Subgroups — C₃×#69119 #11796046

Order 1152.

469 conjugacy classes of subgroups.

The subgroup lattice has 357 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#907	C₂	1	1	yes	maximal, characteristic
2	C₃×D₃×M₃₂	C₂	1	2	yes	maximal, complemented
3	C₃²×#349	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#4553	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#4514	C₂	1	2	yes	maximal, complemented
6	C₃×#2115	—	3	1	no	maximal
7	#69119	C₃	1	1	yes	maximal, characteristic, direct product factor
8	C₂×C₃×#235	C₄	1	1	yes	characteristic
9	C₂×C₃²×M₁₆	C₂²	1	1	yes	characteristic
10	C₂²×C₃×C₃⋊C₈	C₄	1	1	yes	characteristic
11	C₃×#907	—	2	2	no	—
12	C₃×#907	—	2	2	no	—
13	C₂×C₃×ΓL(1,𝔽₂₅)	C₄	1	1	yes	characteristic
14	C₂×C₃×ΓL(1,𝔽₂₅)	C₂²	1	1	yes	characteristic
15	C₂×C₃×D₃×C₈	C₄	1	1	yes	characteristic
16	C₂×C₃×D₃×C₈	C₂²	1	1	yes	characteristic
17	C₃²×M₃₂	C₂²	1	2	yes	—
18	C₃×#812	C₂²	1	2	yes	—
19	C₃×#798	—	2	2	no	—
20	C₃×D₃×C₁₆	—	2	2	no	—
21	C₂×C₃×#131	—	3	1	no	—
22	C₂×#907	C₂×C₃	1	1	yes	characteristic
23	C₂×C₃×M₃₂	—	3	2	no	—
24	D₃×M₃₂	C₂×C₃	1	2	yes	complemented
25	C₃×#349	—	2	1	no	—
26	C₃×#349	—	3	1	no	—
27	C₃×#349	—	3	2	no	—
28	C₃×#349	D₃	1	1	yes	characteristic, complemented
29	C₃×#349	C₂×C₃	1	1	yes	characteristic, complemented
30	#4553	C₂×C₃	1	1	yes	characteristic, complemented
31	#4514	C₂×C₃	1	2	yes	complemented
32	C₂²×C₃²×C₄	C₂×C₄	1	1	yes	characteristic
33	C₂×C₃×#67	C₈	1	1	yes	characteristic
34	C₂×C₃×#67	C₈	1	1	yes	characteristic
35	C₃×#235	—	2	2	no	—
36	C₃×#235	—	2	2	no	—
37	C₂×C₃×D₁₂	C₈	1	1	yes	characteristic
38	C₂×C₃×C₄×D₃	C₂×C₄	1	1	yes	characteristic
39	C₂×C₃×C₄×D₃	C₂×C₄	1	1	yes	characteristic
40	C₂×C₃×Dic₆	C₈	1	1	yes	characteristic
41	C₃²×M₁₆	—	2	2	no	—
42	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
43	C₂×C₃²×C₈	C₂³	1	1	yes	characteristic
44	C₂×C₃×C₃⋊C₈	—	2	1	no	—
45	C₂×C₃×C₃⋊C₈	—	2	1	no	—
46	C₂×C₃×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
