Subgroups — C₃×#70712 #11785452

Order 1152.

597 conjugacy classes of subgroups.

The subgroup lattice has 353 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×ΓL(1,𝔽₂₅)	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃²×#108	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#5126	C₂	1	2	yes	maximal
4	C₂×C₃×#819	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#4558	C₂	1	2	yes	maximal, complemented
6	C₃×#1811	—	3	1	no	maximal
7	#70712	C₃	1	1	yes	maximal, characteristic, direct product factor
8	C₂²×C₃×C₄×D₃	C₂²	1	1	yes	characteristic
9	C₂×C₃²×M₁₆	C₂²	1	2	yes	—
10	C₂²×C₃²×C₈	C₂²	1	1	yes	characteristic
11	C₂×C₃×#208	C₂²	1	2	yes	—
12	C₂²×C₃×C₃⋊C₈	C₂²	1	1	yes	characteristic
13	C₂×C₃×ΓL(1,𝔽₂₅)	—	2	2	no	—
14	C₂×C₃×ΓL(1,𝔽₂₅)	—	2	2	no	—
15	C₂×C₃×ΓL(1,𝔽₂₅)	C₄	1	4	yes	—
16	C₃²×#108	—	2	2	no	—
17	C₃×(C₂×D₃)⋊C₈	—	2	4	no	—
18	C₃×#819	—	2	2	no	—
19	C₂²×C₃×M₁₆	—	3	1	no	—
20	C₂²×ΓL(1,𝔽₂₅)	C₂×C₃	1	1	yes	characteristic
21	C₂×C₃×#108	—	2	1	no	—
22	C₂×C₃×#108	—	3	1	no	—
23	C₂×C₃×#108	D₃	1	1	yes	characteristic, complemented
24	C₂×C₃×#108	C₂×C₃	1	1	yes	characteristic, complemented
25	C₃×#407	—	3	2	no	—
26	#5126	C₂×C₃	1	2	yes	—
27	C₂×#819	C₂×C₃	1	1	yes	characteristic, complemented
28	C₃×#341	—	3	2	no	—
29	#4558	C₂×C₃	1	2	yes	complemented
30	C₂³×C₃×D₃	Q₈	1	1	yes	characteristic
31	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
32	C₂²×C₃×Dic₃	Q₈	1	1	yes	characteristic
33	C₂×C₃×C₄×D₃	—	2	1	no	—
34	C₂×C₃×C₄×D₃	—	2	1	no	—
35	C₂×C₃×C₄×D₃	—	2	1	no	—
36	C₂×C₃×C₄×D₃	—	2	1	no	—
37	C₂×C₃×C₄×D₃	C₂×C₄	1	2	yes	—
38	C₂×C₃×C₄×D₃	D₄	1	2	yes	—
39	C₃²×M₁₆	—	2	2	no	—
40	C₃²×M₁₆	—	2	2	no	—
41	C₂×C₃²×C₈	—	2	2	no	—
42	C₂×C₃²×C₈	C₂×C₄	1	2	yes	—
43	C₂×C₃²×C₈	D₄	1	2	yes	—
44	C₂×C₃²×C₈	D₄	1	2	yes	—
45	C₃×#208	—	2	2	no	—
46	C₃×#208	—	2	2	no	—
47	C₂×C₃×C₃⋊C₈	—	2	1	no	—
48	C₂×C₃×C₃⋊C₈	—	2	1	no	—
49	C₂×C₃×C₃⋊C₈	—	2	2	no	—
50	C₂×C₃×C₃⋊C₈	C₂×C₄	1	2	yes	—
51	C₃×ΓL(1,𝔽₂₅)	—	2	8	no	—
52	#1811	—	3	1	no	Sylow
53	C₂³×C₃×C₄	—	3	1	no	—
54	C₂²×C₄×D₃	C₂²×C₃	1	1	yes	characteristic
55	C₂×C₃×M₁₆	—	2	2	no	—
56	C₂×C₃×M₁₆	—	3	4	no	—
57	C₂×C₃×M₁₆	—	3	2	no	—
58	C₂×C₃×M₁₆	—	6	2	no	—
59	C₂×C₃×M₁₆	—	6	2	no	—
60	C₂×C₃×M₁₆	C₂×D₃	1	2	yes	—
61	C₂×C₃×M₁₆	C₂²×C₃	1	2	yes	—
62	C₂²×C₃×C₈	—	2	1	no	—
63	C₂²×C₃×C₈	—	3	1	no	—
64	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
65	C₂²×C₃×C₈	C₂²×C₃	1	1	yes	characteristic
