Subgroups — C₂×C₃×#553 #87355

Order 384.

1252 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#553	C₂	1	16	yes	maximal, direct product factor
2	C₂×C₃×#126	C₂	1	2	yes	maximal, complemented
3	C₂×C₃×#125	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₃×#123	C₂	1	2	yes	maximal, complemented
5	C₂×C₃×#121	C₂	1	2	yes	maximal, complemented
6	C₂×C₃×#121	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₃×#121	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₃×C₄×D₄	C₂	1	2	yes	maximal, complemented
9	C₂×C₃×C₄×D₄	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₃×C₄×D₄	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₃×#117	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×C₃×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂×#553	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
14	C₂²×C₃×D₄	C₂²	1	2	yes	—
15	C₂²×C₃×D₄	C₂²	1	1	yes	characteristic
16	C₂²×C₃×D₄	C₂²	1	1	yes	characteristic
17	C₂²×C₃×D₄	C₂²	1	2	yes	—
18	C₂³×C₃×C₄	C₂²	1	1	yes	characteristic
19	C₂³×C₃×C₄	C₂²	1	1	yes	characteristic
20	C₂³×C₃×C₄	C₂²	1	2	yes	—
21	C₂³×C₃×C₄	C₂²	1	2	yes	—
22	C₃×#126	C₂²	1	16	yes	complemented
23	C₃×#125	C₂²	1	8	yes	complemented
24	C₃×#123	C₂²	1	16	yes	complemented
25	C₃×#121	C₂²	1	16	yes	complemented
26	C₃×#121	C₂²	1	8	yes	complemented
27	C₃×#121	C₂²	1	8	yes	complemented
28	C₃×C₄×D₄	C₂²	1	16	yes	complemented
29	C₃×C₄×D₄	C₂²	1	8	yes	complemented
30	C₃×C₄×D₄	C₂²	1	8	yes	complemented
31	C₃×#117	C₂²	1	8	yes	complemented
32	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
33	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
34	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
35	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
36	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
37	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
38	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
39	C₂×C₃×C₄⋊C₄	C₂²	1	8	yes	complemented
40	C₂×C₃×C₂²⋊C₄	C₂²	1	2	yes	—
41	C₂×C₃×C₂²⋊C₄	C₂²	1	1	yes	characteristic
42	C₂×C₃×C₂²⋊C₄	C₂²	1	2	yes	—
43	C₂×C₃×C₂²⋊C₄	C₂²	1	2	yes	—
44	C₂×C₃×C₂²⋊C₄	C₂²	1	1	yes	characteristic
45	C₂×C₃×C₂²⋊C₄	C₂²	1	2	yes	—
46	C₂×C₃×C₄²	C₂²	1	2	yes	—
47	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
48	#553	C₂×C₃	1	16	yes	direct product factor
49	C₂×#126	C₂×C₃	1	2	yes	complemented
50	C₂×#125	C₂×C₃	1	1	yes	characteristic, complemented
51	C₂×#123	C₂×C₃	1	2	yes	complemented
52	C₂×#121	C₂×C₃	1	2	yes	complemented
