Subgroups — C₂×#10588823 #11867916

Order 1152.

732 conjugacy classes of subgroups.

The subgroup lattice has 327 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₉×#131	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#63512	C₂	1	2	yes	maximal, complemented
3	C₂²×D₇₂	C₂	1	1	yes	maximal, characteristic, complemented
4	#10588823	C₂	1	8	yes	maximal, direct product factor
5	C₂×#63383	C₂	1	2	yes	maximal, complemented
6	C₂×#63372	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#5142	—	3	1	no	maximal
8	C₂×C₉×#41	C₂²	1	1	yes	characteristic
9	C₂²×D₃₆	C₂²	1	2	yes	—
10	C₉×#131	—	2	2	no	—
11	C₉×#131	C₂²	1	4	yes	complemented
12	C₂×C₉×M₁₆	—	2	1	no	—
13	C₂×C₉×M₁₆	C₂²	1	1	yes	characteristic
14	C₂²×C₈×C₉	—	2	1	no	—
15	C₂²×C₈×C₉	C₂²	1	1	yes	characteristic
16	#63512	—	2	4	no	—
17	#63512	C₂²	1	8	yes	complemented
18	C₂×#3675	—	2	2	no	—
19	C₂×#3673	—	2	2	no	—
20	C₂×#3667	C₂²	1	2	yes	—
21	C₂×D₇₂	—	2	1	no	—
22	C₂×D₇₂	—	2	1	no	—
23	C₂×D₇₂	—	2	2	no	—
24	C₂×D₇₂	C₂²	1	4	yes	complemented
25	#63383	C₂²	1	8	yes	complemented
26	#63372	C₂²	1	4	yes	complemented
27	C₂×C₃×#131	D₃	1	1	yes	characteristic
28	C₂×#949	—	3	2	no	—
29	C₂²×D₂₄	—	3	1	no	—
30	#5142	—	3	8	no	—
31	C₂×#823	—	3	2	no	—
32	C₂×#819	—	3	1	no	—
33	C₂³×D₉	—	2	2	no	—
34	C₉×#41	—	2	1	no	—
35	C₉×#41	—	2	1	no	—
36	C₉×#41	D₄	1	2	yes	—
37	C₉×#41	C₂³	1	2	yes	—
38	C₂×Q₈×C₉	D₄	1	1	yes	characteristic
39	C₂×D₄×C₉	—	2	1	no	—
40	C₂×D₄×C₉	D₄	1	1	yes	characteristic
41	C₂²×C₄×C₉	—	2	1	no	—
42	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
43	C₂×D₃₆	—	2	2	no	—
44	C₂×D₃₆	—	2	2	no	—
45	C₂×D₃₆	—	2	2	no	—
46	C₂×D₃₆	—	2	2	no	—
47	C₂×D₃₆	—	2	2	no	—
48	C₂×D₃₆	C₂³	1	4	yes	—
49	C₉×M₁₆	—	2	1	no	—
50	C₉×M₁₆	—	2	4	no	—
51	C₉×M₁₆	C₂³	1	2	yes	—
52	C₂×C₈×C₉	—	2	1	no	—
53	C₂×C₈×C₉	—	2	4	no	—
54	C₂×C₈×C₉	D₄	1	1	yes	characteristic
55	C₂×C₈×C₉	D₄	1	1	yes	characteristic
56	C₂×C₈×C₉	D₄	1	2	yes	—
57	C₂×C₈×C₉	C₂³	1	2	yes	—
58	#3675	—	2	8	no	—
59	#3673	—	2	8	no	—
60	#3667	—	2	2	no	—
61	#3667	C₂³	1	4	yes	—
62	C₂×C₉⋊C₈	—	2	2	no	—
63	D₇₂	—	2	4	no	—
64	D₇₂	—	2	4	no	—
65	C₂×#472	—	9	1	no	Sylow
66	C₂×C₃×#41	C₂×D₃	1	1	yes	characteristic
67	C₂²×D₁₂	—	3	2	no	—
68	C₃×#131	—	2	2	no	—
69	C₃×#131	C₂×D₃	1	4	yes	—
70	C₂×C₃×M₁₆	—	2	1	no	—
71	C₂×C₃×M₁₆	C₂×D₃	1	1	yes	characteristic
72	C₂²×C₃×C₈	—	2	1	no	—
73	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
74	#949	—	3	8	no	—
75	#949	—	6	4	no	—
76	C₂×#215	—	6	2	no	—
77	C₂×#213	—	6	2	no	—
78	C₂×#208	—	3	2	no	—
79	C₂×D₂₄	—	3	4	no	—
80	C₂×D₂₄	—	6	1	no	—
81	C₂×D₂₄	—	6	1	no	—
82	C₂×D₂₄	—	6	2	no	—
83	#823	—	3	8	no	—
