Subgroups — #19148939

Order 5504.

1004 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14583648	C₂	1	4	yes	maximal, complemented
2	#14583629	C₂	1	1	yes	maximal, characteristic, complemented
3	#14583615	C₂	1	1	yes	maximal, characteristic, complemented
4	#14583610	C₂	1	2	yes	maximal, complemented
5	#14583584	C₂	1	4	yes	maximal, complemented
6	#14583550	C₂	1	4	yes	maximal, complemented
7	#14583526	C₂	1	4	yes	maximal, complemented
8	#14583523	C₂	1	4	yes	maximal, complemented
9	#14583473	C₂	1	4	yes	maximal, complemented
10	#14583433	C₂	1	2	yes	maximal, complemented
11	C₂×C₄⋊Q₈×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#10621898	C₂²	1	4	yes	—
13	#13063324	C₂²	1	4	yes	—
14	#13063323	C₂²	1	4	yes	—
15	#13063323	C₂²	1	4	yes	—
16	#13063320	C₂²	1	8	yes	—
17	#13063319	C₂²	1	4	yes	—
18	#13063317	C₂²	1	2	yes	—
19	#13063316	C₂²	1	4	yes	—
20	#13063315	C₂²	1	8	yes	—
21	#13063314	C₂²	1	4	yes	—
22	#13063314	C₂²	1	4	yes	—
23	#13063313	C₂²	1	2	yes	—
24	#13063310	C₂²	1	8	yes	—
25	#13063308	C₂²	1	8	yes	—
26	#13063308	C₂²	1	4	yes	—
27	#13063305	C₂²	1	4	yes	—
28	#13063304	C₂²	1	2	yes	—
29	#13063303	C₂²	1	8	yes	—
30	#13063301	C₂²	1	8	yes	—
31	#13063301	C₂²	1	8	yes	—
32	#13063300	C₂²	1	4	yes	—
33	#13063299	C₂²	1	2	yes	—
34	C₄×Dic₈₆	C₂²	1	2	yes	—
35	#13063290	C₂²	1	4	yes	—
36	C₄×D₁₇₂	C₂²	1	2	yes	—
37	#13063287	C₂²	1	4	yes	—
38	C₄×C₂²⋊D₄₃	C₂²	1	4	yes	—
39	#13063284	C₂²	1	4	yes	—
40	#13063283	C₂²	1	4	yes	—
41	#13063282	C₂²	1	4	yes	—
42	C₄⋊C₄×D₄₃	C₂²	1	4	yes	—
43	C₂²×Q₈×C₄₃	C₂²	1	2	yes	—
44	C₄⋊Q₈×C₄₃	C₂²	1	1	yes	characteristic
45	C₄⋊Q₈×C₄₃	C₂²	1	1	yes	characteristic
46	C₄⋊Q₈×C₄₃	C₂²	1	2	yes	—
47	C₄⋊Q₈×C₄₃	C₂²	1	4	yes	—
48	C₂×C₄⋊C₄×C₄₃	C₂²	1	4	yes	—
49	C₂×C₄²×C₄₃	C₂²	1	1	yes	characteristic
50	C₂×Q₈×C₄₃	—	4	4	no	—
51	C₂×Q₈×C₄₃	C₂³	1	2	yes	—
52	C₂×Q₈×C₄₃	C₂³	1	4	yes	—
53	C₂×Q₈×C₄₃	C₂³	1	2	yes	—
54	C₂²×C₄×C₄₃	C₂³	1	2	yes	—
55	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
56	C₂²×C₄×C₄₃	C₂³	1	4	yes	—
57	C₂×C₂²⋊D₄₃	C₂³	1	4	yes	—
