Subgroups — C₂×#19135373 #25100135

Order 11008.

918 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13063261	C₂	1	1	yes	maximal, characteristic
2	C₂×#14582939	C₂	1	2	yes	maximal
3	C₂×#14582854	C₂	1	1	yes	maximal, characteristic
4	#19135373	C₂	1	8	yes	maximal, direct product factor
5	C₂×#14582668	C₂	1	2	yes	maximal
6	C₂²×#106×C₄₃	C₂	1	1	yes	maximal, characteristic
7	C₂²×#10621877	C₂²	1	2	yes	—
8	C₂²×#10621877	C₂²	1	1	yes	characteristic
9	C₂×#13063284	C₂²	1	1	yes	characteristic
10	C₂×#13063261	C₄	1	8	yes	complemented
11	C₂×#13063261	C₂²	1	4	yes	—
12	#14582939	—	2	4	no	—
13	#14582939	C₂²	1	8	yes	—
14	C₂×#13063258	—	2	2	no	—
15	C₂×#13063256	—	2	2	no	—
16	#14582854	—	2	2	no	—
17	#14582854	C₂²	1	4	yes	—
18	C₂×#13063235	—	2	1	no	—
19	C₂×C₈×Dic₄₃	—	2	1	no	—
20	#14582668	C₂²	1	8	yes	—
21	C₂³×C₈×C₄₃	C₂²	1	1	yes	characteristic
22	C₂²×C₄⋊C₄×C₄₃	C₂²	1	2	yes	—
23	C₂×#106×C₄₃	—	2	1	no	—
24	C₂×#106×C₄₃	—	2	1	no	—
25	C₂×#106×C₄₃	—	2	2	no	—
26	C₂×#106×C₄₃	C₂²	1	4	yes	—
27	C₂×#10621877	—	2	1	no	—
28	C₂×#10621877	—	2	2	no	—
29	C₂×#10621877	—	2	4	no	—
30	C₂×#10621877	—	2	2	no	—
31	C₂×#10621877	C₂×C₄	1	8	yes	—
32	C₂×#10621877	Q₈	1	4	yes	—
33	C₂×#10621877	C₂³	1	4	yes	—
34	C₂×#10621877	C₂³	1	2	yes	—
35	C₂²×C₄₃⋊C₈	—	2	1	no	—
36	C₂²×C₄₃⋊C₈	—	2	2	no	—
37	#13063284	—	2	1	no	—
38	#13063284	—	2	1	no	—
39	#13063284	D₄	1	2	yes	—
40	#13063284	C₂³	1	2	yes	—
41	C₂×C₄⋊Dic₄₃	D₄	1	2	yes	—
42	C₂×C₂²⋊Dic₄₃	—	2	1	no	—
43	C₂×C₄×Dic₄₃	—	2	1	no	—
44	#13063261	C₂×C₄	1	16	yes	complemented
45	#13063258	—	2	8	no	—
46	#13063256	—	2	8	no	—
47	#13063235	—	2	4	no	—
48	C₈×Dic₄₃	—	2	4	no	—
49	C₂³×C₄×C₄₃	—	2	2	no	—
50	C₂³×C₄×C₄₃	C₂³	1	1	yes	characteristic
51	C₂²×C₈×C₄₃	C₂×C₄	1	4	yes	—
52	C₂²×C₈×C₄₃	D₄	1	1	yes	characteristic
53	C₂²×C₈×C₄₃	D₄	1	1	yes	characteristic
54	C₂²×C₈×C₄₃	D₄	1	2	yes	—
55	C₂²×C₈×C₄₃	Q₈	1	4	yes	—
56	C₂²×C₈×C₄₃	C₂³	1	2	yes	—
57	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
58	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
59	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
60	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
61	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
62	C₂×C₄⋊C₄×C₄₃	C₂³	1	4	yes	—
63	#106×C₄₃	—	2	4	no	—
64	#106×C₄₃	—	2	4	no	—
65	C₂⁴×C₄₃	#41	1	1	yes	characteristic
