Subgroups — #19148221

Order 5504.

1202 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×Dic₈₆	C₂	1	1	yes	maximal, characteristic, complemented
2	#14583614	C₂	1	3	yes	maximal, complemented
3	#14583612	C₂	1	6	yes	maximal, complemented
4	#14583571	C₂	1	1	yes	maximal, characteristic, complemented
5	#14583570	C₂	1	3	yes	maximal, complemented
6	#14583568	C₂	1	2	yes	maximal, complemented
7	#14583449	C₂	1	3	yes	maximal, complemented
8	#14583446	C₂	1	6	yes	maximal, complemented
9	C₄×#10621902	C₂	1	2	yes	maximal, complemented
10	C₄×#10621898	C₂	1	3	yes	maximal, complemented
11	C₄×#41×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#10621902	C₂²	1	2	yes	—
13	C₂×#10621898	C₂²	1	3	yes	—
14	C₂×#10621898	C₂²	1	6	yes	—
15	#13063325	C₂²	1	3	yes	—
16	#13063324	C₂²	1	6	yes	—
17	#13063322	C₂²	1	3	yes	—
18	#13063321	C₂²	1	6	yes	—
19	#13063313	C₂²	1	6	yes	—
20	#13063311	C₂²	1	12	yes	—
21	#13063300	C₂²	1	6	yes	—
22	#13063299	C₂²	1	6	yes	—
23	#13063299	C₂²	1	6	yes	—
24	#13063298	C₂²	1	12	yes	—
25	#13063296	C₂²	1	6	yes	—
26	Q₈×Dic₄₃	C₂²	1	2	yes	—
27	C₄×Dic₈₆	C₂²	1	3	yes	—
28	C₄×Dic₈₆	C₂²	1	1	yes	characteristic
29	C₄×D₁₇₂	C₂²	1	3	yes	—
30	C₄×D₁₇₂	C₂²	1	3	yes	—
31	C₄×D₁₇₂	C₂²	1	6	yes	—
32	#13063287	C₂²	1	6	yes	—
33	C₄×C₂²⋊D₄₃	C₂²	1	6	yes	—
34	C₄×C₂²⋊D₄₃	C₂²	1	6	yes	—
35	#13063284	C₂²	1	3	yes	—
36	#13063283	C₂²	1	6	yes	—
37	#13063282	C₂²	1	6	yes	—
38	C₄²×D₄₃	C₂²	1	6	yes	—
39	C₂×#41×C₄₃	C₂²	1	1	yes	characteristic
40	C₄×Q₈×C₄₃	C₂²	1	1	yes	characteristic
41	C₄×Q₈×C₄₃	C₂²	1	1	yes	characteristic
42	C₄×D₄×C₄₃	C₂²	1	3	yes	—
43	C₄×D₄×C₄₃	C₂²	1	3	yes	—
44	#117×C₄₃	C₂²	1	3	yes	—
45	C₂×C₄²×C₄₃	C₂²	1	3	yes	—
46	#41×C₄₃	—	2	4	no	—
47	C₂×Q₈×C₄₃	C₂³	1	1	yes	characteristic
48	C₂×D₄×C₄₃	C₂³	1	3	yes	—
49	C₂²×C₄×C₄₃	C₂³	1	3	yes	—
50	C₂²×C₄×C₄₃	C₂³	1	3	yes	—
51	C₂²×C₄×C₄₃	C₂³	1	3	yes	—
52	C₂×C₂²⋊D₄₃	C₂³	1	6	yes	—
53	C₂×C₂²⋊D₄₃	C₂³	1	12	yes	—
54	#10621902	—	2	8	no	—
55	#10621898	—	2	12	no	—
56	#10621898	—	4	12	no	—
57	C₂×D₁₇₂	C₂³	1	6	yes	—
58	C₂×D₁₇₂	C₂³	1	3	yes	—
59	C₂×D₁₇₂	C₂³	1	6	yes	—
