Subgroups — #15879726

Order 3328.

1086 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₈×#1055	C₂	1	1	yes	maximal, characteristic
2	Q₈×C₄⋊Dic₁₃	C₂	1	1	yes	maximal, characteristic
3	#13115760	C₂	1	3	yes	maximal
4	#13115639	C₂	1	3	yes	maximal
5	#13114841	C₂	1	3	yes	maximal
6	C₄×#88075	C₂	1	3	yes	maximal
7	C₂×C₁₃×#557	C₂	1	1	yes	maximal, characteristic
8	C₁₃×#557	—	2	4	no	—
9	C₁₃×#557	C₄	1	8	yes	complemented
10	C₂×C₁₃×C₄⋊Q₈	C₂²	1	3	yes	—
11	C₂×C₁₃×#125	—	2	3	no	—
12	C₂×C₄×Q₈×C₁₃	—	2	1	no	—
13	C₂×C₄×Q₈×C₁₃	C₂²	1	3	yes	—
14	C₂×C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic
15	C₂×Q₈×Dic₁₃	—	2	1	no	—
16	C₂×#88075	C₂²	1	3	yes	—
17	C₂×#88075	C₂²	1	3	yes	—
18	C₂×#88075	C₂²	1	6	yes	—
19	Q₈×#1055	—	2	4	no	—
20	#11718646	—	2	3	no	—
21	#11718596	C₂²	1	3	yes	—
22	C₄×C₄⋊Dic₁₃	C₂²	1	3	yes	—
23	C₂×C₄⋊#1055	C₂²	1	3	yes	—
24	C₂×C₄×#1055	C₂²	1	3	yes	—
25	#11718121	C₂²	1	1	yes	characteristic
26	#11718121	C₂²	1	3	yes	—
27	C₂²×Q₈×C₁₃	D₄	1	1	yes	characteristic
28	C₂²×Q₈×C₁₃	C₂³	1	3	yes	—
29	C₁₃×C₄⋊Q₈	—	2	6	no	—
30	C₁₃×C₄⋊Q₈	C₂×C₄	1	12	yes	—
31	C₁₃×#125	—	2	24	no	—
32	C₄×Q₈×C₁₃	—	2	8	no	—
33	C₄×Q₈×C₁₃	—	2	6	no	—
34	C₄×Q₈×C₁₃	C₂×C₄	1	12	yes	—
35	C₄×Q₈×C₁₃	C₂×C₄	1	4	yes	—
36	C₄×Q₈×C₁₃	D₄	1	4	yes	—
37	C₂×C₁₃×C₄⋊C₄	—	2	3	no	—
38	C₂×C₁₃×C₄⋊C₄	—	2	3	no	—
39	C₂×C₁₃×C₄⋊C₄	—	2	3	no	—
40	C₂×C₁₃×C₄⋊C₄	D₄	1	3	yes	—
41	C₂×C₁₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
42	C₂×C₄²×C₁₃	—	2	3	no	—
43	C₂×C₄²×C₁₃	C₂³	1	3	yes	—
44	Q₈×Dic₁₃	—	4	4	no	—
45	C₂×C₄⋊Dic₁₃	—	2	3	no	—
46	C₂×C₄⋊Dic₁₃	—	2	3	no	—
47	C₂×C₄⋊Dic₁₃	C₂³	1	3	yes	—
48	C₂×C₄⋊Dic₁₃	C₂³	1	1	yes	characteristic
49	C₂×C₄×Dic₁₃	—	2	3	no	—
50	C₂²×#1055	C₂³	1	3	yes	—
51	C₂²×#1055	C₂³	1	1	yes	characteristic
52	#88075	—	2	6	no	—
53	#88075	—	2	6	no	—
54	#88075	—	4	6	no	—
55	#88071	—	2	3	no	—
56	C₄⋊#1055	—	2	6	no	—
57	C₄×#1055	—	2	6	no	—
58	#22542	—	13	1	no	maximal, Sylow
59	C₂×Q₈×C₁₃	—	2	3	no	—
60	C₂×Q₈×C₁₃	—	2	3	no	—
61	C₂×Q₈×C₁₃	—	2	3	no	—
62	C₂×Q₈×C₁₃	—	2	3	no	—
63	C₂×Q₈×C₁₃	—	2	3	no	—
