₂²	1	4	yes	—
29	#88190	C₂²	1	4	yes	—
30	#88184	C₂²	1	2	yes	—
31	#88184	C₂²	1	2	yes	—
32	#88184	C₂²	1	4	yes	complemented
33	#88183	C₂²	1	8	yes	complemented
34	C₄×C₂²⋊D₁₃	C₂²	1	8	yes	—
35	C₄×C₂²⋊D₁₃	C₂²	1	8	yes	complemented
36	#88180	C₂²	1	2	yes	—
37	#88180	C₂²	1	4	yes	complemented
38	#88178	C₂²	1	2	yes	—
39	#88178	C₂²	1	2	yes	—
40	#88178	C₂²	1	4	yes	complemented
41	#88147	C₂²	1	4	yes	complemented
42	#88146	C₂²	1	4	yes	complemented
43	C₄⋊C₄×D₁₃	C₂²	1	4	yes	complemented
44	#88141	C₂²	1	4	yes	complemented
45	#88139	C₂²	1	8	yes	complemented
46	#88138	C₂²	1	8	yes	complemented
47	#88137	C₂²	1	8	yes	complemented
48	#88133	C₂²	1	8	yes	complemented
49	#88129	C₂²	1	2	yes	complemented
50	#88128	C₂²	1	1	yes	characteristic, complemented
51	C₄×D₅₂	C₂²	1	2	yes	complemented
52	C₄×D₅₂	C₂²	1	2	yes	complemented
53	#88124	C₂²	1	2	yes	complemented
54	#88123	C₂²	1	1	yes	characteristic, complemented
55	#88123	C₂²	1	1	yes	characteristic, complemented
56	#88123	C₂²	1	1	yes	characteristic, complemented
57	C₄×Dic₂₆	C₂²	1	2	yes	complemented
58	C₄×Dic₂₆	C₂²	1	2	yes	complemented
59	C₂⁴×C₁₃	C₂³	1	2	yes	—
60	C₂×D₄×C₁₃	—	2	4	no	—
61	C₂×D₄×C₁₃	C₂³	1	2	yes	—
62	C₂×D₄×C₁₃	C₂³	1	2	yes	—
63	C₂²×C₄×C₁₃	—	4	4	no	—
64	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
65	C₂²×C₄×C₁₃	C₂³	1	4	yes	—
66	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
67	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
68	C₂²×C₄×C₁₃	C₂³	1	4	yes	—
69	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
70	C₂×C₂²⋊D₁₃	C₂³	1	4	yes	—
71	C₂×C₂²⋊D₁₃	C₂³	1	8	yes	—
72	C₂×C₂²⋊D₁₃	C₂³	1	8	yes	—
73	#6127	—	2	8	no	—
74	#6127	—	4	8	no	—
75	C₂×D₅₂	C₂³	1	2	yes	—
76	C₂×D₅₂	C₂³	1	2	yes	—
77	C₂×D₅₂	C₂³	1	2	yes	—
78	C₂×C₄×D₁₃	C₂³	1	4	yes	—
79	C₂×C₄×D₁₃	C₂³	1	8	yes	—
80	C₂×Dic₂₆	C₂³	1	2	yes	—
81	C₂×Dic₂₆	C₂³	1	2	yes	—
82	C₂×Dic₂₆	C₂³	1	2	yes	—
83	C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
84	C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
85	C₁₃×C₂²⋊C₄	C₂³	1	4	yes	—
86	C₁₃×C₂²⋊C₄	C₂³	1	4	yes	—
87	C₄²×C₁₃	C₂³	1	1	yes	characteristic
88	C₄²×C₁₃	C₂³	1	1	yes	characteristic
89	C₄²×C₁₃	C₂³	1	1	yes	characteristic
90	C₄²×C₁₃	C₂³	1	1	yes	characteristic
91	C₂²⋊Dic₁₃	C₂³	1	8	yes	—
92	C₂²⋊Dic₁₃	C₂³	1	8	yes	—
93	C₂²⋊Dic₁₃	C₂³	1	4	yes	—
94	#6103	C₂³	1	4	yes	—
95	#6103	C₂³	1	4	yes	—
96	#6103	C₂³	1	8	yes	—
97	C₄⋊Dic₁₃	C₂³	1	2	yes	—
98	C₄⋊Dic₁₃	C₂³	1	2	yes	—
99	C₄⋊Dic₁₃	C₂³	1	2	yes	—
100	C₄⋊Dic₁₃	C₂³	1	2	yes	—
101	C₄⋊Dic₁₃	C₂³	1	2	yes	—
102	C₄⋊Dic₁₃	C₂³	1	2	yes	—
103	#6101	C₂³	1	4	yes	—
104	#6101	C₂³	1	4	yes	—
105	#6101	C₂³	1	8	yes	—
106	C₄×Dic₁₃	C₂³	1	4	yes	—