58	#10621898	—	4	8	no	—
59	C₂×D₁₇₂	C₂³	1	2	yes	—
60	C₂×D₁₇₂	C₂³	1	4	yes	—
61	C₂×C₄×D₄₃	C₂³	1	4	yes	—
62	C₂×C₄×D₄₃	C₂³	1	8	yes	—
63	C₂×Dic₈₆	C₂³	1	2	yes	—
64	C₂×Dic₈₆	C₂³	1	4	yes	—
65	C₄⋊C₄×C₄₃	C₂³	1	4	yes	—
66	C₄⋊C₄×C₄₃	C₂³	1	4	yes	—
67	C₄⋊C₄×C₄₃	C₂³	1	8	yes	—
68	C₄²×C₄₃	C₂³	1	2	yes	—
69	C₄²×C₄₃	C₂³	1	1	yes	characteristic
70	C₄²×C₄₃	C₂³	1	1	yes	characteristic
71	C₂²⋊Dic₄₃	C₂³	1	4	yes	—
72	(C₂×D₄₃)⋊C₄	C₂³	1	4	yes	—
73	(C₂×D₄₃)⋊C₄	C₂³	1	4	yes	—
74	(C₂×D₄₃)⋊C₄	C₂³	1	8	yes	—
75	(C₂×D₄₃)⋊C₄	C₂³	1	4	yes	—
76	(C₂×D₄₃)⋊C₄	C₂³	1	8	yes	—
77	(C₂×D₄₃)⋊C₄	C₂³	1	4	yes	—
78	C₄⋊Dic₄₃	C₂³	1	2	yes	—
79	C₄⋊Dic₄₃	C₂³	1	2	yes	—
80	C₄⋊Dic₄₃	C₂³	1	8	yes	—
81	Dic₄₃⋊C₄	C₂³	1	8	yes	—
82	Dic₄₃⋊C₄	C₂³	1	4	yes	—
83	Dic₄₃⋊C₄	C₂³	1	4	yes	—
84	Dic₄₃⋊C₄	C₂³	1	4	yes	—
85	Dic₄₃⋊C₄	C₂³	1	8	yes	—
86	Dic₄₃⋊C₄	C₂³	1	4	yes	—
87	C₄×Dic₄₃	C₂³	1	4	yes	—
88	C₄×Dic₄₃	C₂³	1	4	yes	—
89	C₄×Dic₄₃	C₂³	1	4	yes	—
90	C₂³×C₄₃	C₂⁴	1	1	yes	characteristic
91	C₂²×D₄₃	C₂⁴	1	4	yes	—
92	Q₈×C₄₃	—	4	2	no	—
93	Q₈×C₄₃	—	4	4	no	—
94	Q₈×C₄₃	—	4	2	no	—
95	C₂×C₄×C₄₃	—	2	8	no	—
96	C₂×C₄×C₄₃	—	2	2	no	—
97	C₂×C₄×C₄₃	—	2	4	no	—
98	C₂×C₄×C₄₃	C₂⁴	1	4	yes	—
99	C₂×C₄×C₄₃	C₂⁴	1	2	yes	—
100	C₂×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
101	C₂×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
102	C₂×C₄×C₄₃	C₂⁴	1	4	yes	—
103	C₂×C₄×C₄₃	C₂⁴	1	2	yes	—
104	C₂²⋊D₄₃	—	2	8	no	—
105	C₂×Dic₄₃	C₂⁴	1	4	yes	—
106	C₂×Dic₄₃	C₂⁴	1	4	yes	—
107	C₂×Dic₄₃	C₂⁴	1	4	yes	—
108	D₁₇₂	—	2	4	no	—
109	D₁₇₂	—	4	4	no	—
110	C₄×D₄₃	—	2	8	no	—
111	C₄×D₄₃	—	4	8	no	—
112	Dic₈₆	—	2	4	no	—
113	Dic₈₆	—	4	4	no	—
114	C₂²×C₄₃	—	2	2	no	—
115	C₂²×C₄₃	H(5,𝔽₂)	1	2	yes	—
116	C₂²×C₄₃	C₂⁵	1	1	yes	characteristic
117	C₂×D₄₃	—	2	4	no	—
118	C₂×D₄₃	—	2	4	no	—
119	C₂×D₄₃	—	2	4	no	—
120	C₄×C₄₃	—	2	4	no	—
121	C₄×C₄₃	—	2	4	no	—
122	C₄×C₄₃	—	2	1	no	—
123	C₄×C₄₃	—	2	1	no	—
124	C₄×C₄₃	—	2	2	no	—
125	C₄×C₄₃	2₋¹⁺⁴	1	4	yes	—
126	Dic₄₃	—	2	4	no	—
127	Dic₄₃	—	2	4	no	—
128	Dic₄₃	—	2	4	no	—
129	#3544	—	43	1	no	maximal, Sylow
130	C₂×C₄₃	—	2	2	no	—
