Subgroups — #16807279

Order 3328.

526 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	#13131251	C₂	1	1	yes	maximal, characteristic, complemented
2	Q₈×#6092	C₂	1	1	yes	maximal, characteristic, complemented
3	#13131220	C₂	1	1	yes	maximal, characteristic, complemented
4	#13131216	C₂	1	2	yes	maximal, complemented
5	D₄×#6092	C₂	1	2	yes	maximal, complemented
6	D₄×#6092	C₂	1	1	yes	maximal, characteristic, complemented
7	#13131201	C₂	1	2	yes	maximal, complemented
8	#13131197	C₂	1	1	yes	maximal, characteristic, complemented
9	#13131045	C₂	1	2	yes	maximal, complemented
10	C₂×C₄×#6092	C₂	1	1	yes	maximal, characteristic, complemented
11	#13123812	C₂	1	1	yes	maximal, characteristic
12	C₁₃×#529	C₄	1	1	yes	characteristic
13	C₂×#88106	C₂²	1	2	yes	—
14	C₂²×#6092	C₂²	1	1	yes	characteristic
15	C₂²×#6092	C₂²	1	1	yes	characteristic
16	C₂²×#6092	C₂²	1	2	yes	—
17	#11718685	C₄	1	1	yes	characteristic
18	Q₈×#1055	C₂²	1	1	yes	characteristic
19	#11718673	C₄	1	1	yes	characteristic
20	#11718672	C₄	1	2	yes	—
21	D₄×#1055	C₂²	1	2	yes	—
22	D₄×#1055	C₂²	1	1	yes	characteristic
23	#11718661	C₄	1	2	yes	—
24	#11718658	C₂²	1	2	yes	—
25	#11718587	C₄	1	1	yes	characteristic
26	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
27	C₂²⋊#6092	C₂²	1	2	yes	—
28	C₂²⋊#6092	C₂²	1	2	yes	—
29	C₂²⋊#6092	C₂²	1	2	yes	—
30	#11718231	C₂²	1	2	yes	—
31	#11718231	C₂²	1	2	yes	—
32	C₄⋊#6092	C₂²	1	1	yes	characteristic
33	C₄⋊#6092	C₂²	1	1	yes	characteristic
34	C₄⋊#6092	C₂²	1	2	yes	—
35	C₄⋊#6092	C₂²	1	2	yes	—
36	C₄×#6092	C₂²	1	1	yes	characteristic
37	C₄×#6092	C₂²	1	1	yes	characteristic
38	C₄×#6092	C₂²	1	1	yes	characteristic
39	C₄×#6092	C₂²	1	1	yes	characteristic
40	C₄×#6092	C₂²	1	2	yes	—
41	C₁₃×#126	C₈	1	2	yes	—
42	C₁₃×#125	C₈	1	1	yes	characteristic
43	C₁₃×#124	C₈	1	1	yes	characteristic
44	C₁₃×#123	C₈	1	2	yes	—
45	C₁₃×C₂²⋊Q₈	C₈	1	1	yes	characteristic
46	C₁₃×#121	C₈	1	1	yes	characteristic
47	C₄×Q₈×C₁₃	C₂×C₄	1	1	yes	characteristic
48	C₄×D₄×C₁₃	C₂×C₄	1	2	yes	—
49	C₄×D₄×C₁₃	C₂×C₄	1	1	yes	characteristic
50	C₁₃×#117	C₂×C₄	1	2	yes	—
51	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
52	C₂×#6099	C₂×C₄	1	2	yes	—
53	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
54	C₂²×#1055	C₂³	1	2	yes	—
55	C₂²×#1055	C₂³	1	1	yes	characteristic
56	#88106	—	2	4	no	—
57	C₂×#6092	—	2	2	no	—
58	C₂×#6092	—	2	4	no	—
59	C₂×#6092	—	2	2	no	—
