Subgroups — C₂²×#14583329 #26003082

Order 11008.

2118 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13063347	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#14583329	C₂	1	24	yes	maximal, direct product factor
3	C₂²×#13063271	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×#13063269	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×#13063268	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×#13063266	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×C₄₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×#135×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×#10621898	—	2	1	no	—
10	C₂×#13063347	—	2	6	no	—
11	C₂×#13063347	C₂²	1	12	yes	complemented
12	C₂²×Dic₁₇₂	C₂²	1	1	yes	characteristic
13	C₂²×#10621873	—	2	1	no	—
14	C₂²×D₃₄₄	C₂²	1	1	yes	characteristic
15	#14583329	C₂²	1	64	yes	direct product factor
16	C₂×#13063271	C₂²	1	12	yes	complemented
17	C₂×#13063269	C₂²	1	12	yes	complemented
18	C₂×#13063268	C₂²	1	12	yes	complemented
19	C₂×#13063266	C₂²	1	12	yes	complemented
20	C₂²×C₄₃⋊C₁₆	C₂²	1	1	yes	characteristic
21	C₂²×C₄₃⋊C₁₆	C₂²	1	1	yes	characteristic
22	C₂²×C₄₃⋊C₁₆	C₂²	1	12	yes	complemented
23	C₂²×#41×C₄₃	—	2	1	no	—
24	C₂×#135×C₄₃	—	2	6	no	—
25	C₂×#135×C₄₃	C₂²	1	12	yes	complemented
26	C₂²×Q₁₆×C₄₃	C₂²	1	1	yes	characteristic
27	C₂²×SD₁₆×C₄₃	—	2	1	no	—
28	C₂²×D₈×C₄₃	C₂²	1	1	yes	characteristic
29	C₂³×C₈×C₄₃	C₂²	1	1	yes	characteristic
30	C₂×#10621898	—	2	12	no	—
31	C₂×#10621898	—	4	6	no	—
32	C₂²×Dic₈₆	—	2	1	no	—
33	C₂²×D₁₇₂	—	2	1	no	—
34	C₂²×C₂²⋊D₄₃	—	4	1	no	—
35	C₂²×C₄×D₄₃	—	4	1	no	—
36	#13063347	—	2	24	no	—
37	#13063347	C₂³	1	16	yes	complemented
38	C₂×Dic₁₇₂	—	2	3	no	—
39	C₂×Dic₁₇₂	C₂³	1	6	yes	—
40	C₂×#10621873	—	2	12	no	—
41	C₂×D₃₄₄	—	2	3	no	—
42	C₂×D₃₄₄	C₂³	1	6	yes	—
43	#13063271	C₂³	1	16	yes	complemented
44	#13063269	C₂³	1	16	yes	complemented
45	#13063268	C₂³	1	16	yes	complemented
46	#13063266	C₂³	1	16	yes	complemented
47	C₂×C₄₃⋊C₁₆	C₂³	1	6	yes	—
48	C₂×C₄₃⋊C₁₆	C₂³	1	6	yes	—
49	C₂×C₄₃⋊C₁₆	C₂³	1	16	yes	complemented
50	C₂×#41×C₄₃	—	2	12	no	—
51	C₂×#41×C₄₃	—	4	6	no	—
52	C₂²×Q₈×C₄₃	—	2	1	no	—
53	C₂²×D₄×C₄₃	—	2	1	no	—
54	C₂²×D₄×C₄₃	—	4	1	no	—
