Subgroups — #16785134

Order 3328.

538 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	#13130769	C₂	1	1	yes	maximal, characteristic
2	C₄×#88112	C₂	1	1	yes	maximal, characteristic
3	#13129714	C₂	1	4	yes	maximal
4	#13123233	C₂	1	1	yes	maximal, characteristic
5	C₂×#88142	—	2	1	no	—
6	C₂×#88142	—	2	1	no	—
7	C₂×#88142	C₄	1	2	yes	—
8	C₂×#88112	—	2	1	no	—
9	C₂×#88112	C₂²	1	2	yes	—
10	C₂×#88112	C₂²	1	2	yes	—
11	C₂×C₄⋊#1057	—	2	1	no	—
12	C₂×C₄×#1057	—	2	1	no	—
13	C₁₃×#384	C₄	1	1	yes	characteristic
14	#11718637	—	2	1	no	—
15	#11718637	—	2	1	no	—
16	#11718637	C₂²	1	2	yes	—
17	#11718596	C₄	1	1	yes	characteristic
18	C₄²×Dic₁₃	C₂²	1	1	yes	characteristic
19	#11718290	—	2	1	no	—
20	C₂²×Dic₂₆	—	2	2	no	—
21	C₂²×#1057	—	2	2	no	—
22	C₂²×#1057	—	2	2	no	—
23	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
24	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
25	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
26	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄⋊Dic₁₃	—	2	1	no	—
28	C₂×C₄⋊Dic₁₃	—	2	1	no	—
29	C₂×C₄⋊Dic₁₃	C₂×C₄	1	2	yes	—
30	C₂×#6101	—	2	2	no	—
31	C₂×#6101	—	2	2	no	—
32	C₂×#6101	—	2	2	no	—
33	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄×Dic₁₃	D₄	1	1	yes	characteristic
35	C₂×C₄×Dic₁₃	D₄	1	1	yes	characteristic
36	C₂×C₄×Dic₁₃	D₄	1	1	yes	characteristic
37	C₂×C₄×Dic₁₃	D₄	1	1	yes	characteristic
38	C₂×C₄×Dic₁₃	C₂³	1	1	yes	characteristic
39	#88142	—	4	4	no	—
40	#88142	—	4	2	no	—
41	#88142	—	4	2	no	—
42	#88142	—	4	2	no	—
43	#88142	—	4	2	no	—
44	#88112	—	2	2	no	—
45	#88112	—	2	4	no	—
46	#88112	—	4	2	no	—
47	#88112	Q₈	1	4	yes	complemented
48	C₄⋊#1057	—	4	2	no	—
49	C₄×#1057	—	4	2	no	—
50	#9898	—	13	1	no	maximal, Sylow
51	C₂²×C₄×C₁₃	—	2	2	no	—
52	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
53	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
54	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
55	C₂²×Dic₁₃	—	2	2	no	—
56	C₂²×Dic₁₃	C₂²⋊C₄	1	1	yes	characteristic
57	C₂²×Dic₁₃	C₂²⋊C₄	1	1	yes	characteristic
58	C₂²×Dic₁₃	C₂×D₄	1	1	yes	characteristic
59	C₂²×Dic₁₃	C₂×D₄	1	1	yes	characteristic
60	C₂×Dic₂₆	—	4	4	no	—
61	C₂×Dic₂₆	—	4	4	no	—
62	C₂×Dic₂₆	—	4	4	no	—
63	C₂×#1057	—	2	4	no	—
64	C₂×#1057	—	4	2	no	—
65	C₂×#1057	—	4	2	no	—
66	C₂×#1057	—	4	2	no	—
67	C₂×#1057	—	4	2	no	—
68	C₂×#1057	—	4	2	no	—
69	C₁₃×C₄⋊C₄	—	4	2	no	—
