Subgroups — #25255226

Order 11008.

3086 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	C₂²×#13063284	C₂	1	2	yes	maximal
2	#19137218	C₂	1	8	yes	maximal, complemented
3	#19137213	C₂	1	4	yes	maximal, complemented
4	#19137209	C₂	1	4	yes	maximal
5	#19137079	C₂	1	8	yes	maximal, complemented
6	#19137002	C₂	1	4	yes	maximal, complemented
7	C₂²×#117×C₄₃	C₂	1	1	yes	maximal, characteristic
8	C₂×#13063284	—	2	16	no	—
9	C₂×#13063284	C₂²	1	4	yes	—
10	C₂×#13063284	C₂²	1	4	yes	—
11	C₂×#13063284	C₂²	1	4	yes	—
12	C₂×#13063284	C₂²	1	4	yes	—
13	C₂²×C₄⋊Dic₄₃	C₂²	1	4	yes	—
14	C₂²×C₂²⋊Dic₄₃	C₂²	1	4	yes	—
15	C₂²×C₄×Dic₄₃	C₂²	1	4	yes	—
16	#14582798	C₂²	1	32	yes	complemented
17	#14582774	C₂²	1	32	yes	—
18	#14582769	C₂²	1	8	yes	complemented
19	#14582765	C₂²	1	8	yes	complemented
20	#14582759	C₂²	1	16	yes	—
21	C₄×C₄⋊Dic₄₃	C₂²	1	16	yes	complemented
22	C₂⁴×C₄×C₄₃	C₂²	1	1	yes	characteristic
23	C₂×#117×C₄₃	C₄	1	16	yes	complemented
24	C₂×#117×C₄₃	C₂²	1	4	yes	—
25	C₂×#117×C₄₃	C₂²	1	4	yes	—
26	C₂²×C₄⋊C₄×C₄₃	C₂²	1	2	yes	—
27	C₂²×C₂²⋊C₄×C₄₃	C₂²	1	2	yes	—
28	C₂²×C₄²×C₄₃	C₂²	1	2	yes	—
29	C₂³×Dic₄₃	C₂³	1	8	yes	—
30	#13063284	—	2	16	no	—
31	#13063284	—	2	16	no	—
32	#13063284	—	2	16	no	—
33	#13063284	—	2	16	no	—
34	C₂×C₄⋊Dic₄₃	—	2	16	no	—
35	C₂×C₄⋊Dic₄₃	C₂³	1	8	yes	—
36	C₂×C₄⋊Dic₄₃	C₂³	1	8	yes	—
37	C₂×C₄⋊Dic₄₃	C₂³	1	32	yes	—
38	C₂×C₂²⋊Dic₄₃	—	2	16	no	—
39	C₂×C₂²⋊Dic₄₃	C₂³	1	8	yes	—
40	C₂×C₂²⋊Dic₄₃	C₂³	1	8	yes	—
41	C₂×C₄×Dic₄₃	—	2	16	no	—
42	C₂×C₄×Dic₄₃	C₂³	1	8	yes	—
43	C₂×C₄×Dic₄₃	C₂³	1	8	yes	—
44	C₄₃⋊#95	C₂³	1	32	yes	—
45	C₂⁵×C₄₃	C₂³	1	1	yes	characteristic
46	C₂³×C₄×C₄₃	C₂×C₄	1	8	yes	—
47	C₂³×C₄×C₄₃	D₄	1	8	yes	—
48	C₂³×C₄×C₄₃	Q₈	1	8	yes	—
49	C₂³×C₄×C₄₃	C₂³	1	4	yes	—
50	C₂³×C₄×C₄₃	C₂³	1	1	yes	characteristic
51	C₂³×C₄×C₄₃	C₂³	1	1	yes	characteristic
52	C₂³×C₄×C₄₃	C₂³	1	2	yes	—