60	C₂×#6092	C₂³	1	1	yes	characteristic
61	C₂×#6092	C₂³	1	1	yes	characteristic
62	C₂×#6092	C₂³	1	2	yes	—
63	C₂×#6092	C₂³	1	2	yes	—
64	C₂×#6092	C₂³	1	1	yes	characteristic
65	C₂×#6092	C₂³	1	1	yes	characteristic
66	C₂²⋊#1055	C₂×C₄	1	2	yes	—
67	C₂²⋊#1055	C₂×C₄	1	2	yes	—
68	C₂²⋊#1055	C₂×C₄	1	2	yes	—
69	C₄⋊#1055	C₂×C₄	1	2	yes	—
70	C₄⋊#1055	C₂×C₄	1	2	yes	—
71	C₄⋊#1055	C₂×C₄	1	1	yes	characteristic
72	C₄⋊#1055	C₂×C₄	1	1	yes	characteristic
73	#88043	C₂×C₄	1	2	yes	—
74	#88043	C₂×C₄	1	2	yes	—
75	C₄×#1055	C₂×C₄	1	1	yes	characteristic
76	C₄×#1055	C₂×C₄	1	1	yes	characteristic
77	C₄×#1055	C₂³	1	1	yes	characteristic
78	C₄×#1055	C₂³	1	2	yes	—
79	C₄×#1055	C₂³	1	1	yes	characteristic
80	#33480	—	13	1	no	maximal, Sylow
81	C₂×Q₈×C₁₃	C₂×C₈	1	1	yes	characteristic
82	C₂×D₄×C₁₃	C₂×C₈	1	2	yes	—
83	C₂×D₄×C₁₃	C₂×C₈	1	1	yes	characteristic
84	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
85	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
86	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
87	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
88	C₁₃×C₄⋊C₄	C₂×C₈	1	2	yes	—
89	C₁₃×C₄⋊C₄	C₂×C₈	1	2	yes	—
90	C₁₃×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
91	C₁₃×C₄⋊C₄	C₂×C₈	1	2	yes	—
92	C₁₃×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
93	C₁₃×C₄⋊C₄	C₂×C₈	1	2	yes	—
94	C₁₃×C₂²⋊C₄	C₂×C₈	1	2	yes	—
95	C₁₃×C₂²⋊C₄	C₂×C₈	1	2	yes	—
96	C₁₃×C₂²⋊C₄	C₂×C₈	1	2	yes	—
97	C₁₃×C₂²⋊C₄	C₂×C₈	1	2	yes	—
98	C₁₃×C₂²⋊C₄	C₂×C₈	1	2	yes	—
99	C₄²×C₁₃	C₂×C₈	1	1	yes	characteristic
100	C₄²×C₁₃	C₂×C₈	1	1	yes	characteristic
101	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
102	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
103	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
104	#6099	—	2	4	no	—
105	C₂×#1055	—	2	2	no	—
106	C₂×#1055	—	2	2	no	—
107	C₂×#1055	—	2	1	no	—
108	C₂×#1055	—	2	1	no	—
109	C₂×#1055	—	2	2	no	—
110	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
111	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
112	C₂×#1055	C₂²×C₄	1	2	yes	—
113	C₂×#1055	C₂²×C₄	1	2	yes	—
114	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
115	C₂×#1055	C₂⁴	1	1	yes	characteristic
116	#6092	—	2	1	no	—
117	#6092	—	2	1	no	—
118	#6092	—	2	2	no	—
119	#6092	—	2	2	no	—
120	#6092	#41	1	2	yes	—
121	#6092	#41	1	2	yes	—
122	#2966	—	13	1	no	—
123	#2183	—	13	1	no	—
