Subgroups — #15881064

Order 3328.

982 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718767	C₂	1	2	yes	maximal, complemented
2	#13117785	C₂	1	2	yes	maximal
3	#13115600	C₂	1	2	yes	maximal, complemented
4	#13115037	C₂	1	4	yes	maximal, complemented
5	#13114996	C₂	1	1	yes	maximal, characteristic, complemented
6	#13114865	C₂	1	2	yes	maximal, complemented
7	C₄×#88058	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₈×D₄×C₁₃	C₂	1	1	yes	maximal, characteristic
9	C₂³×C₈×C₁₃	C₂²	1	2	yes	—
10	C₂×C₄×D₄×C₁₃	C₂²	1	1	yes	characteristic
11	C₂×#88180	C₂²	1	4	yes	—
12	C₈×D₄×C₁₃	C₄	1	8	yes	complemented
13	C₂×C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
14	C₂×C₁₃×C₂²⋊C₈	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
16	#11718882	—	2	2	no	—
17	#11718873	C₂²	1	4	yes	—
18	#11718767	—	2	8	no	—
19	C₂×#88058	C₂²	1	2	yes	—
20	C₂×#88058	C₂²	1	1	yes	characteristic
21	C₂×#88058	C₂²	1	2	yes	—
22	C₂×#88058	C₂²	1	1	yes	characteristic
23	C₂×#88057	C₂²	1	4	yes	—
24	C₄⋊C₄×Dic₁₃	—	2	2	no	—
25	#11718612	—	2	4	no	—
26	#11718596	C₂²	1	2	yes	—
27	C₄×C₄⋊Dic₁₃	C₂²	1	2	yes	—
28	#11718143	C₂²	1	2	yes	—
29	#11718143	C₂²	1	4	yes	—
30	C₂²×D₄×C₁₃	D₄	1	1	yes	characteristic
31	C₂³×C₄×C₁₃	C₂³	1	2	yes	—
32	C₂²×C₈×C₁₃	—	2	4	no	—
33	C₂²×C₈×C₁₃	—	2	4	no	—
34	C₂²×C₈×C₁₃	C₂×C₄	1	8	yes	—
35	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
36	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
37	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
38	C₄×D₄×C₁₃	C₂×C₄	1	4	yes	—
39	C₄×D₄×C₁₃	Q₈	1	4	yes	—
40	C₂×C₁₃×C₄⋊C₄	D₄	1	1	yes	characteristic
41	C₂×C₁₃×C₂²⋊C₄	D₄	1	2	yes	—
42	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
43	C₂×C₂²⋊Dic₁₃	—	2	4	no	—
44	C₂×C₂²⋊Dic₁₃	—	2	4	no	—
45	#88180	—	2	8	no	—
46	#88180	—	4	4	no	—
47	C₂×C₄⋊Dic₁₃	—	2	2	no	—
48	C₂×C₄⋊Dic₁₃	—	2	2	no	—
49	C₂×C₄⋊Dic₁₃	C₂³	1	4	yes	—
50	C₂×C₄⋊Dic₁₃	C₂³	1	2	yes	—
51	C₂×C₄⋊Dic₁₃	C₂³	1	2	yes	—
52	C₂×C₄×Dic₁₃	—	2	4	no	—
53	C₂×C₄×Dic₁₃	—	2	2	no	—
54	C₁₃×C₄⋊C₈	C₂×C₄	1	4	yes	—
55	C₁₃×C₂²⋊C₈	C₂×C₄	1	8	yes	—
56	C₄×C₈×C₁₃	C₂×C₄	1	4	yes	—
57	#88071	—	2	2	no	—
58	#88071	—	2	4	no	—
59	#88058	—	2	4	no	—
60	#88058	—	2	2	no	—
61	#88058	—	4	2	no	—
62	#88058	D₄	1	4	yes	complemented
63	#88057	—	2	8	no	—
64	#22658	—	13	1	no	maximal, Sylow
65	C₂⁴×C₁₃	C₂×D₄	1	2	yes	—
66	C₂×D₄×C₁₃	—	2	2	no	—
67	C₂×D₄×C₁₃	—	2	2	no	—
68	C₂×D₄×C₁₃	C₄⋊C₄	1	4	yes	—
69	C₂²×C₄×C₁₃	—	2	2	no	—
70	C₂²×C₄×C₁₃	—	2	2	no	—