55	C₂³×C₄×C₄₃	—	4	1	no	—
56	C₂³×C₄×C₄₃	D₄	1	1	yes	characteristic
57	#135×C₄₃	—	2	24	no	—
58	#135×C₄₃	C₂³	1	16	yes	complemented
59	C₂×Q₁₆×C₄₃	—	2	3	no	—
60	C₂×Q₁₆×C₄₃	C₂³	1	6	yes	—
61	C₂×SD₁₆×C₄₃	—	2	12	no	—
62	C₂×D₈×C₄₃	—	2	3	no	—
63	C₂×D₈×C₄₃	C₂³	1	6	yes	—
64	C₂²×C₈×C₄₃	D₄	1	1	yes	characteristic
65	C₂²×C₈×C₄₃	D₄	1	6	yes	—
66	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
67	C₂²×C₈×C₄₃	C₂³	1	6	yes	—
68	C₂⁴×C₄₃	—	4	1	no	—
69	C₂⁴×C₄₃	D₈	1	1	yes	characteristic
70	C₂³×D₄₃	—	4	1	no	—
71	#41×C₄₃	—	2	16	no	—
72	#41×C₄₃	—	4	24	no	—
73	C₂×Q₈×C₄₃	—	2	6	no	—
74	C₂×Q₈×C₄₃	—	4	3	no	—
75	C₂×D₄×C₄₃	—	2	6	no	—
76	C₂×D₄×C₄₃	—	4	3	no	—
77	C₂×D₄×C₄₃	—	4	12	no	—
78	C₂²×C₄×C₄₃	—	4	12	no	—
79	C₂²×C₄×C₄₃	—	4	1	no	—
80	C₂²×C₄×C₄₃	D₈	1	6	yes	—
81	C₂²×C₄×C₄₃	D₈	1	1	yes	characteristic
82	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₄×C₄₃	C₂×D₄	1	6	yes	—
84	C₂×C₂²⋊D₄₃	—	4	12	no	—
85	C₂²×Dic₄₃	—	4	1	no	—
86	#10621898	—	2	16	no	—
87	#10621898	—	4	24	no	—
88	C₂×D₁₇₂	—	2	6	no	—
89	C₂×D₁₇₂	—	4	3	no	—
90	C₂×C₄×D₄₃	—	4	12	no	—
91	C₂×Dic₈₆	—	2	6	no	—
92	C₂×Dic₈₆	—	4	3	no	—
93	Q₁₆×C₄₃	—	2	6	no	—
94	Q₁₆×C₄₃	C₂⁴	1	4	yes	—
95	SD₁₆×C₄₃	—	2	16	no	—
96	D₈×C₄₃	—	2	6	no	—
97	D₈×C₄₃	C₂⁴	1	4	yes	—
98	C₂×C₈×C₄₃	C₂×D₄	1	3	yes	—
99	C₂×C₈×C₄₃	C₂×D₄	1	6	yes	—
100	C₂×C₈×C₄₃	C₂×D₄	1	12	yes	—
101	C₂×C₈×C₄₃	C₂⁴	1	3	yes	—
102	C₂×C₈×C₄₃	C₂⁴	1	4	yes	—
103	Dic₁₇₂	—	2	6	no	—
104	Dic₁₇₂	C₂⁴	1	4	yes	—
105	D₃₄₄	—	2	6	no	—
106	D₃₄₄	C₂⁴	1	4	yes	—
107	#10621873	—	2	16	no	—
108	C₄₃⋊C₁₆	C₂⁴	1	4	yes	—
109	C₄₃⋊C₁₆	C₂⁴	1	4	yes	—
110	C₂³×C₄₃	—	2	4	no	—
111	C₂³×C₄₃	—	4	6	no	—
112	C₂³×C₄₃	—	8	4	no	—
113	C₂³×C₄₃	C₂×D₈	1	6	yes	—
114	C₂³×C₄₃	C₂×D₈	1	1	yes	characteristic
115	C₂²×D₄₃	—	4	6	no	—
116	C₂²×D₄₃	—	8	4	no	—
117	Q₈×C₄₃	—	2	4	no	—
118	Q₈×C₄₃	—	4	6	no	—
119	D₄×C₄₃	—	2	4	no	—
120	D₄×C₄₃	—	4	16	no	—
121	D₄×C₄₃	—	4	6	no	—
122	C₂×C₄×C₄₃	—	4	6	no	—
123	C₂×C₄×C₄₃	—	4	16	no	—
124	C₂×C₄×C₄₃	C₂×D₈	1	3	yes	—
125	C₂×C₄×C₄₃	C₂×D₈	1	6	yes	—
126	C₂×C₄×C₄₃	C₂×D₈	1	12	yes	—
127	C₂×C₄×C₄₃	C₂²×D₄	1	3	yes	—
128	C₂×C₄×C₄₃	C₂²×D₄	1	4	yes	—
129	C₂²⋊D₄₃	—	4	16	no	—
130	C₂×Dic₄₃	—	4	6	no	—
131	D₁₇₂	—	2	4	no	—
132	D₁₇₂	—	4	6	no	—
133	C₄×D₄₃	—	4	16	no	—
134	Dic₈₆	—	2	4	no	—