53	C₂³×C₄×C₄₃	C₂³	1	2	yes	—
54	#117×C₄₃	C₂×C₄	1	32	yes	complemented
55	#117×C₄₃	C₂×C₄	1	32	yes	complemented
56	C₂×C₄⋊C₄×C₄₃	C₂×C₄	1	16	yes	complemented
57	C₂×C₄⋊C₄×C₄₃	C₂³	1	8	yes	—
58	C₂×C₂²⋊C₄×C₄₃	C₂×C₄	1	16	yes	—
59	C₂×C₂²⋊C₄×C₄₃	C₂³	1	8	yes	—
60	C₂×C₄²×C₄₃	C₂×C₄	1	16	yes	complemented
61	C₂×C₄²×C₄₃	C₂³	1	4	yes	—
62	C₂×C₄²×C₄₃	C₂³	1	4	yes	—
63	C₂⁴×C₄₃	—	4	4	no	—
64	C₂⁴×C₄₃	C₂²×C₄	1	4	yes	—
65	C₂⁴×C₄₃	C₂×D₄	1	4	yes	—
66	C₂⁴×C₄₃	C₂×Q₈	1	4	yes	—
67	C₂⁴×C₄₃	C₂⁴	1	2	yes	—
68	C₂⁴×C₄₃	C₂⁴	1	1	yes	characteristic
69	C₂²×C₄×C₄₃	—	4	8	no	—
70	C₂²×C₄×C₄₃	C₄⋊C₄	1	64	yes	complemented
71	C₂²×C₄×C₄₃	C₂²×C₄	1	16	yes	—
72	C₂²×C₄×C₄₃	C₂²×C₄	1	4	yes	—
73	C₂²×C₄×C₄₃	C₂²×C₄	1	4	yes	—
74	C₂²×C₄×C₄₃	C₂²×C₄	1	8	yes	—
75	C₂²×C₄×C₄₃	C₂²×C₄	1	8	yes	—
76	C₂²×C₄×C₄₃	C₂×D₄	1	8	yes	—
77	C₂²×C₄×C₄₃	C₂×D₄	1	8	yes	—
78	C₂²×C₄×C₄₃	C₂×D₄	1	4	yes	—
79	C₂²×C₄×C₄₃	C₂×D₄	1	4	yes	—
80	C₂²×C₄×C₄₃	C₂×Q₈	1	4	yes	—
81	C₂²×C₄×C₄₃	C₂×Q₈	1	4	yes	—
82	C₂²×C₄×C₄₃	C₂×Q₈	1	8	yes	—
83	C₂²×C₄×C₄₃	C₂×Q₈	1	8	yes	—
84	C₂²×C₄×C₄₃	C₂⁴	1	8	yes	—
85	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
86	C₂²×C₄×C₄₃	C₂⁴	1	2	yes	—
87	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
88	C₂²×Dic₄₃	—	2	16	no	—
89	C₂²×Dic₄₃	—	4	16	no	—
90	C₂²×Dic₄₃	C₂⁴	1	16	yes	—
91	C₄⋊C₄×C₄₃	C₂²×C₄	1	32	yes	—
92	C₂²⋊C₄×C₄₃	C₂²×C₄	1	32	yes	—
93	C₄²×C₄₃	C₂²×C₄	1	16	yes	—
94	C₄²×C₄₃	C₂²×C₄	1	16	yes	—
95	C₂²⋊Dic₄₃	—	2	16	no	—
96	C₂²⋊Dic₄₃	—	2	16	no	—
97	C₄⋊Dic₄₃	—	2	16	no	—
98	C₄⋊Dic₄₃	—	2	16	no	—
99	C₄⋊Dic₄₃	—	4	32	no	—
100	C₄×Dic₄₃	—	2	16	no	—
101	C₄×Dic₄₃	—	2	16	no	—
102	C₂³×C₄₃	—	2	8	no	—
103	C₂³×C₄₃	—	2	4	no	—
104	C₂³×C₄₃	—	4	8	no	—
105	C₂³×C₄₃	—	4	8	no	—
106	C₂³×C₄₃	—	4	8	no	—
107	C₂³×C₄₃	C₂×C₄⋊C₄	1	16	yes	—
108	C₂³×C₄₃	C₂³×C₄	1	4	yes	—
109	C₂³×C₄₃	C₂³×C₄	1	2	yes	—
110	C₂³×C₄₃	C₂²×D₄	1	2	yes	—
111	C₂³×C₄₃	C₂²×D₄	1	4	yes	—