47	C₂×C₃×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
48	C₃×ΓL(1,𝔽₂₅)	—	2	2	no	—
49	C₃×ΓL(1,𝔽₂₅)	—	2	2	no	—
50	C₃×D₃×C₈	—	2	1	no	—
51	C₃×D₃×C₈	—	2	1	no	—
52	C₃×D₃×C₈	D₄	1	2	yes	—
53	C₃×D₃×C₈	D₄	1	2	yes	—
54	C₃²×C₁₆	—	2	2	no	—
55	C₃×C₃⋊C₁₆	—	2	2	no	—
56	#2115	—	3	1	no	Sylow
57	C₂×C₃×#41	—	3	1	no	—
58	C₂×#235	C₃×C₄	1	1	yes	characteristic
59	C₃×#131	—	6	2	no	—
60	C₃×#131	—	6	2	no	—
61	C₂×C₃×M₁₆	—	2	1	no	—
62	C₂×C₃×M₁₆	—	3	1	no	—
63	C₂×C₃×M₁₆	—	3	1	no	—
64	C₂×C₃×M₁₆	C₂×D₃	1	1	yes	characteristic
65	C₂×C₃×M₁₆	C₂²×C₃	1	1	yes	characteristic
66	C₂²×C₃×C₈	—	3	1	no	—
67	C₂²×C₃×C₈	—	3	1	no	—
68	C₂²×C₃×C₈	—	3	1	no	—
69	C₂²×C₃⋊C₈	C₃×C₄	1	1	yes	characteristic
70	#907	—	2	2	no	—
71	#907	—	2	2	no	—
72	C₂×ΓL(1,𝔽₂₅)	C₃×C₄	1	1	yes	characteristic
73	C₂×ΓL(1,𝔽₂₅)	C₂²×C₃	1	1	yes	characteristic
74	C₂×D₃×C₈	C₃×C₄	1	1	yes	characteristic
75	C₂×D₃×C₈	C₂²×C₃	1	1	yes	characteristic
76	C₃×M₃₂	—	2	2	no	—
77	C₃×M₃₂	—	3	2	no	—
78	C₃×M₃₂	—	6	2	no	—
79	C₃×M₃₂	C₂×D₃	1	2	yes	—
80	C₃×M₃₂	C₂²×C₃	1	2	yes	—
81	C₂×C₃×C₁₆	—	6	2	no	—
82	#812	C₂²×C₃	1	2	yes	—
83	#798	—	2	2	no	—
84	D₃×C₁₆	—	2	2	no	—
85	C₂³×C₃²	C₂×C₈	1	1	yes	characteristic
86	C₂²×C₃×D₃	C₂×C₈	1	1	yes	characteristic
87	C₂²×C₃×D₃	C₂×C₈	1	1	yes	characteristic
88	C₂×C₃²×C₄	—	2	1	no	—
89	C₂×C₃²×C₄	—	2	1	no	—
90	C₂×C₃²×C₄	C₂×C₈	1	1	yes	characteristic
91	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
92	C₃×#67	—	2	2	no	—
93	C₃×#67	—	2	2	no	—
94	C₂×C₃×Dic₃	C₂×C₈	1	1	yes	characteristic
95	C₂×C₃×Dic₃	C₂×C₈	1	1	yes	characteristic
96	C₃×D₁₂	—	2	2	no	—
97	C₃×C₄×D₃	—	2	1	no	—
98	C₃×C₄×D₃	—	2	1	no	—
99	C₃×C₄×D₃	C₂²⋊C₄	1	2	yes	—
100	C₃×C₄×D₃	M₁₆	1	2	yes	—
101	C₃×Dic₆	—	2	2	no	—
102	C₃²×C₈	—	2	1	no	—
103	C₃²×C₈	C₂×D₄	1	2	yes	—
104	C₃×C₃⋊C₈	—	2	1	no	—
105	C₃×C₃⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
106	C₃×C₃⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
107	C₂×#131	—	3	1	no	—
108	C₂×M₃₂	—	3	2	no	—
109	#349	—	3	1	no	—
110	#349	—	3	2	no	—