64	C₂×Q₈×C₁₃	C₂²⋊C₄	1	4	yes	—
65	C₂×Q₈×C₁₃	SD₁₆	1	4	yes	—
66	C₂×Q₈×C₁₃	Q₁₆	1	4	yes	—
67	C₂×Q₈×C₁₃	C₂²×C₄	1	6	yes	—
68	C₂²×C₄×C₁₃	—	2	1	no	—
69	C₂²×C₄×C₁₃	—	2	3	no	—
70	C₂²×C₄×C₁₃	C₂×D₄	1	3	yes	—
71	C₂²×C₄×C₁₃	C₂×D₄	1	3	yes	—
72	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
73	C₂²×Dic₁₃	—	2	1	no	—
74	C₂²×Dic₁₃	—	2	3	no	—
75	C₁₃×C₄⋊C₄	—	2	12	no	—
76	C₁₃×C₄⋊C₄	—	2	12	no	—
77	C₁₃×C₄⋊C₄	—	2	12	no	—
78	C₁₃×C₄⋊C₄	—	2	1	no	—
79	C₁₃×C₄⋊C₄	C₂²⋊C₄	1	12	yes	—
80	C₁₃×C₄⋊C₄	C₂²×C₄	1	2	yes	—
81	C₄²×C₁₃	—	2	12	no	—
82	C₄²×C₁₃	C₂²×C₄	1	6	yes	—
83	C₄²×C₁₃	C₂×D₄	1	6	yes	—
84	C₄⋊Dic₁₃	—	2	1	no	—
85	C₄⋊Dic₁₃	—	2	3	no	—
86	C₄⋊Dic₁₃	—	2	3	no	—
87	C₄⋊Dic₁₃	—	4	6	no	—
88	C₄⋊Dic₁₃	—	8	3	no	—
89	C₄⋊Dic₁₃	#41	1	2	yes	—
90	C₄×Dic₁₃	—	4	6	no	—
91	C₂×#1055	—	2	1	no	—
92	C₂×#1055	—	2	1	no	—
93	C₂×#1055	—	2	3	no	—
94	C₂×#1055	—	2	3	no	—
95	C₂×#1055	—	2	3	no	—
96	C₂×#1055	#41	1	2	yes	—
97	C₂×#557	D₁₃	1	1	yes	characteristic
98	C₂×C₈×Q₈	—	13	1	no	—
99	Q₈×C₄⋊C₄	—	13	1	no	—
100	#1981	—	13	3	no	—
101	#1956	—	13	3	no	—
102	#1803	—	13	3	no	—
103	C₄×#103	—	13	3	no	—
104	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
105	Q₈×C₁₃	—	2	6	no	—
106	Q₈×C₁₃	—	2	6	no	—
107	Q₈×C₁₃	—	4	3	no	—
108	Q₈×C₁₃	—	4	3	no	—
109	Q₈×C₁₃	#103	1	16	yes	complemented
110	C₂×C₄×C₁₃	—	2	3	no	—
111	C₂×C₄×C₁₃	—	2	3	no	—
112	C₂×C₄×C₁₃	—	2	2	no	—
113	C₂×C₄×C₁₃	—	2	6	no	—
114	C₂×C₄×C₁₃	—	4	3	no	—
115	C₂×C₄×C₁₃	—	4	3	no	—
116	C₂×C₄×C₁₃	—	4	1	no	—
117	C₂×C₄×C₁₃	—	4	1	no	—
118	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	6	yes	—
119	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	6	yes	—
120	C₂×C₄×C₁₃	C₂×SD₁₆	1	6	yes	—
121	C₂×C₄×C₁₃	C₂×Q₁₆	1	6	yes	—
122	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
123	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
124	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
125	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
126	C₂×C₄×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