107	C₄×Dic₁₃	C₂³	1	8	yes	—
108	#3506	—	13	1	no	maximal, Sylow
109	C₂³×C₁₃	—	2	4	no	—
110	C₂³×C₁₃	—	4	2	no	—
111	C₂³×C₁₃	—	4	2	no	—
112	C₂³×C₁₃	C₂⁴	1	4	yes	—
113	C₂³×C₁₃	C₂⁴	1	1	yes	characteristic
114	C₂²×D₁₃	C₂⁴	1	4	yes	—
115	D₄×C₁₃	—	2	4	no	—
116	D₄×C₁₃	—	2	4	no	—
117	C₂×C₄×C₁₃	—	2	8	no	—
118	C₂×C₄×C₁₃	—	2	2	no	—
119	C₂×C₄×C₁₃	—	4	2	no	—
120	C₂×C₄×C₁₃	—	4	4	no	—
121	C₂×C₄×C₁₃	—	4	1	no	—
122	C₂×C₄×C₁₃	#41	1	4	yes	complemented
123	C₂×C₄×C₁₃	C₂⁴	1	4	yes	—
124	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
125	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
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	C₂²⋊D₁₃	—	2	8	no	—
131	C₂²⋊D₁₃	—	4	8	no	—
132	C₂²⋊D₁₃	—	4	8	no	—
133	C₂×Dic₁₃	C₂⁴	1	4	yes	—
134	C₂×Dic₁₃	C₂⁴	1	4	yes	—
135	C₂×Dic₁₃	C₂⁴	1	4	yes	—
136	D₅₂	—	2	4	no	—
137	D₅₂	—	2	4	no	—
138	D₅₂	—	4	2	no	—
139	C₄×D₁₃	—	2	8	no	—
140	C₄×D₁₃	—	4	8	no	—
141	Dic₂₆	—	2	4	no	—
142	Dic₂₆	—	2	4	no	—
143	Dic₂₆	—	4	2	no	—
144	#559	—	13	4	no	—
145	#556	—	13	2	no	—
146	#555	—	13	2	no	—
147	#553	—	13	4	no	—
148	#550	—	13	2	no	—
149	#547	—	13	2	no	—
150	#538	—	13	4	no	—
151	#535	—	13	2	no	—
152	#533	—	13	1	no	—
153	#532	—	13	1	no	—
154	#525	—	13	4	no	—
155	#520	—	13	2	no	—
156	C₂×C₄×D₄	D₁₃	1	1	yes	characteristic, complemented
157	C₂²×C₁₃	—	2	1	no	—
158	C₂²×C₁₃	—	2	1	no	—
159	C₂²×C₁₃	—	2	4	no	—
160	C₂²×C₁₃	—	2	4	no	—
161	C₂²×C₁₃	—	2	4	no	—
162	C₂²×C₁₃	—	4	8	no	—
163	C₂²×C₁₃	C₂×#41	1	2	yes	—
164	C₂²×C₁₃	C₂⁵	1	1	yes	characteristic
165	C₂×D₁₃	—	2	4	no	—
166	C₂×D₁₃	—	2	4	no	—
167	C₂×D₁₃	—	2	4	no	—
168	C₄×C₁₃	—	2	4	no	—
169	C₄×C₁₃	—	2	1	no	—
170	C₄×C₁₃	—	2	1	no	—
171	C₄×C₁₃	—	2	1	no	—
172	C₄×C₁₃	C₂×#41	1	2	yes	—
173	C₄×C₁₃	C₂×#41	1	2	yes	—
174	C₄×C₁₃	H(5,𝔽₂)	1	2	yes	—
175	Dic₁₃	—	2	4	no	—
176	Dic₁₃	—	2	4	no	—
177	Dic₁₃	—	2	4	no	—
178	C₂×#41	—	13	2	no	—
179	C₂×#41	—	13	4	no	—
180	C₂²×D₄	C₂×D₁₃	1	1	yes	characteristic
181	C₂³×C₄	C₂×D₁₃	1	2	yes	—
182	C₄⋊Q₈	—	13	1	no	—
183	C₄⋊Q₈	—	13	1	no	—
184	C₄⋊Q₈	—	13	1	no	—
185	#127	—	13	4	no	—
186	#127	—	13	1	no	—
187	#126	—	13	8	no	—
188	#125	—	13	2	no	—
189	#124	—	13	4	no	—
190	#124	—	13	8	no	—
191	#124	—	13	2	no	—
192	#123	—	13	8	no	—
193	#123	—	13	8	no	—
194	C₂²⋊Q₈	—	13	2	no	—
195	C₂²⋊Q₈	—	13	4	no	—
196	C₂²⋊Q₈	—	13	2	no	—
197	#121	—	13	4	no	—
198	#121	—	13	4	no	—
199	#121	—	13	2	no	—
200	#121	—	13	4	no	—
201	#121	—	13	2	no	—
202	#121	—	13	8	no	—
203	#120	—	13	8	no	—
204	C₄×Q₈	—	13	4	no	—
205	C₄×Q₈	—	13	2	no	—
206	C₄×Q₈	—	13	2	no	—
207	C₄×D₄	—	13	8	no	—
208	C₄×D₄	—	13	8	no	—
209	C₄×D₄	—	13	4	no	—
210	C₄×D₄	—	13	2	no	—
211	C₄×D₄	—	13	2	no	—
212	C₄×D₄	C₂×D₁₃	1	2	yes	—
213	C₄×D₄	C₂×D₁₃	1	2	yes	—
