Subgroups — C₂×#19147233 #25976157

Order 11008.

2058 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₈×D₁₇₂	C₂	1	1	yes	maximal, characteristic
2	#19147233	C₂	1	16	yes	maximal, direct product factor
3	C₂×#14583249	C₂	1	3	yes	maximal, complemented
4	C₂×#14583184	C₂	1	3	yes	maximal, complemented
5	C₂×#14583101	C₂	1	3	yes	maximal
6	C₂×Q₈×C₄₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₄×#10621884	C₂	1	3	yes	maximal
8	C₂×C₄₃×#557	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×Q₈×D₄₃	—	2	1	no	—
10	C₂²×#10621884	C₂²	1	3	yes	—
11	Q₈×D₁₇₂	—	2	4	no	—
12	Q₈×D₁₇₂	C₂²	1	8	yes	—
13	C₂×#13063325	C₂²	1	3	yes	—
14	C₂×#13063305	—	2	3	no	—
15	C₂×C₄×D₁₇₂	C₂²	1	3	yes	—
16	#14583249	C₂²	1	24	yes	complemented
17	#14583184	C₂²	1	24	yes	complemented
18	#14583101	C₂²	1	24	yes	—
19	Q₈×C₄₃⋊C₈	C₂²	1	8	yes	complemented
20	C₄×#10621884	C₂²	1	24	yes	—
21	C₂×#13063256	C₂²	1	3	yes	—
22	C₂×C₄⋊(C₄₃⋊C₈)	C₂²	1	3	yes	—
23	C₂×#13063249	C₂²	1	6	yes	—
24	C₂×#13063249	C₂²	1	3	yes	—
25	C₂×#13063245	C₂²	1	1	yes	characteristic
26	C₂×C₄×C₄₃⋊C₈	C₂²	1	3	yes	—
27	C₄₃×#557	—	2	4	no	—
28	C₄₃×#557	C₂²	1	8	yes	complemented
29	C₂×C₄⋊Q₈×C₄₃	C₂²	1	3	yes	—
30	C₂×#125×C₄₃	—	2	3	no	—
31	C₂×C₄×Q₈×C₄₃	—	2	1	no	—
32	C₂×C₄×Q₈×C₄₃	C₂²	1	3	yes	—
33	C₂×C₄×Q₈×C₄₃	C₂²	1	1	yes	characteristic
34	C₂²×Dic₈₆	—	2	3	no	—
35	C₂×Q₈×D₄₃	—	2	8	no	—
36	C₂×Q₈×D₄₃	—	4	8	no	—
37	C₂²×D₁₇₂	C₂³	1	1	yes	characteristic
38	C₂²×C₄×D₄₃	—	2	3	no	—
39	C₂×#10621884	—	2	12	no	—
40	C₂×#10621884	C₂³	1	12	yes	—
41	C₂²×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
42	C₂²×C₄₃⋊C₈	C₂³	1	3	yes	—
43	#13063325	—	2	6	no	—
44	#13063325	C₂³	1	12	yes	—
45	#13063305	—	2	24	no	—
46	C₄×D₁₇₂	—	2	6	no	—
47	C₄×D₁₇₂	C₂³	1	12	yes	—
48	C₂×C₄⋊Dic₄₃	—	2	3	no	—
49	C₂×C₄⋊Dic₄₃	C₂³	1	3	yes	—
50	C₂×(C₂×D₄₃)⋊C₄	—	2	3	no	—
51	#13063256	C₂³	1	12	yes	—
52	C₄⋊(C₄₃⋊C₈)	C₂³	1	12	yes	—
53	#13063249	C₂³	1	24	yes	—
54	#13063249	C₂³	1	12	yes	—
55	#13063245	C₂³	1	4	yes	—
56	C₄×C₄₃⋊C₈	C₂³	1	12	yes	—
57	C₂²×Q₈×C₄₃	D₄	1	1	yes	characteristic
58	C₂²×Q₈×C₄₃	C₂³	1	3	yes	—
59	C₄⋊Q₈×C₄₃	—	2	6	no	—
60	C₄⋊Q₈×C₄₃	C₂³	1	12	yes	—
61	#125×C₄₃	—	2	24	no	—
62	C₄×Q₈×C₄₃	—	2	8	no	—