127	C₂×C₄×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
128	C₂×Dic₁₃	—	2	2	no	—
129	C₂×Dic₁₃	—	4	1	no	—
130	C₂×Dic₁₃	—	4	1	no	—
131	C₂×Dic₁₃	—	4	3	no	—
132	C₂×Dic₁₃	—	4	3	no	—
133	C₂×Dic₁₃	—	4	3	no	—
134	#1055	—	2	2	no	—
135	#1055	—	4	3	no	—
136	#557	—	2	4	no	—
137	#557	Dic₁₃	1	8	yes	complemented
138	C₂×C₄⋊Q₈	C₂×D₁₃	1	3	yes	—
139	C₂×#125	—	2	3	no	—
140	C₂×C₄×Q₈	—	2	1	no	—
141	C₂×C₄×Q₈	—	26	1	no	—
142	C₂×C₄×Q₈	C₂×D₁₃	1	3	yes	—
143	C₂×C₄×Q₈	C₂×D₁₃	1	1	yes	characteristic
144	C₈×Q₈	—	26	4	no	—
145	C₂×C₄⋊C₈	—	13	3	no	—
146	C₂×#103	—	13	3	no	—
147	C₂×#103	—	13	3	no	—
148	C₂×#103	—	13	6	no	—
149	C₂×C₄×C₈	—	13	3	no	—
150	#389	—	26	3	no	—
151	#384	—	13	3	no	—
152	C₄×C₄⋊C₄	—	13	3	no	—
153	#340	—	13	3	no	—
154	#340	—	13	1	no	—
155	C₂²×C₁₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
156	C₂²×C₁₃	#548	1	1	yes	characteristic
157	C₂²×C₁₃	#550	1	1	yes	characteristic
158	C₂²×C₁₃	C₂²×SD₁₆	1	1	yes	characteristic
159	C₂²×C₁₃	C₂²×Q₁₆	1	1	yes	characteristic
160	C₂²×C₁₃	#577	1	1	yes	characteristic
161	C₂²×C₁₃	#578	1	1	yes	characteristic
162	C₄×C₁₃	—	2	6	no	—
163	C₄×C₁₃	—	4	6	no	—
164	C₄×C₁₃	—	4	2	no	—
165	C₄×C₁₃	C₂×#103	1	12	yes	—
166	C₄×C₁₃	C₄×#41	1	1	yes	characteristic
167	C₄×C₁₃	#520	1	1	yes	characteristic
168	C₄×C₁₃	#548	1	1	yes	characteristic
169	C₄×C₁₃	#550	1	1	yes	characteristic
170	Dic₁₃	—	4	2	no	—
171	Dic₁₃	—	8	3	no	—
172	C₂²×Q₈	C₂²⋊D₁₃	1	1	yes	characteristic
173	C₂²×Q₈	C₂²×D₁₃	1	3	yes	—
174	C₂²×C₈	—	13	1	no	—
175	C₂²×C₈	—	13	3	no	—
176	C₄⋊Q₈	—	2	6	no	—
177	C₄⋊Q₈	C₂×Dic₁₃	1	12	yes	—
178	#125	—	2	24	no	—
179	C₄×Q₈	—	2	8	no	—
180	C₄×Q₈	—	2	6	no	—
181	C₄×Q₈	—	52	4	no	—
182	C₄×Q₈	C₂×Dic₁₃	1	12	yes	—
183	C₄×Q₈	C₂×Dic₁₃	1	4	yes	—
184	C₄×Q₈	C₂²⋊D₁₃	1	4	yes	—
185	C₂×C₄⋊C₄	—	2	3	no	—
186	C₂×C₄⋊C₄	—	2	3	no	—
187	C₂×C₄⋊C₄	—	2	3	no	—
188	C₂×C₄⋊C₄	—	13	1	no	—
189	C₂×C₄⋊C₄	—	13	3	no	—
190	C₂×C₄⋊C₄	—	26	3	no	—
191	C₂×C₄⋊C₄	—	26	3	no	—
192	C₂×C₄⋊C₄	C₂²⋊D₁₃	1	3	yes	—
193	C₂×C₄⋊C₄	C₂²×D₁₃	1	1	yes	characteristic
194	C₂×C₄²	—	2	3	no	—
195	C₂×C₄²	—	26	3	no	—
