Subgroups — C₂×#19145181 #25966341

Order 11008.

2006 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14583456	C₂	1	2	yes	maximal
2	#19145181	C₂	1	16	yes	maximal, direct product factor
3	C₂×#14583304	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#14583279	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#14583250	C₂	1	2	yes	maximal, complemented
6	C₂×#14583140	C₂	1	4	yes	maximal
7	C₂×C₄×Dic₁₇₂	C₂	1	4	yes	maximal, complemented
8	C₂²×C₄×C₈×C₄₃	C₂	1	1	yes	maximal, characteristic
9	C₂²×Dic₁₇₂	C₂²	1	4	yes	—
10	#14583456	—	2	8	no	—
11	#14583456	C₂²	1	16	yes	—
12	C₂×#13063325	C₂²	1	2	yes	—
13	C₂×#13063325	C₂²	1	2	yes	—
14	C₂×#13063317	—	2	2	no	—
15	C₂×#13063304	—	2	2	no	—
16	C₂×#13063304	C₂²	1	4	yes	—
17	C₂×C₄×Dic₈₆	—	2	2	no	—
18	C₂×C₄×Dic₈₆	C₂²	1	4	yes	—
19	C₂×#13063284	—	2	2	no	—
20	#14583304	C₂²	1	8	yes	complemented
21	#14583279	C₂²	1	8	yes	complemented
22	#14583250	C₂²	1	16	yes	complemented
23	#14583140	C₂²	1	32	yes	—
24	C₄×Dic₁₇₂	C₂²	1	32	yes	complemented
25	C₂×#13063259	C₂²	1	4	yes	—
26	C₂×#13063250	C₂²	1	8	yes	—
27	C₂³×C₈×C₄₃	C₂²	1	2	yes	—
28	C₂²×C₄²×C₄₃	C₂²	1	1	yes	characteristic
29	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
30	C₂×C₄×C₈×C₄₃	C₂²	1	2	yes	—
31	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
32	C₂×C₄×C₈×C₄₃	C₂²	1	8	yes	—
33	C₂²×Dic₈₆	—	2	2	no	—
34	C₂²×Dic₈₆	C₂³	1	4	yes	—
35	C₂×Dic₁₇₂	—	2	16	no	—
36	C₂×Dic₁₇₂	C₂³	1	16	yes	—
37	#13063325	—	2	4	no	—
38	#13063325	—	2	4	no	—
39	#13063325	C₂³	1	8	yes	—
40	#13063325	C₂³	1	8	yes	—
41	#13063317	—	2	16	no	—
42	#13063304	—	2	16	no	—
43	#13063304	—	2	8	no	—
44	#13063304	C₂³	1	16	yes	—
45	C₄×Dic₈₆	—	2	16	no	—
46	C₄×Dic₈₆	—	2	8	no	—
47	C₄×Dic₈₆	C₂³	1	16	yes	—
48	#13063284	—	2	16	no	—
49	C₂×C₄⋊Dic₄₃	—	2	2	no	—
50	C₂×C₄⋊Dic₄₃	—	2	2	no	—
51	C₂×C₄⋊Dic₄₃	—	2	2	no	—
52	C₂×C₄⋊Dic₄₃	C₂³	1	4	yes	—
53	C₂×Dic₄₃⋊C₄	—	2	4	no	—
54	C₂×Dic₄₃⋊C₄	—	2	4	no	—
55	C₂×C₂²⋊Dic₄₃	—	2	4	no	—
56	C₂×C₄×Dic₄₃	—	2	4	no	—
57	#13063259	C₂³	1	16	yes	—
58	#13063250	C₂³	1	32	yes	—
59	C₂³×C₄×C₄₃	D₄	1	1	yes	characteristic
60	C₂³×C₄×C₄₃	D₄	1	1	yes	characteristic
61	C₂³×C₄×C₄₃	C₂³	1	1	yes	characteristic
62	C₂²×C₈×C₄₃	—	2	8	no	—
63	C₂²×C₈×C₄₃	C₂³	1	8	yes	—
64	C₂²×C₈×C₄₃	C₂³	1	2	yes	—
65	C₂²×C₈×C₄₃	C₂³	1	2	yes	—
66	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