112	C₂³×C₄₃	C₂²×Q₈	1	4	yes	—
113	C₂³×C₄₃	C₂²×Q₈	1	2	yes	—
114	C₂³×C₄₃	C₂⁵	1	1	yes	characteristic
115	C₂×C₄×C₄₃	—	2	16	no	—
116	C₂×C₄×C₄₃	—	4	16	no	—
117	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	16	yes	—
118	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	16	yes	—
119	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	32	yes	—
120	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	32	yes	—
121	C₂×C₄×C₄₃	C₂³×C₄	1	16	yes	—
122	C₂×C₄×C₄₃	C₂³×C₄	1	2	yes	—
123	C₂×C₄×C₄₃	C₂³×C₄	1	2	yes	—
124	C₂×C₄×C₄₃	C₂³×C₄	1	4	yes	—
125	C₂×C₄×C₄₃	C₂²×D₄	1	2	yes	—
126	C₂×C₄×C₄₃	C₂²×D₄	1	4	yes	—
127	C₂×C₄×C₄₃	C₂²×D₄	1	2	yes	—
128	C₂×C₄×C₄₃	C₂²×Q₈	1	2	yes	—
129	C₂×C₄×C₄₃	C₂²×Q₈	1	4	yes	—
130	C₂×C₄×C₄₃	C₂²×Q₈	1	2	yes	—
131	C₂×Dic₄₃	—	2	16	no	—
132	C₂×Dic₄₃	—	2	32	no	—
133	C₂×Dic₄₃	—	4	32	no	—
134	#34052	—	43	1	no	maximal, Sylow
135	C₂²×C₄₃	—	2	8	no	—
136	C₂²×C₄₃	—	2	4	no	—
137	C₂²×C₄₃	—	2	4	no	—
138	C₂²×C₄₃	—	2	8	no	—
139	C₂²×C₄₃	—	2	8	no	—
140	C₂²×C₄₃	—	2	4	no	—
141	C₂²×C₄₃	—	4	16	no	—
142	C₂²×C₄₃	C₂²×C₄⋊C₄	1	8	yes	—
143	C₂²×C₄₃	C₂²×C₄⋊C₄	1	4	yes	—
144	C₂²×C₄₃	C₂⁴×C₄	1	1	yes	characteristic
145	C₂²×C₄₃	C₂³×D₄	1	1	yes	characteristic
146	C₂²×C₄₃	C₂³×Q₈	1	1	yes	characteristic
147	C₂²×C₄₃	#585	1	4	yes	—
148	C₄×C₄₃	—	2	16	no	—
149	C₄×C₄₃	C₂²×C₄⋊C₄	1	4	yes	—
150	C₄×C₄₃	C₂²×C₄⋊C₄	1	4	yes	—
151	C₄×C₄₃	C₂²×C₄⋊C₄	1	8	yes	—
152	Dic₄₃	—	4	16	no	—
153	C₂²×#117	—	43	2	no	—
154	C₂²×#117	D₄₃	1	1	yes	characteristic
155	#2317	—	43	8	no	—
156	#2316	—	43	4	no	—
157	#2315	—	43	4	no	—
158	#2296	—	43	8	no	—
159	#2285	—	43	4	no	—
160	C₂×C₄₃	—	2	8	no	—
161	C₂×C₄₃	—	2	4	no	—
162	C₂×C₄₃	C₂³×C₄⋊C₄	1	1	yes	characteristic
163	C₂×C₄₃	#3428	1	2	yes	—
164	C₂×C₄₃	#3449	1	2	yes	—
165	C₂×C₄₃	#3458	1	2	yes	—
166	C₂⁴×C₄	C₂×D₄₃	1	1	yes	characteristic
167	C₂×#117	—	43	4	no	—
168	C₂×#117	—	43	4	no	—