66	C₂×#208	C₂²×C₃	1	2	yes	—
67	C₂²×C₃⋊C₈	C₂²×C₃	1	1	yes	characteristic
68	C₂×ΓL(1,𝔽₂₅)	—	2	2	no	—
69	C₂×ΓL(1,𝔽₂₅)	—	2	2	no	—
70	C₂×ΓL(1,𝔽₂₅)	C₃×C₄	1	4	yes	—
71	C₃×#108	—	2	4	no	—
72	C₃×#108	—	2	2	no	—
73	C₃×#108	—	2	2	no	—
74	C₃×#108	—	6	2	no	—
75	C₃×C₂²⋊C₈	—	6	4	no	—
76	(C₂×D₃)⋊C₈	—	2	4	no	—
77	#819	—	2	2	no	—
78	C₂³×C₃²	C₂×Q₈	1	1	yes	characteristic
79	C₂²×C₃×D₃	—	2	1	no	—
80	C₂²×C₃×D₃	—	2	1	no	—
81	C₂²×C₃×D₃	—	4	2	no	—
82	C₂²×C₃×D₃	C₄⋊C₄	1	2	yes	—
83	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
84	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
85	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
86	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
87	C₂×C₃²×C₄	#41	1	1	yes	characteristic
88	C₂×C₃²×C₄	#41	1	1	yes	characteristic
89	C₂×C₃×Dic₃	—	2	1	no	—
90	C₂×C₃×Dic₃	—	2	1	no	—
91	C₂×C₃×Dic₃	C₄⋊C₄	1	2	yes	—
92	C₃×C₄×D₃	—	2	2	no	—
93	C₃×C₄×D₃	—	2	2	no	—
94	C₃×C₄×D₃	—	2	2	no	—
95	C₃×C₄×D₃	—	2	2	no	—
96	C₃²×C₈	—	4	2	no	—
97	C₃²×C₈	C₂²⋊C₄	1	4	yes	—
98	C₃×C₃⋊C₈	—	2	2	no	—
99	C₃×C₃⋊C₈	—	4	2	no	—
100	C₂²×M₁₆	—	3	1	no	—
101	C₂×#108	—	3	1	no	—
102	C₂×#108	C₃×D₃	1	1	yes	characteristic, complemented
103	#407	—	3	2	no	—
104	#341	—	3	2	no	—
105	C₂⁴×C₃	—	3	1	no	—
106	C₂³×D₃	C₃×Q₈	1	1	yes	characteristic
107	C₂²×C₃×C₄	—	2	1	no	—
108	C₂²×C₃×C₄	—	3	1	no	—
109	C₂²×C₃×C₄	—	3	2	no	—
110	C₂²×C₃×C₄	—	3	2	no	—
111	C₂²×C₃×C₄	—	6	1	no	—
112	C₂²×C₃×C₄	—	6	1	no	—
113	C₂²×C₃×C₄	—	6	1	no	—
114	C₂²×C₃×C₄	—	6	1	no	—
115	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
116	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
117	C₂²×Dic₃	C₃×Q₈	1	1	yes	characteristic
118	C₂×C₄×D₃	—	2	1	no	—
119	C₂×C₄×D₃	—	2	1	no	—
120	C₂×C₄×D₃	—	2	1	no	—
121	C₂×C₄×D₃	—	2	1	no	—
122	C₂×C₄×D₃	C₂×C₃×C₄	1	2	yes	—
123	C₂×C₄×D₃	C₃×D₄	1	2	yes	—
124	C₃×M₁₆	—	2	4	no	—
125	C₃×M₁₆	—	2	2	no	—
126	C₃×M₁₆	—	2	2	no	—
127	C₃×M₁₆	—	2	2	no	—
128	C₃×M₁₆	—	2	2	no	—
129	C₃×M₁₆	—	4	2	no	—
130	C₃×M₁₆	—	6	8	no	—
131	C₃×M₁₆	—	6	2	no	—
132	C₃×M₁₆	—	6	2	no	—
133	C₂×C₃×C₈	—	2	2	no	—
134	C₂×C₃×C₈	—	2	2	no	—
135	C₂×C₃×C₈	—	2	2	no	—
136	C₂×C₃×C₈	—	2	2	no	—
137	C₂×C₃×C₈	—	2	2	no	—
138	C₂×C₃×C₈	—	3	2	no	—
139	C₂×C₃×C₈	—	4	2	no	—
140	C₂×C₃×C₈	—	6	1	no	—
141	C₂×C₃×C₈	—	6	1	no	—
142	C₂×C₃×C₈	—	6	2	no	—
143	C₂×C₃×C₈	C₄×D₃	1	2	yes	—
144	C₂×C₃×C₈	D₁₂	1	2	yes	—
145	C₂×C₃×C₈	#67	1	2	yes	—
146	C₂×C₃×C₈	C₂×C₃×C₄	1	2	yes	—
