Subgroups — C₂²×C₃×#3697 #13164895

Order 1728.

2906 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₃×C₄×C₉	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₃×#611	C₂	1	2	yes	maximal, complemented
3	C₂×C₃×#3697	C₂	1	24	yes	maximal, direct product factor
4	C₂²×C₃×D₃₆	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×C₃×C₄×D₉	C₂	1	2	yes	maximal, complemented
6	C₂²×C₃×Dic₁₈	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×C₃×#235	—	3	1	no	maximal
8	C₂²×#3697	C₃	1	1	yes	maximal, characteristic, direct product factor
9	C₂⁴×C₃×C₉	C₂²	1	1	yes	characteristic
10	C₂³×C₃×D₉	C₂²	1	2	yes	—
11	C₂²×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
12	C₂²×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
13	C₂²×C₃×C₄×C₉	C₂²	1	12	yes	complemented
14	C₂×C₃×#611	C₂²	1	24	yes	complemented
15	C₂²×C₃×Dic₉	C₂²	1	2	yes	—
16	C₃×#3697	C₂²	1	64	yes	direct product factor
17	C₂×C₃×D₃₆	C₂²	1	12	yes	complemented
18	C₂×C₃×C₄×D₉	C₂²	1	24	yes	complemented
19	C₂×C₃×Dic₁₈	C₂²	1	12	yes	complemented
20	C₂³×C₃²×C₄	D₃	1	1	yes	characteristic
21	C₂²×C₃×#67	—	3	2	no	—
22	C₂×C₃×#235	—	3	24	no	—
23	C₂²×C₃×D₁₂	—	3	1	no	—
24	C₂²×C₃×C₄×D₃	—	3	2	no	—
25	C₂²×C₃×Dic₆	—	3	1	no	—
26	C₂³×C₄×C₉	—	2	1	no	—
27	C₂³×C₄×C₉	C₂×C₃	1	1	yes	characteristic, complemented
28	C₂²×#611	C₂×C₃	1	2	yes	complemented
29	C₂×#3697	C₂×C₃	1	24	yes	direct product factor
30	C₂²×D₃₆	C₂×C₃	1	1	yes	characteristic, complemented
31	C₂²×C₄×D₉	C₂×C₃	1	2	yes	complemented
32	C₂²×Dic₁₈	C₂×C₃	1	1	yes	characteristic, complemented
33	C₂³×C₃×C₉	—	2	4	no	—
34	C₂³×C₃×C₉	C₂³	1	6	yes	—
35	C₂³×C₃×C₉	C₂³	1	1	yes	characteristic
36	C₂²×C₃×D₉	—	2	8	no	—
37	C₂²×C₃×D₉	C₂³	1	12	yes	—
38	C₂×C₃×C₄×C₉	C₂³	1	6	yes	—
39	C₂×C₃×C₄×C₉	C₂³	1	6	yes	—
40	C₂×C₃×C₄×C₉	C₂³	1	16	yes	complemented
41	C₃×#611	C₂³	1	32	yes	complemented
42	C₂×C₃×Dic₉	C₂³	1	12	yes	—
43	C₃×D₃₆	C₂³	1	16	yes	complemented
44	C₃×C₄×D₉	C₂³	1	32	yes	complemented
45	C₃×Dic₁₈	C₂³	1	16	yes	complemented
46	C₂²×C₃×#41	—	9	1	no	—