111	#349	C₃×D₃	1	1	yes	characteristic, complemented
112	C₃×#41	—	6	2	no	—
113	C₃×#41	—	6	2	no	—
114	C₂×C₃×Q₈	—	3	1	no	—
115	C₂×C₃×D₄	—	3	1	no	—
116	C₂×C₃×D₄	—	3	1	no	—
117	C₂×C₃×D₄	—	3	1	no	—
118	C₂²×C₃×C₄	—	2	1	no	—
119	C₂²×C₃×C₄	—	3	1	no	—
120	C₂²×C₃×C₄	—	3	1	no	—
121	C₂²×C₃×C₄	C₄×D₃	1	1	yes	characteristic
122	C₂²×C₃×C₄	C₂×C₃×C₄	1	1	yes	characteristic
123	C₂×#67	C₃×C₈	1	1	yes	characteristic
124	C₂×#67	C₃×C₈	1	1	yes	characteristic
125	#235	—	2	2	no	—
126	#235	—	2	2	no	—
127	C₂×D₁₂	C₃×C₈	1	1	yes	characteristic
128	C₂×C₄×D₃	C₂×C₃×C₄	1	1	yes	characteristic
129	C₂×C₄×D₃	C₂×C₃×C₄	1	1	yes	characteristic
130	C₂×Dic₆	C₃×C₈	1	1	yes	characteristic
131	C₃×M₁₆	—	2	2	no	—
132	C₃×M₁₆	—	2	2	no	—
133	C₃×M₁₆	—	4	2	no	—
134	C₃×M₁₆	—	6	2	no	—
135	C₃×M₁₆	—	6	2	no	—
136	C₂×C₃×C₈	—	2	1	no	—
137	C₂×C₃×C₈	—	2	1	no	—
138	C₂×C₃×C₈	—	3	1	no	—
139	C₂×C₃×C₈	—	3	2	no	—
140	C₂×C₃×C₈	—	3	2	no	—
141	C₂×C₃×C₈	—	3	1	no	—
142	C₂×C₃×C₈	—	6	1	no	—
143	C₂×C₃×C₈	—	6	1	no	—
144	C₂×C₃×C₈	—	6	1	no	—
145	C₂×C₃×C₈	—	6	1	no	—
146	C₂×C₃×C₈	C₄×D₃	1	1	yes	characteristic
147	C₂×C₃×C₈	C₂×C₃×C₄	1	1	yes	characteristic
148	C₂×C₃×C₈	C₂²×D₃	1	1	yes	characteristic
149	C₂×C₃×C₈	C₂³×C₃	1	1	yes	characteristic
150	C₂×C₃⋊C₈	—	2	1	no	—
151	C₂×C₃⋊C₈	—	2	1	no	—
152	C₂×C₃⋊C₈	C₂×C₃×C₄	1	1	yes	characteristic
153	C₂×C₃⋊C₈	C₂×C₃×C₄	1	1	yes	characteristic
154	ΓL(1,𝔽₂₅)	—	2	2	no	—
155	ΓL(1,𝔽₂₅)	—	2	2	no	—
156	D₃×C₈	—	2	1	no	—
157	D₃×C₈	—	2	1	no	—
158	D₃×C₈	C₃×D₄	1	2	yes	—
159	D₃×C₈	C₃×D₄	1	2	yes	—
160	C₃×C₁₆	—	2	2	no	—
161	C₃×C₁₆	—	2	2	no	—
162	C₃×C₁₆	—	4	2	no	—
163	C₃×C₁₆	—	6	2	no	—
164	C₃⋊C₁₆	—	2	2	no	—
165	C₂²×C₃²	—	2	1	no	—
166	C₂²×C₃²	—	4	1	no	—
167	C₂²×C₃²	C₂²×C₈	1	1	yes	characteristic
168	C₂×C₃×D₃	—	2	1	no	—
169	C₂×C₃×D₃	—	4	1	no	—
170	C₂×C₃×D₃	—	4	1	no	—
171	C₂×C₃×D₃	C₂²⋊C₈	1	2	yes	—
172	C₃²×C₄	—	2	1	no	—
173	C₃²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
174	C₃²×C₄	C₂×M₁₆	1	1	yes	characteristic
175	C₃×Dic₃	—	2	1	no	—