53	C₂×#121	C₂×C₃	1	1	yes	characteristic, complemented
54	C₂×#121	C₂×C₃	1	1	yes	characteristic, complemented
55	C₂×C₄×D₄	C₂×C₃	1	2	yes	complemented
56	C₂×C₄×D₄	C₂×C₃	1	1	yes	characteristic, complemented
57	C₂×C₄×D₄	C₂×C₃	1	1	yes	characteristic, complemented
58	C₂×#117	C₂×C₃	1	1	yes	characteristic, complemented
59	C₂²×C₄⋊C₄	C₂×C₃	1	1	yes	characteristic, complemented
60	C₂⁴×C₃	C₂³	1	1	yes	characteristic
61	C₂⁴×C₃	C₂³	1	2	yes	—
62	C₂⁴×C₃	C₂³	1	1	yes	characteristic
63	C₂×C₃×D₄	—	2	8	no	—
64	C₂×C₃×D₄	—	2	4	no	—
65	C₂×C₃×D₄	—	2	4	no	—
66	C₂×C₃×D₄	—	4	4	no	—
67	C₂×C₃×D₄	C₂³	1	8	yes	—
68	C₂×C₃×D₄	C₂³	1	4	yes	—
69	C₂×C₃×D₄	C₂³	1	4	yes	—
70	C₂×C₃×D₄	C₂³	1	8	yes	—
71	C₂²×C₃×C₄	—	2	4	no	—
72	C₂²×C₃×C₄	—	2	4	no	—
73	C₂²×C₃×C₄	—	4	4	no	—
74	C₂²×C₃×C₄	—	4	4	no	—
75	C₂²×C₃×C₄	C₂³	1	4	yes	—
76	C₂²×C₃×C₄	C₂³	1	4	yes	—
77	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
78	C₂²×C₃×C₄	C₂³	1	2	yes	—
79	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
80	C₂²×C₃×C₄	C₂³	1	2	yes	—
81	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
82	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
83	C₂²×C₃×C₄	C₂³	1	2	yes	—
84	C₂²×C₃×C₄	C₂³	1	8	yes	—
85	C₂²×C₃×C₄	C₂³	1	8	yes	—
86	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
87	C₃×C₄⋊C₄	C₂³	1	4	yes	—
88	C₃×C₄⋊C₄	C₂³	1	4	yes	—
89	C₃×C₄⋊C₄	C₂³	1	8	yes	—
90	C₃×C₄⋊C₄	C₂³	1	4	yes	—
91	C₃×C₄⋊C₄	C₂³	1	8	yes	—
92	C₃×C₄⋊C₄	C₂³	1	4	yes	—
93	C₃×C₄⋊C₄	C₂³	1	8	yes	—
94	C₃×C₂²⋊C₄	C₂³	1	8	yes	—
95	C₃×C₂²⋊C₄	C₂³	1	4	yes	—
96	C₃×C₂²⋊C₄	C₂³	1	8	yes	—
97	C₃×C₂²⋊C₄	C₂³	1	8	yes	—
98	C₃×C₂²⋊C₄	C₂³	1	4	yes	—
99	C₃×C₂²⋊C₄	C₂³	1	8	yes	—
100	C₃×C₄²	C₂³	1	8	yes	—
101	C₃×C₄²	C₂³	1	4	yes	—
102	C₂²×D₄	C₂²×C₃	1	2	yes	—
103	C₂²×D₄	C₂²×C₃	1	1	yes	characteristic
104	C₂²×D₄	C₂²×C₃	1	1	yes	characteristic
105	C₂²×D₄	C₂²×C₃	1	2	yes	—
106	C₂³×C₄	C₂²×C₃	1	1	yes	characteristic
107	C₂³×C₄	C₂²×C₃	1	1	yes	characteristic
108	C₂³×C₄	C₂²×C₃	1	2	yes	—
109	C₂³×C₄	C₂²×C₃	1	2	yes	—
110	#126	C₂²×C₃	1	16	yes	complemented
111	#125	C₂²×C₃	1	8	yes	complemented
112	#123	C₂²×C₃	1	16	yes	complemented
113	#121	C₂²×C₃	1	16	yes	complemented
114	#121	C₂²×C₃	1	8	yes	complemented
115	#121	C₂²×C₃	1	8	yes	complemented
116	C₄×D₄	C₂²×C₃	1	16	yes	complemented
117	C₄×D₄	C₂²×C₃	1	8	yes	complemented
118	C₄×D₄	C₂²×C₃	1	8	yes	complemented
119	#117	C₂²×C₃	1	8	yes	complemented
120	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
121	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