169	C₂×#117	—	43	4	no	—
170	C₂×#117	—	43	4	no	—
171	C₂×#117	—	86	16	no	—
172	C₂×#117	Dic₄₃	1	16	yes	complemented
173	C₂×#117	C₂×D₄₃	1	4	yes	—
174	C₂×#117	C₂×D₄₃	1	4	yes	—
175	C₂²×C₄⋊C₄	—	43	4	no	—
176	C₂²×C₄⋊C₄	C₂×D₄₃	1	2	yes	—
177	C₂²×C₂²⋊C₄	—	43	4	no	—
178	C₂²×C₂²⋊C₄	C₂×D₄₃	1	2	yes	—
179	C₂²×C₄²	—	43	4	no	—
180	C₂²×C₄²	C₂×D₄₃	1	2	yes	—
181	#389	—	43	32	no	—
182	#385	—	43	32	no	—
183	#384	—	43	8	no	—
184	#382	—	43	8	no	—
185	#380	—	43	16	no	—
186	C₄×C₄⋊C₄	—	43	16	no	—
187	C₄₃	#34052	1	1	yes	characteristic, complemented, Sylow
188	C₂⁵	C₂²×D₄₃	1	1	yes	characteristic
189	C₂³×C₄	—	43	8	no	—
190	C₂³×C₄	Dic₈₆	1	8	yes	—
191	C₂³×C₄	D₁₇₂	1	8	yes	—
192	C₂³×C₄	C₂×Dic₄₃	1	8	yes	—
193	C₂³×C₄	C₂²×D₄₃	1	4	yes	—
194	C₂³×C₄	C₂²×D₄₃	1	2	yes	—
195	C₂³×C₄	C₂²×D₄₃	1	1	yes	characteristic
196	C₂³×C₄	C₂²×D₄₃	1	1	yes	characteristic
197	C₂³×C₄	C₂²×D₄₃	1	2	yes	—
198	#117	—	86	16	no	—
199	#117	—	86	16	no	—
200	#117	—	86	16	no	—
201	#117	—	86	16	no	—
202	#117	C₂×Dic₄₃	1	32	yes	complemented
203	#117	C₂×Dic₄₃	1	32	yes	complemented
204	C₂×C₄⋊C₄	—	43	8	no	—
205	C₂×C₄⋊C₄	—	43	8	no	—
206	C₂×C₄⋊C₄	—	43	32	no	—
207	C₂×C₄⋊C₄	—	86	16	no	—
208	C₂×C₄⋊C₄	C₂×Dic₄₃	1	16	yes	complemented
209	C₂×C₄⋊C₄	C₂²×D₄₃	1	8	yes	—
210	C₂×C₂²⋊C₄	—	43	8	no	—
211	C₂×C₂²⋊C₄	—	43	8	no	—
212	C₂×C₂²⋊C₄	—	86	16	no	—
213	C₂×C₂²⋊C₄	C₂×Dic₄₃	1	16	yes	—
214	C₂×C₂²⋊C₄	C₂²×D₄₃	1	8	yes	—
215	C₂×C₄²	—	43	8	no	—
216	C₂×C₄²	—	43	8	no	—
217	C₂×C₄²	—	86	16	no	—
218	C₂×C₄²	C₂×Dic₄₃	1	16	yes	complemented
219	C₂×C₄²	C₂²×D₄₃	1	4	yes	—
220	C₂×C₄²	C₂²×D₄₃	1	4	yes	—
221	#95	—	43	32	no	—
222	C₂⁴	—	4	4	no	—
223	C₂⁴	C₂×Dic₈₆	1	4	yes	—
224	C₂⁴	C₂×D₁₇₂	1	4	yes	—
225	C₂⁴	C₂²×Dic₄₃	1	4	yes	—
226	C₂⁴	C₂³×D₄₃	1	1	yes	characteristic
227	C₂⁴	C₂³×D₄₃	1	2	yes	—
228	C₂²×C₄	—	4	8	no	—
229	C₂²×C₄	—	43	16	no	—
230	C₂²×C₄	—	86	16	no	—