176	C₃×Dic₃	C₂²⋊C₈	1	2	yes	—
177	C₂×#41	—	3	1	no	—
178	#131	—	6	2	no	—
179	#131	—	6	2	no	—
180	C₂×M₁₆	—	3	1	no	—
181	C₂×M₁₆	—	3	1	no	—
182	C₂×M₁₆	C₂×C₃×D₃	1	1	yes	characteristic
183	C₂²×C₈	—	3	1	no	—
184	C₂²×C₈	—	3	1	no	—
185	C₂²×C₈	—	3	1	no	—
186	M₃₂	—	3	2	no	—
187	M₃₂	—	6	2	no	—
188	M₃₂	C₂×C₃×D₃	1	2	yes	—
189	C₂×C₁₆	—	6	2	no	—
190	C₂³×C₃	—	2	1	no	—
191	C₂³×C₃	—	3	1	no	—
192	C₂³×C₃	—	3	1	no	—
193	C₂³×C₃	D₃×C₈	1	1	yes	characteristic
194	C₂³×C₃	C₂×C₃×C₈	1	1	yes	characteristic
195	C₂²×D₃	C₂×C₃×C₈	1	1	yes	characteristic
196	C₂²×D₃	C₂×C₃×C₈	1	1	yes	characteristic
197	C₃×Q₈	—	6	2	no	—
198	C₃×D₄	—	6	2	no	—
199	C₃×D₄	—	6	2	no	—
200	C₃×D₄	—	6	2	no	—
201	C₂×C₃×C₄	—	2	1	no	—
202	C₂×C₃×C₄	—	2	1	no	—
203	C₂×C₃×C₄	—	2	1	no	—
204	C₂×C₃×C₄	—	2	1	no	—
205	C₂×C₃×C₄	—	2	1	no	—
206	C₂×C₃×C₄	—	2	1	no	—
207	C₂×C₃×C₄	—	3	1	no	—
208	C₂×C₃×C₄	—	3	2	no	—
209	C₂×C₃×C₄	—	3	2	no	—
210	C₂×C₃×C₄	—	3	1	no	—
211	C₂×C₃×C₄	—	4	1	no	—
212	C₂×C₃×C₄	—	4	1	no	—
213	C₂×C₃×C₄	—	6	1	no	—
214	C₂×C₃×C₄	—	6	1	no	—
215	C₂×C₃×C₄	D₃×C₈	1	1	yes	characteristic
216	C₂×C₃×C₄	C₂×C₃×C₈	1	1	yes	characteristic
217	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
218	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
219	#67	—	2	2	no	—
220	#67	—	2	2	no	—
221	C₂×Dic₃	C₂×C₃×C₈	1	1	yes	characteristic
222	C₂×Dic₃	C₂×C₃×C₈	1	1	yes	characteristic
223	D₁₂	—	2	2	no	—
224	C₄×D₃	—	2	1	no	—
225	C₄×D₃	—	2	1	no	—
226	C₄×D₃	C₃×C₂²⋊C₄	1	2	yes	—
227	C₄×D₃	C₃×M₁₆	1	2	yes	—
228	Dic₆	—	2	2	no	—
229	C₃×C₈	—	2	1	no	—
230	C₃×C₈	—	2	1	no	—
231	C₃×C₈	—	2	2	no	—
232	C₃×C₈	—	3	1	no	—
233	C₃×C₈	—	3	1	no	—
234	C₃×C₈	—	4	1	no	—
235	C₃×C₈	—	6	1	no	—
236	C₃×C₈	D₃×D₄	1	2	yes	—
237	C₃×C₈	C₂×C₃×D₄	1	2	yes	—
238	C₃⋊C₈	—	2	1	no	—
239	C₃⋊C₈	C₃×C₂²⋊C₄	1	1	yes	characteristic
240	C₃⋊C₈	C₃×C₂²⋊C₄	1	1	yes	characteristic
241	C₂×C₃²	—	2	1	no	—
242	C₂×C₃²	—	4	1	no	—
243	C₂×C₃²	C₂×C₂²⋊C₈	1	1	yes	characteristic
244	C₃×D₃	—	2	2	no	—
245	C₃×D₃	—	4	1	no	—