67	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
68	C₂×C₄²×C₄₃	D₄	1	4	yes	—
69	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
70	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
71	C₂×C₄²×C₄₃	C₂³	1	4	yes	—
72	C₄×C₈×C₄₃	C₂³	1	4	yes	—
73	C₄×C₈×C₄₃	C₂³	1	4	yes	—
74	C₄×C₈×C₄₃	C₂³	1	8	yes	—
75	C₂⁴×C₄₃	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₄₃	—	2	4	no	—
77	C₂²×C₄×C₄₃	—	2	4	no	—
78	C₂²×C₄×C₄₃	Q₁₆	1	8	yes	complemented
79	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
81	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₄₃	C₂×D₄	1	4	yes	—
83	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₄₃	C₂×D₄	1	4	yes	—
85	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
86	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
87	C₂²×C₄×C₄₃	C₂⁴	1	2	yes	—
88	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
89	C₂²×Dic₄₃	—	2	4	no	—
90	C₂²×Dic₄₃	—	2	4	no	—
91	C₂×Dic₈₆	—	2	16	no	—
92	C₂×Dic₈₆	—	2	4	no	—
93	C₂×Dic₈₆	—	2	8	no	—
94	C₂×Dic₈₆	—	4	8	no	—
95	C₂×Dic₈₆	C₂⁴	1	8	yes	—
96	C₂×C₈×C₄₃	—	2	4	no	—
97	C₂×C₈×C₄₃	—	2	16	no	—
98	C₂×C₈×C₄₃	#41	1	8	yes	—
99	C₂×C₈×C₄₃	C₂⁴	1	4	yes	—
100	C₂×C₈×C₄₃	C₂⁴	1	4	yes	—
101	C₄²×C₄₃	C₂×D₄	1	4	yes	—
102	C₄²×C₄₃	C₂×D₄	1	2	yes	—
103	C₄²×C₄₃	C₂×D₄	1	4	yes	—
104	C₄²×C₄₃	C₂×D₄	1	2	yes	—
105	C₄²×C₄₃	C₂⁴	1	2	yes	—
106	C₄²×C₄₃	C₂⁴	1	2	yes	—
107	C₂²⋊Dic₄₃	—	2	16	no	—
108	C₄⋊Dic₄₃	—	2	8	no	—
109	C₄⋊Dic₄₃	—	2	8	no	—
110	C₄⋊Dic₄₃	—	2	8	no	—
111	C₄⋊Dic₄₃	—	2	4	no	—
112	C₄⋊Dic₄₃	C₂⁴	1	8	yes	—
113	Dic₄₃⋊C₄	—	2	16	no	—
114	Dic₄₃⋊C₄	—	2	16	no	—
115	C₄×Dic₄₃	—	2	16	no	—
116	Dic₁₇₂	—	2	32	no	—
117	C₂³×C₄₃	—	2	2	no	—
118	C₂³×C₄₃	—	2	2	no	—
119	C₂³×C₄₃	C₂×Q₁₆	1	4	yes	—
120	C₂³×C₄₃	C₂²×D₄	1	2	yes	—
121	C₂³×C₄₃	C₂²×D₄	1	1	yes	characteristic
122	C₂×C₄×C₄₃	—	2	2	no	—
123	C₂×C₄×C₄₃	—	2	2	no	—
124	C₂×C₄×C₄₃	—	2	4	no	—
125	C₂×C₄×C₄₃	—	2	8	no	—
126	C₂×C₄×C₄₃	—	2	4	no	—
127	C₂×C₄×C₄₃	C₂×Q₁₆	1	4	yes	—
128	C₂×C₄×C₄₃	C₂×Q₁₆	1	4	yes	—
129	C₂×C₄×C₄₃	C₂×Q₁₆	1	16	yes	complemented
130	C₂×C₄×C₄₃	#135	1	4	yes	—
131	C₂×C₄×C₄₃	C₂²×D₄	1	2	yes	—
132	C₂×C₄×C₄₃	C₂²×D₄	1	2	yes	—
133	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
134	C₂×C₄×C₄₃	C₂²×D₄	1	2	yes	—
135	C₂×C₄×C₄₃	C₂²×D₄	1	2	yes	—
136	C₂×C₄×C₄₃	C₂²×D₄	1	2	yes	—
137	C₂×C₄×C₄₃	C₂×#41	1	4	yes	—
138	C₂×C₄×C₄₃	C₂⁵	1	1	yes	characteristic