66	C₂²×C₄×C₄₃	—	2	4	no	—
67	C₂²×C₄×C₄₃	—	4	2	no	—
68	C₂²×C₄×C₄₃	—	4	2	no	—
69	C₂²×C₄×C₄₃	—	8	4	no	—
70	C₂²×C₄×C₄₃	C₂²×C₄	1	2	yes	—
71	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₄₃	C₂×Q₈	1	2	yes	—
76	C₂²×C₄×C₄₃	C₂×Q₈	1	2	yes	—
77	C₂²×C₄×C₄₃	#41	1	2	yes	—
78	C₂²×C₄×C₄₃	#41	1	1	yes	characteristic
79	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
80	C₂²×Dic₄₃	—	2	2	no	—
81	C₂×C₈×C₄₃	C₄⋊C₄	1	4	yes	—
82	C₂×C₈×C₄₃	C₄⋊C₄	1	4	yes	—
83	C₂×C₈×C₄₃	C₄⋊C₄	1	8	yes	—
84	C₂×C₈×C₄₃	C₂²×C₄	1	4	yes	—
85	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
86	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
87	C₂×C₈×C₄₃	C₂×Q₈	1	4	yes	—
88	C₄⋊C₄×C₄₃	—	2	4	no	—
89	C₄⋊C₄×C₄₃	—	2	4	no	—
90	C₄⋊C₄×C₄₃	—	4	2	no	—
91	C₄⋊C₄×C₄₃	—	4	2	no	—
92	C₂²⋊Dic₄₃	—	2	2	no	—
93	C₂²⋊Dic₄₃	—	2	2	no	—
94	C₄⋊Dic₄₃	—	2	2	no	—
95	C₄⋊Dic₄₃	C₂×D₄	1	4	yes	—
96	C₄×Dic₄₃	—	2	2	no	—
97	C₄×Dic₄₃	—	2	2	no	—
98	#10621877	—	2	8	no	—
99	#10621877	—	2	4	no	—
100	#10621877	—	2	4	no	—
101	#10621877	—	2	2	no	—
102	#10621877	C₂²×C₄	1	8	yes	—
103	#10621877	C₂×Q₈	1	4	yes	—
104	C₂×C₄₃⋊C₈	—	2	8	no	—
105	C₂×C₄₃⋊C₈	—	2	4	no	—
106	C₂×C₄₃⋊C₈	—	2	2	no	—
107	C₂×C₄₃⋊C₈	—	2	4	no	—
108	C₂³×C₄₃	—	2	4	no	—
109	C₂³×C₄₃	C₄×Q₈	1	2	yes	—
110	C₂³×C₄₃	#121	1	1	yes	characteristic
111	C₂³×C₄₃	#121	1	1	yes	characteristic
112	C₂³×C₄₃	#125	1	2	yes	—
113	C₂³×C₄₃	C₂×#41	1	1	yes	characteristic
114	C₂×C₄×C₄₃	—	4	4	no	—
115	C₂×C₄×C₄₃	—	8	4	no	—
116	C₂×C₄×C₄₃	—	8	4	no	—
117	C₂×C₄×C₄₃	—	8	4	no	—
118	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	2	yes	—
119	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	2	yes	—
120	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
121	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
122	C₂×C₄×C₄₃	C₄×Q₈	1	2	yes	—
123	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
124	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
125	C₂×C₄×C₄₃	C₂²⋊Q₈	1	2	yes	—
126	C₂×C₄×C₄₃	C₂²⋊Q₈	1	2	yes	—
127	C₂×C₄×C₄₃	#125	1	2	yes	—
128	C₂×C₄×C₄₃	C₄⋊Q₈	1	2	yes	—
129	C₂×C₄×C₄₃	C₄⋊Q₈	1	2	yes	—
130	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
131	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
132	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
133	C₂×C₄×C₄₃	C₂²×Q₈	1	1	yes	characteristic
