Subgroups — #10693716

Order 768.

356 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#1950	C₂	1	1	yes	maximal, characteristic, complemented
2	#70498	C₂	1	1	yes	maximal, characteristic
3	#70431	C₂	1	1	yes	maximal, characteristic
4	#69867	C₂	1	1	yes	maximal, characteristic
5	#69847	C₂	1	1	yes	maximal, characteristic, complemented
6	#67715	C₂	1	1	yes	maximal, characteristic, complemented
7	#67704	C₂	1	1	yes	maximal, characteristic, complemented
8	#9592	—	3	1	no	maximal, Sylow
9	#5477	C₂²	1	1	yes	characteristic
10	C₂×C₃×#108	—	2	1	no	—
11	C₃×#423	C₂²	1	1	yes	characteristic
12	C₂×C₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
13	C₃×C₄×M₁₆	C₂²	1	1	yes	characteristic
14	C₂×#825	—	2	1	no	—
15	C₂×#825	—	2	1	no	—
16	C₄×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
17	#4925	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₃⋊C₈	C₂²	1	1	yes	characteristic
19	C₃×#341	—	2	1	no	—
20	#4485	C₄	1	1	yes	characteristic, complemented
21	#4485	C₄	1	1	yes	characteristic, complemented
22	#4483	—	2	1	no	—
23	#4483	—	2	1	no	—
24	#4482	C₄	1	1	yes	characteristic, complemented
25	#4482	C₄	1	1	yes	characteristic, complemented
26	#4454	—	2	1	no	—
27	#4454	—	2	1	no	—
28	#1980	—	3	1	no	—
29	#1950	D₃	1	1	yes	characteristic, complemented
30	#1540	—	3	1	no	—
31	#1492	—	3	1	no	—
32	#1451	—	3	1	no	—
33	#1303	—	3	1	no	—
34	#1279	—	3	1	no	—
35	C₂×#235	—	2	1	no	—
36	C₂×C₃×M₁₆	—	2	1	no	—
37	C₂×C₃×M₁₆	D₄	1	1	yes	characteristic
38	C₂×C₃×M₁₆	D₄	1	1	yes	characteristic
39	C₂²×C₃×C₈	—	2	1	no	—
40	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
41	#930	—	2	1	no	—
42	#930	—	2	1	no	—
43	C₂×C₄⋊Dic₃	—	2	1	no	—
44	C₂×C₄×Dic₃	—	2	1	no	—
45	C₂²×C₃⋊C₈	—	2	1	no	—
46	#875	—	2	1	no	—
47	#874	C₂×C₄	1	1	yes	characteristic
48	C₄×D₁₂	—	2	1	no	—
49	C₄×D₁₂	—	2	1	no	—
50	#869	C₂×C₄	1	1	yes	characteristic
51	C₃×#108	—	4	2	no	—
52	C₃×C₄⋊C₈	—	2	1	no	—
53	C₃×C₄⋊C₈	—	2	1	no	—
54	C₃×C₄⋊C₈	—	2	1	no	—
55	C₃×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
56	C₃×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
57	C₃×#97	—	2	1	no	—
58	C₃×C₄×C₈	—	2	1	no	—
59	C₃⋊#95	—	2	1	no	—
60	#830	—	2	1	no	—
61	#825	—	4	2	no	—
62	#825	—	4	2	no	—
63	C₄⋊(C₃⋊C₈)	C₂×C₄	1	1	yes	characteristic
64	C₄⋊(C₃⋊C₈)	C₂×C₄	1	1	yes	characteristic
65	C₄×C₃⋊C₈	—	2	1	no	—
66	C₄×C₃⋊C₈	—	2	1	no	—
67	C₄×C₃⋊C₈	D₄	1	1	yes	characteristic
68	C₄×C₃⋊C₈	D₄	1	1	yes	characteristic
69	#532	—	3	1	no	—
70	#431	—	3	1	no	—
71	C₂×#108	—	2	1	no	—
72	#423	C₂×D₃	1	1	yes	characteristic
73	C₂×C₄⋊C₈	C₂×D₃	1	1	yes	characteristic
74	C₂×#104	—	6	1	no	—
75	C₂×#104	—	6	1	no	—