63	C₄×Q₈×C₄₃	—	2	6	no	—
64	C₄×Q₈×C₄₃	D₄	1	4	yes	—
65	C₄×Q₈×C₄₃	C₂³	1	12	yes	—
66	C₄×Q₈×C₄₃	C₂³	1	4	yes	—
67	C₂×C₄⋊C₄×C₄₃	—	2	3	no	—
68	C₂×C₄⋊C₄×C₄₃	—	2	3	no	—
69	C₂×C₄⋊C₄×C₄₃	—	2	3	no	—
70	C₂×C₄⋊C₄×C₄₃	D₄	1	3	yes	—
71	C₂×C₄⋊C₄×C₄₃	C₂³	1	1	yes	characteristic
72	C₂×C₄²×C₄₃	—	2	3	no	—
73	C₂×C₄²×C₄₃	C₂³	1	3	yes	—
74	C₂³×D₄₃	—	2	1	no	—
75	C₂×Q₈×C₄₃	—	2	3	no	—
76	C₂×Q₈×C₄₃	—	2	6	no	—
77	C₂×Q₈×C₄₃	—	2	6	no	—
78	C₂×Q₈×C₄₃	SD₁₆	1	8	yes	complemented
79	C₂×Q₈×C₄₃	C₂×D₄	1	4	yes	—
80	C₂×Q₈×C₄₃	C₂⁴	1	6	yes	—
81	C₂²×C₄×C₄₃	—	2	1	no	—
82	C₂²×C₄×C₄₃	—	2	3	no	—
83	C₂²×C₄×C₄₃	C₂×D₄	1	3	yes	—
84	C₂²×C₄×C₄₃	C₂×D₄	1	3	yes	—
85	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
86	C₂²×Dic₄₃	—	2	3	no	—
87	Q₈×D₄₃	—	4	32	no	—
88	C₂×D₁₇₂	—	2	1	no	—
89	C₂×D₁₇₂	—	2	2	no	—
90	C₂×D₁₇₂	#41	1	4	yes	—
91	C₂×D₁₇₂	C₂⁴	1	2	yes	—
92	C₂×C₄×D₄₃	—	2	12	no	—
93	C₂×C₄×D₄₃	—	4	12	no	—
94	C₂×Dic₈₆	—	2	12	no	—
95	C₂×Dic₈₆	—	4	12	no	—
96	C₄⋊C₄×C₄₃	—	2	12	no	—
97	C₄⋊C₄×C₄₃	—	2	12	no	—
98	C₄⋊C₄×C₄₃	—	2	12	no	—
99	C₄⋊C₄×C₄₃	—	2	1	no	—
100	C₄⋊C₄×C₄₃	C₂×D₄	1	12	yes	—
101	C₄⋊C₄×C₄₃	C₂⁴	1	2	yes	—
102	C₄²×C₄₃	—	2	12	no	—
103	C₄²×C₄₃	C₂×D₄	1	6	yes	—
104	C₄²×C₄₃	C₂⁴	1	6	yes	—
105	#10621884	—	2	24	no	—
106	(C₂×D₄₃)⋊C₄	—	2	12	no	—
107	C₄⋊Dic₄₃	—	2	12	no	—
108	C₄⋊Dic₄₃	—	2	3	no	—
109	C₄⋊Dic₄₃	C₂⁴	1	6	yes	—
110	C₂×C₄₃⋊C₈	—	2	6	no	—
111	C₂×C₄₃⋊C₈	—	2	2	no	—
112	C₂×C₄₃⋊C₈	C₂⁴	1	2	yes	—
113	C₂×C₄₃⋊C₈	C₂⁴	1	6	yes	—
114	C₂³×C₄₃	C₂²×D₄	1	1	yes	characteristic
115	C₂²×D₄₃	—	2	4	no	—
116	C₂²×D₄₃	—	2	2	no	—
117	C₂²×D₄₃	—	4	2	no	—
118	C₂²×D₄₃	—	4	2	no	—
119	Q₈×C₄₃	—	2	6	no	—
120	Q₈×C₄₃	—	2	6	no	—
121	Q₈×C₄₃	—	4	6	no	—
122	Q₈×C₄₃	C₂×SD₁₆	1	16	yes	complemented
123	C₂×C₄×C₄₃	—	2	6	no	—
124	C₂×C₄×C₄₃	—	2	2	no	—
125	C₂×C₄×C₄₃	—	2	6	no	—
126	C₂×C₄×C₄₃	—	4	6	no	—
127	C₂×C₄×C₄₃	—	4	2	no	—
128	C₂×C₄×C₄₃	C₂×SD₁₆	1	12	yes	—
129	C₂×C₄×C₄₃	C₂²×D₄	1	6	yes	—
130	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
131	C₂×C₄×C₄₃	C₂²×D₄	1	6	yes	—
132	C₂×C₄×C₄₃	C₂×#41	1	2	yes	—
133	C₂×C₄×C₄₃	2₋¹⁺⁴	1	2	yes	—
134	C₂×C₄×C₄₃	C₂⁵	1	1	yes	characteristic
135	C₂×Dic₄₃	—	2	6	no	—
136	C₂×Dic₄₃	—	4	6	no	—
137	D₁₇₂	—	2	4	no	—
138	D₁₇₂	—	2	2	no	—
139	D₁₇₂	C₂×#41	1	4	yes	—