84	#819	—	3	4	no	—
85	C₂³×C₉	—	2	1	no	—
86	C₂³×C₉	#41	1	1	yes	characteristic
87	C₂²×D₉	—	2	4	no	—
88	C₂²×D₉	—	4	2	no	—
89	C₂²×D₉	—	4	2	no	—
90	C₂²×D₉	—	8	4	no	—
91	Q₈×C₉	—	2	1	no	—
92	Q₈×C₉	C₂×D₄	1	2	yes	—
93	D₄×C₉	—	2	2	no	—
94	D₄×C₉	—	2	2	no	—
95	D₄×C₉	—	2	1	no	—
96	D₄×C₉	C₂×D₄	1	2	yes	—
97	C₂×C₄×C₉	—	2	1	no	—
98	C₂×C₄×C₉	—	2	2	no	—
99	C₂×C₄×C₉	—	2	2	no	—
100	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
101	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
102	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
103	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
104	C₂×C₄×C₉	#41	1	1	yes	characteristic
105	C₂×C₄×C₉	C₂⁴	1	1	yes	characteristic
106	D₃₆	—	2	4	no	—
107	D₃₆	—	2	4	no	—
108	D₃₆	—	4	2	no	—
109	D₃₆	—	4	2	no	—
110	C₈×C₉	—	2	2	no	—
111	C₈×C₉	C₂×D₄	1	2	yes	—
112	C₈×C₉	C₂×D₄	1	2	yes	—
113	C₉⋊C₈	—	2	4	no	—
114	C₂×AGL(1,ℤ/8)	—	9	2	no	—
115	C₂²×D₈	—	9	1	no	—
116	C₂×#131	D₉	1	1	yes	characteristic, complemented
117	#472	—	9	8	no	—
118	C₂×#108	—	9	1	no	—
119	C₂×#100	—	9	2	no	—
120	C₂³×D₃	—	6	2	no	—
121	C₃×#41	—	2	1	no	—
122	C₃×#41	—	2	1	no	—
123	C₃×#41	D₁₂	1	2	yes	—
124	C₃×#41	C₂²×D₃	1	2	yes	—
125	C₂×C₃×Q₈	D₁₂	1	1	yes	characteristic
126	C₂×C₃×D₄	—	2	1	no	—
127	C₂×C₃×D₄	D₁₂	1	1	yes	characteristic
128	C₂²×C₃×C₄	—	2	1	no	—
129	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
130	C₂×D₁₂	—	3	4	no	—
131	C₂×D₁₂	—	6	2	no	—
132	C₂×D₁₂	—	6	2	no	—
133	C₂×D₁₂	—	6	2	no	—
134	C₂×D₁₂	—	6	2	no	—
135	C₂×D₁₂	—	6	2	no	—
136	C₃×M₁₆	—	2	1	no	—
137	C₃×M₁₆	—	2	4	no	—
138	C₃×M₁₆	C₂²×D₃	1	2	yes	—
139	C₂×C₃×C₈	—	2	1	no	—
140	C₂×C₃×C₈	—	2	4	no	—
141	C₂×C₃×C₈	#67	1	1	yes	characteristic
142	C₂×C₃×C₈	#67	1	1	yes	characteristic
143	C₂×C₃×C₈	#67	1	2	yes	—
144	C₂×C₃×C₈	C₂²×D₃	1	2	yes	—
145	#215	—	6	8	no	—
146	#213	—	6	8	no	—
147	#208	—	3	4	no	—
148	#208	—	6	2	no	—
149	C₂×C₃⋊C₈	—	6	2	no	—
150	D₂₄	—	6	4	no	—
151	D₂₄	—	6	4	no	—
152	C₂²×C₉	—	2	2	no	—
153	C₂²×C₉	—	2	2	no	—
154	C₂²×C₉	—	4	2	no	—
155	C₂²×C₉	#121	1	1	yes	characteristic
156	C₂²×C₉	#121	1	1	yes	characteristic
157	C₂²×C₉	C₂×#41	1	1	yes	characteristic
158	C₂×D₉	—	4	4	no	—
159	C₂×D₉	—	8	4	no	—
160	C₂×D₉	—	8	4	no	—
161	C₂×D₉	—	8	4	no	—
162	C₄×C₉	—	2	2	no	—
163	C₄×C₉	#121	1	1	yes	characteristic
164	C₄×C₉	#121	1	1	yes	characteristic
165	C₄×C₉	C₂²×D₄	1	1	yes	characteristic
166	C₄×C₉	C₂²×D₄	1	1	yes	characteristic
167	C₂×#41	C₂×D₉	1	1	yes	characteristic
168	C₂²×D₄	—	9	2	no	—
169	AGL(1,ℤ/8)	—	9	8	no	—
170	AGL(1,ℤ/8)	—	18	4	no	—
171	C₂×SD₁₆	—	18	2	no	—
172	C₂×D₈	—	9	4	no	—
173	C₂×D₈	—	18	2	no	—
174	C₂×D₈	—	18	1	no	—