70	C₁₃×C₄⋊C₄	—	4	2	no	—
71	C₁₃×C₄⋊C₄	—	4	2	no	—
72	C₄²×C₁₃	—	2	2	no	—
73	C₄⋊Dic₁₃	—	4	2	no	—
74	C₄⋊Dic₁₃	—	4	2	no	—
75	C₄⋊Dic₁₃	—	4	2	no	—
76	#6101	—	4	4	no	—
77	#6101	—	4	2	no	—
78	#6101	—	4	2	no	—
79	#6101	—	4	2	no	—
80	#6101	—	4	2	no	—
81	C₄×Dic₁₃	—	2	2	no	—
82	C₄×Dic₁₃	—	2	2	no	—
83	C₄×Dic₁₃	D₈	1	2	yes	—
84	C₄×Dic₁₃	D₈	1	2	yes	—
85	C₄×Dic₁₃	SD₁₆	1	2	yes	—
86	C₄×Dic₁₃	SD₁₆	1	2	yes	—
87	C₄×Dic₁₃	SD₁₆	1	2	yes	—
88	C₄×Dic₁₃	Q₁₆	1	2	yes	—
89	C₄×Dic₁₃	C₂×Q₈	1	2	yes	—
90	C₄×Dic₁₃	#41	1	2	yes	—
91	#2868	—	13	1	no	—
92	#1842	—	13	1	no	—
93	C₄×C₄⋊C₈	—	13	1	no	—
94	#1296	—	13	4	no	—
95	C₂³×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
96	C₂×C₄×C₁₃	—	2	1	no	—
97	C₂×C₄×C₁₃	—	2	1	no	—
98	C₂×C₄×C₁₃	—	2	1	no	—
99	C₂×C₄×C₁₃	—	2	1	no	—
100	C₂×C₄×C₁₃	—	4	2	no	—
101	C₂×C₄×C₁₃	—	4	2	no	—
102	C₂×C₄×C₁₃	—	4	2	no	—
103	C₂×C₄×C₁₃	#101	1	2	yes	—
104	C₂×C₄×C₁₃	#102	1	2	yes	—
105	C₂×C₄×C₁₃	#102	1	2	yes	—
106	C₂×C₄×C₁₃	#103	1	2	yes	—
107	C₂×C₄×C₁₃	C₄×Q₈	1	2	yes	—
108	C₂×Dic₁₃	—	2	1	no	—
109	C₂×Dic₁₃	—	2	1	no	—
110	C₂×Dic₁₃	—	4	2	no	—
111	C₂×Dic₁₃	—	4	2	no	—
112	C₂×Dic₁₃	—	4	2	no	—
113	C₂×Dic₁₃	#100	1	2	yes	—
114	C₂×Dic₁₃	#102	1	2	yes	—
115	C₂×Dic₁₃	#102	1	2	yes	—
116	C₂×Dic₁₃	#103	1	2	yes	—
117	C₂×Dic₁₃	C₂²⋊Q₈	1	2	yes	—
118	C₂×Dic₁₃	#124	1	1	yes	characteristic
119	C₂×Dic₁₃	C₄⋊Q₈	1	1	yes	characteristic
120	C₂×Dic₁₃	C₂×D₈	1	1	yes	characteristic
121	C₂×Dic₁₃	C₂×D₈	1	1	yes	characteristic
122	C₂×Dic₁₃	C₂×SD₁₆	1	1	yes	characteristic
123	C₂×Dic₁₃	C₂×SD₁₆	1	1	yes	characteristic
124	C₂×Dic₁₃	C₂×SD₁₆	1	1	yes	characteristic
125	C₂×Dic₁₃	C₂×Q₁₆	1	1	yes	characteristic
126	C₂×Dic₁₃	#137	1	2	yes	—
127	Dic₂₆	—	8	8	no	—
128	#1057	—	4	4	no	—
129	#1057	—	8	2	no	—
130	C₂×C₄⋊Q₈	—	13	2	no	—
131	C₂×C₄⋊Q₈	—	26	1	no	—
132	C₂×C₄⋊Q₈	—	26	1	no	—
133	C₂×C₄⋊C₈	—	13	2	no	—
134	C₂×C₄⋊C₈	—	13	2	no	—
135	C₂×C₄⋊C₈	—	26	1	no	—
136	C₂×C₄⋊C₈	—	26	1	no	—
137	C₂×C₄×C₈	—	26	1	no	—
138	#384	—	13	2	no	—
139	#384	—	13	1	no	—
140	#384	—	26	1	no	—
141	#384	—	26	1	no	—
142	#384	#261	1	1	yes	characteristic
143	C₄³	—	13	1	no	—
144	#336	—	26	1	no	—
145	C₂²×C₁₃	#391	1	1	yes	characteristic
146	C₂²×C₁₃	C₂×#100	1	1	yes	characteristic
147	C₂²×C₁₃	C₂×#101	1	1	yes	characteristic
148	C₂²×C₁₃	C₂×#102	1	1	yes	characteristic
149	C₂²×C₁₃	C₂×#102	1	1	yes	characteristic
150	C₂²×C₁₃	C₂×#103	1	1	yes	characteristic