147	C₂×C₃×C₈	C₃×D₄	1	2	yes	—
148	C₂×C₃×C₈	C₃×D₄	1	2	yes	—
149	#208	—	2	2	no	—
150	#208	—	2	2	no	—
151	C₂×C₃⋊C₈	—	2	1	no	—
152	C₂×C₃⋊C₈	—	2	1	no	—
153	C₂×C₃⋊C₈	—	2	2	no	—
154	C₂×C₃⋊C₈	C₂×C₃×C₄	1	2	yes	—
155	ΓL(1,𝔽₂₅)	—	2	8	no	—
156	C₂²×C₃²	—	2	2	no	—
157	C₂²×C₃²	C₂×C₄⋊C₄	1	1	yes	characteristic
158	C₂²×C₃²	C₂²⋊Q₈	1	1	yes	characteristic
159	C₂²×C₃²	C₂²⋊Q₈	1	1	yes	characteristic
160	C₂×C₃×D₃	—	2	2	no	—
161	C₂×C₃×D₃	—	2	4	no	—
162	C₂×C₃×D₃	—	4	2	no	—
163	C₂×C₃×D₃	—	4	2	no	—
164	C₃²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
165	C₃²×C₄	#117	1	1	yes	characteristic
166	C₃²×C₄	#121	1	1	yes	characteristic
167	C₃²×C₄	#121	1	1	yes	characteristic
168	C₃×Dic₃	—	2	2	no	—
169	C₂³×C₄	—	3	1	no	—
170	C₂×M₁₆	—	3	4	no	—
171	C₂×M₁₆	—	3	2	no	—
172	C₂×M₁₆	—	6	2	no	—
173	C₂×M₁₆	—	6	2	no	—
174	C₂×M₁₆	C₂×C₃×D₃	1	2	yes	—
175	C₂²×C₈	—	3	1	no	—
176	C₂²×C₈	C₂×C₃×D₃	1	1	yes	characteristic
177	#108	—	2	2	no	—
178	#108	—	6	2	no	—
179	C₂²⋊C₈	—	6	4	no	—
180	C₂³×C₃	—	2	1	no	—
181	C₂³×C₃	—	3	2	no	—
182	C₂³×C₃	—	6	1	no	—
183	C₂³×C₃	—	6	1	no	—
184	C₂³×C₃	—	12	2	no	—
185	C₂³×C₃	D₃×Q₈	1	1	yes	characteristic
186	C₂³×C₃	C₂×C₃×Q₈	1	1	yes	characteristic
187	C₂²×D₃	—	2	1	no	—
188	C₂²×D₃	—	2	1	no	—
189	C₂²×D₃	—	4	2	no	—
190	C₂²×D₃	C₃×C₄⋊C₄	1	2	yes	—
191	C₂×C₃×C₄	—	2	1	no	—
192	C₂×C₃×C₄	—	2	1	no	—
193	C₂×C₃×C₄	—	2	1	no	—
194	C₂×C₃×C₄	—	2	1	no	—
195	C₂×C₃×C₄	—	2	1	no	—
196	C₂×C₃×C₄	—	2	1	no	—
197	C₂×C₃×C₄	—	3	2	no	—
198	C₂×C₃×C₄	—	6	1	no	—
199	C₂×C₃×C₄	—	6	1	no	—
200	C₂×C₃×C₄	—	6	2	no	—
201	C₂×C₃×C₄	—	6	2	no	—
202	C₂×C₃×C₄	—	6	2	no	—
203	C₂×C₃×C₄	—	6	2	no	—
204	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
205	C₂×C₃×C₄	C₂×D₁₂	1	1	yes	characteristic
206	C₂×C₃×C₄	D₃×D₄	1	1	yes	characteristic
207	C₂×C₃×C₄	#237	1	1	yes	characteristic
208	C₂×C₃×C₄	#239	1	1	yes	characteristic
209	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
210	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
211	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
212	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
213	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
214	C₂×C₃×C₄	C₃×#41	1	1	yes	characteristic
215	C₂×C₃×C₄	C₃×#41	1	1	yes	characteristic
216	C₂×Dic₃	—	2	1	no	—
217	C₂×Dic₃	—	2	1	no	—
218	C₂×Dic₃	C₃×C₄⋊C₄	1	2	yes	—
219	C₄×D₃	—	2	2	no	—
220	C₄×D₃	—	2	2	no	—
221	C₄×D₃	—	2	2	no	—