76	C₄×M₁₆	C₂×D₃	1	1	yes	characteristic
77	C₂×C₄×C₈	—	3	1	no	—
78	C₄×C₄⋊C₄	—	3	1	no	—
79	#341	—	2	1	no	—
80	#332	—	3	1	no	—
81	#332	—	3	1	no	—
82	#331	—	3	1	no	—
83	#331	—	3	1	no	—
84	D₄⋊C₈	—	6	1	no	—
85	D₄⋊C₈	—	6	1	no	—
86	#322	—	6	1	no	—
87	#322	—	6	1	no	—
88	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
89	C₂²×C₃×C₄	#41	1	1	yes	characteristic
90	C₂²×C₃×C₄	#41	1	1	yes	characteristic
91	C₂×#67	—	4	1	no	—
92	C₂²×Dic₃	—	4	1	no	—
93	#235	—	4	1	no	—
94	#235	—	4	1	no	—
95	#235	—	4	1	no	—
96	#235	—	4	1	no	—
97	C₂×D₁₂	—	2	1	no	—
98	C₂×D₁₂	—	2	1	no	—
99	C₂×D₁₂	—	2	1	no	—
100	C₂×C₄×D₃	—	4	1	no	—
101	C₂×Dic₆	—	2	1	no	—
102	C₃×M₁₆	—	2	1	no	—
103	C₃×M₁₆	—	2	1	no	—
104	C₃×M₁₆	—	2	1	no	—
105	C₃×M₁₆	—	2	1	no	—
106	C₃×M₁₆	—	4	1	no	—
107	C₃×M₁₆	—	4	1	no	—
108	C₂×C₃×C₈	—	2	1	no	—
109	C₂×C₃×C₈	—	2	1	no	—
110	C₂×C₃×C₈	—	2	1	no	—
111	C₂×C₃×C₈	—	2	1	no	—
112	C₂×C₃×C₈	—	4	1	no	—
113	C₂×C₃×C₈	—	4	1	no	—
114	C₂×C₃×C₈	—	4	1	no	—
115	C₃×C₄²	C₂²×C₄	1	1	yes	characteristic
116	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
117	C₃×C₄²	#41	1	1	yes	characteristic
118	C₃×C₄²	#41	1	1	yes	characteristic
119	(C₂×D₃)⋊C₄	—	4	1	no	—
120	(C₂×D₃)⋊C₄	—	4	1	no	—
121	C₄⋊Dic₃	—	2	2	no	—
122	C₄⋊Dic₃	—	2	1	no	—
123	C₄⋊Dic₃	—	2	1	no	—
124	C₄⋊Dic₃	—	4	1	no	—
125	C₄×Dic₃	—	4	1	no	—
126	C₄×Dic₃	—	4	1	no	—
127	C₂×C₃⋊C₈	—	2	1	no	—
128	C₂×C₃⋊C₈	—	2	1	no	—
129	C₂×C₃⋊C₈	—	2	1	no	—
130	C₂×C₃⋊C₈	—	2	1	no	—
131	C₂×C₃⋊C₈	—	4	2	no	—
132	C₂×#41	—	6	1	no	—
133	C₂×M₁₆	—	2	1	no	—
134	C₂×M₁₆	#67	1	1	yes	characteristic
135	C₂×M₁₆	#67	1	1	yes	characteristic
136	C₂²×C₈	—	2	1	no	—
137	C₂²×C₈	—	6	1	no	—
138	C₄⋊Q₈	—	3	1	no	—
139	#127	—	3	1	no	—
140	#124	—	6	1	no	—
141	#121	—	6	1	no	—
142	#121	—	6	1	no	—
143	C₄×D₄	—	6	1	no	—
144	C₄×D₄	—	6	1	no	—
145	C₂×C₄⋊C₄	—	6	1	no	—
146	C₂×C₄²	—	6	1	no	—
147	C₂×C₄²	C₂²×D₃	1	1	yes	characteristic
148	#108	—	4	2	no	—
149	C₄⋊C₈	—	2	1	no	—
150	C₄⋊C₈	—	2	1	no	—
151	C₄⋊C₈	—	2	1	no	—
152	C₄⋊C₈	—	3	1	no	—
153	C₄⋊C₈	—	3	1	no	—
154	C₄⋊C₈	C₄×D₃	1	1	yes	characteristic
155	C₄⋊C₈	C₄×D₃	1	1	yes	characteristic
156	#104	—	12	2	no	—
157	#104	—	12	2	no	—
158	C₂²⋊C₈	—	6	1	no	—
159	#97	—	2	1	no	—
160	C₄×C₈	—	2	1	no	—
161	C₄×C₈	—	3	1	no	—
162	C₄×C₈	—	3	1	no	—
163	C₄×C₈	—	6	1	no	—
164	C₄×C₈	—	6	1	no	—
165	#95	—	6	1	no	—
166	C₂³×C₃	#124	1	1	yes	characteristic
167	C₂²×D₃	—	4	1	no	—
168	C₂×C₃×C₄	—	2	1	no	—
169	C₂×C₃×C₄	—	2	1	no	—
170	C₂×C₃×C₄	—	4	1	no	—
171	C₂×C₃×C₄	#117	1	1	yes	characteristic
172	C₂×C₃×C₄	C₄×D₄	1	1	yes	characteristic
173	C₂×C₃×C₄	C₄×D₄	1	1	yes	characteristic
174	C₂×C₃×C₄	#121	1	1	yes	characteristic
175	C₂×C₃×C₄	#123	1	1	yes	characteristic