131	C₂×C₄₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
132	C₂×C₄₃	C₂×2₋¹⁺⁴	1	2	yes	—
133	D₄₃	—	4	4	no	—
134	#563	—	43	4	no	—
135	Q₈²	—	43	1	no	—
136	#555	—	43	1	no	—
137	#554	—	43	2	no	—
138	#552	—	43	4	no	—
139	#547	—	43	4	no	—
140	#544	—	43	4	no	—
141	#542	—	43	4	no	—
142	#536	—	43	4	no	—
143	C₂×C₄⋊Q₈	D₄₃	1	1	yes	characteristic, complemented
144	#529	—	43	2	no	—
145	C₄₃	#3544	1	1	yes	characteristic, complemented, Sylow
146	C₂×#41	—	43	4	no	—
147	C₂²×Q₈	C₂×D₄₃	1	2	yes	—
148	C₄⋊Q₈	—	43	4	no	—
149	C₄⋊Q₈	—	43	4	no	—
150	C₄⋊Q₈	—	43	4	no	—
151	C₄⋊Q₈	C₂×D₄₃	1	4	yes	—
152	C₄⋊Q₈	C₂×D₄₃	1	1	yes	characteristic
153	C₄⋊Q₈	C₂×D₄₃	1	1	yes	characteristic
154	C₄⋊Q₈	C₂×D₄₃	1	2	yes	—
155	#126	—	43	8	no	—
156	#126	—	43	4	no	—
157	#125	—	43	4	no	—
158	#125	—	43	8	no	—
159	#125	—	43	2	no	—
160	#124	—	43	4	no	—
161	#124	—	43	4	no	—
162	#124	—	43	2	no	—
163	#123	—	43	4	no	—
164	#123	—	43	8	no	—
165	#123	—	43	8	no	—
166	C₂²⋊Q₈	—	43	8	no	—
167	C₂²⋊Q₈	—	43	8	no	—
168	C₂²⋊Q₈	—	43	2	no	—
169	C₂²⋊Q₈	—	43	4	no	—
170	C₂²⋊Q₈	—	43	8	no	—
171	#121	—	43	2	no	—
172	#121	—	43	4	no	—
173	C₄×Q₈	—	43	4	no	—
174	C₄×Q₈	—	43	2	no	—
175	C₄×D₄	—	43	4	no	—
176	C₄×D₄	—	43	4	no	—
177	C₄×D₄	—	43	2	no	—
178	#117	—	43	4	no	—
179	#117	—	43	4	no	—
180	#117	—	43	4	no	—
181	C₂×C₄⋊C₄	—	43	4	no	—
182	C₂×C₄⋊C₄	C₂×D₄₃	1	4	yes	—
183	C₂×C₄²	C₂×D₄₃	1	1	yes	characteristic
184	#41	—	172	8	no	—
185	C₂×Q₈	—	4	4	no	—
186	C₂×Q₈	—	43	2	no	—
187	C₂×Q₈	—	43	4	no	—
188	C₂×Q₈	C₂²×D₄₃	1	2	yes	—
189	C₂×Q₈	C₂²×D₄₃	1	4	yes	—
190	C₂×Q₈	C₂²×D₄₃	1	2	yes	—
191	C₂×D₄	—	43	4	no	—
192	C₂×D₄	—	43	2	no	—
193	C₂×D₄	—	43	4	no	—
194	C₂²×C₄	—	43	4	no	—
195	C₂²×C₄	—	43	8	no	—
196	C₂²×C₄	C₂²×D₄₃	1	2	yes	—
197	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
198	C₂²×C₄	C₂²×D₄₃	1	4	yes	—
199	C₄⋊C₄	—	43	2	no	—
200	C₄⋊C₄	—	43	8	no	—
201	C₄⋊C₄	—	43	2	no	—
202	C₄⋊C₄	—	43	8	no	—
203	C₄⋊C₄	—	43	4	no	—
204	C₄⋊C₄	—	43	4	no	—
205	C₄⋊C₄	—	43	4	no	—
206	C₄⋊C₄	—	43	4	no	—
207	C₄⋊C₄	—	43	8	no	—
208	C₄⋊C₄	C₂²×D₄₃	1	4	yes	—