151	C₂²×C₁₃	#419	1	1	yes	characteristic
152	C₄×C₁₃	—	2	2	no	—
153	C₄×C₁₃	—	2	2	no	—
154	C₄×C₁₃	—	8	2	no	—
155	C₄×C₁₃	#332	1	4	yes	—
156	Dic₁₃	—	2	2	no	—
157	Dic₁₃	—	8	2	no	—
158	Dic₁₃	#331	1	4	yes	—
159	Dic₁₃	#474	1	2	yes	—
160	Dic₁₃	#475	1	2	yes	—
161	Dic₁₃	#486	1	1	yes	characteristic
162	Dic₁₃	#487	1	1	yes	characteristic
163	Dic₁₃	#498	1	1	yes	characteristic
164	Dic₁₃	#500	1	1	yes	characteristic
165	C₂²×Q₈	—	26	2	no	—
166	C₂²×C₈	—	26	2	no	—
167	C₂²×C₈	—	26	2	no	—
168	C₄⋊Q₈	—	52	2	no	—
169	C₄⋊Q₈	—	52	2	no	—
170	C₄⋊Q₈	—	52	2	no	—
171	C₄⋊Q₈	—	52	2	no	—
172	C₄⋊Q₈	—	52	4	no	—
173	C₂×C₄⋊C₄	—	2	1	no	—
174	C₂×C₄⋊C₄	—	2	1	no	—
175	C₂×C₄⋊C₄	—	13	2	no	—
176	C₂×C₄⋊C₄	—	26	2	no	—
177	C₂×C₄⋊C₄	—	26	2	no	—
178	C₂×C₄⋊C₄	—	26	2	no	—
179	C₂×C₄⋊C₄	—	26	1	no	—
180	C₂×C₄⋊C₄	—	26	1	no	—
181	C₂×C₄⋊C₄	C₂×#261	1	2	yes	—
182	C₂×C₄²	—	13	1	no	—
183	C₂×C₄²	—	13	1	no	—
184	C₂×C₄²	—	13	1	no	—
185	C₂×C₄²	—	13	1	no	—
186	C₂×C₄²	—	13	1	no	—
187	C₂×C₄²	—	13	1	no	—
188	C₂×C₄²	C₂×#261	1	1	yes	characteristic
189	C₄⋊C₈	—	13	4	no	—
190	C₄⋊C₈	—	26	2	no	—
191	C₄⋊C₈	—	26	4	no	—
192	C₄⋊C₈	—	52	2	no	—
193	C₄⋊C₈	—	52	2	no	—
194	C₄×C₈	—	52	2	no	—
195	C₂×C₁₃	C₂×#331	1	1	yes	characteristic
196	C₂×C₁₃	C₂×#332	1	1	yes	characteristic
197	C₂×C₁₃	#1546	1	1	yes	characteristic
198	C₂×C₁₃	#1548	1	1	yes	characteristic
199	C₂×C₁₃	#1988	1	1	yes	characteristic
200	C₂×C₁₃	#1990	1	1	yes	characteristic
201	C₂×C₁₃	#1997	1	1	yes	characteristic
202	C₂×Q₈	—	52	4	no	—
203	C₂×Q₈	—	52	4	no	—
204	C₂×Q₈	—	52	4	no	—
205	C₂²×C₄	—	2	2	no	—
206	C₂²×C₄	—	13	1	no	—
207	C₂²×C₄	—	13	1	no	—
208	C₂²×C₄	—	13	1	no	—
209	C₂²×C₄	—	13	1	no	—
210	C₂²×C₄	—	26	2	no	—
211	C₂²×C₄	C₂²⋊#261	1	1	yes	characteristic
212	C₂²×C₄	C₂²⋊#261	1	1	yes	characteristic
213	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
214	C₂×C₈	—	26	4	no	—
215	C₂×C₈	—	52	2	no	—
216	C₂×C₈	—	52	2	no	—
217	C₂×C₈	—	52	2	no	—
218	C₂×C₈	—	52	2	no	—
219	C₂×C₈	—	52	2	no	—
220	C₄⋊C₄	—	4	2	no	—
221	C₄⋊C₄	—	4	2	no	—
222	C₄⋊C₄	—	4	2	no	—
223	C₄⋊C₄	—	52	2	no	—
224	C₄⋊C₄	—	52	2	no	—
225	C₄⋊C₄	—	52	2	no	—
226	C₄⋊C₄	—	52	2	no	—
227	C₄⋊C₄	—	52	2	no	—
228	C₄⋊C₄	—	52	4	no	—
229	C₄⋊C₄	—	52	2	no	—
230	C₄⋊C₄	—	52	2	no	—
231	C₄²	—	2	2	no	—
232	C₄²	—	13	2	no	—
233	C₄²	—	13	2	no	—
234	C₄²	—	13	2	no	—
235	C₄²	—	13	2	no	—
236	C₄²	—	13	2	no	—
237	C₄²	—	13	2	no	—
238	C₄²	—	13	2	no	—
239	C₄²	—	13	2	no	—