176	C₂×C₃×C₄	#126	1	1	yes	characteristic
177	C₂×C₃×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
178	C₂×C₃×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
179	C₂×C₃×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
180	C₂×C₃×C₄	#137	1	1	yes	characteristic
181	#67	—	8	1	no	—
182	#67	—	8	1	no	—
183	C₂×Dic₃	—	4	2	no	—
184	C₂×Dic₃	—	4	1	no	—
185	C₂×Dic₃	—	8	1	no	—
186	C₂×Dic₃	—	8	1	no	—
187	D₁₂	—	4	1	no	—
188	D₁₂	—	4	1	no	—
189	D₁₂	—	4	1	no	—
190	D₁₂	—	4	1	no	—
191	D₁₂	—	8	1	no	—
192	C₄×D₃	—	8	1	no	—
193	C₄×D₃	—	8	1	no	—
194	Dic₆	—	4	1	no	—
195	Dic₆	—	4	1	no	—
196	C₃×C₈	—	4	1	no	—
197	C₃×C₈	—	4	1	no	—
198	C₃×C₈	—	4	1	no	—
199	C₃×C₈	—	4	1	no	—
200	C₃×C₈	—	8	1	no	—
201	C₃⋊C₈	—	2	2	no	—
202	C₃⋊C₈	—	4	1	no	—
203	C₃⋊C₈	—	4	1	no	—
204	C₃⋊C₈	—	4	1	no	—
205	#41	—	12	1	no	—
206	#41	—	12	1	no	—
207	#41	—	12	1	no	—
208	#41	—	12	1	no	—
209	C₂×Q₈	—	6	1	no	—
210	C₂×D₄	—	6	1	no	—
211	C₂×D₄	—	6	1	no	—
212	C₂×D₄	—	6	1	no	—
213	C₂×D₄	—	12	1	no	—
214	C₂²×C₄	—	12	1	no	—
215	C₂²×C₄	—	12	1	no	—
216	C₂²×C₄	#239	1	1	yes	characteristic
217	C₂²×C₄	#239	1	1	yes	characteristic
218	C₂²×C₄	C₂×#67	1	1	yes	characteristic
219	M₁₆	—	2	1	no	—
220	M₁₆	—	2	1	no	—
221	M₁₆	—	2	1	no	—
222	M₁₆	—	2	1	no	—
223	M₁₆	—	4	1	no	—
224	M₁₆	—	4	1	no	—
225	C₂×C₈	—	2	1	no	—
226	C₂×C₈	—	2	1	no	—
227	C₂×C₈	—	2	1	no	—
228	C₂×C₈	—	2	1	no	—
229	C₂×C₈	—	4	1	no	—
230	C₂×C₈	—	4	1	no	—
231	C₂×C₈	—	4	1	no	—
232	C₂×C₈	—	6	1	no	—
233	C₂×C₈	—	6	1	no	—
234	C₂×C₈	—	6	1	no	—
235	C₂×C₈	—	6	1	no	—
236	C₂×C₈	—	12	2	no	—
237	C₄⋊C₄	—	6	2	no	—
238	C₄⋊C₄	—	6	1	no	—
239	C₄⋊C₄	—	6	1	no	—
240	C₄⋊C₄	—	12	1	no	—
241	C₂²⋊C₄	—	12	1	no	—
242	C₂²⋊C₄	—	12	1	no	—
243	C₄²	—	12	1	no	—
244	C₄²	—	12	1	no	—
245	C₄²	C₂×C₄×D₃	1	1	yes	characteristic
246	C₄²	#235	1	1	yes	characteristic
247	C₄²	D₃×D₄	1	1	yes	characteristic
248	C₄²	#237	1	1	yes	characteristic
249	C₂²×C₃	—	2	1	no	—
250	C₂²×C₃	—	2	1	no	—
251	C₂²×C₃	#388	1	1	yes	characteristic
252	C₂²×C₃	#489	1	1	yes	characteristic
253	C₂²×C₃	#490	1	1	yes	characteristic
254	C₂×D₃	—	8	1	no	—
255	C₂×D₃	—	8	1	no	—
256	C₂×D₃	—	8	1	no	—
257	C₃×C₄	—	2	1	no	—
258	C₃×C₄	—	2	1	no	—
259	C₃×C₄	—	2	1	no	—
260	C₃×C₄	#440	1	1	yes	characteristic
261	C₃×C₄	#442	1	1	yes	characteristic
262	C₃×C₄	#443	1	1	yes	characteristic
263	C₃×C₄	#444	1	1	yes	characteristic
264	C₃×C₄	#468	1	1	yes	characteristic
265	C₃×C₄	#469	1	1	yes	characteristic
266	Dic₃	—	8	2	no	—
267	Dic₃	—	8	1	no	—
268	C₂³	—	12	1	no	—
269	C₂³	#947	1	1	yes	characteristic
270	Q₈	—	12	1	no	—
271	Q₈	—	12	1	no	—
272	D₄	—	12	1	no	—
273	D₄	—	12	1	no	—
274	D₄	—	12	1	no	—