47	C₂²×#235	—	3	1	no	—
48	C₂⁴×C₃²	C₂×D₃	1	1	yes	characteristic
49	C₂³×C₃×D₃	—	3	2	no	—
50	C₂²×C₃²×C₄	C₂×D₃	1	1	yes	characteristic
51	C₂²×C₃²×C₄	C₂×D₃	1	1	yes	characteristic
52	C₂²×C₃²×C₄	C₂×D₃	1	12	yes	—
53	C₂×C₃×#67	—	3	24	no	—
54	C₂²×C₃×Dic₃	—	3	2	no	—
55	C₃×#235	—	3	64	no	—
56	C₂×C₃×D₁₂	—	3	12	no	—
57	C₂×C₃×C₄×D₃	—	3	24	no	—
58	C₂×C₃×Dic₆	—	3	12	no	—
59	C₂⁴×C₉	—	2	1	no	—
60	C₂⁴×C₉	C₂²×C₃	1	1	yes	characteristic
61	C₂³×D₉	C₂²×C₃	1	2	yes	—
62	C₂²×C₄×C₉	—	2	1	no	—
63	C₂²×C₄×C₉	—	2	1	no	—
64	C₂²×C₄×C₉	—	2	12	no	—
65	C₂²×C₄×C₉	C₂²×C₃	1	1	yes	characteristic
66	C₂²×C₄×C₉	C₂²×C₃	1	1	yes	characteristic
67	C₂²×C₄×C₉	C₂²×C₃	1	12	yes	complemented
68	C₂×#611	C₂²×C₃	1	24	yes	complemented
69	C₂²×Dic₉	C₂²×C₃	1	2	yes	—
70	#3697	C₂²×C₃	1	64	yes	direct product factor
71	C₂×D₃₆	C₂²×C₃	1	12	yes	complemented
72	C₂×C₄×D₉	C₂²×C₃	1	24	yes	complemented
73	C₂×Dic₁₈	C₂²×C₃	1	12	yes	complemented
74	C₂²×C₃×C₉	—	2	12	no	—
75	C₂²×C₃×C₉	#41	1	4	yes	complemented
76	C₂²×C₃×C₉	C₂⁴	1	4	yes	—
77	C₂²×C₃×C₉	C₂⁴	1	3	yes	—
78	C₂×C₃×D₉	—	2	24	no	—
79	C₂×C₃×D₉	C₂⁴	1	8	yes	—
80	C₃×C₄×C₉	C₂⁴	1	4	yes	—
81	C₃×C₄×C₉	C₂⁴	1	4	yes	—
82	C₃×Dic₉	C₂⁴	1	8	yes	—
83	C₂×C₃×#41	—	9	24	no	—
84	C₂²×C₃×Q₈	—	9	1	no	—
85	C₂²×C₃×D₄	—	9	2	no	—
86	C₂²×C₃×D₄	—	9	1	no	—
87	C₂³×C₃×C₄	—	2	1	no	—
88	C₂³×C₃×C₄	—	9	2	no	—
89	C₂³×C₃×C₄	D₉	1	1	yes	characteristic, complemented
90	C₂³×C₃×C₄	C₃×D₃	1	1	yes	characteristic
91	C₂²×#67	—	3	2	no	—
92	C₂×#235	—	3	24	no	—
93	C₂²×D₁₂	—	3	1	no	—
94	C₂²×C₄×D₃	—	3	2	no	—
95	C₂²×Dic₆	—	3	1	no	—
96	C₂³×C₃²	—	2	4	no	—
97	C₂³×C₃²	C₂²×D₃	1	6	yes	—
98	C₂³×C₃²	C₂²×D₃	1	1	yes	characteristic
99	C₂²×C₃×D₃	—	3	12	no	—
100	C₂²×C₃×D₃	—	6	8	no	—
101	C₂×C₃²×C₄	C₂²×D₃	1	6	yes	—
102	C₂×C₃²×C₄	C₂²×D₃	1	6	yes	—
103	C₂×C₃²×C₄	C₂²×D₃	1	16	yes	—
104	C₃×#67	—	3	32	no	—
105	C₂×C₃×Dic₃	—	3	12	no	—
106	C₃×D₁₂	—	3	16	no	—
107	C₃×C₄×D₃	—	3	32	no	—
108	C₃×Dic₆	—	3	16	no	—
109	C₂³×C₉	—	2	8	no	—
110	C₂³×C₉	—	2	4	no	—
111	C₂³×C₉	—	2	6	no	—
112	C₂³×C₉	—	2	1	no	—