134	C₂×C₄×C₄₃	C₂²×Q₈	1	1	yes	characteristic
135	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
136	C₂×Dic₄₃	—	2	4	no	—
137	C₂×Dic₄₃	—	4	4	no	—
138	C₈×C₄₃	C₂×C₄⋊C₄	1	4	yes	—
139	C₈×C₄₃	C₂×C₄⋊C₄	1	4	yes	—
140	C₄₃⋊C₈	—	2	8	no	—
141	C₄₃⋊C₈	—	2	4	no	—
142	C₂×#1951	—	43	1	no	maximal, Sylow
143	C₂²×C₄₃	—	2	4	no	—
144	C₂²×C₄₃	—	2	4	no	—
145	C₂²×C₄₃	—	2	2	no	—
146	C₂²×C₄₃	—	2	2	no	—
147	C₂²×C₄₃	#389	1	2	yes	—
148	C₂²×C₄₃	#389	1	2	yes	—
149	C₂²×C₄₃	#471	1	4	yes	—
150	C₂²×C₄₃	C₂×C₄×Q₈	1	1	yes	characteristic
151	C₂²×C₄₃	C₂×#121	1	1	yes	characteristic
152	C₂²×C₄₃	C₂×#125	1	1	yes	characteristic
153	C₄×C₄₃	—	8	4	no	—
154	C₄×C₄₃	#389	1	2	yes	—
155	C₄×C₄₃	#389	1	2	yes	—
156	C₄×C₄₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
157	C₄×C₄₃	C₂×C₄×D₄	1	1	yes	characteristic
158	C₄×C₄₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
159	C₄×C₄₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
160	Dic₄₃	—	4	4	no	—
161	C₂²×#108	—	43	1	no	—
162	C₂×#428	—	43	2	no	—
163	C₂²×#106	D₄₃	1	1	yes	characteristic
164	C₂×#405	—	43	1	no	—
165	#1951	—	43	8	no	—
166	C₂×#343	—	43	2	no	—
167	C₂×C₄₃	—	2	1	no	—
168	C₂×C₄₃	—	2	1	no	—
169	C₂×C₄₃	—	2	2	no	—
170	C₂×C₄₃	#1951	1	4	yes	complemented
171	C₂×C₄₃	C₂×#389	1	1	yes	characteristic
172	C₂×C₄₃	C₂×#471	1	2	yes	—
173	C₂²×M₁₆	—	43	1	no	—
174	C₂²×M₁₆	—	43	2	no	—
175	C₂³×C₈	C₂×D₄₃	1	1	yes	characteristic
176	C₂×#117	—	43	1	no	—
177	C₂²×C₄⋊C₄	C₂×D₄₃	1	2	yes	—
178	C₂×#108	—	43	4	no	—
179	C₂×#108	—	43	8	no	—
180	#428	—	43	8	no	—
181	#428	—	86	4	no	—
182	C₂×#107	—	86	2	no	—
183	C₂×#106	—	2	1	no	—
184	C₂×#106	—	2	1	no	—
185	C₂×#106	—	2	2	no	—
186	C₂×#106	—	86	2	no	—
187	C₂×#106	C₂×D₄₃	1	4	yes	—
188	#405	—	43	4	no	—
189	#405	—	86	2	no	—
190	C₂×#97	—	86	1	no	—
191	C₂×C₄×C₈	—	86	1	no	—
192	#343	—	43	8	no	—
193	C₄₃	C₂×#1951	1	1	yes	characteristic, complemented, Sylow
194	C₂³×C₄	—	2	2	no	—
195	C₂³×C₄	C₂²×D₄₃	1	1	yes	characteristic
196	C₂×M₁₆	—	43	8	no	—
197	C₂×M₁₆	—	43	4	no	—
198	C₂×M₁₆	—	43	2	no	—
199	C₂×M₁₆	—	43	4	no	—
200	C₂×M₁₆	—	86	2	no	—
201	C₂×M₁₆	—	86	4	no	—
202	C₂×M₁₆	—	86	2	no	—
203	C₂×M₁₆	—	86	1	no	—
204	C₂²×C₈	—	86	2	no	—
205	C₂²×C₈	—	86	1	no	—
206	C₂²×C₈	Dic₈₆	1	4	yes	—
207	C₂²×C₈	C₄×D₄₃	1	4	yes	—
208	C₂²×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