139	C₂×Dic₄₃	—	2	8	no	—
140	C₂×Dic₄₃	—	2	8	no	—
141	C₂×Dic₄₃	—	4	8	no	—
142	C₂×Dic₄₃	—	4	8	no	—
143	Dic₈₆	—	2	16	no	—
144	Dic₈₆	—	4	8	no	—
145	Dic₈₆	—	4	16	no	—
146	C₈×C₄₃	—	2	4	no	—
147	C₈×C₄₃	C₂×#41	1	8	yes	—
148	C₂×#3170	—	43	1	no	maximal, Sylow
149	C₂²×C₄₃	—	2	8	no	—
150	C₂²×C₄₃	—	2	2	no	—
151	C₂²×C₄₃	—	2	2	no	—
152	C₂²×C₄₃	#532	1	2	yes	—
153	C₂²×C₄₃	C₂²×Q₁₆	1	4	yes	—
154	C₂²×C₄₃	C₂²×Q₁₆	1	2	yes	—
155	C₂²×C₄₃	C₂×#135	1	2	yes	—
156	C₂²×C₄₃	C₂³×D₄	1	1	yes	characteristic
157	C₄×C₄₃	—	2	2	no	—
158	C₄×C₄₃	—	2	2	no	—
159	C₄×C₄₃	#532	1	2	yes	—
160	C₄×C₄₃	C₂²×Q₁₆	1	4	yes	—
161	C₄×C₄₃	C₂²×Q₁₆	1	4	yes	—
162	C₄×C₄₃	C₂×#135	1	4	yes	—
163	C₄×C₄₃	C₂²×#41	1	2	yes	—
164	Dic₄₃	—	4	8	no	—
165	Dic₄₃	—	4	8	no	—
166	C₂×#533	—	43	2	no	—
167	#3170	—	43	16	no	—
168	C₂×#500	—	43	1	no	—
169	C₂×#494	—	43	1	no	—
170	C₂×#487	—	43	2	no	—
171	C₂×#467	—	43	4	no	—
172	C₂×C₄×Q₁₆	—	43	4	no	—
173	C₂²×C₄×C₈	D₄₃	1	1	yes	characteristic
174	C₂×C₄₃	—	2	4	no	—
175	C₂×C₄₃	#3170	1	4	yes	complemented
176	C₂×C₄₃	C₂×#532	1	1	yes	characteristic
177	C₂×C₄₃	C₂³×Q₁₆	1	1	yes	characteristic
178	C₂×C₄₃	C₂²×#135	1	1	yes	characteristic
179	C₂²×Q₁₆	—	43	4	no	—
180	C₂³×C₈	C₂×D₄₃	1	2	yes	—
181	#533	—	43	16	no	—
182	#533	—	86	8	no	—
183	C₂×C₄⋊Q₈	—	43	2	no	—
184	C₂×C₄⋊Q₈	—	43	2	no	—
185	C₂×#125	—	86	2	no	—
186	C₂×C₂²⋊Q₈	—	43	4	no	—
187	C₂×C₂²⋊Q₈	—	86	2	no	—
188	C₂×C₄×Q₈	—	43	4	no	—
189	C₂×C₄×Q₈	—	86	2	no	—
190	C₂×#117	—	86	2	no	—
191	C₂²×C₄²	C₂×D₄₃	1	1	yes	characteristic
192	#500	—	43	8	no	—
193	#494	—	43	8	no	—
194	#487	—	43	16	no	—
195	#467	—	43	32	no	—
196	C₄×Q₁₆	—	43	32	no	—
197	C₂×#107	—	43	4	no	—
198	C₂×#103	—	43	8	no	—
199	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
200	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
201	C₂×C₄×C₈	C₂×D₄₃	1	8	yes	—
202	C₂×C₄×C₈	C₂×D₄₃	1	2	yes	—
203	C₄₃	C₂×#3170	1	1	yes	characteristic, complemented, Sylow
204	C₂²×Q₈	—	43	4	no	—
205	C₂²×Q₈	—	86	2	no	—
206	C₂³×C₄	D₁₇₂	1	1	yes	characteristic
207	C₂³×C₄	D₁₇₂	1	1	yes	characteristic
208	C₂³×C₄	C₂²×D₄₃	1	1	yes	characteristic
209	C₂×Q₁₆	—	43	16	no	—
210	C₂×Q₁₆	—	86	16	no	—
211	C₂²×C₈	—	2	8	no	—
212	C₂²×C₈	C₂²×D₄₃	1	2	yes	—
213	C₂²×C₈	C₂²×D₄₃	1	8	yes	—
214	C₂²×C₈	C₂²×D₄₃	1	2	yes	—
215	C₄⋊Q₈	—	43	8	no	—
216	C₄⋊Q₈	—	43	8	no	—
217	C₄⋊Q₈	—	86	4	no	—
218	C₄⋊Q₈	—	86	4	no	—