60	C₂×C₄×D₄₃	C₂³	1	6	yes	—
61	C₂×C₄×D₄₃	C₂³	1	6	yes	—
62	C₂×C₄×D₄₃	C₂³	1	6	yes	—
63	C₂×Dic₈₆	C₂³	1	6	yes	—
64	C₂×Dic₈₆	C₂³	1	1	yes	characteristic
65	C₄⋊C₄×C₄₃	C₂³	1	3	yes	—
66	C₄⋊C₄×C₄₃	C₂³	1	3	yes	—
67	C₂²⋊C₄×C₄₃	C₂³	1	6	yes	—
68	C₄²×C₄₃	C₂³	1	1	yes	characteristic
69	C₄²×C₄₃	C₂³	1	3	yes	—
70	C₄²×C₄₃	C₂³	1	3	yes	—
71	C₄²×C₄₃	C₂³	1	3	yes	—
72	C₂²⋊Dic₄₃	C₂³	1	6	yes	—
73	(C₂×D₄₃)⋊C₄	C₂³	1	6	yes	—
74	(C₂×D₄₃)⋊C₄	C₂³	1	6	yes	—
75	(C₂×D₄₃)⋊C₄	C₂³	1	12	yes	—
76	C₄⋊Dic₄₃	C₂³	1	6	yes	—
77	C₄⋊Dic₄₃	C₂³	1	1	yes	characteristic
78	C₄⋊Dic₄₃	C₂³	1	3	yes	—
79	Dic₄₃⋊C₄	C₂³	1	2	yes	—
80	Dic₄₃⋊C₄	C₂³	1	6	yes	—
81	C₄×Dic₄₃	C₂³	1	2	yes	—
82	C₄×Dic₄₃	C₂³	1	6	yes	—
83	C₄×Dic₄₃	C₂³	1	6	yes	—
84	C₂³×C₄₃	C₂⁴	1	3	yes	—
85	C₂²×D₄₃	C₂⁴	1	6	yes	—
86	Q₈×C₄₃	#41	1	4	yes	complemented
87	D₄×C₄₃	—	2	6	no	—
88	C₂×C₄×C₄₃	—	2	6	no	—
89	C₂×C₄×C₄₃	—	2	6	no	—
90	C₂×C₄×C₄₃	—	2	6	no	—
91	C₂×C₄×C₄₃	C₂⁴	1	3	yes	—
92	C₂×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
93	C₂×C₄×C₄₃	C₂⁴	1	3	yes	—
94	C₂×C₄×C₄₃	C₂⁴	1	3	yes	—
95	C₂×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
96	C₂×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
97	C₂²⋊D₄₃	—	2	12	no	—
98	C₂²⋊D₄₃	—	4	12	no	—
99	C₂×Dic₄₃	C₂⁴	1	2	yes	—
100	C₂×Dic₄₃	C₂⁴	1	6	yes	—
101	C₂×Dic₄₃	C₂⁴	1	2	yes	—
102	D₁₇₂	—	2	6	no	—
103	D₁₇₂	—	2	12	no	—
104	D₁₇₂	—	4	6	no	—
105	C₄×D₄₃	—	2	12	no	—
106	C₄×D₄₃	—	2	12	no	—
107	C₄×D₄₃	—	2	12	no	—
108	Dic₈₆	—	4	6	no	—
109	Dic₈₆	#41	1	4	yes	complemented
110	C₂²×C₄₃	—	2	3	no	—
111	C₂²×C₄₃	—	2	3	no	—
112	C₂²×C₄₃	—	2	3	no	—
113	C₂²×C₄₃	C₂⁵	1	1	yes	characteristic
114	C₂×D₄₃	—	2	6	no	—
115	C₂×D₄₃	—	2	6	no	—
116	C₂×D₄₃	—	2	6	no	—
117	C₄×C₄₃	—	2	3	no	—
118	C₄×C₄₃	—	2	1	no	—
119	C₄×C₄₃	—	2	3	no	—
120	C₄×C₄₃	C₂×#41	1	2	yes	—
121	C₄×C₄₃	C₂×#41	1	6	yes	—
122	C₄×C₄₃	H(5,𝔽₂)	1	2	yes	—
123	Dic₄₃	—	2	2	no	—
124	Dic₄₃	—	2	6	no	—
125	Dic₄₃	C₂×#41	1	4	yes	—
126	#3483	—	43	1	no	maximal, Sylow
127	C₂×C₄₃	—	4	3	no	—
128	C₂×C₄₃	C₂²×#41	1	1	yes	characteristic
129	C₂×C₄₃	C₂²×#41	1	1	yes	characteristic
130	C₂×C₄₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