222	C₄×D₃	—	2	2	no	—
223	C₃×C₈	—	2	4	no	—
224	C₃×C₈	—	4	4	no	—
225	C₃×C₈	—	4	2	no	—
226	C₃×C₈	—	4	2	no	—
227	C₃×C₈	—	6	2	no	—
228	C₃×C₈	—	12	2	no	—
229	C₃×C₈	(C₂×D₃)⋊C₄	1	4	yes	—
230	C₃×C₈	C₃×C₂²⋊C₄	1	4	yes	—
231	C₃⋊C₈	—	2	2	no	—
232	C₃⋊C₈	—	4	2	no	—
233	C₂×C₃²	—	2	1	no	—
234	C₂×C₃²	—	2	1	no	—
235	C₂×C₃²	#380	1	1	yes	characteristic
236	C₂×C₃²	#430	1	2	yes	—
237	C₃×D₃	—	4	2	no	—
238	C₂⁴	—	3	1	no	—
239	C₂²×C₄	—	3	1	no	—
240	C₂²×C₄	—	3	2	no	—
241	C₂²×C₄	—	3	2	no	—
242	C₂²×C₄	—	6	1	no	—
243	C₂²×C₄	—	6	1	no	—
244	C₂²×C₄	—	6	1	no	—
245	C₂²×C₄	—	6	1	no	—
246	C₂²×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
247	M₁₆	—	2	2	no	—
248	M₁₆	—	2	2	no	—
249	M₁₆	—	6	8	no	—
250	M₁₆	—	6	2	no	—
251	M₁₆	—	6	2	no	—
252	C₂×C₈	—	2	2	no	—
253	C₂×C₈	—	3	2	no	—
254	C₂×C₈	—	6	1	no	—
255	C₂×C₈	—	6	1	no	—
256	C₂×C₈	—	6	2	no	—
257	C₂×C₈	C₃×C₄×D₃	1	2	yes	—
258	C₂×C₈	C₃×D₁₂	1	2	yes	—
259	C₂×C₈	C₃×#67	1	2	yes	—
260	C₂²×C₃	—	2	2	no	—
261	C₂²×C₃	—	2	2	no	—
262	C₂²×C₃	—	2	1	no	—
263	C₂²×C₃	—	2	1	no	—
264	C₂²×C₃	—	2	1	no	—
265	C₂²×C₃	—	4	2	no	—
266	C₂²×C₃	—	6	4	no	—
267	C₂²×C₃	—	6	2	no	—
268	C₂²×C₃	—	12	2	no	—
269	C₂²×C₃	—	12	2	no	—
270	C₂²×C₃	D₃×C₄⋊C₄	1	1	yes	characteristic
271	C₂²×C₃	#897	1	1	yes	characteristic
272	C₂²×C₃	#946	1	1	yes	characteristic
273	C₂²×C₃	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
274	C₂²×C₃	C₃×C₂²⋊Q₈	1	1	yes	characteristic
275	C₂²×C₃	C₃×C₂²⋊Q₈	1	1	yes	characteristic
276	C₂×D₃	—	2	2	no	—
277	C₂×D₃	—	2	4	no	—
278	C₂×D₃	—	4	2	no	—
279	C₂×D₃	—	4	2	no	—
280	C₃×C₄	—	2	1	no	—
281	C₃×C₄	—	2	1	no	—
282	C₃×C₄	—	2	1	no	—
283	C₃×C₄	—	2	1	no	—
284	C₃×C₄	—	6	2	no	—
285	C₃×C₄	#892	1	1	yes	characteristic
286	C₃×C₄	#895	1	1	yes	characteristic
287	C₃×C₄	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
288	C₃×C₄	#938	1	1	yes	characteristic
289	C₃×C₄	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
290	C₃×C₄	C₃×#117	1	1	yes	characteristic
291	C₃×C₄	C₃×#121	1	1	yes	characteristic
292	C₃×C₄	C₃×#121	1	1	yes	characteristic
293	Dic₃	—	2	2	no	—
294	C₃²	#1811	1	1	yes	characteristic, complemented, Sylow
295	C₂³	—	3	2	no	—
296	C₂³	—	6	1	no	—
297	C₂³	—	6	1	no	—
298	C₂³	—	12	2	no	—
299	C₂³	C₃×D₃×Q₈	1	1	yes	characteristic
300	C₂×C₄	—	3	2	no	—
301	C₂×C₄	—	6	1	no	—
302	C₂×C₄	—	6	1	no	—
303	C₂×C₄	—	6	2	no	—
304	C₂×C₄	—	6	2	no	—
305	C₂×C₄	—	6	2	no	—
306	C₂×C₄	—	6	2	no	—
