Subgroups — #19148680

Order 5504.

1104 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14583634	C₂	1	3	yes	maximal, complemented
2	Q₈×Dic₈₆	C₂	1	1	yes	maximal, characteristic, complemented
3	#14583609	C₂	1	3	yes	maximal, complemented
4	#14583571	C₂	1	1	yes	maximal, characteristic, complemented
5	#14583560	C₂	1	2	yes	maximal, complemented
6	#14583550	C₂	1	6	yes	maximal, complemented
7	#14583521	C₂	1	6	yes	maximal, complemented
8	#14583449	C₂	1	3	yes	maximal, complemented
9	#14583433	C₂	1	3	yes	maximal, complemented
10	#14583381	C₂	1	2	yes	maximal, complemented
11	C₂×C₄×Q₈×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#10621902	C₂²	1	2	yes	—
13	C₂×#10621898	C₂²	1	6	yes	—
14	#13063325	C₂²	1	3	yes	—
15	#13063324	C₂²	1	6	yes	—
16	#13063322	C₂²	1	3	yes	—
17	#13063319	C₂²	1	6	yes	—
18	#13063317	C₂²	1	3	yes	—
19	#13063314	C₂²	1	6	yes	—
20	#13063313	C₂²	1	3	yes	—
21	#13063313	C₂²	1	6	yes	—
22	#13063310	C₂²	1	12	yes	—
23	#13063308	C₂²	1	6	yes	—
24	#13063304	C₂²	1	1	yes	characteristic
25	#13063303	C₂²	1	12	yes	—
26	#13063301	C₂²	1	6	yes	—
27	#13063300	C₂²	1	6	yes	—
28	#13063299	C₂²	1	6	yes	—
29	#13063299	C₂²	1	3	yes	—
30	Q₈×Dic₄₃	C₂²	1	2	yes	—
31	C₄×Dic₈₆	C₂²	1	3	yes	—
32	C₄×D₁₇₂	C₂²	1	3	yes	—
33	C₄×D₁₇₂	C₂²	1	6	yes	—
34	C₄×D₁₇₂	C₂²	1	6	yes	—
35	#13063287	C₂²	1	6	yes	—
36	C₄×C₂²⋊D₄₃	C₂²	1	6	yes	—
37	#13063282	C₂²	1	6	yes	—
38	C₄⋊C₄×D₄₃	C₂²	1	6	yes	—
39	C₂²×Q₈×C₄₃	C₂²	1	1	yes	characteristic
40	C₄×Q₈×C₄₃	C₂²	1	1	yes	characteristic
41	C₄×Q₈×C₄₃	C₂²	1	3	yes	—
42	C₄×Q₈×C₄₃	C₂²	1	1	yes	characteristic
43	C₄×Q₈×C₄₃	C₂²	1	3	yes	—
44	C₂×C₄⋊C₄×C₄₃	C₂²	1	3	yes	—
45	C₂×C₄²×C₄₃	C₂²	1	3	yes	—
46	C₂×Q₈×C₄₃	—	2	4	no	—
47	C₂×Q₈×C₄₃	C₂³	1	3	yes	—
48	C₂×Q₈×C₄₃	C₂³	1	1	yes	characteristic
49	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
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	#10621902	—	2	8	no	—
54	#10621898	—	4	12	no	—
55	C₂×D₁₇₂	C₂³	1	6	yes	—
56	C₂×D₁₇₂	C₂³	1	3	yes	—
57	C₂×D₁₇₂	C₂³	1	6	yes	—
58	C₂×C₄×D₄₃	C₂³	1	6	yes	—
59	C₂×C₄×D₄₃	C₂³	1	12	yes	—
60	C₂×Dic₈₆	C₂³	1	1	yes	characteristic
61	C₂×Dic₈₆	C₂³	1	6	yes	—
62	C₄⋊C₄×C₄₃	C₂³	1	3	yes	—
63	C₄⋊C₄×C₄₃	C₂³	1	3	yes	—
64	C₄⋊C₄×C₄₃	C₂³	1	6	yes	—
65	C₄²×C₄₃	C₂³	1	3	yes	—
66	C₄²×C₄₃	C₂³	1	3	yes	—
67	C₄²×C₄₃	C₂³	1	3	yes	—
68	C₄²×C₄₃	C₂³	1	3	yes	—
69	C₂²⋊Dic₄₃	C₂³	1	2	yes	—
70	(C₂×D₄₃)⋊C₄	C₂³	1	6	yes	—
71	(C₂×D₄₃)⋊C₄	C₂³	1	6	yes	—
72	(C₂×D₄₃)⋊C₄	C₂³	1	6	yes	—
73	(C₂×D₄₃)⋊C₄	C₂³	1	12	yes	—
74	(C₂×D₄₃)⋊C₄	C₂³	1	6	yes	—
75	C₄⋊Dic₄₃	C₂³	1	6	yes	—
76	C₄⋊Dic₄₃	C₂³	1	1	yes	characteristic
77	C₄⋊Dic₄₃	C₂³	1	3	yes	—
78	Dic₄₃⋊C₄	C₂³	1	6	yes	—
79	Dic₄₃⋊C₄	C₂³	1	2	yes	—
80	Dic₄₃⋊C₄	C₂³	1	6	yes	—
81	Dic₄₃⋊C₄	C₂³	1	6	yes	—
82	C₄×Dic₄₃	C₂³	1	6	yes	—
83	C₂³×C₄₃	C₂⁴	1	1	yes	characteristic
84	C₂²×D₄₃	C₂⁴	1	6	yes	—
85	Q₈×C₄₃	—	2	6	no	—