122	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
123	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
124	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
125	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
126	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
127	C₂×C₄⋊C₄	C₂²×C₃	1	8	yes	complemented
128	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
129	C₂×C₂²⋊C₄	C₂²×C₃	1	1	yes	characteristic
130	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
131	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
132	C₂×C₂²⋊C₄	C₂²×C₃	1	1	yes	characteristic
133	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
134	C₂×C₄²	C₂²×C₃	1	2	yes	—
135	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic
136	C₂³×C₃	—	2	2	no	—
137	C₂³×C₃	—	2	2	no	—
138	C₂³×C₃	—	2	2	no	—
139	C₂³×C₃	—	2	2	no	—
140	C₂³×C₃	—	2	4	no	—
141	C₂³×C₃	—	2	4	no	—
142	C₂³×C₃	—	2	4	no	—
143	C₂³×C₃	—	2	2	no	—
144	C₂³×C₃	#41	1	4	yes	—
145	C₂³×C₃	C₂⁴	1	2	yes	—
146	C₂³×C₃	C₂⁴	1	4	yes	—
147	C₂³×C₃	C₂⁴	1	2	yes	—
148	C₂³×C₃	C₂⁴	1	1	yes	characteristic
149	C₃×D₄	—	2	16	no	—
150	C₃×D₄	—	2	8	no	—
151	C₃×D₄	—	2	8	no	—
152	C₃×D₄	—	4	8	no	—
153	C₂×C₃×C₄	—	2	2	no	—
154	C₂×C₃×C₄	—	2	2	no	—
155	C₂×C₃×C₄	—	2	4	no	—
156	C₂×C₃×C₄	—	2	2	no	—
157	C₂×C₃×C₄	—	2	2	no	—
158	C₂×C₃×C₄	—	2	4	no	—
159	C₂×C₃×C₄	—	2	4	no	—
160	C₂×C₃×C₄	—	2	8	no	—
161	C₂×C₃×C₄	—	2	8	no	—
162	C₂×C₃×C₄	—	4	8	no	—
163	C₂×C₃×C₄	—	4	8	no	—
164	C₂×C₃×C₄	#41	1	4	yes	—
165	C₂×C₃×C₄	C₂⁴	1	2	yes	—
166	C₂×C₃×C₄	C₂⁴	1	4	yes	—
167	C₂×C₃×C₄	C₂⁴	1	2	yes	—
168	C₂×C₃×C₄	C₂⁴	1	4	yes	—
169	C₂×C₃×C₄	C₂⁴	1	2	yes	—
170	C₂×C₃×C₄	C₂⁴	1	2	yes	—
171	C₂×C₃×C₄	C₂⁴	1	4	yes	—
172	C₂×C₃×C₄	C₂⁴	1	2	yes	—
173	C₂⁴	C₂³×C₃	1	1	yes	characteristic
174	C₂⁴	C₂³×C₃	1	2	yes	—
175	C₂⁴	C₂³×C₃	1	1	yes	characteristic
176	C₂×D₄	—	2	8	no	—
177	C₂×D₄	—	2	4	no	—
178	C₂×D₄	—	2	4	no	—
179	C₂×D₄	—	4	4	no	—
180	C₂×D₄	C₂³×C₃	1	8	yes	—
181	C₂×D₄	C₂³×C₃	1	4	yes	—
182	C₂×D₄	C₂³×C₃	1	4	yes	—
183	C₂×D₄	C₂³×C₃	1	8	yes	—
184	C₂²×C₄	—	2	4	no	—
185	C₂²×C₄	—	2	4	no	—
186	C₂²×C₄	—	4	4	no	—
187	C₂²×C₄	—	4	4	no	—
188	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
189	C₂²×C₄	C₂³×C₃	1	2	yes	—
190	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
191	C₂²×C₄	C₂³×C₃	1	2	yes	—
192	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
193	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
194	C₂²×C₄	C₂³×C₃	1	2	yes	—
195	C₂²×C₄	C₂³×C₃	1	4	yes	—