135	Dic₈₆	—	4	6	no	—
136	C₈×C₄₃	C₂²×D₄	1	4	yes	—
137	C₈×C₄₃	C₂²×D₄	1	3	yes	—
138	C₈×C₄₃	C₂⁵	1	1	yes	characteristic
139	C₂²×#508	—	43	1	no	maximal, Sylow
140	C₂²×C₄₃	—	2	12	no	—
141	C₂²×C₄₃	—	4	4	no	—
142	C₂²×C₄₃	—	8	12	no	—
143	C₂²×C₄₃	#508	1	4	yes	complemented
144	C₂²×C₄₃	C₂²×D₈	1	4	yes	—
145	C₂²×C₄₃	C₂²×D₈	1	3	yes	—
146	C₂×D₄₃	—	4	4	no	—
147	C₂×D₄₃	—	8	12	no	—
148	C₄×C₄₃	—	4	4	no	—
149	C₄×C₄₃	C₂²×D₈	1	3	yes	—
150	C₄×C₄₃	C₂²×D₈	1	4	yes	—
151	C₄×C₄₃	C₂³×D₄	1	1	yes	characteristic
152	Dic₄₃	—	4	4	no	—
153	C₂²×#135	—	43	1	no	—
154	C₂²×#135	D₄₃	1	1	yes	characteristic, complemented
155	C₂×#508	—	43	24	no	—
156	C₂²×Q₃₂	—	43	1	no	—
157	C₂²×SD₃₂	—	43	1	no	—
158	C₂²×SD₃₂	—	43	1	no	—
159	C₂²×D₁₆	—	43	1	no	—
160	C₂³×C₁₆	—	43	1	no	—
161	C₂×C₄₃	—	2	4	no	—
162	C₂×C₄₃	—	8	4	no	—
163	C₂×C₄₃	C₂×#508	1	6	yes	complemented
164	C₂×C₄₃	C₂³×D₈	1	1	yes	characteristic
165	D₄₃	—	8	4	no	—
166	C₂²×#41	—	2	1	no	—
167	C₂²×#41	—	86	1	no	—
168	C₂×#135	—	2	6	no	—
169	C₂×#135	—	43	12	no	—
170	C₂×#135	—	86	6	no	—
171	C₂×#135	C₂×D₄₃	1	12	yes	complemented
172	C₂²×Q₁₆	—	43	1	no	—
173	C₂²×Q₁₆	C₂×D₄₃	1	1	yes	characteristic
174	C₂²×SD₁₆	—	2	1	no	—
175	C₂²×SD₁₆	—	86	1	no	—
176	C₂²×D₈	—	43	1	no	—
177	C₂²×D₈	C₂×D₄₃	1	1	yes	characteristic
178	C₂³×C₈	C₂×D₄₃	1	1	yes	characteristic
179	#508	—	43	64	no	—
180	C₂×Q₃₂	—	43	12	no	—
181	C₂×SD₃₂	—	43	12	no	—
182	C₂×SD₃₂	—	43	12	no	—
183	C₂×D₁₆	—	43	12	no	—
184	C₂²×C₁₆	—	43	1	no	—
185	C₂²×C₁₆	—	43	1	no	—
186	C₂²×C₁₆	—	43	12	no	—
187	C₄₃	C₂²×#508	1	1	yes	characteristic, complemented, Sylow
188	C₂×#41	—	2	12	no	—
189	C₂×#41	—	4	6	no	—
190	C₂×#41	—	86	12	no	—
191	C₂×#41	—	172	6	no	—
192	C₂²×Q₈	—	2	1	no	—
193	C₂²×Q₈	—	86	1	no	—
194	C₂²×D₄	—	2	1	no	—
195	C₂²×D₄	—	4	1	no	—
196	C₂²×D₄	—	86	1	no	—
197	C₂²×D₄	—	172	1	no	—
198	C₂³×C₄	—	4	1	no	—
199	C₂³×C₄	—	172	1	no	—
200	C₂³×C₄	C₂²⋊D₄₃	1	1	yes	characteristic
201	#135	—	2	24	no	—
202	#135	—	43	16	no	—
203	#135	—	86	24	no	—
204	#135	C₂²×D₄₃	1	16	yes	complemented
205	C₂×Q₁₆	—	2	3	no	—
206	C₂×Q₁₆	—	43	6	no	—
207	C₂×Q₁₆	—	86	3	no	—
208	C₂×Q₁₆	C₂²×D₄₃	1	6	yes	—
209	C₂×SD₁₆	—	2	12	no	—
210	C₂×SD₁₆	—	86	12	no	—
211	C₂×D₈	—	2	3	no	—
212	C₂×D₈	—	43	6	no	—