124	Q₈×C₁₆	—	13	1	no	—
125	#2169	—	13	1	no	—
126	#2168	—	13	2	no	—
127	D₄×C₁₆	—	13	1	no	—
128	D₄×C₁₆	—	13	2	no	—
129	#2164	—	13	2	no	—
130	#2163	—	13	1	no	—
131	#2108	—	13	2	no	—
132	C₂×C₄×C₁₆	—	13	1	no	—
133	C₂³×C₁₃	C₂²×C₈	1	2	yes	—
134	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
135	Q₈×C₁₃	—	2	2	no	—
136	D₄×C₁₃	—	2	4	no	—
137	D₄×C₁₃	—	2	2	no	—
138	C₂×C₄×C₁₃	—	2	2	no	—
139	C₂×C₄×C₁₃	—	2	1	no	—
140	C₂×C₄×C₁₃	—	2	2	no	—
141	C₂×C₄×C₁₃	—	2	1	no	—
142	C₂×C₄×C₁₃	—	2	2	no	—
143	C₂×C₄×C₁₃	—	2	2	no	—
144	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
145	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
146	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
147	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
148	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
149	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
150	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
151	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
152	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
153	#1055	—	2	1	no	—
154	#1055	—	2	2	no	—
155	#1055	—	2	1	no	—
156	#1055	—	2	2	no	—
157	#1055	#131	1	2	yes	—
158	#1055	C₂×#41	1	1	yes	characteristic
159	#1055	C₂×#41	1	1	yes	characteristic
160	#529	#261	1	1	yes	characteristic
161	C₂×M₃₂	—	13	2	no	—
162	C₂²×C₁₆	—	13	1	no	—
163	C₂²×C₁₆	—	13	1	no	—
164	C₂²×C₁₆	—	13	2	no	—
165	#446	—	13	1	no	—
166	C₈×Q₈	—	13	1	no	—
167	#436	—	13	1	no	—
168	#435	—	13	2	no	—
169	C₈×D₄	—	13	2	no	—
170	C₈×D₄	—	13	1	no	—
171	#433	—	13	2	no	—
172	#432	—	13	1	no	—
173	#405	—	13	2	no	—
174	C₂×C₄×C₈	—	13	1	no	—
175	C₄⋊C₁₆	—	13	1	no	—
176	C₄⋊C₁₆	—	13	1	no	—
177	C₄⋊C₁₆	—	13	2	no	—
178	C₄⋊C₁₆	—	13	2	no	—
179	C₂²⋊C₁₆	—	13	2	no	—
180	C₂²⋊C₁₆	—	13	2	no	—
181	C₂²⋊C₁₆	—	13	2	no	—
182	#346	—	13	2	no	—
183	#346	—	13	2	no	—
184	C₄×C₁₆	—	13	1	no	—
185	C₄×C₁₆	—	13	1	no	—
186	C₄×C₁₆	—	13	1	no	—
187	C₄×C₁₆	—	13	1	no	—
188	C₄×C₁₆	—	13	2	no	—
189	C₂²×C₁₃	—	2	1	no	—
190	C₂²×C₁₃	—	2	1	no	—
191	C₂²×C₁₃	—	2	2	no	—
192	C₂²×C₁₃	—	2	2	no	—
193	C₂²×C₁₃	—	2	2	no	—
194	C₂²×C₁₃	#504	1	2	yes	—
195	C₂²×C₁₃	C₂³×C₈	1	1	yes	characteristic
196	C₄×C₁₃	—	2	1	no	—
197	C₄×C₁₃	—	2	1	no	—
198	C₄×C₁₃	—	2	1	no	—
199	C₄×C₁₃	—	2	2	no	—
200	C₄×C₁₃	—	2	2	no	—
201	C₄×C₁₃	—	2	1	no	—
202	C₄×C₁₃	—	2	2	no	—
203	C₄×C₁₃	#504	1	2	yes	—
204	C₄×C₁₃	C₄×#41	1	1	yes	characteristic