71	C₂²×C₄×C₁₃	—	2	2	no	—
72	C₂²×C₄×C₁₃	—	2	2	no	—
73	C₂²×C₄×C₁₃	C₂²×C₄	1	4	yes	—
74	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
75	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₁₃	C₂×Q₈	1	4	yes	—
78	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
79	C₂²×Dic₁₃	—	2	2	no	—
80	C₂²×Dic₁₃	—	2	2	no	—
81	C₂²×Dic₁₃	—	2	4	no	—
82	C₂×C₈×C₁₃	—	2	2	no	—
83	C₂×C₈×C₁₃	—	2	2	no	—
84	C₂×C₈×C₁₃	—	2	1	no	—
85	C₂×C₈×C₁₃	—	2	1	no	—
86	C₂×C₈×C₁₃	—	2	16	no	—
87	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
88	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
89	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
90	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
91	C₂×C₈×C₁₃	#41	1	2	yes	—
92	C₁₃×C₄⋊C₄	C₄⋊C₄	1	4	yes	—
93	C₁₃×C₂²⋊C₄	C₄⋊C₄	1	8	yes	—
94	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
95	C₄²×C₁₃	C₂×Q₈	1	2	yes	—
96	C₂²⋊Dic₁₃	—	4	8	no	—
97	C₂²⋊Dic₁₃	—	8	4	no	—
98	C₄⋊Dic₁₃	—	2	2	no	—
99	C₄⋊Dic₁₃	—	2	4	no	—
100	C₄⋊Dic₁₃	—	2	2	no	—
101	C₄⋊Dic₁₃	—	2	4	no	—
102	C₄⋊Dic₁₃	—	2	2	no	—
103	C₄⋊Dic₁₃	—	4	4	no	—
104	C₄⋊Dic₁₃	—	8	2	no	—
105	C₄⋊Dic₁₃	C₂×D₄	1	4	yes	—
106	C₄×Dic₁₃	—	4	8	no	—
107	C₄×Dic₁₃	—	4	4	no	—
108	C₂×C₈×D₄	D₁₃	1	1	yes	characteristic
109	C₂×#427	—	13	2	no	—
110	#2354	—	13	2	no	—
111	#1948	—	13	2	no	—
112	#1854	—	13	4	no	—
113	#1845	—	13	1	no	—
114	#1808	—	13	2	no	—
115	C₄×#107	—	13	1	no	—
116	C₂³×C₁₃	—	2	2	no	—
117	C₂³×C₁₃	—	2	2	no	—
118	C₂³×C₁₃	—	2	2	no	—
119	C₂³×C₁₃	—	2	2	no	—
120	C₂³×C₁₃	—	2	2	no	—
121	C₂³×C₁₃	—	2	2	no	—
122	C₂³×C₁₃	C₂×C₄⋊C₄	1	4	yes	—
123	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
124	D₄×C₁₃	—	2	8	no	—
125	C₂×C₄×C₁₃	—	2	2	no	—
126	C₂×C₄×C₁₃	—	2	2	no	—
127	C₂×C₄×C₁₃	—	2	1	no	—
128	C₂×C₄×C₁₃	—	2	1	no	—
129	C₂×C₄×C₁₃	—	2	4	no	—
130	C₂×C₄×C₁₃	—	2	4	no	—
131	C₂×C₄×C₁₃	—	2	4	no	—
132	C₂×C₄×C₁₃	—	2	4	no	—
133	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
134	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	4	yes	—
135	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
136	C₂×C₄×C₁₃	#135	1	2	yes	—
137	C₂×C₄×C₁₃	#135	1	2	yes	—
138	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
139	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
140	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
141	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
142	C₂×C₄×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
