Subgroups — C₂²×#14583619 #26003612

Order 11008.

7846 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14583619	C₂	1	48	yes	maximal, direct product factor
2	C₂²×#13063325	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×#13063318	C₂	1	4	yes	maximal, complemented
4	C₂²×#13063312	C₂	1	2	yes	maximal
5	C₂²×#13063304	C₂	1	2	yes	maximal, complemented
6	C₂²×C₄×Dic₈₆	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×#13063290	C₂	1	2	yes	maximal, complemented
8	C₂²×#13063284	C₂	1	2	yes	maximal, complemented
9	C₂²×C₄×D₄×C₄₃	C₂	1	1	yes	maximal, characteristic
10	C₂³×Dic₈₆	C₂²	1	1	yes	characteristic
11	C₂³×Dic₈₆	C₂²	1	2	yes	—
12	#14583619	C₂²	1	256	yes	direct product factor
13	C₂×#13063325	C₂²	1	24	yes	complemented
14	C₂×#13063318	C₂²	1	96	yes	complemented
15	C₂×#13063312	C₂²	1	48	yes	—
16	C₂×#13063304	C₂²	1	48	yes	complemented
17	C₂×C₄×Dic₈₆	C₂²	1	24	yes	complemented
18	C₂×#13063290	C₂²	1	48	yes	complemented
19	C₂×#13063284	C₂²	1	48	yes	complemented
20	C₂²×C₄⋊Dic₄₃	C₂²	1	2	yes	—
21	C₂²×C₄⋊Dic₄₃	C₂²	1	2	yes	—
22	C₂²×C₄⋊Dic₄₃	C₂²	1	1	yes	characteristic
23	C₂²×Dic₄₃⋊C₄	C₂²	1	4	yes	—
24	C₂²×Dic₄₃⋊C₄	C₂²	1	2	yes	—
25	C₂²×C₂²⋊Dic₄₃	C₂²	1	4	yes	—
26	C₂²×C₂²⋊Dic₄₃	C₂²	1	4	yes	—
27	C₂²×C₄×Dic₄₃	C₂²	1	2	yes	—
28	C₂²×C₄×Dic₄₃	C₂²	1	4	yes	—
29	C₂³×D₄×C₄₃	C₂²	1	1	yes	characteristic
30	C₂⁴×C₄×C₄₃	C₂²	1	2	yes	—
31	C₂×C₄×D₄×C₄₃	C₂²	1	24	yes	—
32	C₂²×C₄⋊C₄×C₄₃	C₂²	1	1	yes	characteristic
33	C₂²×C₂²⋊C₄×C₄₃	C₂²	1	2	yes	—
34	C₂²×C₄²×C₄₃	C₂²	1	1	yes	characteristic
35	C₂²×Dic₈₆	—	2	16	no	—
36	C₂²×Dic₈₆	—	2	8	no	—
37	C₂²×Dic₈₆	C₂³	1	24	yes	—
38	C₂²×Dic₈₆	C₂³	1	12	yes	—
39	C₂³×Dic₄₃	C₂³	1	2	yes	—
40	C₂³×Dic₄₃	C₂³	1	2	yes	—
41	C₂³×Dic₄₃	C₂³	1	4	yes	—
42	#13063325	C₂³	1	64	yes	complemented
43	#13063318	C₂³	1	256	yes	complemented
44	#13063312	C₂³	1	128	yes	—