131	D₄₃	—	4	6	no	—
132	Q₈²	—	43	1	no	—
133	#555	—	43	3	no	—
134	#555	—	43	6	no	—
135	#550	—	43	2	no	—
136	#550	—	43	1	no	—
137	#550	—	43	3	no	—
138	#532	—	43	6	no	—
139	#532	—	43	3	no	—
140	C₄×#41	—	43	2	no	—
141	C₄×#41	—	43	3	no	—
142	C₄×#41	D₄₃	1	1	yes	characteristic, complemented
143	C₄₃	#3483	1	1	yes	characteristic, complemented, Sylow
144	C₂×#41	—	43	2	no	—
145	C₂×#41	—	43	3	no	—
146	C₂×#41	—	43	6	no	—
147	C₂×#41	C₂×D₄₃	1	1	yes	characteristic
148	C₄⋊Q₈	—	43	6	no	—
149	C₄⋊Q₈	—	43	3	no	—
150	#127	—	43	6	no	—
151	#127	—	43	3	no	—
152	#124	—	43	12	no	—
153	#124	—	43	6	no	—
154	#121	—	43	6	no	—
155	#121	—	43	6	no	—
156	#121	—	43	6	no	—
157	#121	—	43	6	no	—
158	#121	—	43	12	no	—
159	C₄×Q₈	—	43	2	no	—
160	C₄×Q₈	—	43	1	no	—
161	C₄×Q₈	—	43	3	no	—
162	C₄×Q₈	C₂×D₄₃	1	1	yes	characteristic
163	C₄×Q₈	C₂×D₄₃	1	1	yes	characteristic
164	C₄×D₄	—	43	6	no	—
165	C₄×D₄	—	43	6	no	—
166	C₄×D₄	—	43	6	no	—
167	C₄×D₄	—	43	6	no	—
168	C₄×D₄	—	43	3	no	—
169	C₄×D₄	—	43	3	no	—
170	C₄×D₄	C₂×D₄₃	1	3	yes	—
171	C₄×D₄	C₂×D₄₃	1	3	yes	—
172	#117	—	43	3	no	—
173	#117	—	43	6	no	—
174	#117	—	43	6	no	—
175	#117	C₂×D₄₃	1	3	yes	—
176	C₂×C₄²	—	43	6	no	—
177	C₂×C₄²	C₂×D₄₃	1	3	yes	—
178	#41	—	2	4	no	—
179	#41	—	86	12	no	—
180	#41	—	86	8	no	—
181	#41	—	172	12	no	—
182	C₂×Q₈	—	43	1	no	—
183	C₂×Q₈	—	43	6	no	—
184	C₂×Q₈	C₂²×D₄₃	1	1	yes	characteristic
185	C₂×D₄	—	43	6	no	—
186	C₂×D₄	—	43	12	no	—
187	C₂×D₄	—	43	3	no	—
188	C₂×D₄	—	43	6	no	—
189	C₂×D₄	—	43	6	no	—
190	C₂×D₄	C₂²×D₄₃	1	3	yes	—
191	C₂²×C₄	—	43	6	no	—
192	C₂²×C₄	—	43	6	no	—
193	C₂²×C₄	—	43	6	no	—
194	C₂²×C₄	C₂²×D₄₃	1	3	yes	—
195	C₂²×C₄	C₂²×D₄₃	1	3	yes	—
196	C₂²×C₄	C₂²×D₄₃	1	3	yes	—
197	C₄⋊C₄	—	43	2	no	—
198	C₄⋊C₄	—	43	1	no	—
199	C₄⋊C₄	—	43	6	no	—
200	C₄⋊C₄	—	43	3	no	—
201	C₄⋊C₄	—	43	6	no	—
202	C₄⋊C₄	C₂²×D₄₃	1	3	yes	—
203	C₄⋊C₄	C₂²×D₄₃	1	3	yes	—
204	C₂²⋊C₄	—	43	6	no	—
205	C₂²⋊C₄	—	43	6	no	—
206	C₂²⋊C₄	—	43	12	no	—
207	C₂²⋊C₄	—	43	6	no	—
208	C₂²⋊C₄	C₂²×D₄₃	1	6	yes	—
209	C₄²	—	43	6	no	—
210	C₄²	—	43	6	no	—
211	C₄²	—	43	2	no	—
212	C₄²	C₂²×D₄₃	1	3	yes	—