143	C₂×C₄×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
144	C₂×Dic₁₃	—	2	4	no	—
145	C₂×Dic₁₃	—	4	2	no	—
146	C₂×Dic₁₃	—	4	2	no	—
147	C₂×Dic₁₃	—	4	4	no	—
148	C₂×Dic₁₃	—	4	2	no	—
149	C₂×Dic₁₃	—	4	2	no	—
150	C₂×Dic₁₃	—	4	4	no	—
151	C₂×Dic₁₃	—	4	4	no	—
152	C₂×Dic₁₃	—	4	2	no	—
153	C₈×C₁₃	—	2	4	no	—
154	C₈×C₁₃	—	2	2	no	—
155	C₈×C₁₃	C₄×D₄	1	4	yes	—
156	C₂³×C₈	C₂×D₁₃	1	2	yes	—
157	C₂×C₄×D₄	C₂×D₁₃	1	1	yes	characteristic
158	C₂×#117	—	13	4	no	—
159	C₈×D₄	Dic₁₃	1	8	yes	complemented
160	#427	—	26	8	no	—
161	C₂×#107	—	13	1	no	—
162	C₂×#107	—	13	1	no	—
163	C₂×#107	—	13	2	no	—
164	C₂×#107	—	13	2	no	—
165	C₂×#106	—	13	4	no	—
166	C₂×C₄⋊C₈	C₂×D₁₃	1	1	yes	characteristic
167	C₂×C₂²⋊C₈	C₂×D₁₃	1	2	yes	—
168	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
169	#390	—	26	2	no	—
170	#388	—	26	4	no	—
171	#384	—	13	2	no	—
172	#380	—	13	4	no	—
173	C₄×C₄⋊C₄	—	13	2	no	—
174	C₄×C₄⋊C₄	—	26	2	no	—
175	#340	—	13	4	no	—
176	#340	—	13	2	no	—
177	C₂²×C₁₃	—	2	4	no	—
178	C₂²×C₁₃	—	2	4	no	—
179	C₂²×C₁₃	—	2	4	no	—
180	C₂²×C₁₃	—	4	2	no	—
181	C₂²×C₁₃	—	4	2	no	—
182	C₂²×C₁₃	—	4	2	no	—
183	C₂²×C₁₃	—	4	2	no	—
184	C₂²×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
185	C₂²×C₁₃	D₄²	1	1	yes	characteristic
186	C₂²×C₁₃	#547	1	1	yes	characteristic
187	C₂²×C₁₃	C₂×#135	1	1	yes	characteristic
188	C₂²×C₁₃	C₂×#135	1	1	yes	characteristic
189	C₂²×C₁₃	#576	1	1	yes	characteristic
190	C₂²×C₁₃	#578	1	1	yes	characteristic
191	C₄×C₁₃	—	2	4	no	—
192	C₄×C₁₃	—	2	2	no	—
193	C₄×C₁₃	#427	1	4	yes	complemented
194	C₄×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
195	C₄×C₁₃	#520	1	1	yes	characteristic
196	C₄×C₁₃	D₄×Q₈	1	1	yes	characteristic
197	C₄×C₁₃	#554	1	1	yes	characteristic
198	Dic₁₃	—	4	4	no	—
199	Dic₁₃	—	8	2	no	—
200	Dic₁₃	—	8	4	no	—
201	C₂²×D₄	D₅₂	1	1	yes	characteristic
202	C₂³×C₄	C₂²×D₁₃	1	2	yes	—
203	C₂²×C₈	—	2	4	no	—
204	C₂²×C₈	—	2	4	no	—
205	C₂²×C₈	C₂×Dic₁₃	1	8	yes	—
206	C₂²×C₈	C₂²×D₁₃	1	2	yes	—
207	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
208	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
209	C₄×D₄	Dic₂₆	1	4	yes	—
210	C₄×D₄	C₂×Dic₁₃	1	4	yes	—
211	#117	—	26	8	no	—
212	#117	—	52	4	no	—
213	C₂×C₄⋊C₄	—	13	2	no	—
214	C₂×C₄⋊C₄	—	13	4	no	—
215	C₂×C₄⋊C₄	—	13	2	no	—
216	C₂×C₄⋊C₄	—	26	2	no	—
217	C₂×C₄⋊C₄	—	26	2	no	—
218	C₂×C₄⋊C₄	D₅₂	1	1	yes	characteristic
219	C₂×C₂²⋊C₄	—	26	4	no	—
220	C₂×C₂²⋊C₄	—	26	4	no	—
221	C₂×C₂²⋊C₄	D₅₂	1	2	yes	—
222	C₂×C₄²	—	26	4	no	—
223	C₂×C₄²	—	26	2	no	—