196	C₂²×C₄	C₂³×C₃	1	4	yes	—
197	C₂²×C₄	C₂³×C₃	1	8	yes	—
198	C₂²×C₄	C₂³×C₃	1	8	yes	—
199	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
200	C₄⋊C₄	C₂³×C₃	1	4	yes	—
201	C₄⋊C₄	C₂³×C₃	1	4	yes	—
202	C₄⋊C₄	C₂³×C₃	1	8	yes	—
203	C₄⋊C₄	C₂³×C₃	1	4	yes	—
204	C₄⋊C₄	C₂³×C₃	1	8	yes	—
205	C₄⋊C₄	C₂³×C₃	1	4	yes	—
206	C₄⋊C₄	C₂³×C₃	1	8	yes	—
207	C₂²⋊C₄	C₂³×C₃	1	8	yes	—
208	C₂²⋊C₄	C₂³×C₃	1	4	yes	—
209	C₂²⋊C₄	C₂³×C₃	1	8	yes	—
210	C₂²⋊C₄	C₂³×C₃	1	8	yes	—
211	C₂²⋊C₄	C₂³×C₃	1	4	yes	—
212	C₂²⋊C₄	C₂³×C₃	1	8	yes	—
213	C₄²	C₂³×C₃	1	8	yes	—
214	C₄²	C₂³×C₃	1	4	yes	—
215	C₂²×C₃	—	2	8	no	—
216	C₂²×C₃	—	2	2	no	—
217	C₂²×C₃	—	2	2	no	—
218	C₂²×C₃	—	2	2	no	—
219	C₂²×C₃	—	2	2	no	—
220	C₂²×C₃	—	2	4	no	—
221	C₂²×C₃	—	2	4	no	—
222	C₂²×C₃	—	2	4	no	—
223	C₂²×C₃	—	2	2	no	—
224	C₂²×C₃	—	4	4	no	—
225	C₂²×C₃	—	4	8	no	—
226	C₂²×C₃	C₂×#41	1	4	yes	—
227	C₂²×C₃	C₂×#41	1	2	yes	—
228	C₂²×C₃	C₂×#41	1	2	yes	—
229	C₂²×C₃	H(5,𝔽₂)	1	2	yes	—
230	C₂²×C₃	C₂⁵	1	1	yes	characteristic
231	C₃×C₄	—	2	2	no	—
232	C₃×C₄	—	2	2	no	—
233	C₃×C₄	—	2	4	no	—
234	C₃×C₄	—	2	2	no	—
235	C₃×C₄	—	2	2	no	—
236	C₃×C₄	—	2	4	no	—
237	C₃×C₄	—	2	4	no	—
238	C₃×C₄	C₂×#41	1	4	yes	—
239	C₂³	—	2	2	no	—
240	C₂³	—	2	2	no	—
241	C₂³	—	2	2	no	—
242	C₂³	—	2	2	no	—
243	C₂³	—	2	4	no	—
244	C₂³	—	2	4	no	—
245	C₂³	—	2	4	no	—
246	C₂³	—	2	2	no	—
247	C₂³	C₃×#41	1	4	yes	—
248	C₂³	C₂⁴×C₃	1	2	yes	—
249	C₂³	C₂⁴×C₃	1	4	yes	—
250	C₂³	C₂⁴×C₃	1	2	yes	—
251	C₂³	C₂⁴×C₃	1	1	yes	characteristic
252	D₄	—	2	16	no	—
253	D₄	—	2	8	no	—
254	D₄	—	2	8	no	—
255	D₄	—	4	8	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	2	4	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	4	no	—
262	C₂×C₄	—	2	4	no	—
263	C₂×C₄	—	2	8	no	—
264	C₂×C₄	—	2	8	no	—
265	C₂×C₄	—	4	8	no	—
266	C₂×C₄	—	4	8	no	—
267	C₂×C₄	C₃×#41	1	4	yes	—
268	C₂×C₄	C₂⁴×C₃	1	2	yes	—
269	C₂×C₄	C₂⁴×C₃	1	4	yes	—
270	C₂×C₄	C₂⁴×C₃	1	2	yes	—
271	C₂×C₄	C₂⁴×C₃	1	4	yes	—
272	C₂×C₄	C₂⁴×C₃	1	2	yes	—
273	C₂×C₄	C₂⁴×C₃	1	2	yes	—
274	C₂×C₄	C₂⁴×C₃	1	4	yes	—
275	C₂×C₄	C₂⁴×C₃	1	2	yes	—
276	C₂×C₃	—	2	4	no	—
277	C₂×C₃	—	4	2	no	—
278	C₂×C₃	—	4	4	no	—
279	C₂×C₃	#553	1	4	yes	direct product factor
280	C₂×C₃	C₂²×#41	1	1	yes	characteristic
281	C₂×C₃	C₂²×#41	1	1	yes	characteristic
282	C₂×C₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
283	C₂²	—	2	2	no	—
284	C₂²	—	2	2	no	—
285	C₂²	—	2	8	no	—
286	C₂²	—	2	2	no	—
287	C₂²	—	2	2	no	—
288	C₂²	—	2	4	no	—
289	C₂²	—	2	4	no	—
290	C₂²	—	2	4	no	—
291	C₂²	—	2	2	no	—
292	C₂²	—	4	4	no	—