205	C₄×C₁₃	C₂×#131	1	1	yes	characteristic
206	C₂×M₁₆	—	13	2	no	—
207	C₂²×C₈	—	13	1	no	—
208	C₂²×C₈	—	13	2	no	—
209	C₂²×C₈	—	13	1	no	—
210	#126	#1057	1	2	yes	—
211	#125	#1057	1	1	yes	characteristic
212	#124	#1057	1	1	yes	characteristic
213	#123	#1057	1	2	yes	—
214	C₂²⋊Q₈	#1057	1	1	yes	characteristic
215	#121	#1057	1	1	yes	characteristic
216	C₄×Q₈	C₂×#261	1	1	yes	characteristic
217	C₄×D₄	C₂×#261	1	2	yes	—
218	C₄×D₄	C₂×#261	1	1	yes	characteristic
219	#117	C₂×#261	1	2	yes	—
220	C₂×C₄²	C₂×#261	1	1	yes	characteristic
221	M₃₂	—	26	4	no	—
222	C₂×C₁₆	—	13	1	no	—
223	C₂×C₁₆	—	13	1	no	—
224	C₂×C₁₆	—	13	1	no	—
225	C₂×C₁₆	—	13	1	no	—
226	C₂×C₁₆	—	13	2	no	—
227	C₂×C₁₆	—	13	2	no	—
228	C₂×C₁₆	—	26	2	no	—
229	C₂×C₁₆	—	26	4	no	—
230	C₂×C₁₆	—	26	2	no	—
231	C₄⋊C₈	—	13	1	no	—
232	C₄⋊C₈	—	13	1	no	—
233	C₄⋊C₈	—	13	2	no	—
234	C₄⋊C₈	—	13	2	no	—
235	C₂²⋊C₈	—	13	2	no	—
236	C₂²⋊C₈	—	13	2	no	—
237	C₂²⋊C₈	—	13	2	no	—
238	#97	—	13	2	no	—
239	#97	—	13	2	no	—
240	C₄×C₈	—	13	1	no	—
241	C₄×C₈	—	13	1	no	—
242	C₄×C₈	—	13	2	no	—
243	C₄×C₈	—	13	1	no	—
244	C₄×C₈	—	13	1	no	—
245	C₂×C₁₃	—	2	2	no	—
246	C₂×C₁₃	—	4	2	no	—
247	C₂×C₁₃	C₈×#41	1	1	yes	characteristic
248	C₂×C₁₃	C₂×#504	1	1	yes	characteristic
249	C₂×C₁₃	C₂×#504	1	1	yes	characteristic
250	C₂×Q₈	C₂×#1057	1	1	yes	characteristic
251	C₂×D₄	C₂×#1057	1	2	yes	—
252	C₂×D₄	C₂×#1057	1	1	yes	characteristic
253	C₂²×C₄	C₂×#1057	1	1	yes	characteristic
254	C₂²×C₄	C₂×#1057	1	1	yes	characteristic
255	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
256	C₂²×C₄	C₂²×#261	1	2	yes	—
257	M₁₆	—	26	4	no	—
258	C₂×C₈	—	13	1	no	—
259	C₂×C₈	—	13	1	no	—
260	C₂×C₈	—	13	1	no	—
261	C₂×C₈	—	13	1	no	—
262	C₂×C₈	—	13	2	no	—
263	C₂×C₈	—	13	2	no	—
264	C₂×C₈	—	26	1	no	—
265	C₂×C₈	—	26	1	no	—
266	C₂×C₈	—	26	2	no	—
267	C₂×C₈	—	26	2	no	—
268	C₂×C₈	—	26	2	no	—
269	C₄⋊C₄	C₂×#1057	1	2	yes	—
270	C₄⋊C₄	C₂×#1057	1	2	yes	—
271	C₄⋊C₄	C₂×#1057	1	1	yes	characteristic
272	C₄⋊C₄	C₂×#1057	1	2	yes	—
273	C₄⋊C₄	C₂×#1057	1	1	yes	characteristic
274	C₄⋊C₄	C₂×#1057	1	2	yes	—
275	C₂²⋊C₄	C₂×#1057	1	2	yes	—
276	C₂²⋊C₄	C₂×#1057	1	2	yes	—
277	C₂²⋊C₄	C₂×#1057	1	2	yes	—
278	C₂²⋊C₄	C₂×#1057	1	2	yes	—
279	C₂²⋊C₄	C₂×#1057	1	2	yes	—
280	C₄²	C₂×#1057	1	1	yes	characteristic
281	C₄²	C₂×#1057	1	1	yes	characteristic
282	C₄²	C₂²×#261	1	1	yes	characteristic
283	C₄²	C₂²×#261	1	1	yes	characteristic
284	C₄²	C₂²×#261	1	2	yes	—