224	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
225	#107	—	13	4	no	—
226	#107	—	26	4	no	—
227	#107	—	26	2	no	—
228	#107	—	52	2	no	—
229	#106	—	26	8	no	—
230	C₄⋊C₈	C₂×Dic₁₃	1	4	yes	—
231	C₂²⋊C₈	C₂×Dic₁₃	1	8	yes	—
232	C₄×C₈	C₂×Dic₁₃	1	4	yes	—
233	#95	—	26	2	no	—
234	#95	—	26	4	no	—
235	C₂×C₁₃	—	4	4	no	—
236	C₂×C₁₃	D₄×C₄⋊C₄	1	1	yes	characteristic
237	C₂×C₁₃	C₂×#427	1	1	yes	characteristic
238	C₂×C₁₃	#2913	1	1	yes	characteristic
239	C₂×C₁₃	#3283	1	1	yes	characteristic
240	C₂×C₁₃	#3287	1	1	yes	characteristic
241	C₂×C₁₃	#3341	1	1	yes	characteristic
242	C₂×C₁₃	#3342	1	1	yes	characteristic
243	C₂⁴	C₂×D₅₂	1	2	yes	—
244	C₂×D₄	—	2	2	no	—
245	C₂×D₄	—	2	2	no	—
246	C₂×D₄	C₄⋊Dic₁₃	1	4	yes	—
247	C₂²×C₄	—	2	2	no	—
248	C₂²×C₄	—	2	2	no	—
249	C₂²×C₄	—	2	2	no	—
250	C₂²×C₄	—	2	2	no	—
251	C₂²×C₄	—	26	2	no	—
252	C₂²×C₄	—	26	2	no	—
253	C₂²×C₄	—	26	4	no	—
254	C₂²×C₄	C₂×Dic₂₆	1	4	yes	—
255	C₂²×C₄	C₂×D₅₂	1	2	yes	—
256	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
257	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
258	C₂²×C₄	C₂²×Dic₁₃	1	4	yes	—
259	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
260	C₂×C₈	—	2	2	no	—
261	C₂×C₈	—	2	2	no	—
262	C₂×C₈	—	2	1	no	—
263	C₂×C₈	—	2	1	no	—
264	C₂×C₈	—	2	16	no	—
265	C₂×C₈	D₄×D₁₃	1	2	yes	—
266	C₂×C₈	#6129	1	2	yes	—
267	C₂×C₈	C₂²×Dic₁₃	1	2	yes	—
268	C₂×C₈	C₂²×Dic₁₃	1	4	yes	—
269	C₂×C₈	C₂²×Dic₁₃	1	2	yes	—
270	C₄⋊C₄	—	13	4	no	—
271	C₄⋊C₄	—	26	2	no	—
272	C₄⋊C₄	—	26	4	no	—
273	C₄⋊C₄	—	26	2	no	—
274	C₄⋊C₄	—	26	4	no	—
275	C₄⋊C₄	—	26	2	no	—
276	C₄⋊C₄	—	52	4	no	—
277	C₄⋊C₄	—	104	2	no	—
278	C₄⋊C₄	C₄⋊Dic₁₃	1	4	yes	—
279	C₂²⋊C₄	—	52	8	no	—
280	C₂²⋊C₄	—	104	4	no	—
281	C₂²⋊C₄	C₄⋊Dic₁₃	1	8	yes	—
282	C₄²	—	52	8	no	—
283	C₄²	—	52	4	no	—
284	C₄²	C₂×Dic₂₆	1	2	yes	—
285	C₄²	C₂²×Dic₁₃	1	2	yes	—
286	C₁₃	#22658	1	1	yes	characteristic, complemented, Sylow
287	C₂³	—	2	2	no	—
288	C₂³	—	2	2	no	—
289	C₂³	—	2	2	no	—
290	C₂³	—	2	2	no	—
291	C₂³	—	2	2	no	—
292	C₂³	—	2	2	no	—
293	C₂³	C₂×C₄⋊Dic₁₃	1	4	yes	—
294	C₂³	C₂²×D₅₂	1	1	yes	characteristic
295	D₄	—	2	8	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	2	2	no	—
299	C₂×C₄	—	2	2	no	—
300	C₂×C₄	—	2	4	no	—
301	C₂×C₄	—	2	4	no	—
302	C₂×C₄	—	2	4	no	—
303	C₂×C₄	—	2	4	no	—
304	C₂×C₄	—	26	4	no	—
305	C₂×C₄	—	52	2	no	—
306	C₂×C₄	—	52	2	no	—
307	C₂×C₄	—	52	2	no	—
308	C₂×C₄	—	52	4	no	—
309	C₂×C₄	—	52	2	no	—
310	C₂×C₄	—	52	4	no	—
311	C₂×C₄	—	52	4	no	—
312	C₂×C₄	—	52	2	no	—