45	#13063304	C₂³	1	128	yes	complemented
46	C₄×Dic₈₆	C₂³	1	64	yes	complemented
47	#13063290	C₂³	1	128	yes	complemented
48	#13063284	C₂³	1	128	yes	complemented
49	C₂×C₄⋊Dic₄₃	C₂³	1	12	yes	—
50	C₂×C₄⋊Dic₄₃	C₂³	1	24	yes	—
51	C₂×C₄⋊Dic₄₃	C₂³	1	24	yes	—
52	C₂×Dic₄₃⋊C₄	C₂³	1	48	yes	—
53	C₂×Dic₄₃⋊C₄	C₂³	1	24	yes	—
54	C₂×C₂²⋊Dic₄₃	C₂³	1	48	yes	—
55	C₂×C₂²⋊Dic₄₃	C₂³	1	48	yes	—
56	C₂×C₄×Dic₄₃	C₂³	1	48	yes	—
57	C₂×C₄×Dic₄₃	C₂³	1	24	yes	—
58	C₂⁵×C₄₃	C₂³	1	2	yes	—
59	C₂²×D₄×C₄₃	—	2	8	no	—
60	C₂²×D₄×C₄₃	C₂³	1	12	yes	—
61	C₂³×C₄×C₄₃	—	2	16	no	—
62	C₂³×C₄×C₄₃	C₂³	1	1	yes	characteristic
63	C₂³×C₄×C₄₃	C₂³	1	2	yes	—
64	C₂³×C₄×C₄₃	C₂³	1	24	yes	—
65	C₂³×C₄×C₄₃	C₂³	1	1	yes	characteristic
66	C₂³×C₄×C₄₃	C₂³	1	1	yes	characteristic
67	C₄×D₄×C₄₃	C₂³	1	64	yes	—
68	C₂×C₄⋊C₄×C₄₃	C₂³	1	12	yes	—
69	C₂×C₂²⋊C₄×C₄₃	C₂³	1	24	yes	—
70	C₂×C₄²×C₄₃	C₂³	1	12	yes	—
71	C₂⁴×C₄₃	—	2	8	no	—
72	C₂⁴×C₄₃	—	2	8	no	—
73	C₂⁴×C₄₃	—	2	8	no	—
74	C₂⁴×C₄₃	C₂⁴	1	12	yes	—
75	C₂⁴×C₄₃	C₂⁴	1	1	yes	characteristic
76	C₂×D₄×C₄₃	—	2	48	no	—
77	C₂×D₄×C₄₃	C₂⁴	1	16	yes	—
78	C₂²×C₄×C₄₃	—	2	96	no	—
79	C₂²×C₄×C₄₃	—	2	8	no	—
80	C₂²×C₄×C₄₃	—	2	4	no	—
81	C₂²×C₄×C₄₃	#41	1	8	yes	—
82	C₂²×C₄×C₄₃	#41	1	8	yes	—
83	C₂²×C₄×C₄₃	C₂⁴	1	6	yes	—
84	C₂²×C₄×C₄₃	C₂⁴	1	12	yes	—
85	C₂²×C₄×C₄₃	C₂⁴	1	32	yes	—
86	C₂²×C₄×C₄₃	C₂⁴	1	6	yes	—
87	C₂²×C₄×C₄₃	C₂⁴	1	6	yes	—
88	C₂²×Dic₄₃	—	2	8	no	—
89	C₂²×Dic₄₃	—	2	16	no	—
90	C₂²×Dic₄₃	—	2	8	no	—
91	C₂²×Dic₄₃	C₂⁴	1	12	yes	—
92	C₂²×Dic₄₃	C₂⁴	1	12	yes	—
93	C₂²×Dic₄₃	C₂⁴	1	24	yes	—
94	C₂×Dic₈₆	—	2	48	no	—
95	C₂×Dic₈₆	—	2	96	no	—
96	C₂×Dic₈₆	C₂⁴	1	32	yes	—
97	C₂×Dic₈₆	C₂⁴	1	16	yes	—
98	C₄⋊C₄×C₄₃	C₂⁴	1	16	yes	—
99	C₂²⋊C₄×C₄₃	C₂⁴	1	32	yes	—
100	C₄²×C₄₃	C₂⁴	1	16	yes	—
101	C₂²⋊Dic₄₃	C₂⁴	1	64	yes	—
102	C₂²⋊Dic₄₃	C₂⁴	1	64	yes	—
103	C₄⋊Dic₄₃	C₂⁴	1	16	yes	—
104	C₄⋊Dic₄₃	C₂⁴	1	32	yes	—
105	C₄⋊Dic₄₃	C₂⁴	1	32	yes	—
106	Dic₄₃⋊C₄	C₂⁴	1	64	yes	—