307	C₂×C₄	C₂×C₃×C₄×D₃	1	1	yes	characteristic
308	C₂×C₄	C₂×C₃×D₁₂	1	1	yes	characteristic
309	C₂×C₄	C₃×D₃×D₄	1	1	yes	characteristic
310	C₂×C₄	C₃×#237	1	1	yes	characteristic
311	C₂×C₄	C₃×#239	1	1	yes	characteristic
312	C₂×C₄	C₂×C₃×#67	1	1	yes	characteristic
313	C₈	—	4	2	no	—
314	C₈	—	6	2	no	—
315	C₈	—	12	2	no	—
316	C₈	C₃×(C₂×D₃)⋊C₄	1	4	yes	—
317	C₂×C₃	—	2	1	no	—
318	C₂×C₃	—	2	1	no	—
319	C₂×C₃	—	2	1	no	—
320	C₂×C₃	—	2	1	no	—
321	C₂×C₃	—	2	1	no	—
322	C₂×C₃	—	2	2	no	—
323	C₂×C₃	—	4	1	no	—
324	C₂×C₃	—	4	1	no	—
325	C₂×C₃	—	12	2	no	—
326	C₂×C₃	#4893	1	2	yes	—
327	C₂×C₃	#4987	1	1	yes	characteristic
328	C₂×C₃	C₃×#380	1	1	yes	characteristic
329	C₂×C₃	C₃×#430	1	2	yes	—
330	D₃	—	4	2	no	—
331	C₂²	—	2	2	no	—
332	C₂²	—	6	2	no	—
333	C₂²	—	6	4	no	—
334	C₂²	—	12	2	no	—
335	C₂²	—	12	2	no	—
336	C₂²	C₃×D₃×C₄⋊C₄	1	1	yes	characteristic
337	C₂²	C₃×#897	1	1	yes	characteristic
338	C₂²	C₃×#946	1	1	yes	characteristic
339	C₄	—	6	2	no	—
340	C₄	C₃×#892	1	1	yes	characteristic
341	C₄	C₃×#895	1	1	yes	characteristic
342	C₄	C₂×C₃×(C₂×D₃)⋊C₄	1	1	yes	characteristic
343	C₄	C₃×#938	1	1	yes	characteristic
344	C₃	—	2	1	no	—
345	C₃	#70712	1	1	yes	characteristic, direct product factor
346	C₃	C₃×#1811	1	1	yes	characteristic, complemented
347	C₂	—	2	1	no	—
348	C₂	—	2	1	no	—
349	C₂	—	12	2	no	—
350	C₂	C₃×#4893	1	2	yes	—
351	C₂	C₃×#4987	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₄	D₃×D₄
commutator	C₂×C₃×C₄	C₂²×C₃×C₄
Frattini	C₂²×C₄	C₂²×C₃×D₃
Fitting	C₂×C₃²×#108	C₂
socle	C₂²×C₃²	C₂×C₄⋊C₄
prime radical	C₂³×C₃×D₃	Q₈
cosocle	C₂²×C₃×C₄	C₂³×C₃
hypercenter	C₂×C₃×#108	D₃
hypocenter	C₃	C₃×#1811
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₂²×C₃×C₄ ◁ C₂×C₃×#108

Complements

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

N	H ≅ G/N	type
2. C₂×C₃²×#108	C₂	semidirect
4. C₂×C₃×#819	C₂	semidirect
5. C₃×#4558	C₂	semidirect
7. #70712	C₃	direct (N × H)
23. C₂×C₃×#108	D₃	semidirect
24. C₂×C₃×#108	C₂×C₃	semidirect
27. C₂×#819	C₂×C₃	semidirect
29. #4558	C₂×C₃	semidirect
102. C₂×#108	C₃×D₃	semidirect
294. C₃²	#1811	semidirect
345. C₃	#70712	direct (N × H)
346. C₃	C₃×#1811	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₃×ΓL(1,𝔽₂₅)	2
C₂×C₃²×#108	2
C₃×#5126	2
C₂×C₃×#819	2
C₃×#4558	2
C₃×#1811	3
#70712	3

Subgroups — C3×#70712 #11785452

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#70712 #11785452