231	C₂²×C₄	—	172	16	no	—
232	C₂²×C₄	C₄⋊Dic₄₃	1	64	yes	complemented
233	C₂²×C₄	C₂×Dic₈₆	1	8	yes	—
234	C₂²×C₄	C₂×Dic₈₆	1	4	yes	—
235	C₂²×C₄	C₂×Dic₈₆	1	4	yes	—
236	C₂²×C₄	C₂×Dic₈₆	1	8	yes	—
237	C₂²×C₄	C₂×D₁₇₂	1	8	yes	—
238	C₂²×C₄	C₂×D₁₇₂	1	8	yes	—
239	C₂²×C₄	C₂×D₁₇₂	1	4	yes	—
240	C₂²×C₄	C₂×D₁₇₂	1	4	yes	—
241	C₂²×C₄	C₂²×Dic₄₃	1	16	yes	—
242	C₂²×C₄	C₂²×Dic₄₃	1	4	yes	—
243	C₂²×C₄	C₂²×Dic₄₃	1	4	yes	—
244	C₂²×C₄	C₂²×Dic₄₃	1	8	yes	—
245	C₂²×C₄	C₂²×Dic₄₃	1	8	yes	—
246	C₂²×C₄	C₂³×D₄₃	1	8	yes	—
247	C₂²×C₄	C₂³×D₄₃	1	2	yes	—
248	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
249	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
250	C₄⋊C₄	—	86	16	no	—
251	C₄⋊C₄	—	86	16	no	—
252	C₄⋊C₄	—	172	32	no	—
253	C₄⋊C₄	C₂²×Dic₄₃	1	32	yes	—
254	C₂²⋊C₄	—	86	16	no	—
255	C₂²⋊C₄	—	86	16	no	—
256	C₂²⋊C₄	C₂²×Dic₄₃	1	32	yes	—
257	C₄²	—	86	16	no	—
258	C₄²	—	86	16	no	—
259	C₄²	C₂²×Dic₄₃	1	16	yes	—
260	C₄²	C₂²×Dic₄₃	1	16	yes	—
261	C₂³	—	2	4	no	—
262	C₂³	—	2	8	no	—
263	C₂³	—	4	8	no	—
264	C₂³	—	4	8	no	—
265	C₂³	—	4	8	no	—
266	C₂³	C₂×C₄⋊Dic₄₃	1	16	yes	—
267	C₂³	C₂³×Dic₄₃	1	4	yes	—
268	C₂³	C₂³×Dic₄₃	1	2	yes	—
269	C₂³	C₂²×D₁₇₂	1	2	yes	—
270	C₂³	C₂²×D₁₇₂	1	4	yes	—
271	C₂³	C₂²×Dic₈₆	1	2	yes	—
272	C₂³	C₂²×Dic₈₆	1	4	yes	—
273	C₂³	C₂⁴×D₄₃	1	1	yes	characteristic
274	C₂×C₄	—	2	16	no	—
275	C₂×C₄	—	4	16	no	—
276	C₂×C₄	—	86	16	no	—
277	C₂×C₄	—	86	32	no	—
278	C₂×C₄	—	172	32	no	—
279	C₂×C₄	C₂×C₄⋊Dic₄₃	1	32	yes	—
280	C₂×C₄	C₂×C₄⋊Dic₄₃	1	32	yes	—
281	C₂×C₄	C₂×C₄⋊Dic₄₃	1	16	yes	—
282	C₂×C₄	C₂×C₄⋊Dic₄₃	1	16	yes	—
283	C₂×C₄	C₂³×Dic₄₃	1	16	yes	—
284	C₂×C₄	C₂³×Dic₄₃	1	2	yes	—
285	C₂×C₄	C₂³×Dic₄₃	1	2	yes	—
286	C₂×C₄	C₂³×Dic₄₃	1	4	yes	—
287	C₂×C₄	C₂²×D₁₇₂	1	2	yes	—
288	C₂×C₄	C₂²×D₁₇₂	1	4	yes	—
289	C₂×C₄	C₂²×D₁₇₂	1	2	yes	—
290	C₂×C₄	C₂²×Dic₈₆	1	4	yes	—
291	C₂×C₄	C₂²×Dic₈₆	1	2	yes	—
292	C₂×C₄	C₂²×Dic₈₆	1	2	yes	—