107	Dic₄₃⋊C₄	C₂⁴	1	32	yes	—
108	C₄×Dic₄₃	C₂⁴	1	64	yes	—
109	C₄×Dic₄₃	C₂⁴	1	32	yes	—
110	C₂³×C₄₃	—	2	24	no	—
111	C₂³×C₄₃	—	2	24	no	—
112	C₂³×C₄₃	—	2	24	no	—
113	C₂³×C₄₃	—	4	32	no	—
114	C₂³×C₄₃	C₂×#41	1	4	yes	—
115	C₂³×C₄₃	C₂×#41	1	4	yes	—
116	C₂³×C₄₃	2₋¹⁺⁴	1	4	yes	—
117	C₂³×C₄₃	C₂⁵	1	8	yes	—
118	C₂³×C₄₃	C₂⁵	1	3	yes	—
119	D₄×C₄₃	—	2	32	no	—
120	C₂×C₄×C₄₃	—	2	24	no	—
121	C₂×C₄×C₄₃	—	2	12	no	—
122	C₂×C₄×C₄₃	—	2	64	no	—
123	C₂×C₄×C₄₃	C₂×#41	1	24	yes	—
124	C₂×C₄×C₄₃	C₂×#41	1	24	yes	—
125	C₂×C₄×C₄₃	C₂⁵	1	8	yes	—
126	C₂×C₄×C₄₃	C₂⁵	1	4	yes	—
127	C₂×C₄×C₄₃	C₂⁵	1	4	yes	—
128	C₂×C₄×C₄₃	C₂⁵	1	4	yes	—
129	C₂×Dic₄₃	—	2	24	no	—
130	C₂×Dic₄₃	—	2	48	no	—
131	C₂×Dic₄₃	—	2	24	no	—
132	C₂×Dic₄₃	C₂⁵	1	8	yes	—
133	C₂×Dic₄₃	C₂⁵	1	8	yes	—
134	C₂×Dic₄₃	C₂⁵	1	16	yes	—
135	Dic₈₆	—	2	32	no	—
136	Dic₈₆	—	2	64	no	—
137	C₂²×#556	—	43	1	no	maximal, Sylow
138	C₂²×C₄₃	—	2	8	no	—
139	C₂²×C₄₃	—	2	8	no	—
140	C₂²×C₄₃	—	2	8	no	—
141	C₂²×C₄₃	—	4	48	no	—
142	C₂²×C₄₃	#556	1	16	yes	complemented
143	C₂²×C₄₃	C₂²×#41	1	6	yes	—
144	C₂²×C₄₃	C₂²×#41	1	6	yes	—
145	C₂²×C₄₃	C₂×2₋¹⁺⁴	1	6	yes	—
146	C₂²×C₄₃	C₂⁶	1	1	yes	characteristic
147	C₄×C₄₃	—	2	8	no	—
148	C₄×C₄₃	—	2	4	no	—
149	C₄×C₄₃	C₂²×#41	1	8	yes	—
150	C₄×C₄₃	C₂²×#41	1	8	yes	—
151	Dic₄₃	—	2	8	no	—
152	Dic₄₃	—	2	16	no	—
153	Dic₄₃	—	2	8	no	—
154	C₂×#556	—	43	48	no	—
155	C₂²×C₄⋊Q₈	—	43	1	no	—
156	C₂²×#126	—	43	4	no	—
157	C₂²×#124	—	43	2	no	—
158	C₂²×C₂²⋊Q₈	—	43	2	no	—
159	C₂²×C₄×Q₈	—	43	1	no	—
160	C₂²×C₄×Q₈	—	43	2	no	—
161	C₂²×C₄×D₄	D₄₃	1	1	yes	characteristic
162	C₂²×#117	—	43	2	no	—
163	C₂×C₄₃	—	4	8	no	—
164	C₂×C₄₃	C₂×#556	1	12	yes	complemented
165	C₂×C₄₃	C₂³×#41	1	1	yes	characteristic
166	C₂×C₄₃	C₂³×#41	1	1	yes	characteristic
167	C₂×C₄₃	C₂²×2₋¹⁺⁴	1	1	yes	characteristic
168	C₂³×Q₈	—	43	1	no	—
169	C₂³×Q₈	—	43	2	no	—
170	C₂³×D₄	C₂×D₄₃	1	1	yes	characteristic