175	C₂×D₈	—	18	1	no	—
176	C₂×D₈	—	18	2	no	—
177	#131	—	2	2	no	—
178	#131	C₂×D₉	1	4	yes	complemented
179	C₂×M₁₆	—	2	1	no	—
180	C₂×M₁₆	—	9	2	no	—
181	C₂×M₁₆	C₂×D₉	1	1	yes	characteristic
182	C₂²×C₈	—	2	1	no	—
183	C₂²×C₈	C₂×D₉	1	1	yes	characteristic
184	#108	—	9	4	no	—
185	#100	—	9	8	no	—
186	C₂³×C₃	—	2	1	no	—
187	C₂³×C₃	#235	1	1	yes	characteristic
188	C₂²×D₃	—	6	4	no	—
189	C₂²×D₃	—	12	2	no	—
190	C₂²×D₃	—	12	2	no	—
191	C₂²×D₃	—	24	4	no	—
192	C₃×Q₈	—	2	1	no	—
193	C₃×Q₈	C₂×D₁₂	1	2	yes	—
194	C₃×D₄	—	2	2	no	—
195	C₃×D₄	—	2	2	no	—
196	C₃×D₄	—	2	1	no	—
197	C₃×D₄	C₂×D₁₂	1	2	yes	—
198	C₂×C₃×C₄	—	2	1	no	—
199	C₂×C₃×C₄	—	2	2	no	—
200	C₂×C₃×C₄	—	2	2	no	—
201	C₂×C₃×C₄	C₂×D₁₂	1	1	yes	characteristic
202	C₂×C₃×C₄	C₂×D₁₂	1	1	yes	characteristic
203	C₂×C₃×C₄	#235	1	1	yes	characteristic
204	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
205	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
206	C₂×C₃×C₄	C₂³×D₃	1	1	yes	characteristic
207	D₁₂	—	6	4	no	—
208	D₁₂	—	6	4	no	—
209	D₁₂	—	12	2	no	—
210	D₁₂	—	12	2	no	—
211	C₃×C₈	—	2	2	no	—
212	C₃×C₈	C₂×#67	1	2	yes	—
213	C₃×C₈	C₂×#67	1	2	yes	—
214	C₃⋊C₈	—	6	4	no	—
215	C₂×C₉	—	2	1	no	—
216	C₂×C₉	—	2	1	no	—
217	C₂×C₉	—	4	2	no	—
218	C₂×C₉	#472	1	2	yes	complemented
219	C₂×C₉	C₂×#121	1	1	yes	characteristic
220	D₉	—	8	4	no	—
221	C₂⁴	—	18	2	no	—
222	#41	—	2	1	no	—
223	#41	—	2	1	no	—
224	#41	D₃₆	1	2	yes	—
225	#41	C₂²×D₉	1	2	yes	—
226	C₂×Q₈	D₃₆	1	1	yes	characteristic
227	C₂×D₄	—	2	1	no	—
228	C₂×D₄	—	9	4	no	—
229	C₂×D₄	—	18	2	no	—
230	C₂×D₄	—	18	2	no	—
231	C₂×D₄	—	18	2	no	—
232	C₂×D₄	—	18	2	no	—
233	C₂×D₄	—	18	2	no	—
234	C₂×D₄	D₃₆	1	1	yes	characteristic
235	C₂²×C₄	—	2	1	no	—
236	C₂²×C₄	C₂²×D₉	1	1	yes	characteristic
237	SD₁₆	—	18	8	no	—
238	D₈	—	18	8	no	—
239	D₈	—	18	4	no	—
240	D₈	—	18	4	no	—
241	M₁₆	—	2	1	no	—
242	M₁₆	—	2	4	no	—
243	M₁₆	—	9	4	no	—
244	M₁₆	—	18	2	no	—
245	M₁₆	C₂²×D₉	1	2	yes	—
246	C₂×C₈	—	2	1	no	—
247	C₂×C₈	—	2	4	no	—
248	C₂×C₈	—	18	2	no	—
249	C₂×C₈	#611	1	1	yes	characteristic
250	C₂×C₈	#611	1	1	yes	characteristic
251	C₂×C₈	#611	1	2	yes	—
252	C₂×C₈	C₂²×D₉	1	2	yes	—
253	C₂²×C₃	—	2	2	no	—
254	C₂²×C₃	—	2	2	no	—
255	C₂²×C₃	—	4	2	no	—
256	C₂²×C₃	#930	1	1	yes	characteristic
257	C₂²×C₃	#930	1	1	yes	characteristic
258	C₂²×C₃	C₂×#235	1	1	yes	characteristic
259	C₂×D₃	—	12	4	no	—
260	C₂×D₃	—	24	4	no	—
261	C₂×D₃	—	24	4	no	—
262	C₂×D₃	—	24	4	no	—
263	C₃×C₄	—	2	2	no	—
264	C₃×C₄	#930	1	1	yes	characteristic
265	C₃×C₄	#930	1	1	yes	characteristic
266	C₃×C₄	C₂²×D₁₂	1	1	yes	characteristic
267	C₃×C₄	C₂²×#67	1	1	yes	characteristic