209	C₂²×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
210	C₂²×C₈	C₂²⋊D₄₃	1	2	yes	—
211	C₂²×C₈	C₂²×D₄₃	1	2	yes	—
212	#117	—	43	2	no	—
213	#117	—	43	2	no	—
214	#117	—	86	1	no	—
215	#117	—	86	1	no	—
216	C₂×C₄⋊C₄	—	2	2	no	—
217	C₂×C₄⋊C₄	—	2	2	no	—
218	C₂×C₄⋊C₄	—	2	2	no	—
219	C₂×C₄⋊C₄	—	2	2	no	—
220	C₂×C₄⋊C₄	—	2	2	no	—
221	C₂×C₄⋊C₄	—	43	2	no	—
222	C₂×C₄⋊C₄	C₂²×D₄₃	1	4	yes	—
223	C₂×C₂²⋊C₄	—	86	1	no	—
224	C₂×C₄²	—	86	1	no	—
225	#108	—	43	16	no	—
226	#107	—	86	8	no	—
227	#106	—	2	4	no	—
228	#106	—	2	4	no	—
229	#106	—	86	8	no	—
230	#97	—	86	4	no	—
231	C₄×C₈	—	86	4	no	—
232	C₂⁴	#10621900	1	1	yes	characteristic
233	C₂²×C₄	—	2	4	no	—
234	C₂²×C₄	—	4	2	no	—
235	C₂²×C₄	—	4	2	no	—
236	C₂²×C₄	—	8	4	no	—
237	C₂²×C₄	—	86	2	no	—
238	C₂²×C₄	C₂×Dic₈₆	1	2	yes	—
239	C₂²×C₄	C₂×C₄×D₄₃	1	2	yes	—
240	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
241	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
242	C₂²×C₄	#10621900	1	1	yes	characteristic
243	C₂²×C₄	Q₈×D₄₃	1	2	yes	—
244	C₂²×C₄	#10621902	1	2	yes	—
245	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
246	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
247	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
248	M₁₆	—	43	8	no	—
249	M₁₆	—	43	4	no	—
250	M₁₆	—	86	8	no	—
251	M₁₆	—	86	4	no	—
252	M₁₆	—	86	4	no	—
253	M₁₆	—	86	2	no	—
254	C₂×C₈	—	86	8	no	—
255	C₂×C₈	—	86	4	no	—
256	C₂×C₈	—	86	4	no	—
257	C₂×C₈	—	86	2	no	—
258	C₂×C₈	Dic₄₃⋊C₄	1	4	yes	—
259	C₂×C₈	Dic₄₃⋊C₄	1	4	yes	—
260	C₂×C₈	Dic₄₃⋊C₄	1	8	yes	—
261	C₂×C₈	C₂×Dic₈₆	1	4	yes	—
262	C₂×C₈	C₂×C₄×D₄₃	1	4	yes	—
263	C₂×C₈	C₂×C₂²⋊D₄₃	1	2	yes	—
264	C₂×C₈	C₂×C₂²⋊D₄₃	1	2	yes	—
265	C₄⋊C₄	—	2	4	no	—
266	C₄⋊C₄	—	2	4	no	—
267	C₄⋊C₄	—	4	2	no	—
268	C₄⋊C₄	—	4	2	no	—
269	C₄⋊C₄	—	43	4	no	—
270	C₄⋊C₄	—	86	2	no	—
271	C₂²⋊C₄	—	86	2	no	—
272	C₂²⋊C₄	—	86	2	no	—
273	C₄²	—	86	2	no	—
274	C₄²	—	86	2	no	—
275	C₂³	—	2	4	no	—
276	C₂³	#13063290	1	2	yes	—
277	C₂³	#13063296	1	1	yes	characteristic
278	C₂³	#13063296	1	1	yes	characteristic
279	C₂³	#13063316	1	2	yes	—
280	C₂³	C₂×#10621900	1	1	yes	characteristic
281	C₂×C₄	—	4	4	no	—
282	C₂×C₄	—	8	4	no	—
283	C₂×C₄	—	8	4	no	—
284	C₂×C₄	—	8	4	no	—
285	C₂×C₄	—	86	4	no	—
286	C₂×C₄	—	172	4	no	—
287	C₂×C₄	C₂×Dic₄₃⋊C₄	1	2	yes	—
288	C₂×C₄	C₂×Dic₄₃⋊C₄	1	2	yes	—
289	C₂×C₄	#13063287	1	2	yes	—