219	#125	—	86	16	no	—
220	C₂²⋊Q₈	—	43	16	no	—
221	C₂²⋊Q₈	—	86	8	no	—
222	C₂²⋊Q₈	—	86	16	no	—
223	C₄×Q₈	—	43	16	no	—
224	C₄×Q₈	—	86	8	no	—
225	C₄×Q₈	—	86	16	no	—
226	#117	—	86	16	no	—
227	C₂×C₄⋊C₄	—	43	4	no	—
228	C₂×C₄⋊C₄	—	86	2	no	—
229	C₂×C₄⋊C₄	—	86	2	no	—
230	C₂×C₄⋊C₄	—	86	4	no	—
231	C₂×C₄⋊C₄	—	86	4	no	—
232	C₂×C₄⋊C₄	—	86	2	no	—
233	C₂×C₂²⋊C₄	—	86	4	no	—
234	C₂×C₄²	—	86	4	no	—
235	C₂×C₄²	D₁₇₂	1	1	yes	characteristic
236	C₂×C₄²	D₁₇₂	1	4	yes	—
237	C₂×C₄²	D₁₇₂	1	1	yes	characteristic
238	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
239	C₂×C₄²	C₂²×D₄₃	1	4	yes	—
240	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
241	#107	—	43	16	no	—
242	#103	—	43	32	no	—
243	C₄×C₈	C₂²×D₄₃	1	8	yes	—
244	C₄×C₈	C₂²×D₄₃	1	4	yes	—
245	C₄×C₈	C₂²×D₄₃	1	4	yes	—
246	C₂⁴	C₂×D₁₇₂	1	1	yes	characteristic
247	C₂×Q₈	—	43	8	no	—
248	C₂×Q₈	—	86	8	no	—
249	C₂×Q₈	—	86	4	no	—
250	C₂×Q₈	—	86	16	no	—
251	C₂×Q₈	—	172	8	no	—
252	C₂²×C₄	—	2	4	no	—
253	C₂²×C₄	—	2	4	no	—
254	C₂²×C₄	—	86	4	no	—
255	C₂²×C₄	—	86	4	no	—
256	C₂²×C₄	Dic₁₇₂	1	8	yes	complemented
257	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
258	C₂²×C₄	C₂×D₁₇₂	1	4	yes	—
259	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
260	C₂²×C₄	C₂×D₁₇₂	1	2	yes	—
261	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
262	C₂²×C₄	C₂×D₁₇₂	1	4	yes	—
263	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
264	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
265	C₂²×C₄	C₂³×D₄₃	1	2	yes	—
266	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
267	Q₁₆	—	86	32	no	—
268	C₂×C₈	—	2	4	no	—
269	C₂×C₈	—	2	16	no	—
270	C₂×C₈	#10621898	1	8	yes	—
271	C₂×C₈	C₂³×D₄₃	1	4	yes	—
272	C₂×C₈	C₂³×D₄₃	1	4	yes	—
273	C₄⋊C₄	—	43	8	no	—
274	C₄⋊C₄	—	86	4	no	—
275	C₄⋊C₄	—	86	8	no	—
276	C₄⋊C₄	—	86	8	no	—
277	C₄⋊C₄	—	86	16	no	—
278	C₄⋊C₄	—	86	16	no	—
279	C₄⋊C₄	—	86	8	no	—
280	C₂²⋊C₄	—	86	16	no	—
281	C₄²	—	86	16	no	—
282	C₄²	C₂×D₁₇₂	1	4	yes	—
283	C₄²	C₂×D₁₇₂	1	2	yes	—
284	C₄²	C₂×D₁₇₂	1	4	yes	—
285	C₄²	C₂×D₁₇₂	1	2	yes	—
286	C₄²	C₂³×D₄₃	1	2	yes	—
287	C₄²	C₂³×D₄₃	1	2	yes	—
288	C₂³	—	2	2	no	—
289	C₂³	—	2	2	no	—
290	C₂³	C₂×Dic₁₇₂	1	4	yes	—
291	C₂³	C₂²×D₁₇₂	1	2	yes	—
292	C₂³	C₂²×D₁₇₂	1	1	yes	characteristic
293	Q₈	—	86	16	no	—
294	Q₈	—	172	16	no	—
295	Q₈	—	172	8	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	2	8	no	—