140	C₄×D₄₃	—	4	24	no	—
141	Dic₈₆	—	4	24	no	—
142	C₄₃⋊C₈	—	2	6	no	—
143	C₄₃⋊C₈	—	2	2	no	—
144	C₂×#3359	—	43	1	no	maximal, Sylow
145	C₂²×C₄₃	#548	1	2	yes	—
146	C₂²×C₄₃	C₂²×SD₁₆	1	2	yes	—
147	C₂²×C₄₃	#578	1	2	yes	—
148	C₂²×C₄₃	C₂³×D₄	1	1	yes	characteristic
149	C₂×D₄₃	—	2	4	no	—
150	C₂×D₄₃	—	4	8	no	—
151	C₂×D₄₃	—	4	2	no	—
152	C₂×D₄₃	—	4	2	no	—
153	C₄×C₄₃	—	2	6	no	—
154	C₄×C₄₃	—	4	6	no	—
155	C₄×C₄₃	—	4	2	no	—
156	C₄×C₄₃	#548	1	2	yes	—
157	C₄×C₄₃	C₂²×SD₁₆	1	12	yes	—
158	C₄×C₄₃	C₂²×#41	1	1	yes	characteristic
159	C₄×C₄₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
160	Dic₄₃	—	4	6	no	—
161	C₂×#557	D₄₃	1	1	yes	characteristic, complemented
162	C₂×D₄×Q₈	—	43	1	no	—
163	#3359	—	43	16	no	—
164	C₂×#487	—	43	3	no	—
165	C₂×#475	—	43	3	no	—
166	C₂×#461	—	43	3	no	—
167	C₂×C₈×Q₈	—	43	1	no	—
168	C₂×C₄×SD₁₆	—	43	3	no	—
169	C₂×C₄₃	#3359	1	4	yes	complemented
170	C₂×C₄₃	C₂×#548	1	1	yes	characteristic
171	C₂×C₄₃	C₂³×SD₁₆	1	1	yes	characteristic
172	C₂×C₄₃	C₂×#578	1	1	yes	characteristic
173	D₄₃	—	4	4	no	—
174	C₂³×Q₈	—	86	1	no	—
175	C₂²×SD₁₆	—	43	3	no	—
176	#557	—	2	4	no	—
177	#557	C₂×D₄₃	1	8	yes	complemented
178	D₄×Q₈	—	43	8	no	—
179	D₄×Q₈	—	86	4	no	—
180	C₂×C₄⋊Q₈	—	43	3	no	—
181	C₂×C₄⋊Q₈	C₂×D₄₃	1	3	yes	—
182	C₂×#125	—	2	3	no	—
183	C₂×C₂²⋊Q₈	—	86	3	no	—
184	C₂×C₄×Q₈	—	2	1	no	—
185	C₂×C₄×Q₈	C₂×D₄₃	1	3	yes	—
186	C₂×C₄×Q₈	C₂×D₄₃	1	1	yes	characteristic
187	C₂×C₄×D₄	—	43	3	no	—
188	#487	—	43	24	no	—
189	#475	—	43	24	no	—
190	#461	—	43	24	no	—
191	C₈×Q₈	—	43	8	no	—
192	C₄×SD₁₆	—	43	24	no	—
193	C₂×#106	—	43	3	no	—
194	C₂×C₄⋊C₈	—	43	3	no	—
195	C₂×#103	—	43	6	no	—
196	C₂×#103	—	43	3	no	—
197	C₂×#102	—	43	1	no	—
198	C₂×C₄×C₈	—	43	3	no	—
199	C₄₃	C₂×#3359	1	1	yes	characteristic, complemented, Sylow
200	C₂²×Q₈	—	86	8	no	—
201	C₂²×Q₈	—	86	3	no	—
202	C₂²×Q₈	—	172	8	no	—
203	C₂²×Q₈	C₂²⋊D₄₃	1	1	yes	characteristic
204	C₂²×Q₈	C₂²×D₄₃	1	3	yes	—
205	C₂²×D₄	—	43	1	no	—
206	C₂³×C₄	—	86	3	no	—
207	C₂×SD₁₆	—	43	12	no	—
208	C₂×SD₁₆	—	86	12	no	—
209	C₂²×C₈	—	43	3	no	—
210	C₂²×C₈	—	43	1	no	—
211	C₄⋊Q₈	—	2	6	no	—
212	C₄⋊Q₈	—	43	12	no	—
213	C₄⋊Q₈	—	86	6	no	—
214	C₄⋊Q₈	C₂²×D₄₃	1	12	yes	—
215	#125	—	2	24	no	—
216	C₂²⋊Q₈	—	86	24	no	—