113	C₂³×C₉	C₂³×C₃	1	6	yes	—
114	C₂³×C₉	C₂³×C₃	1	1	yes	characteristic
115	C₂²×D₉	—	2	8	no	—
116	C₂²×D₉	C₂³×C₃	1	12	yes	—
117	C₂×C₄×C₉	—	2	6	no	—
118	C₂×C₄×C₉	—	2	6	no	—
119	C₂×C₄×C₉	—	2	16	no	—
120	C₂×C₄×C₉	C₂³×C₃	1	6	yes	—
121	C₂×C₄×C₉	C₂³×C₃	1	6	yes	—
122	C₂×C₄×C₉	C₂³×C₃	1	16	yes	complemented
123	#611	C₂³×C₃	1	32	yes	complemented
124	C₂×Dic₉	C₂³×C₃	1	12	yes	—
125	D₃₆	C₂³×C₃	1	16	yes	complemented
126	C₄×D₉	C₂³×C₃	1	32	yes	complemented
127	Dic₁₈	C₂³×C₃	1	16	yes	complemented
128	C₂²×#41	—	9	1	no	Sylow
129	C₂×C₃×C₉	—	2	4	no	—
130	C₂×C₃×C₉	C₂×#41	1	6	yes	complemented
131	C₂×C₃×C₉	C₂⁵	1	1	yes	characteristic
132	C₃×D₉	—	2	8	no	—
133	C₂⁴×C₃	—	2	1	no	—
134	C₂⁴×C₃	—	9	2	no	—
135	C₂⁴×C₃	C₂×D₉	1	1	yes	characteristic
136	C₂⁴×C₃	C₂×C₃×D₃	1	1	yes	characteristic
137	C₂³×D₃	—	3	2	no	—
138	C₃×#41	—	9	64	no	—
139	C₂×C₃×Q₈	—	9	12	no	—
140	C₂×C₃×D₄	—	9	24	no	—
141	C₂×C₃×D₄	—	9	12	no	—
142	C₂²×C₃×C₄	—	2	1	no	—
143	C₂²×C₃×C₄	—	2	1	no	—
144	C₂²×C₃×C₄	—	2	12	no	—
145	C₂²×C₃×C₄	—	9	2	no	—
146	C₂²×C₃×C₄	—	9	24	no	—
147	C₂²×C₃×C₄	C₂×D₉	1	1	yes	characteristic
148	C₂²×C₃×C₄	C₂×D₉	1	1	yes	characteristic
149	C₂²×C₃×C₄	C₂×D₉	1	12	yes	complemented
150	C₂²×C₃×C₄	C₂×C₃×D₃	1	1	yes	characteristic
151	C₂²×C₃×C₄	C₂×C₃×D₃	1	1	yes	characteristic
152	C₂²×C₃×C₄	C₂×C₃×D₃	1	12	yes	—
153	C₂×#67	—	3	24	no	—
154	C₂²×Dic₃	—	3	2	no	—
155	#235	—	3	64	no	—
156	C₂×D₁₂	—	3	12	no	—
157	C₂×C₄×D₃	—	3	24	no	—
158	C₂×Dic₆	—	3	12	no	—
159	C₂²×C₃²	—	2	12	no	—
160	C₂²×C₃²	#235	1	4	yes	—
161	C₂²×C₃²	C₂³×D₃	1	4	yes	—
162	C₂²×C₃²	C₂³×D₃	1	3	yes	—
163	C₂×C₃×D₃	—	3	8	no	—
164	C₂×C₃×D₃	—	6	24	no	—
165	C₃²×C₄	C₂³×D₃	1	4	yes	—
166	C₃²×C₄	C₂³×D₃	1	4	yes	—
167	C₃×Dic₃	—	3	8	no	—
168	C₂²×C₉	—	2	24	no	—
169	C₂²×C₉	—	2	12	no	—
170	C₂²×C₉	—	2	4	no	—
171	C₂²×C₉	—	2	3	no	—
172	C₂²×C₉	—	2	4	no	—
173	C₂²×C₉	C₃×#41	1	4	yes	complemented
174	C₂²×C₉	C₂⁴×C₃	1	4	yes	—
175	C₂²×C₉	C₂⁴×C₃	1	3	yes	—
176	C₂×D₉	—	2	24	no	—
177	C₂×D₉	C₂⁴×C₃	1	8	yes	—
178	C₄×C₉	—	2	4	no	—
179	C₄×C₉	—	2	4	no	—