213	C₂×D₈	—	86	3	no	—
214	C₂×D₈	C₂²×D₄₃	1	6	yes	—
215	C₂²×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
216	C₂²×C₈	C₂²⋊D₄₃	1	6	yes	—
217	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
218	C₂²×C₈	C₂²×D₄₃	1	6	yes	—
219	Q₃₂	—	43	16	no	—
220	SD₃₂	—	43	16	no	—
221	SD₃₂	—	43	16	no	—
222	D₁₆	—	43	16	no	—
223	C₂×C₁₆	—	43	6	no	—
224	C₂×C₁₆	—	43	6	no	—
225	C₂×C₁₆	—	43	16	no	—
226	C₂⁴	—	4	1	no	—
227	C₂⁴	—	172	1	no	—
228	C₂⁴	#10621882	1	1	yes	characteristic
229	#41	—	2	16	no	—
230	#41	—	4	24	no	—
231	#41	—	86	16	no	—
232	#41	—	172	24	no	—
233	C₂×Q₈	—	2	6	no	—
234	C₂×Q₈	—	4	3	no	—
235	C₂×Q₈	—	86	6	no	—
236	C₂×Q₈	—	172	3	no	—
237	C₂×D₄	—	2	6	no	—
238	C₂×D₄	—	4	12	no	—
239	C₂×D₄	—	4	3	no	—
240	C₂×D₄	—	86	6	no	—
241	C₂×D₄	—	172	12	no	—
242	C₂×D₄	—	172	3	no	—
243	C₂²×C₄	—	4	1	no	—
244	C₂²×C₄	—	4	12	no	—
245	C₂²×C₄	—	172	1	no	—
246	C₂²×C₄	—	172	12	no	—
247	C₂²×C₄	#10621882	1	6	yes	—
248	C₂²×C₄	#10621882	1	1	yes	characteristic
249	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
250	C₂²×C₄	C₂×C₂²⋊D₄₃	1	6	yes	—
251	Q₁₆	—	2	6	no	—
252	Q₁₆	—	43	4	no	—
253	Q₁₆	—	86	6	no	—
254	Q₁₆	C₂³×D₄₃	1	4	yes	—
255	SD₁₆	—	2	16	no	—
256	SD₁₆	—	86	16	no	—
257	D₈	—	2	6	no	—
258	D₈	—	43	4	no	—
259	D₈	—	86	6	no	—
260	D₈	C₂³×D₄₃	1	4	yes	—
261	C₂×C₈	C₂×C₂²⋊D₄₃	1	3	yes	—
262	C₂×C₈	C₂×C₂²⋊D₄₃	1	6	yes	—
263	C₂×C₈	C₂×C₂²⋊D₄₃	1	12	yes	—
264	C₂×C₈	C₂³×D₄₃	1	3	yes	—
265	C₂×C₈	C₂³×D₄₃	1	4	yes	—
266	C₁₆	—	43	4	no	—
267	C₁₆	—	43	4	no	—
268	C₂³	—	2	4	no	—
269	C₂³	—	4	6	no	—
270	C₂³	—	8	4	no	—
271	C₂³	—	344	4	no	—
272	C₂³	—	172	6	no	—
273	C₂³	C₂×#10621882	1	6	yes	—
274	C₂³	C₂×#10621882	1	1	yes	characteristic
275	Q₈	—	2	4	no	—
276	Q₈	—	4	6	no	—
277	Q₈	—	86	4	no	—
278	Q₈	—	172	6	no	—
279	D₄	—	2	4	no	—
280	D₄	—	4	16	no	—
281	D₄	—	4	6	no	—
282	D₄	—	86	4	no	—
283	D₄	—	172	16	no	—
284	D₄	—	172	6	no	—
285	C₂×C₄	—	4	6	no	—
286	C₂×C₄	—	4	16	no	—
287	C₂×C₄	—	172	6	no	—
288	C₂×C₄	—	172	16	no	—
289	C₂×C₄	C₂×#10621882	1	6	yes	—
290	C₂×C₄	C₂×#10621882	1	3	yes	—
291	C₂×C₄	C₂×#10621882	1	12	yes	—
292	C₂×C₄	C₂²×C₂²⋊D₄₃	1	3	yes	—
293	C₂×C₄	C₂²×C₂²⋊D₄₃	1	4	yes	—
294	C₈	C₂²×C₂²⋊D₄₃	1	3	yes	—
295	C₈	C₂²×C₂²⋊D₄₃	1	4	yes	—
