Subgroups — #19148612

Order 5504.

1010 conjugacy classes of subgroups.

The subgroup lattice has 252 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	#14583619	C₂	1	3	yes	maximal, complemented
3	#14583617	C₂	1	6	yes	maximal, complemented
4	#14583614	C₂	1	3	yes	maximal, complemented
5	Q₈×D₁₇₂	C₂	1	1	yes	maximal, characteristic, complemented
6	#14583556	C₂	1	2	yes	maximal, complemented
7	#14583514	C₂	1	6	yes	maximal, complemented
8	#14583456	C₂	1	3	yes	maximal, complemented
9	#14583433	C₂	1	3	yes	maximal, complemented
10	#14583375	C₂	1	2	yes	maximal, complemented
11	C₄×#41×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₈×D₄₃	C₂²	1	2	yes	—
13	#13063325	C₂²	1	3	yes	—
14	#13063325	C₂²	1	3	yes	—
15	#13063324	C₂²	1	6	yes	—
16	#13063319	C₂²	1	6	yes	—
17	#13063318	C₂²	1	12	yes	—
18	#13063317	C₂²	1	3	yes	—
19	#13063313	C₂²	1	1	yes	characteristic
20	#13063312	C₂²	1	6	yes	—
21	#13063311	C₂²	1	12	yes	—
22	#13063308	C₂²	1	6	yes	—
23	#13063305	C₂²	1	6	yes	—
24	#13063304	C₂²	1	6	yes	—
25	#13063304	C₂²	1	3	yes	—
26	#13063304	C₂²	1	6	yes	—
27	#13063299	C₂²	1	3	yes	—
28	#13063296	C₂²	1	6	yes	—
29	Q₈×Dic₄₃	C₂²	1	2	yes	—
30	C₄×Dic₈₆	C₂²	1	6	yes	—
31	C₄×Dic₈₆	C₂²	1	3	yes	—
32	C₄×Dic₈₆	C₂²	1	6	yes	—
33	#13063290	C₂²	1	6	yes	—
34	C₄×D₁₇₂	C₂²	1	3	yes	—
35	C₄×C₂²⋊D₄₃	C₂²	1	6	yes	—
36	#13063284	C₂²	1	6	yes	—
37	#13063283	C₂²	1	6	yes	—
38	#13063282	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	Q₈×D₄₃	—	2	8	no	—
54	C₂×D₁₇₂	C₂³	1	1	yes	characteristic
55	C₂×C₄×D₄₃	C₂³	1	6	yes	—
56	C₂×Dic₈₆	C₂³	1	6	yes	—
57	C₂×Dic₈₆	C₂³	1	1	yes	characteristic
58	C₂×Dic₈₆	C₂³	1	6	yes	—
59	C₄⋊C₄×C₄₃	C₂³	1	3	yes	—
60	C₄⋊C₄×C₄₃	C₂³	1	3	yes	—
61	C₂²⋊C₄×C₄₃	C₂³	1	6	yes	—
62	C₄²×C₄₃	C₂³	1	1	yes	characteristic
63	C₄²×C₄₃	C₂³	1	3	yes	—
64	C₄²×C₄₃	C₂³	1	3	yes	—
65	C₄²×C₄₃	C₂³	1	3	yes	—
66	C₂²⋊Dic₄₃	C₂³	1	12	yes	—
67	C₂²⋊Dic₄₃	C₂³	1	6	yes	—
68	(C₂×D₄₃)⋊C₄	C₂³	1	2	yes	—
69	(C₂×D₄₃)⋊C₄	C₂³	1	6	yes	—
70	(C₂×D₄₃)⋊C₄	C₂³	1	2	yes	—
71	(C₂×D₄₃)⋊C₄	C₂³	1	6	yes	—
72	C₄⋊Dic₄₃	C₂³	1	6	yes	—
73	C₄⋊Dic₄₃	C₂³	1	3	yes	—
74	C₄⋊Dic₄₃	C₂³	1	3	yes	—
75	C₄⋊Dic₄₃	C₂³	1	6	yes	—
76	Dic₄₃⋊C₄	C₂³	1	6	yes	—
77	Dic₄₃⋊C₄	C₂³	1	6	yes	—
78	Dic₄₃⋊C₄	C₂³	1	6	yes	—
79	Dic₄₃⋊C₄	C₂³	1	6	yes	—
80	C₄×Dic₄₃	C₂³	1	6	yes	—
81	C₄×Dic₄₃	C₂³	1	12	yes	—
82	C₂³×C₄₃	C₂⁴	1	3	yes	—
83	C₂²×D₄₃	C₂⁴	1	2	yes	—
84	Q₈×C₄₃	#41	1	4	yes	—
85	D₄×C₄₃	—	2	6	no	—
86	C₂×C₄×C₄₃	—	2	6	no	—
87	C₂×C₄×C₄₃	—	2	6	no	—
88	C₂×C₄×C₄₃	—	4	3	no	—