209	C₄⋊C₄	C₂²×D₄₃	1	4	yes	—
210	C₄⋊C₄	C₂²×D₄₃	1	8	yes	—
211	C₂²⋊C₄	—	43	4	no	—
212	C₂²⋊C₄	—	43	4	no	—
213	C₂²⋊C₄	—	43	8	no	—
214	C₂²⋊C₄	—	43	4	no	—
215	C₂²⋊C₄	—	43	8	no	—
216	C₂²⋊C₄	—	43	4	no	—
217	C₂²⋊C₄	—	43	4	no	—
218	C₄²	—	43	4	no	—
219	C₄²	—	43	4	no	—
220	C₄²	—	43	4	no	—
221	C₄²	C₂²×D₄₃	1	2	yes	—
222	C₄²	C₂²×D₄₃	1	1	yes	characteristic
223	C₄²	C₂²×D₄₃	1	1	yes	characteristic
224	C₂³	—	43	4	no	—
225	C₂³	C₂³×D₄₃	1	1	yes	characteristic
226	Q₈	—	4	2	no	—
227	Q₈	—	4	4	no	—
228	Q₈	—	4	2	no	—
229	Q₈	—	86	4	no	—
230	Q₈	—	172	4	no	—
231	D₄	—	86	8	no	—
232	D₄	—	86	4	no	—
233	D₄	—	172	4	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	8	no	—
236	C₂×C₄	—	2	4	no	—
237	C₂×C₄	—	43	4	no	—
238	C₂×C₄	—	43	4	no	—
239	C₂×C₄	—	43	4	no	—
240	C₂×C₄	—	86	8	no	—
241	C₂×C₄	—	172	8	no	—
242	C₂×C₄	C₂³×D₄₃	1	4	yes	—
243	C₂×C₄	C₂³×D₄₃	1	2	yes	—
244	C₂×C₄	C₂³×D₄₃	1	1	yes	characteristic
245	C₂×C₄	C₂³×D₄₃	1	1	yes	characteristic
246	C₂×C₄	C₂³×D₄₃	1	4	yes	—
247	C₂×C₄	C₂³×D₄₃	1	2	yes	—
248	C₂²	—	2	2	no	—
249	C₂²	—	86	4	no	—
250	C₂²	—	86	4	no	—
251	C₂²	—	86	4	no	—
252	C₂²	#13063370	1	2	yes	—
253	C₂²	C₂⁴×D₄₃	1	1	yes	characteristic
254	C₄	—	2	4	no	—
255	C₄	—	2	4	no	—
256	C₄	—	2	1	no	—
257	C₄	—	2	1	no	—
258	C₄	—	2	2	no	—
259	C₄	—	86	4	no	—
260	C₄	—	86	4	no	—
261	C₄	—	86	4	no	—
262	C₄	#13063373	1	4	yes	—
263	C₂	—	2	2	no	—
264	C₂	—	172	4	no	—
265	C₂	C₂×#13063370	1	1	yes	characteristic
266	C₂	C₂×#13063373	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₄₃
commutator, socle	C₂²×C₄₃	C₂⁵
Fitting	C₂×C₄⋊Q₈×C₄₃	C₂
hypercenter	C₂×C₄⋊Q₈	D₄₃
hypocenter	C₄₃	#3544
LCS	G ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
1. #14583648	C₂	semidirect
2. #14583629	C₂	semidirect
3. #14583615	C₂	semidirect
4. #14583610	C₂	semidirect
5. #14583584	C₂	semidirect
6. #14583550	C₂	semidirect
7. #14583526	C₂	semidirect
8. #14583523	C₂	semidirect
9. #14583473	C₂	semidirect
10. #14583433	C₂	semidirect
11. C₂×C₄⋊Q₈×C₄₃	C₂	semidirect
143. C₂×C₄⋊Q₈	D₄₃	semidirect
145. C₄₃	#3544	semidirect