285	C₁₆	—	13	2	no	—
286	C₁₆	—	13	2	no	—
287	C₁₆	—	26	1	no	—
288	C₁₆	—	26	1	no	—
289	C₁₆	—	26	2	no	—
290	C₁₆	—	26	2	no	—
291	C₁₃	#33480	1	1	yes	characteristic, complemented, Sylow
292	C₂³	C₂²×#1057	1	2	yes	—
293	C₂³	C₂²×#1057	1	1	yes	characteristic
294	Q₈	—	2	2	no	—
295	D₄	—	2	4	no	—
296	D₄	—	2	2	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	2	2	no	—
300	C₂×C₄	—	2	1	no	—
301	C₂×C₄	—	2	2	no	—
302	C₂×C₄	—	2	2	no	—
303	C₂×C₄	C₂²×#1057	1	1	yes	characteristic
304	C₂×C₄	C₂²×#1057	1	1	yes	characteristic
305	C₂×C₄	C₂²×#1057	1	1	yes	characteristic
306	C₂×C₄	C₂²×#1057	1	1	yes	characteristic
307	C₂×C₄	C₂²×#1057	1	2	yes	—
308	C₂×C₄	C₂²×#1057	1	2	yes	—
309	C₂×C₄	C₂²×#1057	1	2	yes	—
310	C₂×C₄	C₂²×#1057	1	1	yes	characteristic
311	C₂×C₄	C₂³×#261	1	1	yes	characteristic
312	C₈	—	13	2	no	—
313	C₈	—	13	1	no	—
314	C₈	—	13	1	no	—
315	C₈	—	26	1	no	—
316	C₈	—	26	2	no	—
317	C₈	—	26	1	no	—
318	C₈	—	26	2	no	—
319	C₂²	—	2	1	no	—
320	C₂²	—	2	1	no	—
321	C₂²	—	2	2	no	—
322	C₂²	—	2	2	no	—
323	C₂²	—	2	2	no	—
324	C₂²	#11719162	1	2	yes	—
325	C₂²	C₂³×#1057	1	1	yes	characteristic
326	C₄	—	2	1	no	—
327	C₄	—	2	1	no	—
328	C₄	—	2	1	no	—
329	C₄	—	2	2	no	—
330	C₄	—	2	2	no	—
331	C₄	—	2	1	no	—
332	C₄	—	2	2	no	—
333	C₄	#11719162	1	2	yes	—
334	C₄	D₁₃⋊#131	1	1	yes	characteristic
335	C₄	#41×#261	1	1	yes	characteristic
336	C₂	—	2	2	no	—
337	C₂	—	4	2	no	—
338	C₂	#41×#1057	1	1	yes	characteristic
339	C₂	C₂×#11719162	1	1	yes	characteristic
340	C₂	C₂×#11719162	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×#261
commutator, socle	C₂²×C₁₃	C₂³×C₈
Fitting	C₁₃×#529	C₄
prime radical	C₁₃×#121	C₈
cosocle	C₂×#1055	C₂⁴
hypercenter	#529	#261
hypocenter	C₁₃	#33480
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ #529

Complements

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

N	H ≅ G/N	type
1. #13131251	C₂	semidirect
2. Q₈×#6092	C₂	semidirect
3. #13131220	C₂	semidirect
4. #13131216	C₂	semidirect
5. D₄×#6092	C₂	semidirect
6. D₄×#6092	C₂	semidirect
7. #13131201	C₂	semidirect
8. #13131197	C₂	semidirect
9. #13131045	C₂	semidirect
10. C₂×C₄×#6092	C₂	semidirect
291. C₁₃	#33480	semidirect

Maximal subgroups

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

subgroup	index
#13131251	2
Q₈×#6092	2
#13131220	2
#13131216	2
D₄×#6092	2
#13131201	2
#13131197	2
#13131045	2
C₂×C₄×#6092	2
#13123812	2
#33480	13