214	C₄×D₄	C₂×D₁₃	1	2	yes	—
215	C₄×D₄	C₂×D₁₃	1	2	yes	—
216	#117	—	13	2	no	—
217	#117	—	13	4	no	—
218	#117	—	13	4	no	—
219	C₂×C₄⋊C₄	—	13	4	no	—
220	C₂×C₄⋊C₄	C₂×D₁₃	1	1	yes	characteristic
221	C₂×C₂²⋊C₄	C₂×D₁₃	1	2	yes	—
222	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
223	C₂×C₁₃	—	2	2	no	—
224	C₂×C₁₃	—	4	4	no	—
225	C₂×C₁₃	C₂²×#41	1	1	yes	characteristic
226	C₂×C₁₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
227	C₂×C₁₃	#585	1	1	yes	characteristic
228	D₁₃	—	4	4	no	—
229	C₂⁴	C₂²×D₁₃	1	2	yes	—
230	#41	—	26	8	no	—
231	#41	—	52	8	no	—
232	C₂×Q₈	—	13	2	no	—
233	C₂×Q₈	—	13	2	no	—
234	C₂×Q₈	—	13	2	no	—
235	C₂×D₄	—	2	4	no	—
236	C₂×D₄	—	13	4	no	—
237	C₂×D₄	—	13	8	no	—
238	C₂×D₄	—	13	8	no	—
239	C₂×D₄	—	13	2	no	—
240	C₂×D₄	—	13	2	no	—
241	C₂×D₄	—	13	2	no	—
242	C₂×D₄	C₂²×D₁₃	1	2	yes	—
243	C₂×D₄	C₂²×D₁₃	1	2	yes	—
244	C₂²×C₄	—	4	4	no	—
245	C₂²×C₄	—	13	4	no	—
246	C₂²×C₄	—	13	8	no	—
247	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
248	C₂²×C₄	C₂²×D₁₃	1	4	yes	—
249	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
250	C₂²×C₄	C₂²×D₁₃	1	2	yes	—
251	C₂²×C₄	C₂²×D₁₃	1	4	yes	—
252	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
253	C₄⋊C₄	—	13	2	no	—
254	C₄⋊C₄	—	13	8	no	—
255	C₄⋊C₄	—	13	2	no	—
256	C₄⋊C₄	—	13	2	no	—
257	C₄⋊C₄	—	13	2	no	—
258	C₄⋊C₄	—	13	4	no	—
259	C₄⋊C₄	—	13	2	no	—
260	C₄⋊C₄	—	13	4	no	—
261	C₄⋊C₄	—	13	2	no	—
262	C₄⋊C₄	C₂²×D₁₃	1	2	yes	—
263	C₄⋊C₄	C₂²×D₁₃	1	2	yes	—
264	C₂²⋊C₄	—	13	4	no	—
265	C₂²⋊C₄	—	13	8	no	—
266	C₂²⋊C₄	—	13	8	no	—
267	C₂²⋊C₄	—	13	8	no	—
268	C₂²⋊C₄	—	13	4	no	—
269	C₂²⋊C₄	—	13	4	no	—
270	C₂²⋊C₄	C₂²×D₁₃	1	4	yes	—
271	C₂²⋊C₄	C₂²×D₁₃	1	4	yes	—
272	C₄²	—	13	8	no	—
273	C₄²	—	13	4	no	—
274	C₄²	C₂²×D₁₃	1	1	yes	characteristic
275	C₄²	C₂²×D₁₃	1	1	yes	characteristic
276	C₄²	C₂²×D₁₃	1	1	yes	characteristic
277	C₄²	C₂²×D₁₃	1	1	yes	characteristic
278	C₁₃	#3506	1	1	yes	characteristic, complemented, Sylow
279	C₂³	—	2	4	no	—
280	C₂³	—	4	2	no	—
281	C₂³	—	4	2	no	—
282	C₂³	—	13	4	no	—
283	C₂³	C₂³×D₁₃	1	4	yes	—
284	C₂³	C₂³×D₁₃	1	1	yes	characteristic
285	Q₈	—	26	4	no	—
286	Q₈	—	26	4	no	—
287	Q₈	—	52	2	no	—
288	D₄	—	2	4	no	—
289	D₄	—	2	4	no	—
290	D₄	—	26	8	no	—
291	D₄	—	26	4	no	—
292	D₄	—	26	4	no	—
293	D₄	—	52	2	no	—
294	D₄	—	52	8	no	—
295	D₄	—	52	8	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	2	8	no	—
298	C₂×C₄	—	4	4	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	—	4	1	no	—
301	C₂×C₄	—	13	4	no	—
302	C₂×C₄	—	13	4	no	—
303	C₂×C₄	—	13	4	no	—
304	C₂×C₄	—	26	8	no	—
305	C₂×C₄	—	52	8	no	—
306	C₂×C₄	#6127	1	4	yes	complemented
307	C₂×C₄	C₂³×D₁₃	1	4	yes	—