89	C₂×C₄×C₄₃	C₂⁴	1	3	yes	—
90	C₂×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
91	C₂×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
92	C₂×C₄×C₄₃	C₂⁴	1	3	yes	—
93	C₂×C₄×C₄₃	C₂⁴	1	3	yes	—
94	C₂×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
95	C₂²⋊D₄₃	—	4	6	no	—
96	C₂×Dic₄₃	C₂⁴	1	2	yes	—
97	C₂×Dic₄₃	C₂⁴	1	6	yes	—
98	C₂×Dic₄₃	C₂⁴	1	6	yes	—
99	D₁₇₂	—	2	2	no	—
100	C₄×D₄₃	—	2	12	no	—
101	Dic₈₆	—	2	12	no	—
102	Dic₈₆	—	2	2	no	—
103	Dic₈₆	—	4	6	no	—
104	C₂²×C₄₃	—	2	3	no	—
105	C₂²×C₄₃	—	2	3	no	—
106	C₂²×C₄₃	—	2	3	no	—
107	C₂²×C₄₃	C₂⁵	1	1	yes	characteristic
108	C₂×D₄₃	—	2	2	no	—
109	C₂×D₄₃	—	2	2	no	—
110	C₂×D₄₃	—	2	2	no	—
111	C₄×C₄₃	—	2	1	no	—
112	C₄×C₄₃	—	2	3	no	—
113	C₄×C₄₃	—	2	3	no	—
114	C₄×C₄₃	—	2	1	no	—
115	C₄×C₄₃	C₂×#41	1	6	yes	—
116	C₄×C₄₃	2₋¹⁺⁴	1	2	yes	—
117	Dic₄₃	—	2	6	no	—
118	Dic₄₃	—	2	2	no	—
119	Dic₄₃	—	2	6	no	—
120	#3518	—	43	1	no	maximal, Sylow
121	C₂×C₄₃	—	4	3	no	—
122	C₂×C₄₃	C₂²×#41	1	1	yes	characteristic
123	C₂×C₄₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
124	C₂×C₄₃	#585	1	1	yes	characteristic
125	D₄₃	—	4	2	no	—
126	Q₈²	—	43	1	no	—
127	#556	—	43	3	no	—
128	#556	—	43	6	no	—
129	#555	—	43	3	no	—
130	D₄×Q₈	—	43	1	no	—
131	#548	—	43	2	no	—
132	#542	—	43	6	no	—
133	#533	—	43	3	no	—
134	#529	—	43	3	no	—
135	#519	—	43	2	no	—
136	C₄×#41	D₄₃	1	1	yes	characteristic, complemented
137	C₄₃	#3518	1	1	yes	characteristic, complemented, Sylow
138	C₂×#41	C₂×D₄₃	1	1	yes	characteristic
139	C₂²×Q₈	—	43	2	no	—
140	C₄⋊Q₈	—	43	6	no	—
141	C₄⋊Q₈	—	43	3	no	—
142	C₄⋊Q₈	—	43	3	no	—
143	#126	—	43	12	no	—
144	#126	—	43	6	no	—
145	#125	—	43	3	no	—
146	#124	—	43	6	no	—
147	#124	—	43	12	no	—
148	#124	—	43	1	no	—
149	#123	—	43	6	no	—
150	C₂²⋊Q₈	—	43	3	no	—
151	C₂²⋊Q₈	—	43	6	no	—
152	C₂²⋊Q₈	—	43	6	no	—
153	C₂²⋊Q₈	—	43	6	no	—
154	#121	—	43	6	no	—
155	#121	—	43	3	no	—
156	C₄×Q₈	—	43	2	no	—
157	C₄×Q₈	—	43	6	no	—
158	C₄×Q₈	—	43	6	no	—
159	C₄×Q₈	—	43	3	no	—
160	C₄×Q₈	—	43	6	no	—
161	C₄×Q₈	C₂×D₄₃	1	1	yes	characteristic
162	C₄×Q₈	C₂×D₄₃	1	1	yes	characteristic
163	C₄×D₄	—	43	6	no	—
164	C₄×D₄	—	43	3	no	—
165	C₄×D₄	C₂×D₄₃	1	3	yes	—
166	C₄×D₄	C₂×D₄₃	1	3	yes	—
167	#117	—	43	6	no	—
168	#117	—	43	6	no	—
169	#117	—	43	6	no	—
170	#117	C₂×D₄₃	1	3	yes	—
171	C₂×C₄²	C₂×D₄₃	1	3	yes	—
172	#41	—	2	4	no	—
173	C₂×Q₈	—	43	1	no	—
174	C₂×Q₈	—	43	6	no	—
175	C₂×Q₈	—	43	6	no	—
176	C₂×Q₈	—	86	8	no	—
177	C₂×Q₈	C₂²×D₄₃	1	1	yes	characteristic
178	C₂×D₄	—	43	6	no	—
179	C₂×D₄	—	43	1	no	—