196	C₂×C₄²	C₂²×D₁₃	1	3	yes	—
197	C₄⋊C₈	—	26	6	no	—
198	#103	—	26	6	no	—
199	#103	—	26	6	no	—
200	#103	—	52	6	no	—
201	C₄×C₈	—	26	6	no	—
202	#95	—	26	3	no	—
203	C₂×C₁₃	#2341	1	1	yes	characteristic
204	C₂×C₁₃	C₂²×#103	1	1	yes	characteristic
205	C₂×C₁₃	#2898	1	1	yes	characteristic
206	C₂×C₁₃	#3359	1	1	yes	characteristic
207	C₂×C₁₃	#3363	1	1	yes	characteristic
208	C₂×C₁₃	#3392	1	1	yes	characteristic
209	C₂×C₁₃	#3395	1	1	yes	characteristic
210	C₂×Q₈	—	2	3	no	—
211	C₂×Q₈	—	2	3	no	—
212	C₂×Q₈	—	2	3	no	—
213	C₂×Q₈	—	2	3	no	—
214	C₂×Q₈	—	2	3	no	—
215	C₂×Q₈	#6106	1	4	yes	—
216	C₂×Q₈	#6107	1	4	yes	—
217	C₂×Q₈	C₂²⋊Dic₁₃	1	4	yes	—
218	C₂×Q₈	C₂²×Dic₁₃	1	6	yes	—
219	C₂²×C₄	—	2	1	no	—
220	C₂²×C₄	—	2	3	no	—
221	C₂²×C₄	—	26	1	no	—
222	C₂²×C₄	—	26	3	no	—
223	C₂²×C₄	C₂×C₂²⋊D₁₃	1	3	yes	—
224	C₂²×C₄	C₂×C₂²⋊D₁₃	1	3	yes	—
225	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
226	C₂×C₈	—	13	2	no	—
227	C₂×C₈	—	26	1	no	—
228	C₂×C₈	—	26	1	no	—
229	C₂×C₈	—	26	3	no	—
230	C₂×C₈	—	26	3	no	—
231	C₂×C₈	—	26	3	no	—
232	C₄⋊C₄	—	2	12	no	—
233	C₄⋊C₄	—	2	12	no	—
234	C₄⋊C₄	—	2	12	no	—
235	C₄⋊C₄	—	2	1	no	—
236	C₄⋊C₄	—	13	2	no	—
237	C₄⋊C₄	—	26	1	no	—
238	C₄⋊C₄	—	26	3	no	—
239	C₄⋊C₄	—	26	3	no	—
240	C₄⋊C₄	—	52	6	no	—
241	C₄⋊C₄	—	104	3	no	—
242	C₄⋊C₄	C₂²⋊Dic₁₃	1	12	yes	—
243	C₄⋊C₄	C₂²×Dic₁₃	1	2	yes	—
244	C₄²	—	2	12	no	—
245	C₄²	—	52	6	no	—
246	C₄²	C₂²×Dic₁₃	1	6	yes	—
247	C₄²	C₂×C₂²⋊D₁₃	1	6	yes	—
248	C₁₃	#22542	1	1	yes	characteristic, complemented, Sylow
249	C₂³	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
250	Q₈	—	2	6	no	—
251	Q₈	—	2	6	no	—
252	Q₈	—	4	3	no	—
253	Q₈	—	4	3	no	—
254	Q₈	#88075	1	16	yes	complemented
255	C₂×C₄	—	2	3	no	—
256	C₂×C₄	—	2	3	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	2	6	no	—
259	C₂×C₄	—	4	3	no	—
260	C₂×C₄	—	4	3	no	—
261	C₂×C₄	—	4	1	no	—
262	C₂×C₄	—	4	1	no	—
263	C₂×C₄	—	26	2	no	—
264	C₂×C₄	—	52	1	no	—
265	C₂×C₄	—	52	1	no	—
266	C₂×C₄	—	52	3	no	—
267	C₂×C₄	—	52	3	no	—
268	C₂×C₄	—	52	3	no	—
269	C₂×C₄	C₂×#6106	1	6	yes	—