213	C₄²	C₂²×D₄₃	1	3	yes	—
214	C₄²	C₂²×D₄₃	1	3	yes	—
215	C₄²	C₂²×D₄₃	1	1	yes	characteristic
216	C₂³	—	43	6	no	—
217	C₂³	C₂³×D₄₃	1	3	yes	—
218	Q₈	—	43	4	no	—
219	Q₈	—	172	6	no	—
220	Q₈	#10621898	1	4	yes	complemented
221	D₄	—	2	6	no	—
222	D₄	—	86	12	no	—
223	D₄	—	86	6	no	—
224	D₄	—	86	12	no	—
225	D₄	—	172	6	no	—
226	D₄	—	172	12	no	—
227	C₂×C₄	—	2	6	no	—
228	C₂×C₄	—	2	6	no	—
229	C₂×C₄	—	2	6	no	—
230	C₂×C₄	—	43	2	no	—
231	C₂×C₄	—	43	2	no	—
232	C₂×C₄	—	43	6	no	—
233	C₂×C₄	—	86	12	no	—
234	C₂×C₄	—	86	12	no	—
235	C₂×C₄	—	86	12	no	—
236	C₂×C₄	C₂³×D₄₃	1	3	yes	—
237	C₂×C₄	C₂³×D₄₃	1	1	yes	characteristic
238	C₂×C₄	C₂³×D₄₃	1	3	yes	—
239	C₂×C₄	C₂³×D₄₃	1	1	yes	characteristic
240	C₂×C₄	C₂³×D₄₃	1	3	yes	—
241	C₂×C₄	C₂³×D₄₃	1	1	yes	characteristic
242	C₂²	—	2	3	no	—
243	C₂²	—	2	3	no	—
244	C₂²	—	2	3	no	—
245	C₂²	—	86	6	no	—
246	C₂²	—	86	6	no	—
247	C₂²	—	86	6	no	—
248	C₂²	C₂⁴×D₄₃	1	1	yes	characteristic
249	C₄	—	2	3	no	—
250	C₄	—	2	1	no	—
251	C₄	—	2	3	no	—
252	C₄	—	43	4	no	—
253	C₄	—	86	2	no	—
254	C₄	—	86	6	no	—
255	C₄	#41×D₄₃	1	2	yes	—
256	C₄	C₂×#10621898	1	6	yes	—
257	C₄	#13063371	1	2	yes	—
258	C₂	—	4	3	no	—
259	C₂	—	172	6	no	—
260	C₂	C₂×#41×D₄₃	1	1	yes	characteristic
261	C₂	C₂²×#10621898	1	1	yes	characteristic
262	C₂	C₂×#13063371	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂³×D₄₃
commutator, socle	C₂²×C₄₃	C₂⁵
Frattini	C₂²	C₂⁴×D₄₃
Fitting	C₄×#41×C₄₃	C₂
hypercenter	C₄×#41	D₄₃
hypocenter	C₄₃	#3483
LCS	G ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×#41

Complements

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

N	H ≅ G/N	type
1. Q₈×Dic₈₆	C₂	semidirect
2. #14583614	C₂	semidirect
3. #14583612	C₂	semidirect
4. #14583571	C₂	semidirect
5. #14583570	C₂	semidirect
6. #14583568	C₂	semidirect
7. #14583449	C₂	semidirect
8. #14583446	C₂	semidirect
9. C₄×#10621902	C₂	semidirect
10. C₄×#10621898	C₂	semidirect
11. C₄×#41×C₄₃	C₂	semidirect
86. Q₈×C₄₃	#41	semidirect
109. Dic₈₆	#41	semidirect
142. C₄×#41	D₄₃	semidirect
143. C₄₃	#3483	semidirect
220. Q₈	#10621898	semidirect