180	C₂×D₄	C₂²×D₄₃	1	3	yes	—
181	C₂²×C₄	—	43	6	no	—
182	C₂²×C₄	C₂²×D₄₃	1	3	yes	—
183	C₂²×C₄	C₂²×D₄₃	1	3	yes	—
184	C₂²×C₄	C₂²×D₄₃	1	3	yes	—
185	C₄⋊C₄	—	43	3	no	—
186	C₄⋊C₄	—	43	3	no	—
187	C₄⋊C₄	—	43	6	no	—
188	C₄⋊C₄	—	43	6	no	—
189	C₄⋊C₄	—	43	6	no	—
190	C₄⋊C₄	—	43	6	no	—
191	C₄⋊C₄	—	43	6	no	—
192	C₄⋊C₄	—	43	6	no	—
193	C₄⋊C₄	C₂²×D₄₃	1	3	yes	—
194	C₄⋊C₄	C₂²×D₄₃	1	3	yes	—
195	C₂²⋊C₄	—	43	12	no	—
196	C₂²⋊C₄	—	43	6	no	—
197	C₂²⋊C₄	—	43	6	no	—
198	C₂²⋊C₄	—	43	2	no	—
199	C₂²⋊C₄	—	43	2	no	—
200	C₂²⋊C₄	—	43	6	no	—
201	C₂²⋊C₄	C₂²×D₄₃	1	6	yes	—
202	C₄²	—	43	12	no	—
203	C₄²	—	43	6	no	—
204	C₄²	C₂²×D₄₃	1	1	yes	characteristic
205	C₄²	C₂²×D₄₃	1	3	yes	—
206	C₄²	C₂²×D₄₃	1	3	yes	—
207	C₄²	C₂²×D₄₃	1	3	yes	—
208	C₂³	—	43	2	no	—
209	C₂³	C₂³×D₄₃	1	3	yes	—
210	Q₈	—	86	12	no	—
211	Q₈	—	86	2	no	—
212	Q₈	—	172	6	no	—
213	Q₈	#10621898	1	4	yes	—
214	D₄	—	2	6	no	—
215	D₄	—	86	2	no	—
216	D₄	—	172	6	no	—
217	C₂×C₄	—	2	6	no	—
218	C₂×C₄	—	2	6	no	—
219	C₂×C₄	—	4	3	no	—
220	C₂×C₄	—	43	2	no	—
221	C₂×C₄	—	43	6	no	—
222	C₂×C₄	—	43	6	no	—
223	C₂×C₄	—	86	12	no	—
224	C₂×C₄	C₂³×D₄₃	1	3	yes	—
225	C₂×C₄	C₂³×D₄₃	1	1	yes	characteristic
226	C₂×C₄	C₂³×D₄₃	1	1	yes	characteristic
227	C₂×C₄	C₂³×D₄₃	1	3	yes	—
228	C₂×C₄	C₂³×D₄₃	1	3	yes	—
229	C₂×C₄	C₂³×D₄₃	1	1	yes	characteristic
230	C₂²	—	2	3	no	—
231	C₂²	—	2	3	no	—
232	C₂²	—	2	3	no	—
233	C₂²	—	86	2	no	—
234	C₂²	—	86	2	no	—
235	C₂²	—	86	2	no	—
236	C₂²	C₂⁴×D₄₃	1	1	yes	characteristic
237	C₄	—	2	1	no	—
238	C₄	—	2	3	no	—
239	C₄	—	2	1	no	—
240	C₄	—	2	3	no	—
241	C₄	—	86	6	no	—
242	C₄	—	86	2	no	—
243	C₄	—	86	6	no	—
244	C₄	C₂×#10621898	1	6	yes	—
245	C₄	#13063372	1	2	yes	—
246	C₂	—	4	3	no	—
247	C₂	—	172	2	no	—
248	C₂	C₂²×#10621898	1	1	yes	characteristic
249	C₂	C₂×#13063372	1	1	yes	characteristic
250	C₂	#14583735	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₄₃
commutator, socle	C₂²×C₄₃	C₂⁵
Fitting	C₄×#41×C₄₃	C₂
hypercenter	C₄×#41	D₄₃
hypocenter	C₄₃	#3518
LCS	G ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₄×#41

Complements

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

N	H ≅ G/N	type
1. Q₈×Dic₈₆	C₂	semidirect
2. #14583619	C₂	semidirect
3. #14583617	C₂	semidirect
4. #14583614	C₂	semidirect
5. Q₈×D₁₇₂	C₂	semidirect
6. #14583556	C₂	semidirect
7. #14583514	C₂	semidirect
8. #14583456	C₂	semidirect
9. #14583433	C₂	semidirect
10. #14583375	C₂	semidirect
11. C₄×#41×C₄₃	C₂	semidirect
136. C₄×#41	D₄₃	semidirect
137. C₄₃	#3518	semidirect