293	C₂²	—	4	8	no	—
294	C₂²	C₂×C₃×#41	1	4	yes	—
295	C₂²	C₂×C₃×#41	1	2	yes	—
296	C₂²	C₂×C₃×#41	1	2	yes	—
297	C₂²	C₃×H(5,𝔽₂)	1	2	yes	—
298	C₂²	C₂⁵×C₃	1	1	yes	characteristic
299	C₄	—	2	2	no	—
300	C₄	—	2	2	no	—
301	C₄	—	2	4	no	—
302	C₄	—	2	2	no	—
303	C₄	—	2	2	no	—
304	C₄	—	2	4	no	—
305	C₄	—	2	4	no	—
306	C₄	C₂×C₃×#41	1	4	yes	—
307	C₃	C₂×#553	1	1	yes	characteristic, direct product factor, Sylow
308	C₂	—	2	4	no	—
309	C₂	—	4	2	no	—
310	C₂	—	4	4	no	—
311	C₂	C₃×#553	1	4	yes	direct product factor
312	C₂	C₂²×C₃×#41	1	1	yes	characteristic
313	C₂	C₂²×C₃×#41	1	1	yes	characteristic
314	C₂	C₂×C₃×H(5,𝔽₂)	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁵×C₃
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃×#553	C₂	direct (N × H)
2. C₂×C₃×#126	C₂	semidirect
3. C₂×C₃×#125	C₂	semidirect
4. C₂×C₃×#123	C₂	semidirect
5. C₂×C₃×#121	C₂	semidirect
6. C₂×C₃×#121	C₂	semidirect
7. C₂×C₃×#121	C₂	semidirect
8. C₂×C₃×C₄×D₄	C₂	semidirect
9. C₂×C₃×C₄×D₄	C₂	semidirect
10. C₂×C₃×C₄×D₄	C₂	semidirect
11. C₂×C₃×#117	C₂	semidirect
12. C₂²×C₃×C₄⋊C₄	C₂	semidirect
13. C₂×#553	C₃	direct (N × H)
22. C₃×#126	C₂²	semidirect
23. C₃×#125	C₂²	semidirect
24. C₃×#123	C₂²	semidirect
25. C₃×#121	C₂²	semidirect
26. C₃×#121	C₂²	semidirect
27. C₃×#121	C₂²	semidirect
28. C₃×C₄×D₄	C₂²	semidirect
29. C₃×C₄×D₄	C₂²	semidirect
30. C₃×C₄×D₄	C₂²	semidirect
31. C₃×#117	C₂²	semidirect
39. C₂×C₃×C₄⋊C₄	C₂²	semidirect
48. #553	C₂×C₃	direct (N × H)
49. C₂×#126	C₂×C₃	semidirect
50. C₂×#125	C₂×C₃	semidirect
51. C₂×#123	C₂×C₃	semidirect
52. C₂×#121	C₂×C₃	semidirect
53. C₂×#121	C₂×C₃	semidirect
54. C₂×#121	C₂×C₃	semidirect
55. C₂×C₄×D₄	C₂×C₃	semidirect
56. C₂×C₄×D₄	C₂×C₃	semidirect
57. C₂×C₄×D₄	C₂×C₃	semidirect
58. C₂×#117	C₂×C₃	semidirect
59. C₂²×C₄⋊C₄	C₂×C₃	semidirect
110. #126	C₂²×C₃	semidirect
111. #125	C₂²×C₃	semidirect
112. #123	C₂²×C₃	semidirect
113. #121	C₂²×C₃	semidirect
114. #121	C₂²×C₃	semidirect
115. #121	C₂²×C₃	semidirect
116. C₄×D₄	C₂²×C₃	semidirect
117. C₄×D₄	C₂²×C₃	semidirect
118. C₄×D₄	C₂²×C₃	semidirect
119. #117	C₂²×C₃	semidirect
127. C₂×C₄⋊C₄	C₂²×C₃	semidirect
279. C₂×C₃	#553	direct (N × H)
307. C₃	C₂×#553	direct (N × H)
311. C₂	C₃×#553	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×#553	2
C₂×C₃×#126	2
C₂×C₃×#125	2
C₂×C₃×#123	2
C₂×C₃×#121	2
C₂×C₃×C₄×D₄	2
C₂×C₃×#117	2
C₂²×C₃×C₄⋊C₄	2
C₂×#553	3

Subgroups — C2×C3×#553 #87355

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#553 #87355