308	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
309	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
310	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
311	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
312	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
313	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
314	C₂²	—	2	1	no	—
315	C₂²	—	2	1	no	—
316	C₂²	—	2	4	no	—
317	C₂²	—	2	4	no	—
318	C₂²	—	2	4	no	—
319	C₂²	—	4	8	no	—
320	C₂²	—	26	4	no	—
321	C₂²	—	26	4	no	—
322	C₂²	—	26	4	no	—
323	C₂²	C₂×#6127	1	2	yes	—
324	C₂²	C₂⁴×D₁₃	1	1	yes	characteristic
325	C₄	—	2	1	no	—
326	C₄	—	2	4	no	—
327	C₄	—	2	1	no	—
328	C₄	—	2	1	no	—
329	C₄	—	26	4	no	—
330	C₄	—	26	4	no	—
331	C₄	—	26	4	no	—
332	C₄	C₂×#6127	1	2	yes	—
333	C₄	C₂×#6127	1	2	yes	—
334	C₄	#88251	1	2	yes	—
335	C₂	—	2	2	no	—
336	C₂	—	4	4	no	—
337	C₂	—	52	4	no	—
338	C₂	C₂²×#6127	1	1	yes	characteristic
339	C₂	C₂×#88251	1	1	yes	characteristic
340	C₂	#11719648	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₁₃
commutator, socle, cosocle	C₂²×C₁₃	C₂⁵
Fitting	C₂×C₄×D₄×C₁₃	C₂
hypercenter	C₂×C₄×D₄	D₁₃
hypocenter	C₁₃	#3506
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄×D₄

Complements

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

N	H ≅ G/N	type
1. C₂×C₄×D₄×C₁₃	C₂	semidirect
2. #11719510	C₂	semidirect
3. #11719496	C₂	semidirect
4. #11719268	C₂	semidirect
5. #11719264	C₂	semidirect
6. #11719263	C₂	semidirect
7. #11719258	C₂	semidirect
8. #11719256	C₂	semidirect
9. #11719247	C₂	semidirect
10. #11719209	C₂	semidirect
11. #11719199	C₂	semidirect
12. #11719182	C₂	semidirect
13. #11719175	C₂	semidirect
17. C₂×#6127	C₂²	semidirect
32. #88184	C₂²	semidirect
33. #88183	C₂²	semidirect
35. C₄×C₂²⋊D₁₃	C₂²	semidirect
37. #88180	C₂²	semidirect
40. #88178	C₂²	semidirect
41. #88147	C₂²	semidirect
42. #88146	C₂²	semidirect
43. C₄⋊C₄×D₁₃	C₂²	semidirect
44. #88141	C₂²	semidirect
45. #88139	C₂²	semidirect
46. #88138	C₂²	semidirect
47. #88137	C₂²	semidirect
48. #88133	C₂²	semidirect
49. #88129	C₂²	semidirect
50. #88128	C₂²	semidirect
51. C₄×D₅₂	C₂²	semidirect
52. C₄×D₅₂	C₂²	semidirect
53. #88124	C₂²	semidirect
54. #88123	C₂²	semidirect
55. #88123	C₂²	semidirect
56. #88123	C₂²	semidirect
57. C₄×Dic₂₆	C₂²	semidirect
58. C₄×Dic₂₆	C₂²	semidirect
122. C₂×C₄×C₁₃	#41	semidirect
156. C₂×C₄×D₄	D₁₃	semidirect
278. C₁₃	#3506	semidirect
306. C₂×C₄	#6127	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×D₄×C₁₃	2
#11719510	2
#11719496	2
#11719268	2
#11719264	2
#11719263	2
#11719258	2
#11719256	2
#11719247	2
#11719209	2
#11719199	2
#11719182	2
#11719175	2
#3506	13