299	C₂×C₄	—	2	4	no	—
300	C₂×C₄	—	2	4	no	—
301	C₂×C₄	—	86	8	no	—
302	C₂×C₄	—	86	8	no	—
303	C₂×C₄	—	172	8	no	—
304	C₂×C₄	—	172	8	no	—
305	C₂×C₄	C₂×Dic₁₇₂	1	4	yes	—
306	C₂×C₄	C₂×Dic₁₇₂	1	4	yes	—
307	C₂×C₄	C₂×Dic₁₇₂	1	16	yes	complemented
308	C₂×C₄	#13063347	1	4	yes	—
309	C₂×C₄	C₂²×D₁₇₂	1	2	yes	—
310	C₂×C₄	C₂²×D₁₇₂	1	2	yes	—
311	C₂×C₄	C₂²×D₁₇₂	1	2	yes	—
312	C₂×C₄	C₂²×D₁₇₂	1	2	yes	—
313	C₂×C₄	C₂²×D₁₇₂	1	2	yes	—
314	C₂×C₄	C₂²×D₁₇₂	1	1	yes	characteristic
315	C₂×C₄	C₂×#10621898	1	4	yes	—
316	C₂×C₄	C₂⁴×D₄₃	1	1	yes	characteristic
317	C₈	—	2	4	no	—
318	C₈	C₂×#10621898	1	8	yes	—
319	C₂²	—	2	2	no	—
320	C₂²	—	2	2	no	—
321	C₂²	—	2	8	no	—
322	C₂²	#14583449	1	2	yes	—
323	C₂²	C₂²×Dic₁₇₂	1	4	yes	—
324	C₂²	C₂²×Dic₁₇₂	1	2	yes	—
325	C₂²	C₂×#13063347	1	2	yes	—
326	C₂²	C₂³×D₁₇₂	1	1	yes	characteristic
327	C₄	—	2	2	no	—
328	C₄	—	2	2	no	—
329	C₄	—	172	8	no	—
330	C₄	—	172	8	no	—
331	C₄	#14583449	1	2	yes	—
332	C₄	C₂²×Dic₁₇₂	1	4	yes	—
333	C₄	C₂²×Dic₁₇₂	1	4	yes	—
334	C₄	C₂×#13063347	1	4	yes	—
335	C₄	C₂²×#10621898	1	2	yes	—
336	C₂	—	2	4	no	—
337	C₂	#19145181	1	4	yes	direct product factor
338	C₂	C₂×#14583449	1	1	yes	characteristic
339	C₂	C₂³×Dic₁₇₂	1	1	yes	characteristic
340	C₂	C₂²×#13063347	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₁₇₂
commutator, cosocle	C₂×C₄×C₄₃	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₄₃
Fitting	C₂²×C₄×C₈×C₄₃	C₂
socle	C₂³×C₄₃	C₂²×D₄
prime radical	C₂⁴×C₄₃	C₂×D₄
hypercenter	C₂²×C₄×C₈	D₄₃
hypocenter	C₄₃	C₂×#3170
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
2. #19145181	C₂	direct (N × H)
3. C₂×#14583304	C₂	semidirect
4. C₂×#14583279	C₂	semidirect
5. C₂×#14583250	C₂	semidirect
7. C₂×C₄×Dic₁₇₂	C₂	semidirect
20. #14583304	C₂²	semidirect
21. #14583279	C₂²	semidirect
22. #14583250	C₂²	semidirect
24. C₄×Dic₁₇₂	C₂²	semidirect
78. C₂²×C₄×C₄₃	Q₁₆	semidirect
129. C₂×C₄×C₄₃	C₂×Q₁₆	semidirect
175. C₂×C₄₃	#3170	semidirect
203. C₄₃	C₂×#3170	semidirect
256. C₂²×C₄	Dic₁₇₂	semidirect
307. C₂×C₄	C₂×Dic₁₇₂	semidirect
337. C₂	#19145181	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×#14583456	2
#19145181	2
C₂×#14583304	2
C₂×#14583279	2
C₂×#14583250	2
C₂×#14583140	2
C₂×C₄×Dic₁₇₂	2
C₂²×C₄×C₈×C₄₃	2
C₂×#3170	43

Subgroups — C2×#19145181 #25966341

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#19145181 #25966341