180	C₄×C₉	C₂⁴×C₃	1	4	yes	—
181	C₄×C₉	C₂⁴×C₃	1	4	yes	—
182	Dic₉	C₂⁴×C₃	1	8	yes	—
183	C₂×#41	—	9	24	no	—
184	C₂²×Q₈	—	9	1	no	—
185	C₂²×D₄	—	9	2	no	—
186	C₂²×D₄	—	9	1	no	—
187	C₂³×C₄	—	9	2	no	—
188	C₂³×C₄	C₃×D₉	1	1	yes	characteristic, complemented
189	C₃×C₉	C₂²×#41	1	1	yes	characteristic, complemented, Sylow
190	C₂³×C₃	—	2	8	no	—
191	C₂³×C₃	—	2	4	no	—
192	C₂³×C₃	—	2	4	no	—
193	C₂³×C₃	—	2	6	no	—
194	C₂³×C₃	—	2	1	no	—
195	C₂³×C₃	—	9	12	no	—
196	C₂³×C₃	—	18	8	no	—
197	C₂³×C₃	C₂²×D₉	1	6	yes	—
198	C₂³×C₃	C₂²×D₉	1	1	yes	characteristic
199	C₂³×C₃	C₂²×C₃×D₃	1	6	yes	—
200	C₂³×C₃	C₂²×C₃×D₃	1	1	yes	characteristic
201	C₂²×D₃	—	3	12	no	—
202	C₂²×D₃	—	6	8	no	—
203	C₃×Q₈	—	9	16	no	—
204	C₃×D₄	—	9	32	no	—
205	C₃×D₄	—	9	16	no	—
206	C₂×C₃×C₄	—	2	6	no	—
207	C₂×C₃×C₄	—	2	6	no	—
208	C₂×C₃×C₄	—	2	16	no	—
209	C₂×C₃×C₄	—	9	12	no	—
210	C₂×C₃×C₄	—	9	32	no	—
211	C₂×C₃×C₄	C₂²×D₉	1	6	yes	—
212	C₂×C₃×C₄	C₂²×D₉	1	6	yes	—
213	C₂×C₃×C₄	C₂²×D₉	1	16	yes	complemented
214	C₂×C₃×C₄	C₂²×C₃×D₃	1	6	yes	—
215	C₂×C₃×C₄	C₂²×C₃×D₃	1	6	yes	—
216	C₂×C₃×C₄	C₂²×C₃×D₃	1	16	yes	—
217	#67	—	3	32	no	—
218	C₂×Dic₃	—	3	12	no	—
219	D₁₂	—	3	16	no	—
220	C₄×D₃	—	3	32	no	—
221	Dic₆	—	3	16	no	—
222	C₂×C₃²	—	2	4	no	—
223	C₂×C₃²	C₂×#235	1	6	yes	—
224	C₂×C₃²	C₂⁴×D₃	1	1	yes	characteristic
225	C₃×D₃	—	6	8	no	—
226	C₂×C₉	—	2	8	no	—
227	C₂×C₉	—	2	4	no	—
228	C₂×C₉	—	2	6	no	—
229	C₂×C₉	—	2	1	no	—
230	C₂×C₉	C₂×C₃×#41	1	6	yes	complemented
231	C₂×C₉	C₂⁵×C₃	1	1	yes	characteristic
232	D₉	—	2	8	no	—
233	C₂⁴	—	9	2	no	—
234	C₂⁴	C₂×C₃×D₉	1	1	yes	characteristic
235	#41	—	9	64	no	—
236	C₂×Q₈	—	9	12	no	—
237	C₂×D₄	—	9	24	no	—
238	C₂×D₄	—	9	12	no	—
239	C₂²×C₄	—	9	2	no	—
240	C₂²×C₄	—	9	24	no	—
241	C₂²×C₄	C₂×C₃×D₉	1	1	yes	characteristic
242	C₂²×C₄	C₂×C₃×D₉	1	1	yes	characteristic
243	C₂²×C₄	C₂×C₃×D₉	1	12	yes	complemented
244	C₂²×C₃	—	2	24	no	—
245	C₂²×C₃	—	2	12	no	—
246	C₂²×C₃	—	2	12	no	—
247	C₂²×C₃	—	2	4	no	—
248	C₂²×C₃	—	2	3	no	—
249	C₂²×C₃	—	2	4	no	—
250	C₂²×C₃	—	9	8	no	—
251	C₂²×C₃	—	18	24	no	—