171	C₂⁴×C₄	C₂×D₄₃	1	2	yes	—
172	#556	—	43	256	no	—
173	C₂×C₄⋊Q₈	—	43	24	no	—
174	C₂×#126	—	43	96	no	—
175	C₂×#124	—	43	48	no	—
176	C₂×C₂²⋊Q₈	—	43	48	no	—
177	C₂×C₄×Q₈	—	43	48	no	—
178	C₂×C₄×Q₈	—	43	24	no	—
179	C₂×C₄×D₄	C₂×D₄₃	1	24	yes	—
180	C₂×#117	—	43	48	no	—
181	C₂²×C₄⋊C₄	—	43	4	no	—
182	C₂²×C₄⋊C₄	—	43	2	no	—
183	C₂²×C₄⋊C₄	—	43	1	no	—
184	C₂²×C₄⋊C₄	—	43	2	no	—
185	C₂²×C₄⋊C₄	—	43	2	no	—
186	C₂²×C₄⋊C₄	C₂×D₄₃	1	1	yes	characteristic
187	C₂²×C₂²⋊C₄	—	43	4	no	—
188	C₂²×C₂²⋊C₄	—	43	4	no	—
189	C₂²×C₂²⋊C₄	C₂×D₄₃	1	2	yes	—
190	C₂²×C₄²	—	43	4	no	—
191	C₂²×C₄²	—	43	2	no	—
192	C₂²×C₄²	C₂×D₄₃	1	1	yes	characteristic
193	C₄₃	C₂²×#556	1	1	yes	characteristic, complemented, Sylow
194	C₂⁵	C₂²×D₄₃	1	2	yes	—
195	C₂²×Q₈	—	43	12	no	—
196	C₂²×Q₈	—	43	24	no	—
197	C₂²×Q₈	—	86	16	no	—
198	C₂²×Q₈	—	86	8	no	—
199	C₂²×D₄	—	2	8	no	—
200	C₂²×D₄	C₂²×D₄₃	1	12	yes	—
201	C₂³×C₄	—	2	16	no	—
202	C₂³×C₄	—	43	2	no	—
203	C₂³×C₄	—	43	4	no	—
204	C₂³×C₄	—	43	2	no	—
205	C₂³×C₄	C₂²×D₄₃	1	24	yes	—
206	C₂³×C₄	C₂²×D₄₃	1	1	yes	characteristic
207	C₂³×C₄	C₂²×D₄₃	1	2	yes	—
208	C₂³×C₄	C₂²×D₄₃	1	1	yes	characteristic
209	C₂³×C₄	C₂²×D₄₃	1	1	yes	characteristic
210	C₄⋊Q₈	—	43	64	no	—
211	#126	—	43	256	no	—
212	#124	—	43	128	no	—
213	C₂²⋊Q₈	—	43	128	no	—
214	C₄×Q₈	—	43	128	no	—
215	C₄×Q₈	—	43	64	no	—
216	C₄×D₄	C₂²×D₄₃	1	64	yes	—
217	#117	—	43	128	no	—
218	C₂×C₄⋊C₄	—	43	12	no	—
219	C₂×C₄⋊C₄	—	43	24	no	—
220	C₂×C₄⋊C₄	—	43	48	no	—
221	C₂×C₄⋊C₄	—	43	24	no	—
222	C₂×C₄⋊C₄	—	43	24	no	—
223	C₂×C₄⋊C₄	C₂²×D₄₃	1	12	yes	—
224	C₂×C₂²⋊C₄	—	43	48	no	—
225	C₂×C₂²⋊C₄	—	43	48	no	—
226	C₂×C₂²⋊C₄	C₂²×D₄₃	1	24	yes	—
227	C₂×C₄²	—	43	48	no	—
228	C₂×C₄²	—	43	24	no	—
229	C₂×C₄²	C₂²×D₄₃	1	12	yes	—
230	C₂⁴	—	2	8	no	—
231	C₂⁴	—	2	8	no	—
232	C₂⁴	—	2	8	no	—
233	C₂⁴	C₂³×D₄₃	1	12	yes	—
234	C₂⁴	C₂³×D₄₃	1	1	yes	characteristic