268	C₉	C₂×#472	1	1	yes	characteristic, complemented, Sylow
269	C₂³	—	2	1	no	—
270	C₂³	—	18	4	no	—
271	C₂³	—	36	2	no	—
272	C₂³	—	36	2	no	—
273	C₂³	—	72	4	no	—
274	C₂³	#3697	1	1	yes	characteristic
275	Q₈	—	2	1	no	—
276	Q₈	C₂×D₃₆	1	2	yes	—
277	D₄	—	2	2	no	—
278	D₄	—	2	2	no	—
279	D₄	—	2	1	no	—
280	D₄	—	18	4	no	—
281	D₄	—	18	4	no	—
282	D₄	—	36	2	no	—
283	D₄	—	36	2	no	—
284	D₄	C₂×D₃₆	1	2	yes	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	C₂×D₃₆	1	1	yes	characteristic
289	C₂×C₄	C₂×D₃₆	1	1	yes	characteristic
290	C₂×C₄	#3697	1	1	yes	characteristic
291	C₂×C₄	C₂×#611	1	1	yes	characteristic
292	C₂×C₄	C₂×#611	1	1	yes	characteristic
293	C₂×C₄	C₂³×D₉	1	1	yes	characteristic
294	C₈	—	2	2	no	—
295	C₈	—	18	4	no	—
296	C₈	C₂×#611	1	2	yes	—
297	C₈	C₂×#611	1	2	yes	—
298	C₂×C₃	—	2	1	no	—
299	C₂×C₃	—	2	1	no	—
300	C₂×C₃	—	4	2	no	—
301	C₂×C₃	#5142	1	2	yes	—
302	C₂×C₃	C₂×#930	1	1	yes	characteristic
303	D₃	—	24	4	no	—
304	C₂²	—	2	2	no	—
305	C₂²	—	2	2	no	—
306	C₂²	—	4	2	no	—
307	C₂²	—	36	4	no	—
308	C₂²	—	72	4	no	—
309	C₂²	—	72	4	no	—
310	C₂²	—	72	4	no	—
311	C₂²	#63492	1	1	yes	characteristic
312	C₂²	#63492	1	1	yes	characteristic
313	C₂²	C₂×#3697	1	1	yes	characteristic
314	C₄	—	2	2	no	—
315	C₄	#63492	1	1	yes	characteristic
316	C₄	#63492	1	1	yes	characteristic
317	C₄	C₂²×D₃₆	1	1	yes	characteristic
318	C₄	C₂²×#611	1	1	yes	characteristic
319	C₃	C₂×#5142	1	1	yes	characteristic
320	C₂	—	2	1	no	—
321	C₂	—	2	1	no	—
322	C₂	—	4	2	no	—
323	C₂	—	72	4	no	—
324	C₂	#10588823	1	2	yes	direct product factor
325	C₂	C₂×#63492	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#63492
commutator, cosocle	C₂×C₄×C₉	C₂⁴
Frattini	C₂×C₃×C₄	C₂³×D₃
Fitting	C₂×C₉×#131	C₂
socle	C₂²×C₃	#930
hypercenter	C₂×#131	D₉
hypocenter	C₉	C₂×#472
LCS	G ▷ C₂×C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#131

Complements

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

N	H ≅ G/N	type
1. C₂×C₉×#131	C₂	semidirect
2. C₂×#63512	C₂	semidirect
3. C₂²×D₇₂	C₂	semidirect
4. #10588823	C₂	direct (N × H)
5. C₂×#63383	C₂	semidirect
6. C₂×#63372	C₂	semidirect
11. C₉×#131	C₂²	semidirect
17. #63512	C₂²	semidirect
24. C₂×D₇₂	C₂²	semidirect
25. #63383	C₂²	semidirect
26. #63372	C₂²	semidirect
116. C₂×#131	D₉	semidirect
178. #131	C₂×D₉	semidirect
218. C₂×C₉	#472	semidirect
268. C₉	C₂×#472	semidirect
324. C₂	#10588823	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₉×#131	2
C₂×#63512	2
C₂²×D₇₂	2
#10588823	2
C₂×#63383	2
C₂×#63372	2
C₂×#5142	3

Subgroups — C2×#10588823 #11867916

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#10588823 #11867916