293	C₂²	—	2	4	no	—
294	C₂²	—	2	8	no	—
295	C₂²	—	2	8	no	—
296	C₂²	—	2	4	no	—
297	C₂²	—	2	4	no	—
298	C₂²	—	2	8	no	—
299	C₂²	—	4	16	no	—
300	C₂²	C₂²×C₄⋊Dic₄₃	1	8	yes	—
301	C₂²	C₂²×C₄⋊Dic₄₃	1	4	yes	—
302	C₂²	C₂⁴×Dic₄₃	1	1	yes	characteristic
303	C₂²	C₂³×D₁₇₂	1	1	yes	characteristic
304	C₂²	C₂³×Dic₈₆	1	1	yes	characteristic
305	C₂²	#14583735	1	4	yes	—
306	C₄	—	2	16	no	—
307	C₄	—	172	16	no	—
308	C₄	C₂²×C₄⋊Dic₄₃	1	4	yes	—
309	C₄	C₂²×C₄⋊Dic₄₃	1	4	yes	—
310	C₄	C₂²×C₄⋊Dic₄₃	1	8	yes	—
311	C₂	—	2	4	no	—
312	C₂	—	2	8	no	—
313	C₂	C₂³×C₄⋊Dic₄₃	1	1	yes	characteristic
314	C₂	#19147798	1	2	yes	—
315	C₂	#19147928	1	2	yes	—
316	C₂	#19148000	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂³×D₄₃
commutator	C₂²×C₄₃	C₂⁴×C₄
Frattini	C₂³	C₂⁴×D₄₃
Fitting	C₂²×#117×C₄₃	C₂
socle, cosocle	C₂³×C₄₃	C₂⁵
prime radical	C₂⁵×C₄₃	C₂³
hypercenter	C₂²×#117	D₄₃
hypocenter	C₄₃	#34052
LCS	G ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×#117

Complements

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

N	H ≅ G/N	type
2. #19137218	C₂	semidirect
3. #19137213	C₂	semidirect
5. #19137079	C₂	semidirect
6. #19137002	C₂	semidirect
16. #14582798	C₂²	semidirect
18. #14582769	C₂²	semidirect
19. #14582765	C₂²	semidirect
21. C₄×C₄⋊Dic₄₃	C₂²	semidirect
23. C₂×#117×C₄₃	C₄	semidirect
54. #117×C₄₃	C₂×C₄	semidirect
55. #117×C₄₃	C₂×C₄	semidirect
56. C₂×C₄⋊C₄×C₄₃	C₂×C₄	semidirect
60. C₂×C₄²×C₄₃	C₂×C₄	semidirect
70. C₂²×C₄×C₄₃	C₄⋊C₄	semidirect
172. C₂×#117	Dic₄₃	semidirect
187. C₄₃	#34052	semidirect
202. #117	C₂×Dic₄₃	semidirect
203. #117	C₂×Dic₄₃	semidirect
208. C₂×C₄⋊C₄	C₂×Dic₄₃	semidirect
218. C₂×C₄²	C₂×Dic₄₃	semidirect
232. C₂²×C₄	C₄⋊Dic₄₃	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#13063284	2
#19137218	2
#19137213	2
#19137209	2
#19137079	2
#19137002	2
C₂²×#117×C₄₃	2
#34052	43