217	C₄×Q₈	—	2	8	no	—
218	C₄×Q₈	—	2	6	no	—
219	C₄×Q₈	C₂²⋊D₄₃	1	4	yes	—
220	C₄×Q₈	C₂²×D₄₃	1	12	yes	—
221	C₄×Q₈	C₂²×D₄₃	1	4	yes	—
222	C₄×D₄	—	43	12	no	—
223	C₄×D₄	—	86	6	no	—
224	C₂×C₄⋊C₄	—	2	3	no	—
225	C₂×C₄⋊C₄	—	2	3	no	—
226	C₂×C₄⋊C₄	—	2	3	no	—
227	C₂×C₄⋊C₄	—	43	3	no	—
228	C₂×C₄⋊C₄	—	86	3	no	—
229	C₂×C₄⋊C₄	C₂²⋊D₄₃	1	3	yes	—
230	C₂×C₄⋊C₄	C₂²×D₄₃	1	1	yes	characteristic
231	C₂×C₂²⋊C₄	—	86	3	no	—
232	C₂×C₄²	—	2	3	no	—
233	C₂×C₄²	C₂²×D₄₃	1	3	yes	—
234	#106	—	43	12	no	—
235	C₄⋊C₈	—	43	12	no	—
236	#103	—	43	12	no	—
237	#103	—	43	24	no	—
238	#102	—	43	4	no	—
239	C₄×C₈	—	43	12	no	—
240	C₂⁴	—	86	1	no	—
241	C₂×Q₈	—	2	3	no	—
242	C₂×Q₈	—	2	6	no	—
243	C₂×Q₈	—	2	6	no	—
244	C₂×Q₈	—	86	12	no	—
245	C₂×Q₈	—	172	32	no	—
246	C₂×Q₈	—	172	12	no	—
247	C₂×Q₈	#10621884	1	8	yes	complemented
248	C₂×Q₈	C₂×C₂²⋊D₄₃	1	4	yes	—
249	C₂×Q₈	C₂³×D₄₃	1	6	yes	—
250	C₂×D₄	—	43	4	no	—
251	C₂×D₄	—	43	2	no	—
252	C₂×D₄	—	86	1	no	—
253	C₂×D₄	—	86	2	no	—
254	C₂²×C₄	—	2	1	no	—
255	C₂²×C₄	—	2	3	no	—
256	C₂²×C₄	—	86	3	no	—
257	C₂²×C₄	—	86	12	no	—
258	C₂²×C₄	—	172	12	no	—
259	C₂²×C₄	C₂×C₂²⋊D₄₃	1	3	yes	—
260	C₂²×C₄	C₂×C₂²⋊D₄₃	1	3	yes	—
261	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
262	SD₁₆	—	86	24	no	—
263	C₂×C₈	—	43	2	no	—
264	C₂×C₈	—	43	6	no	—
265	C₂×C₈	—	86	6	no	—
266	C₂×C₈	—	86	2	no	—
267	C₄⋊C₄	—	2	12	no	—
268	C₄⋊C₄	—	2	12	no	—
269	C₄⋊C₄	—	2	12	no	—
270	C₄⋊C₄	—	2	1	no	—
271	C₄⋊C₄	—	43	6	no	—
272	C₄⋊C₄	—	86	3	no	—
273	C₄⋊C₄	—	86	12	no	—
274	C₄⋊C₄	C₂×C₂²⋊D₄₃	1	12	yes	—
275	C₄⋊C₄	C₂³×D₄₃	1	2	yes	—
276	C₂²⋊C₄	—	86	12	no	—
277	C₄²	—	2	12	no	—
278	C₄²	C₂×C₂²⋊D₄₃	1	6	yes	—
279	C₄²	C₂³×D₄₃	1	6	yes	—
280	C₂³	—	86	4	no	—
281	C₂³	—	86	2	no	—
282	C₂³	—	172	2	no	—
283	C₂³	—	172	2	no	—
284	C₂³	C₂²×C₂²⋊D₄₃	1	1	yes	characteristic
285	Q₈	—	2	6	no	—
286	Q₈	—	2	6	no	—
287	Q₈	—	4	6	no	—
288	Q₈	—	172	24	no	—
289	Q₈	C₂×#10621884	1	16	yes	complemented
290	D₄	—	43	4	no	—
291	D₄	—	86	4	no	—
292	D₄	—	86	2	no	—
293	C₂×C₄	—	2	6	no	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	2	6	no	—
296	C₂×C₄	—	4	6	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	86	6	no	—
299	C₂×C₄	—	172	6	no	—
300	C₂×C₄	—	172	24	no	—