296	C₈	C₂⁴×D₄₃	1	1	yes	characteristic
297	C₂²	—	2	12	no	—
298	C₂²	—	4	4	no	—
299	C₂²	—	8	12	no	—
300	C₂²	—	344	12	no	—
301	C₂²	—	172	4	no	—
302	C₂²	#14583329	1	4	yes	direct product factor
303	C₂²	C₂²×#10621882	1	4	yes	—
304	C₂²	C₂²×#10621882	1	3	yes	—
305	C₄	—	4	4	no	—
306	C₄	—	172	4	no	—
307	C₄	C₂²×#10621882	1	4	yes	—
308	C₄	C₂²×#10621882	1	3	yes	—
309	C₄	C₂³×C₂²⋊D₄₃	1	1	yes	characteristic
310	C₂	—	2	4	no	—
311	C₂	—	8	4	no	—
312	C₂	—	344	4	no	—
313	C₂	C₂×#14583329	1	6	yes	direct product factor
314	C₂	C₂³×#10621882	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	#10621882
commutator, cosocle	C₈×C₄₃	C₂⁵
Frattini	C₈	C₂⁴×D₄₃
Fitting	C₂²×#135×C₄₃	C₂
socle	C₂³×C₄₃	C₂×D₈
hypercenter	C₂²×#135	D₄₃
hypocenter	C₄₃	C₂²×#508
LCS	G ▷ C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂²×#135

Complements

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

N	H ≅ G/N	type
1. C₂²×#13063347	C₂	semidirect
2. C₂×#14583329	C₂	direct (N × H)
3. C₂²×#13063271	C₂	semidirect
4. C₂²×#13063269	C₂	semidirect
5. C₂²×#13063268	C₂	semidirect
6. C₂²×#13063266	C₂	semidirect
7. C₂³×C₄₃⋊C₁₆	C₂	semidirect
8. C₂²×#135×C₄₃	C₂	semidirect
11. C₂×#13063347	C₂²	semidirect
15. #14583329	C₂²	direct (N × H)
16. C₂×#13063271	C₂²	semidirect
17. C₂×#13063269	C₂²	semidirect
18. C₂×#13063268	C₂²	semidirect
19. C₂×#13063266	C₂²	semidirect
22. C₂²×C₄₃⋊C₁₆	C₂²	semidirect
25. C₂×#135×C₄₃	C₂²	semidirect
37. #13063347	C₂³	semidirect
43. #13063271	C₂³	semidirect
44. #13063269	C₂³	semidirect
45. #13063268	C₂³	semidirect
46. #13063266	C₂³	semidirect
49. C₂×C₄₃⋊C₁₆	C₂³	semidirect
58. #135×C₄₃	C₂³	semidirect
143. C₂²×C₄₃	#508	semidirect
154. C₂²×#135	D₄₃	semidirect
163. C₂×C₄₃	C₂×#508	semidirect
171. C₂×#135	C₂×D₄₃	semidirect
187. C₄₃	C₂²×#508	semidirect
204. #135	C₂²×D₄₃	semidirect
302. C₂²	#14583329	direct (N × H)
313. C₂	C₂×#14583329	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×#13063347	2
C₂×#14583329	2
C₂²×#13063271	2
C₂²×#13063269	2
C₂²×#13063268	2
C₂²×#13063266	2
C₂³×C₄₃⋊C₁₆	2
C₂²×#135×C₄₃	2
C₂²×#508	43

Subgroups — C22×#14583329 #26003082

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#14583329 #26003082