290	C₂×C₄	#13063287	1	2	yes	—
291	C₂×C₄	#13063290	1	2	yes	—
292	C₂×C₄	#13063296	1	1	yes	characteristic
293	C₂×C₄	#13063296	1	1	yes	characteristic
294	C₂×C₄	#13063301	1	2	yes	—
295	C₂×C₄	#13063301	1	2	yes	—
296	C₂×C₄	#13063316	1	2	yes	—
297	C₂×C₄	#13063324	1	2	yes	—
298	C₂×C₄	#13063324	1	2	yes	—
299	C₂×C₄	C₂²×C₄×D₄₃	1	1	yes	characteristic
300	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
301	C₂×C₄	C₂²×C₂²⋊D₄₃	1	1	yes	characteristic
302	C₂×C₄	C₂×Q₈×D₄₃	1	1	yes	characteristic
303	C₂×C₄	C₂²×Dic₈₆	1	1	yes	characteristic
304	C₂×C₄	C₂×#10621902	1	1	yes	characteristic
305	C₈	—	86	8	no	—
306	C₈	—	86	4	no	—
307	C₈	C₂×Dic₄₃⋊C₄	1	4	yes	—
308	C₈	C₂×Dic₄₃⋊C₄	1	4	yes	—
309	C₂²	—	2	2	no	—
310	C₂²	—	2	2	no	—
311	C₂²	—	2	4	no	—
312	C₂²	—	2	4	no	—
313	C₂²	#14582795	1	2	yes	—
314	C₂²	#14582795	1	2	yes	—
315	C₂²	#14583159	1	4	yes	—
316	C₂²	C₂×#13063290	1	1	yes	characteristic
317	C₂²	C₂×#13063296	1	1	yes	characteristic
318	C₂²	C₂×#13063316	1	1	yes	characteristic
319	C₄	—	8	4	no	—
320	C₄	—	172	4	no	—
321	C₄	#14582795	1	2	yes	—
322	C₄	#14582795	1	2	yes	—
323	C₄	C₂²×Dic₄₃⋊C₄	1	1	yes	characteristic
324	C₄	C₂×#13063287	1	1	yes	characteristic
325	C₄	C₂×#13063301	1	1	yes	characteristic
326	C₄	C₂×#13063324	1	1	yes	characteristic
327	C₂	—	2	2	no	—
328	C₂	—	2	1	no	—
329	C₂	—	2	1	no	—
330	C₂	#19135373	1	4	yes	direct product factor
331	C₂	C₂×#14582795	1	1	yes	characteristic
332	C₂	C₂×#14583159	1	2	yes	—

	subgroup	G/H
center	C₂³	#13063296
commutator	C₂×C₄×C₄₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₃
Fitting	C₂²×#106×C₄₃	C₂
socle	C₂³×C₄₃	#121
prime radical	C₂⁴×C₄₃	#41
cosocle	C₂²×C₄×C₄₃	C₂⁴
hypercenter	C₂²×#106	D₄₃
hypocenter	C₄₃	C₂×#1951
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂²×#106

Complements

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

N	H ≅ G/N	type
4. #19135373	C₂	direct (N × H)
10. C₂×#13063261	C₄	semidirect
44. #13063261	C₂×C₄	semidirect
170. C₂×C₄₃	#1951	semidirect
193. C₄₃	C₂×#1951	semidirect
330. C₂	#19135373	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#13063261	2
C₂×#14582939	2
C₂×#14582854	2
#19135373	2
C₂×#14582668	2
C₂²×#106×C₄₃	2
C₂×#1951	43

Subgroups — C2×#19135373 #25100135

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#19135373 #25100135