301	C₂×C₄	C₂×#10621884	1	12	yes	—
302	C₂×C₄	C₂²×C₂²⋊D₄₃	1	6	yes	—
303	C₂×C₄	C₂²×C₂²⋊D₄₃	1	1	yes	characteristic
304	C₂×C₄	C₂²×C₂²⋊D₄₃	1	6	yes	—
305	C₂×C₄	#41×D₄₃	1	2	yes	—
306	C₂×C₄	#13063372	1	2	yes	—
307	C₂×C₄	C₂⁴×D₄₃	1	1	yes	characteristic
308	C₈	—	86	6	no	—
309	C₈	—	86	2	no	—
310	C₂²	—	86	4	no	—
311	C₂²	—	172	2	no	—
312	C₂²	—	172	2	no	—
313	C₂²	—	172	8	no	—
314	C₂²	#14583556	1	2	yes	—
315	C₂²	C₂²×#10621884	1	2	yes	—
316	C₂²	#14583714	1	2	yes	—
317	C₂²	C₂³×C₂²⋊D₄₃	1	1	yes	characteristic
318	C₄	—	2	6	no	—
319	C₄	—	4	6	no	—
320	C₄	—	4	2	no	—
321	C₄	—	172	6	no	—
322	C₄	#14583556	1	2	yes	—
323	C₄	C₂²×#10621884	1	12	yes	—
324	C₄	C₂×#41×D₄₃	1	1	yes	characteristic
325	C₄	C₂×#13063372	1	1	yes	characteristic
326	C₂	—	172	4	no	—
327	C₂	#19147233	1	4	yes	direct product factor
328	C₂	C₂×#14583556	1	1	yes	characteristic
329	C₂	C₂³×#10621884	1	1	yes	characteristic
330	C₂	C₂×#14583714	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²×C₂²⋊D₄₃
commutator, cosocle	C₂×C₄×C₄₃	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₄₃
Fitting	C₂×C₄₃×#557	C₂
socle	C₂³×C₄₃	C₂²×D₄
prime radical	C₂²×D₁₇₂	C₂³
hypercenter	C₂×#557	D₄₃
hypocenter	C₄₃	C₂×#3359
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×Q₈ ◁ C₂×#557

Complements

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

N	H ≅ G/N	type
2. #19147233	C₂	direct (N × H)
3. C₂×#14583249	C₂	semidirect
4. C₂×#14583184	C₂	semidirect
6. C₂×Q₈×C₄₃⋊C₈	C₂	semidirect
8. C₂×C₄₃×#557	C₂	semidirect
16. #14583249	C₂²	semidirect
17. #14583184	C₂²	semidirect
19. Q₈×C₄₃⋊C₈	C₂²	semidirect
28. C₄₃×#557	C₂²	semidirect
78. C₂×Q₈×C₄₃	SD₁₆	semidirect
122. Q₈×C₄₃	C₂×SD₁₆	semidirect
161. C₂×#557	D₄₃	semidirect
169. C₂×C₄₃	#3359	semidirect
177. #557	C₂×D₄₃	semidirect
199. C₄₃	C₂×#3359	semidirect
247. C₂×Q₈	#10621884	semidirect
289. Q₈	C₂×#10621884	semidirect
327. C₂	#19147233	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×Q₈×D₁₇₂	2
#19147233	2
C₂×#14583249	2
C₂×#14583184	2
C₂×#14583101	2
C₂×Q₈×C₄₃⋊C₈	2
C₂×C₄×#10621884	2
C₂×C₄₃×#557	2
C₂×#3359	43

Subgroups — C2×#19147233 #25976157

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#19147233 #25976157