235	C₂×Q₈	—	43	16	no	—
236	C₂×Q₈	—	43	32	no	—
237	C₂×Q₈	—	86	48	no	—
238	C₂×Q₈	—	86	96	no	—
239	C₂×D₄	—	2	48	no	—
240	C₂×D₄	C₂³×D₄₃	1	16	yes	—
241	C₂²×C₄	—	2	8	no	—
242	C₂²×C₄	—	2	4	no	—
243	C₂²×C₄	—	2	96	no	—
244	C₂²×C₄	—	43	12	no	—
245	C₂²×C₄	—	43	12	no	—
246	C₂²×C₄	—	43	24	no	—
247	C₂²×C₄	—	86	8	no	—
248	C₂²×C₄	—	86	16	no	—
249	C₂²×C₄	—	86	8	no	—
250	C₂²×C₄	#10621898	1	8	yes	—
251	C₂²×C₄	#10621900	1	8	yes	—
252	C₂²×C₄	C₂³×D₄₃	1	12	yes	—
253	C₂²×C₄	C₂³×D₄₃	1	6	yes	—
254	C₂²×C₄	C₂³×D₄₃	1	32	yes	—
255	C₂²×C₄	C₂³×D₄₃	1	6	yes	—
256	C₂²×C₄	C₂³×D₄₃	1	6	yes	—
257	C₄⋊C₄	—	43	16	no	—
258	C₄⋊C₄	—	43	32	no	—
259	C₄⋊C₄	—	43	64	no	—
260	C₄⋊C₄	—	43	32	no	—
261	C₄⋊C₄	—	43	32	no	—
262	C₄⋊C₄	C₂³×D₄₃	1	16	yes	—
263	C₂²⋊C₄	—	43	64	no	—
264	C₂²⋊C₄	—	43	64	no	—
265	C₂²⋊C₄	C₂³×D₄₃	1	32	yes	—
266	C₄²	—	43	64	no	—
267	C₄²	—	43	32	no	—
268	C₄²	C₂³×D₄₃	1	16	yes	—
269	C₂³	—	2	24	no	—
270	C₂³	—	2	24	no	—
271	C₂³	—	2	24	no	—
272	C₂³	—	4	32	no	—
273	C₂³	C₂×#10621900	1	4	yes	—
274	C₂³	C₂×#10621898	1	4	yes	—
275	C₂³	#13063372	1	4	yes	—
276	C₂³	C₂⁴×D₄₃	1	8	yes	—
277	C₂³	C₂⁴×D₄₃	1	3	yes	—
278	Q₈	—	86	32	no	—
279	Q₈	—	86	64	no	—
280	D₄	—	2	32	no	—
281	C₂×C₄	—	2	24	no	—
282	C₂×C₄	—	2	12	no	—
283	C₂×C₄	—	2	64	no	—
284	C₂×C₄	—	43	8	no	—
285	C₂×C₄	—	43	8	no	—
286	C₂×C₄	—	43	16	no	—
287	C₂×C₄	—	86	24	no	—
288	C₂×C₄	—	86	48	no	—
289	C₂×C₄	—	86	24	no	—
290	C₂×C₄	C₂×#10621900	1	24	yes	—
291	C₂×C₄	C₂×#10621898	1	24	yes	—
292	C₂×C₄	C₂⁴×D₄₃	1	8	yes	—
293	C₂×C₄	C₂⁴×D₄₃	1	4	yes	—
294	C₂×C₄	C₂⁴×D₄₃	1	4	yes	—
295	C₂×C₄	C₂⁴×D₄₃	1	4	yes	—
296	C₂²	—	2	8	no	—
297	C₂²	—	2	8	no	—
298	C₂²	—	2	8	no	—
299	C₂²	—	4	48	no	—
300	C₂²	#14583619	1	16	yes	direct product factor
301	C₂²	C₂²×#10621900	1	6	yes	—
302	C₂²	C₂²×#10621898	1	6	yes	—
303	C₂²	C₂×#13063372	1	6	yes	—
304	C₂²	C₂⁵×D₄₃	1	1	yes	characteristic
305	C₄	—	2	8	no	—