246	#41	—	6	2	no	—
247	#41	—	6	2	no	—
248	C₂×Q₈	—	3	1	no	—
249	C₂×D₄	—	3	1	no	—
250	C₂×D₄	—	3	1	no	—
251	C₂×D₄	—	3	1	no	—
252	C₂²×C₄	—	3	1	no	—
253	C₂²×C₄	—	3	1	no	—
254	C₂²×C₄	C₃×C₄×D₃	1	1	yes	characteristic
255	M₁₆	—	2	2	no	—
256	M₁₆	—	6	2	no	—
257	M₁₆	—	6	2	no	—
258	C₂×C₈	—	3	1	no	—
259	C₂×C₈	—	3	2	no	—
260	C₂×C₈	—	3	2	no	—
261	C₂×C₈	—	3	1	no	—
262	C₂×C₈	—	6	1	no	—
263	C₂×C₈	—	6	1	no	—
264	C₂×C₈	—	6	1	no	—
265	C₂×C₈	—	6	1	no	—
266	C₂×C₈	C₃×C₄×D₃	1	1	yes	characteristic
267	C₂×C₈	C₂²×C₃×D₃	1	1	yes	characteristic
268	C₁₆	—	2	2	no	—
269	C₁₆	—	6	2	no	—
270	C₂²×C₃	—	2	1	no	—
271	C₂²×C₃	—	2	1	no	—
272	C₂²×C₃	—	2	1	no	—
273	C₂²×C₃	—	3	2	no	—
274	C₂²×C₃	—	4	1	no	—
275	C₂²×C₃	—	4	1	no	—
276	C₂²×C₃	—	4	1	no	—
277	C₂²×C₃	—	6	1	no	—
278	C₂²×C₃	—	8	1	no	—
279	C₂²×C₃	—	12	1	no	—
280	C₂²×C₃	—	12	1	no	—
281	C₂²×C₃	C₂×D₃×C₈	1	1	yes	characteristic
282	C₂²×C₃	C₂²×C₃×C₈	1	1	yes	characteristic
283	C₂×D₃	—	2	1	no	—
284	C₂×D₃	—	4	1	no	—
285	C₂×D₃	—	4	1	no	—
286	C₂×D₃	C₃×C₂²⋊C₈	1	2	yes	—
287	C₃×C₄	—	2	1	no	—
288	C₃×C₄	—	2	1	no	—
289	C₃×C₄	—	2	1	no	—
290	C₃×C₄	—	2	1	no	—
291	C₃×C₄	—	3	2	no	—
292	C₃×C₄	—	4	1	no	—
293	C₃×C₄	—	6	1	no	—
294	C₃×C₄	D₃×C₂²⋊C₄	1	1	yes	characteristic
295	C₃×C₄	D₃×M₁₆	1	1	yes	characteristic
296	C₃×C₄	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
297	C₃×C₄	C₂×C₃×M₁₆	1	1	yes	characteristic
298	Dic₃	—	2	1	no	—
299	Dic₃	C₃×C₂²⋊C₈	1	2	yes	—
300	C₃²	#2115	1	1	yes	characteristic, complemented, Sylow
301	C₂³	—	3	1	no	—
302	C₂³	—	3	1	no	—
303	C₂³	C₃×D₃×C₈	1	1	yes	characteristic
304	Q₈	—	6	2	no	—
305	D₄	—	6	2	no	—
306	D₄	—	6	2	no	—
307	D₄	—	6	2	no	—
308	C₂×C₄	—	2	1	no	—
309	C₂×C₄	—	2	1	no	—
310	C₂×C₄	—	3	1	no	—
311	C₂×C₄	—	3	2	no	—
312	C₂×C₄	—	3	2	no	—
313	C₂×C₄	—	3	1	no	—
314	C₂×C₄	—	6	1	no	—
315	C₂×C₄	—	6	1	no	—
316	C₂×C₄	C₃×D₃×C₈	1	1	yes	characteristic
317	C₂×C₄	C₂×C₃×C₄×D₃	1	1	yes	characteristic
318	C₈	—	2	1	no	—
319	C₈	—	3	1	no	—
320	C₈	—	3	1	no	—
321	C₈	—	6	1	no	—