240	C₄²	—	26	2	no	—
241	C₄²	—	26	2	no	—
242	C₁₃	#9898	1	1	yes	characteristic, complemented, Sylow
243	C₂³	C₂×C₂²⋊#261	1	1	yes	characteristic
244	Q₈	—	104	8	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	2	1	no	—
249	C₂×C₄	—	4	2	no	—
250	C₂×C₄	—	4	2	no	—
251	C₂×C₄	—	4	2	no	—
252	C₂×C₄	—	13	1	no	—
253	C₂×C₄	—	13	1	no	—
254	C₂×C₄	—	13	1	no	—
255	C₂×C₄	—	13	1	no	—
256	C₂×C₄	—	13	2	no	—
257	C₂×C₄	—	13	2	no	—
258	C₂×C₄	—	13	2	no	—
259	C₂×C₄	—	13	1	no	—
260	C₂×C₄	—	13	1	no	—
261	C₂×C₄	—	13	2	no	—
262	C₂×C₄	—	13	2	no	—
263	C₂×C₄	—	13	1	no	—
264	C₂×C₄	—	13	1	no	—
265	C₂×C₄	—	13	2	no	—
266	C₂×C₄	—	26	1	no	—
267	C₂×C₄	—	26	1	no	—
268	C₂×C₄	—	52	2	no	—
269	C₂×C₄	—	52	2	no	—
270	C₂×C₄	—	52	2	no	—
271	C₂×C₄	#88113	1	2	yes	—
272	C₂×C₄	#88115	1	2	yes	—
273	C₂×C₄	#88115	1	2	yes	—
274	C₂×C₄	#88117	1	2	yes	—
275	C₂×C₄	Q₈×#261	1	2	yes	—
276	C₈	—	52	4	no	—
277	C₈	—	104	2	no	—
278	C₂²	C₂×#88113	1	1	yes	characteristic
279	C₂²	C₂×#88115	1	1	yes	characteristic
280	C₂²	C₂×#88115	1	1	yes	characteristic
281	C₂²	D₁₃⋊#100	1	1	yes	characteristic
282	C₂²	C₂×#88117	1	1	yes	characteristic
283	C₂²	#11719153	1	1	yes	characteristic
284	C₂²	(C₂×#261)⋊Q₈	1	1	yes	characteristic
285	C₄	—	2	2	no	—
286	C₄	—	2	2	no	—
287	C₄	—	8	2	no	—
288	C₄	—	13	2	no	—
289	C₄	—	13	2	no	—
290	C₄	—	13	1	no	—
291	C₄	—	13	1	no	—
292	C₄	—	13	1	no	—
293	C₄	—	13	1	no	—
294	C₄	—	13	4	no	—
295	C₄	—	26	2	no	—
296	C₄	—	104	2	no	—
297	C₄	#11718245	1	4	yes	—
298	C₂	D₁₃⋊#331	1	1	yes	characteristic
299	C₂	C₂×#11718245	1	1	yes	characteristic
300	C₂	#13130345	1	1	yes	characteristic
301	C₂	#13130350	1	1	yes	characteristic
302	C₂	#13131015	1	1	yes	characteristic
303	C₂	#13131017	1	1	yes	characteristic
304	C₂	#13131029	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊#261
commutator	C₂²×C₄×C₁₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×#261
Fitting	C₁₃×#384	C₄
socle, prime radical	C₂³×C₁₃	C₂×C₂²⋊C₄
cosocle	C₂×C₄×Dic₁₃	C₂³
hypercenter	#384	#261
hypocenter	C₁₃	#9898
LCS	G ▷ C₂²×C₄×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

Complements

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

N	H ≅ G/N	type
47. #88112	Q₈	semidirect
242. C₁₃	#9898	semidirect

Maximal subgroups

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

subgroup	index
#13130769	2
C₄×#88112	2
#13129714	2
#13123233	2
#9898	13