270	C₂×C₄	C₂×#6107	1	6	yes	—
271	C₂×C₄	C₂×C₂²⋊Dic₁₃	1	6	yes	—
272	C₂×C₄	C₂×C₂²⋊Dic₁₃	1	6	yes	—
273	C₂×C₄	#41×D₁₃	1	1	yes	characteristic
274	C₂×C₄	#41×D₁₃	1	1	yes	characteristic
275	C₂×C₄	#88256	1	1	yes	characteristic
276	C₂×C₄	#88257	1	1	yes	characteristic
277	C₂×C₄	C₂³×Dic₁₃	1	1	yes	characteristic
278	C₂×C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
279	C₈	—	26	2	no	—
280	C₈	—	52	3	no	—
281	C₂²	C₂²×#6106	1	1	yes	characteristic
282	C₂²	C₂²×#6107	1	1	yes	characteristic
283	C₂²	#11719536	1	1	yes	characteristic
284	C₂²	#11719539	1	1	yes	characteristic
285	C₂²	#11719542	1	1	yes	characteristic
286	C₂²	#11719543	1	1	yes	characteristic
287	C₂²	C₂²×C₂²⋊Dic₁₃	1	1	yes	characteristic
288	C₄	—	2	6	no	—
289	C₄	—	4	2	no	—
290	C₄	—	4	6	no	—
291	C₄	—	52	2	no	—
292	C₄	—	104	3	no	—
293	C₄	C₂×#88075	1	12	yes	—
294	C₄	#41×Dic₁₃	1	1	yes	characteristic
295	C₄	#11719532	1	1	yes	characteristic
296	C₄	#11719536	1	1	yes	characteristic
297	C₄	#11719539	1	1	yes	characteristic
298	C₂	#13117677	1	1	yes	characteristic
299	C₂	C₂²×#88075	1	1	yes	characteristic
300	C₂	#13123369	1	1	yes	characteristic
301	C₂	#13127474	1	1	yes	characteristic
302	C₂	#13127510	1	1	yes	characteristic
303	C₂	#13127813	1	1	yes	characteristic
304	C₂	#13127839	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²×C₂²⋊D₁₃
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₁₃
Fitting	C₂×C₁₃×#557	C₂
socle, prime radical	C₂³×C₁₃	C₂²×D₄
cosocle	C₂²×C₄×C₁₃	C₂⁴
hypercenter	C₂×#557	D₁₃
hypocenter	C₁₃	#22542
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×Q₈ ◁ C₂×#557

Complements

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

N	H ≅ G/N	type
9. C₁₃×#557	C₄	semidirect
109. Q₈×C₁₃	#103	semidirect
137. #557	Dic₁₃	semidirect
248. C₁₃	#22542	semidirect
254. Q₈	#88075	semidirect

Maximal subgroups

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

subgroup	index
C₂×Q₈×#1055	2
Q₈×C₄⋊Dic₁₃	2
#13115760	2
#13115639	2
#13114841	2
C₄×#88075	2
C₂×C₁₃×#557	2
#22542	13