252	C₂²×C₃	#3697	1	4	yes	direct product factor
253	C₂²×C₃	C₂³×D₉	1	4	yes	—
254	C₂²×C₃	C₂³×D₉	1	3	yes	—
255	C₂²×C₃	C₃×#235	1	4	yes	—
256	C₂²×C₃	C₂³×C₃×D₃	1	4	yes	—
257	C₂²×C₃	C₂³×C₃×D₃	1	3	yes	—
258	C₂×D₃	—	3	8	no	—
259	C₂×D₃	—	6	24	no	—
260	C₃×C₄	—	2	4	no	—
261	C₃×C₄	—	2	4	no	—
262	C₃×C₄	—	9	8	no	—
263	C₃×C₄	C₂³×D₉	1	4	yes	—
264	C₃×C₄	C₂³×D₉	1	4	yes	—
265	C₃×C₄	C₂³×C₃×D₃	1	4	yes	—
266	C₃×C₄	C₂³×C₃×D₃	1	4	yes	—
267	Dic₃	—	3	8	no	—
268	C₃²	C₂²×#235	1	1	yes	characteristic
269	C₉	—	2	1	no	—
270	C₉	C₂²×C₃×#41	1	1	yes	characteristic, complemented
271	C₂³	—	2	4	no	—
272	C₂³	—	9	12	no	—
273	C₂³	—	18	8	no	—
274	C₂³	C₂²×C₃×D₉	1	6	yes	—
275	C₂³	C₂²×C₃×D₉	1	1	yes	characteristic
276	Q₈	—	9	16	no	—
277	D₄	—	9	32	no	—
278	D₄	—	9	16	no	—
279	C₂×C₄	—	9	12	no	—
280	C₂×C₄	—	9	32	no	—
281	C₂×C₄	C₂²×C₃×D₉	1	6	yes	—
282	C₂×C₄	C₂²×C₃×D₉	1	6	yes	—
283	C₂×C₄	C₂²×C₃×D₉	1	16	yes	complemented
284	C₂×C₃	—	2	8	no	—
285	C₂×C₃	—	2	4	no	—
286	C₂×C₃	—	2	4	no	—
287	C₂×C₃	—	2	1	no	—
288	C₂×C₃	—	2	6	no	—
289	C₂×C₃	—	18	8	no	—
290	C₂×C₃	C₂×#3697	1	6	yes	direct product factor
291	C₂×C₃	C₂⁴×D₉	1	1	yes	characteristic
292	C₂×C₃	C₂×C₃×#235	1	6	yes	—
293	C₂×C₃	C₂⁴×C₃×D₃	1	1	yes	characteristic
294	D₃	—	6	8	no	—
295	C₂²	—	2	12	no	—
296	C₂²	—	9	8	no	—
297	C₂²	—	18	24	no	—
298	C₂²	C₃×#3697	1	4	yes	direct product factor
299	C₂²	C₂³×C₃×D₉	1	4	yes	—
300	C₂²	C₂³×C₃×D₉	1	3	yes	—
301	C₄	—	9	8	no	—
302	C₄	C₂³×C₃×D₉	1	4	yes	—
303	C₄	C₂³×C₃×D₉	1	4	yes	—
304	C₃	—	2	1	no	—
305	C₃	C₂²×#3697	1	1	yes	characteristic, direct product factor
306	C₃	C₂²×C₃×#235	1	1	yes	characteristic
307	C₂	—	2	4	no	—
308	C₂	—	18	8	no	—
309	C₂	C₂×C₃×#3697	1	6	yes	direct product factor
310	C₂	C₂⁴×C₃×D₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃×C₄	C₂×D₉
commutator, cosocle	C₂×C₉	C₂⁵×C₃
Frattini	C₂×C₃	C₂⁴×C₃×D₃
Fitting	C₂³×C₃×C₄×C₉	C₂
socle	C₂³×C₃²	C₂²×D₃
hypercenter	C₂³×C₃×C₄	D₉
hypocenter	C₉	C₂²×C₃×#41
LCS	G ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂²×C₃×C₄ ◁ C₂³×C₃×C₄