313	C₂×C₄	#88158	1	2	yes	—
314	C₂×C₄	#88158	1	2	yes	—
315	C₂×C₄	C₂×C₄⋊Dic₁₃	1	2	yes	—
316	C₂×C₄	C₂×C₄⋊Dic₁₃	1	4	yes	—
317	C₂×C₄	C₂×C₄⋊Dic₁₃	1	2	yes	—
318	C₂×C₄	C₂²×Dic₂₆	1	1	yes	characteristic
319	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
320	C₂×C₄	C₂×#6129	1	1	yes	characteristic
321	C₂×C₄	#88256	1	1	yes	characteristic
322	C₂×C₄	#88257	1	1	yes	characteristic
323	C₂×C₄	C₂³×Dic₁₃	1	1	yes	characteristic
324	C₈	—	2	4	no	—
325	C₈	—	2	2	no	—
326	C₈	D₄×Dic₁₃	1	4	yes	—
327	C₂²	—	2	4	no	—
328	C₂²	—	2	4	no	—
329	C₂²	—	2	4	no	—
330	C₂²	—	4	2	no	—
331	C₂²	—	4	2	no	—
332	C₂²	—	4	2	no	—
333	C₂²	—	4	2	no	—
334	C₂²	D₄×D₅₂	1	1	yes	characteristic
335	C₂²	#11719260	1	1	yes	characteristic
336	C₂²	C₂×#88158	1	1	yes	characteristic
337	C₂²	C₂×#88158	1	1	yes	characteristic
338	C₂²	#11719457	1	1	yes	characteristic
339	C₂²	#11719458	1	1	yes	characteristic
340	C₂²	C₂²×C₄⋊Dic₁₃	1	1	yes	characteristic
341	C₄	—	2	4	no	—
342	C₄	—	2	2	no	—
343	C₄	—	52	4	no	—
344	C₄	—	104	2	no	—
345	C₄	—	104	4	no	—
346	C₄	#11718767	1	4	yes	complemented
347	C₄	D₄×Dic₂₆	1	1	yes	characteristic
348	C₄	#11719248	1	1	yes	characteristic
349	C₄	C₂×D₄×Dic₁₃	1	1	yes	characteristic
350	C₄	#11719532	1	1	yes	characteristic
351	C₂	—	4	4	no	—
352	C₂	D₄×C₄⋊Dic₁₃	1	1	yes	characteristic
353	C₂	C₂×#11718767	1	1	yes	characteristic
354	C₂	#13123436	1	1	yes	characteristic
355	C₂	#13126558	1	1	yes	characteristic
356	C₂	#13126596	1	1	yes	characteristic
357	C₂	#13127259	1	1	yes	characteristic
358	C₂	#13127267	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²×D₅₂
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₁₃
Fitting	C₂×C₈×D₄×C₁₃	C₂
socle	C₂³×C₁₃	C₂²×D₄
prime radical	C₂²×D₄×C₁₃	D₄
cosocle	C₂²×C₄×C₁₃	C₂⁴
hypercenter	C₂×C₈×D₄	D₁₃
hypocenter	C₁₃	#22658
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×D₄ ◁ C₂×C₈×D₄

Complements

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

N	H ≅ G/N	type
1. C₂×#11718767	C₂	semidirect
3. #13115600	C₂	semidirect
4. #13115037	C₂	semidirect
5. #13114996	C₂	semidirect
6. #13114865	C₂	semidirect
7. C₄×#88058	C₂	semidirect
12. C₈×D₄×C₁₃	C₄	semidirect
62. #88058	D₄	semidirect
159. C₈×D₄	Dic₁₃	semidirect
193. C₄×C₁₃	#427	semidirect
286. C₁₃	#22658	semidirect
346. C₄	#11718767	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718767	2
#13117785	2
#13115600	2
#13115037	2
#13114996	2
#13114865	2
C₄×#88058	2
C₂×C₈×D₄×C₁₃	2
#22658	13