275	D₄	—	12	1	no	—
276	D₄	—	24	1	no	—
277	D₄	—	24	1	no	—
278	D₄	—	24	1	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	4	1	no	—
282	C₂×C₄	—	12	2	no	—
283	C₂×C₄	—	12	1	no	—
284	C₂×C₄	—	24	1	no	—
285	C₂×C₄	—	24	1	no	—
286	C₂×C₄	—	24	1	no	—
287	C₂×C₄	—	24	1	no	—
288	C₂×C₄	#892	1	1	yes	characteristic
289	C₂×C₄	#893	1	1	yes	characteristic
290	C₂×C₄	#894	1	1	yes	characteristic
291	C₂×C₄	#898	1	1	yes	characteristic
292	C₂×C₄	C₄×#67	1	1	yes	characteristic
293	C₂×C₄	#932	1	1	yes	characteristic
294	C₂×C₄	#939	1	1	yes	characteristic
295	C₂×C₄	#942	1	1	yes	characteristic
296	C₂×C₄	#949	1	1	yes	characteristic
297	C₂×C₄	#949	1	1	yes	characteristic
298	C₈	—	4	1	no	—
299	C₈	—	4	1	no	—
300	C₈	—	4	1	no	—
301	C₈	—	4	1	no	—
302	C₈	—	6	2	no	—
303	C₈	—	8	1	no	—
304	C₈	—	12	1	no	—
305	C₈	—	12	1	no	—
306	C₈	—	12	1	no	—
307	C₂×C₃	—	2	1	no	—
308	C₂×C₃	—	2	1	no	—
309	C₂×C₃	#1929	1	1	yes	characteristic
310	C₂×C₃	#1932	1	1	yes	characteristic
311	C₂×C₃	#1936	1	1	yes	characteristic
312	D₃	—	16	1	no	—
313	C₂²	—	2	1	no	—
314	C₂²	—	2	1	no	—
315	C₂²	—	24	1	no	—
316	C₂²	—	24	1	no	—
317	C₂²	—	24	1	no	—
318	C₂²	#4990	1	1	yes	characteristic
319	C₂²	#5067	1	1	yes	characteristic
320	C₂²	#5089	1	1	yes	characteristic
321	C₄	—	2	1	no	—
322	C₄	—	2	1	no	—
323	C₄	—	2	1	no	—
324	C₄	—	24	2	no	—
325	C₄	—	24	1	no	—
326	C₄	#4894	1	1	yes	characteristic
327	C₄	#4895	1	1	yes	characteristic
328	C₄	#5014	1	1	yes	characteristic
329	C₄	#5026	1	1	yes	characteristic
330	C₄	#5039	1	1	yes	characteristic
331	C₄	#5049	1	1	yes	characteristic
332	C₃	#9592	1	1	yes	characteristic, complemented, Sylow
333	C₂	—	2	1	no	—
334	C₂	—	2	1	no	—
335	C₂	—	48	1	no	—
336	C₂	#69617	1	1	yes	characteristic
337	C₂	#70699	1	1	yes	characteristic
338	C₂	#70727	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#939
commutator	C₃×C₄²	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₃
Fitting	C₃×#1950	C₂
socle	C₂²×C₃	#388
prime radical	#5477	C₂²
cosocle	C₂×C₃×C₄²	C₂³
hypercenter	#1950	D₃
hypocenter	C₃	#9592
LCS	G ▷ C₃×C₄² ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1950

Complements

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

N	H ≅ G/N	type
1. C₃×#1950	C₂	semidirect
5. #69847	C₂	semidirect
6. #67715	C₂	semidirect
7. #67704	C₂	semidirect
20. #4485	C₄	semidirect
21. #4485	C₄	semidirect
24. #4482	C₄	semidirect
25. #4482	C₄	semidirect
29. #1950	D₃	semidirect
332. C₃	#9592	semidirect

Maximal subgroups

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

subgroup	index
C₃×#1950	2
#70498	2
#70431	2
#69867	2
#69847	2
#67715	2
#67704	2
#9592	3