Complements

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

N	H ≅ G/N	type
1. C₂³×C₃×C₄×C₉	C₂	semidirect
2. C₂²×C₃×#611	C₂	semidirect
3. C₂×C₃×#3697	C₂	direct (N × H)
4. C₂²×C₃×D₃₆	C₂	semidirect
5. C₂²×C₃×C₄×D₉	C₂	semidirect
6. C₂²×C₃×Dic₁₈	C₂	semidirect
8. C₂²×#3697	C₃	direct (N × H)
13. C₂²×C₃×C₄×C₉	C₂²	semidirect
14. C₂×C₃×#611	C₂²	semidirect
16. C₃×#3697	C₂²	direct (N × H)
17. C₂×C₃×D₃₆	C₂²	semidirect
18. C₂×C₃×C₄×D₉	C₂²	semidirect
19. C₂×C₃×Dic₁₈	C₂²	semidirect
27. C₂³×C₄×C₉	C₂×C₃	semidirect
28. C₂²×#611	C₂×C₃	semidirect
29. C₂×#3697	C₂×C₃	direct (N × H)
30. C₂²×D₃₆	C₂×C₃	semidirect
31. C₂²×C₄×D₉	C₂×C₃	semidirect
32. C₂²×Dic₁₈	C₂×C₃	semidirect
40. C₂×C₃×C₄×C₉	C₂³	semidirect
41. C₃×#611	C₂³	semidirect
43. C₃×D₃₆	C₂³	semidirect
44. C₃×C₄×D₉	C₂³	semidirect
45. C₃×Dic₁₈	C₂³	semidirect
67. C₂²×C₄×C₉	C₂²×C₃	semidirect
68. C₂×#611	C₂²×C₃	semidirect
70. #3697	C₂²×C₃	direct (N × H)
71. C₂×D₃₆	C₂²×C₃	semidirect
72. C₂×C₄×D₉	C₂²×C₃	semidirect
73. C₂×Dic₁₈	C₂²×C₃	semidirect
75. C₂²×C₃×C₉	#41	semidirect
89. C₂³×C₃×C₄	D₉	semidirect
122. C₂×C₄×C₉	C₂³×C₃	semidirect
123. #611	C₂³×C₃	semidirect
125. D₃₆	C₂³×C₃	semidirect
126. C₄×D₉	C₂³×C₃	semidirect
127. Dic₁₈	C₂³×C₃	semidirect
130. C₂×C₃×C₉	C₂×#41	semidirect
149. C₂²×C₃×C₄	C₂×D₉	semidirect
173. C₂²×C₉	C₃×#41	semidirect
188. C₂³×C₄	C₃×D₉	semidirect
189. C₃×C₉	C₂²×#41	semidirect
213. C₂×C₃×C₄	C₂²×D₉	semidirect
230. C₂×C₉	C₂×C₃×#41	semidirect
243. C₂²×C₄	C₂×C₃×D₉	semidirect
252. C₂²×C₃	#3697	direct (N × H)
270. C₉	C₂²×C₃×#41	semidirect
283. C₂×C₄	C₂²×C₃×D₉	semidirect
290. C₂×C₃	C₂×#3697	direct (N × H)
298. C₂²	C₃×#3697	direct (N × H)
305. C₃	C₂²×#3697	direct (N × H)
309. C₂	C₂×C₃×#3697	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₃×C₄×C₉	2
C₂²×C₃×#611	2
C₂×C₃×#3697	2
C₂²×C₃×D₃₆	2
C₂²×C₃×C₄×D₉	2
C₂²×C₃×Dic₁₈	2
C₂²×C₃×#235	3
C₂²×#3697	3

Subgroups — C22×C3×#3697 #13164895

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃×#3697 #13164895