322	C₈	C₃×D₃×D₄	1	2	yes	—
323	C₂×C₃	—	2	1	no	—
324	C₂×C₃	—	2	1	no	—
325	C₂×C₃	—	2	1	no	—
326	C₂×C₃	—	4	1	no	—
327	C₂×C₃	—	4	1	no	—
328	C₂×C₃	—	4	1	no	—
329	C₂×C₃	—	6	2	no	—
330	C₂×C₃	—	8	1	no	—
331	C₂×C₃	—	12	1	no	—
332	C₂×C₃	D₃×C₂²⋊C₈	1	1	yes	characteristic
333	C₂×C₃	C₂×C₃×C₂²⋊C₈	1	1	yes	characteristic
334	D₃	—	2	2	no	—
335	D₃	—	4	1	no	—
336	C₂²	—	2	1	no	—
337	C₂²	—	3	2	no	—
338	C₂²	—	4	1	no	—
339	C₂²	—	6	1	no	—
340	C₂²	—	12	1	no	—
341	C₂²	—	12	1	no	—
342	C₂²	C₂×C₃×D₃×C₈	1	1	yes	characteristic
343	C₄	—	2	1	no	—
344	C₄	—	3	2	no	—
345	C₄	—	6	1	no	—
346	C₄	C₃×D₃×C₂²⋊C₄	1	1	yes	characteristic
347	C₄	C₃×D₃×M₁₆	1	1	yes	characteristic
348	C₃	—	2	1	no	—
349	C₃	#69119	1	1	yes	characteristic, direct product factor
350	C₃	C₃×#2115	1	1	yes	characteristic, complemented
351	C₂	—	2	1	no	—
352	C₂	—	4	1	no	—
353	C₂	—	6	2	no	—
354	C₂	—	12	1	no	—
355	C₂	C₃×D₃×C₂²⋊C₈	1	1	yes	characteristic

	subgroup	G/H
center	C₃×C₄	D₃×C₂²⋊C₄
commutator	C₂²×C₃	C₂²×C₃×C₈
Frattini	C₂×C₈	C₂²×C₃×D₃
Fitting	C₃²×#349	C₂
socle	C₂×C₃²	C₂×C₂²⋊C₈
prime radical	C₂×C₃×#235	C₄
cosocle	C₂×C₃×C₈	C₂³×C₃
hypercenter	C₃×#349	D₃
hypocenter	C₃	C₃×#2115
LCS	G ▷ C₂²×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₃×C₄ ◁ C₂×C₃×C₈ ◁ C₃×#349

Complements

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

N	H ≅ G/N	type
2. C₃×D₃×M₃₂	C₂	semidirect
3. C₃²×#349	C₂	semidirect
4. C₃×#4553	C₂	semidirect
5. C₃×#4514	C₂	semidirect
7. #69119	C₃	direct (N × H)
24. D₃×M₃₂	C₂×C₃	semidirect
28. C₃×#349	D₃	semidirect
29. C₃×#349	C₂×C₃	semidirect
30. #4553	C₂×C₃	semidirect
31. #4514	C₂×C₃	semidirect
111. #349	C₃×D₃	semidirect
300. C₃²	#2115	semidirect
349. C₃	#69119	direct (N × H)
350. C₃	C₃×#2115	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×#907	2
C₃×D₃×M₃₂	2
C₃²×#349	2
C₃×#4553	2
C₃×#4514	2
C₃×#2115	3
#69119	3

Subgroups — C3×#69119 #11796046

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#69119 #11796046