306	C₄	—	2	4	no	—
307	C₄	—	86	8	no	—
308	C₄	—	86	16	no	—
309	C₄	—	86	8	no	—
310	C₄	C₂²×#10621900	1	8	yes	—
311	C₄	C₂²×#10621898	1	8	yes	—
312	C₂	—	4	8	no	—
313	C₂	C₂×#14583619	1	12	yes	direct product factor
314	C₂	C₂³×#10621900	1	1	yes	characteristic
315	C₂	C₂³×#10621898	1	1	yes	characteristic
316	C₂	C₂²×#13063372	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂³×D₄₃
commutator, cosocle	C₂²×C₄₃	C₂⁶
Frattini	C₂²	C₂⁵×D₄₃
Fitting	C₂²×C₄×D₄×C₄₃	C₂
socle	C₂⁴×C₄₃	C₂⁴
prime radical	C₂³×D₄×C₄₃	C₂²
hypercenter	C₂²×C₄×D₄	D₄₃
hypocenter	C₄₃	C₂²×#556
LCS	G ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄×D₄

Complements

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

N	H ≅ G/N	type
1. C₂×#14583619	C₂	direct (N × H)
2. C₂²×#13063325	C₂	semidirect
3. C₂²×#13063318	C₂	semidirect
5. C₂²×#13063304	C₂	semidirect
6. C₂²×C₄×Dic₈₆	C₂	semidirect
7. C₂²×#13063290	C₂	semidirect
8. C₂²×#13063284	C₂	semidirect
12. #14583619	C₂²	direct (N × H)
13. C₂×#13063325	C₂²	semidirect
14. C₂×#13063318	C₂²	semidirect
16. C₂×#13063304	C₂²	semidirect
17. C₂×C₄×Dic₈₆	C₂²	semidirect
18. C₂×#13063290	C₂²	semidirect
19. C₂×#13063284	C₂²	semidirect
42. #13063325	C₂³	semidirect
43. #13063318	C₂³	semidirect
45. #13063304	C₂³	semidirect
46. C₄×Dic₈₆	C₂³	semidirect
47. #13063290	C₂³	semidirect
48. #13063284	C₂³	semidirect
142. C₂²×C₄₃	#556	semidirect
164. C₂×C₄₃	C₂×#556	semidirect
193. C₄₃	C₂²×#556	semidirect
300. C₂²	#14583619	direct (N × H)
313. C₂	C₂×#14583619	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×#14583619	2
C₂²×#13063325	2
C₂²×#13063318	2
C₂²×#13063312	2
C₂²×#13063304	2
C₂²×C₄×Dic₈₆	2
C₂²×#13063290	2
C₂²×#13063284	2
C₂²×C₄×D₄×C₄₃	2
C₂²×#556	43

Subgroups — C22×#14583619 #26003612

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#14583619 #26003612