86	Q₈×C₄₃	#41	1	4	yes	complemented
87	C₂×C₄×C₄₃	—	2	6	no	—
88	C₂×C₄×C₄₃	—	2	6	no	—
89	C₂×C₄×C₄₃	—	2	2	no	—
90	C₂×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
91	C₂×C₄×C₄₃	C₂⁴	1	3	yes	—
92	C₂×C₄×C₄₃	C₂⁴	1	3	yes	—
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	C₂²⋊D₄₃	—	2	12	no	—
97	C₂×Dic₄₃	C₂⁴	1	2	yes	—
98	C₂×Dic₄₃	C₂⁴	1	6	yes	—
99	C₂×Dic₄₃	C₂⁴	1	2	yes	—
100	D₁₇₂	—	2	12	no	—
101	D₁₇₂	—	2	6	no	—
102	D₁₇₂	—	4	6	no	—
103	C₄×D₄₃	—	2	12	no	—
104	C₄×D₄₃	—	4	12	no	—
105	Dic₈₆	—	2	2	no	—
106	Dic₈₆	—	4	6	no	—
107	C₂²×C₄₃	—	2	1	no	—
108	C₂²×C₄₃	—	2	1	no	—
109	C₂²×C₄₃	H(5,𝔽₂)	1	2	yes	—
110	C₂²×C₄₃	C₂⁵	1	1	yes	characteristic
111	C₂×D₄₃	—	2	6	no	—
112	C₂×D₄₃	—	2	6	no	—
113	C₂×D₄₃	—	2	6	no	—
114	C₄×C₄₃	—	2	3	no	—
115	C₄×C₄₃	—	2	3	no	—
116	C₄×C₄₃	—	2	3	no	—
117	C₄×C₄₃	—	2	1	no	—
118	C₄×C₄₃	C₂×#41	1	6	yes	—
119	C₄×C₄₃	2₋¹⁺⁴	1	2	yes	—
120	Dic₄₃	—	2	2	no	—
121	Dic₄₃	—	2	6	no	—
122	Dic₄₃	—	2	2	no	—
123	#3522	—	43	1	no	maximal, Sylow
124	C₂×C₄₃	—	2	2	no	—
125	C₂×C₄₃	C₂²×#41	1	1	yes	characteristic
126	C₂×C₄₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
127	C₂×C₄₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
128	D₄₃	—	4	6	no	—
129	#559	—	43	3	no	—
130	Q₈²	—	43	1	no	—
131	#554	—	43	3	no	—
132	#550	—	43	1	no	—
133	#548	—	43	2	no	—
134	#547	—	43	6	no	—
135	#542	—	43	6	no	—
136	#532	—	43	3	no	—
137	#529	—	43	3	no	—
138	#520	—	43	2	no	—
139	C₂×C₄×Q₈	D₄₃	1	1	yes	characteristic, complemented
140	C₄₃	#3522	1	1	yes	characteristic, complemented, Sylow
141	C₂×#41	—	43	2	no	—
142	C₂×#41	—	43	6	no	—
143	C₂²×Q₈	C₂×D₄₃	1	1	yes	characteristic
144	C₄⋊Q₈	—	43	6	no	—
145	C₄⋊Q₈	—	43	3	no	—
146	#127	—	43	3	no	—
147	#126	—	43	6	no	—
148	#125	—	43	3	no	—
149	#124	—	43	6	no	—
150	#124	—	43	3	no	—
151	#124	—	43	6	no	—
152	#123	—	43	6	no	—
153	#123	—	43	12	no	—
154	C₂²⋊Q₈	—	43	6	no	—
155	C₂²⋊Q₈	—	43	1	no	—
156	C₂²⋊Q₈	—	43	12	no	—
157	#121	—	43	3	no	—
158	#121	—	43	6	no	—
159	#121	—	43	6	no	—
160	C₄×Q₈	—	43	2	no	—
161	C₄×Q₈	—	43	3	no	—
162	C₄×Q₈	C₂×D₄₃	1	1	yes	characteristic
163	C₄×Q₈	C₂×D₄₃	1	3	yes	—
164	C₄×Q₈	C₂×D₄₃	1	1	yes	characteristic
165	C₄×Q₈	C₂×D₄₃	1	3	yes	—
166	C₄×D₄	—	43	6	no	—
167	C₄×D₄	—	43	6	no	—
168	C₄×D₄	—	43	6	no	—
169	C₄×D₄	—	43	3	no	—
170	C₄×D₄	—	43	6	no	—
171	#117	—	43	6	no	—
172	C₂×C₄⋊C₄	—	43	6	no	—
173	C₂×C₄⋊C₄	C₂×D₄₃	1	3	yes	—
174	C₂×C₄²	C₂×D₄₃	1	3	yes	—
175	#41	—	86	8	no	—
176	#41	—	172	12	no	—
177	C₂×Q₈	—	2	4	no	—
178	C₂×Q₈	—	43	1	no	—
179	C₂×Q₈	—	43	6	no	—
180	C₂×Q₈	C₂²×D₄₃	1	3	yes	—
181	C₂×Q₈	C₂²×D₄₃	1	1	yes	characteristic
182	C₂×D₄	—	43	6	no	—
183	C₂×D₄	—	43	3	no	—
184	C₂×D₄	—	43	6	no	—
185	C₂×D₄	—	43	6	no	—
186	C₂²×C₄	—	43	6	no	—
187	C₂²×C₄	—	43	12	no	—
188	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
189	C₂²×C₄	C₂²×D₄₃	1	3	yes	—
190	C₂²×C₄	C₂²×D₄₃	1	3	yes	—
191	C₄⋊C₄	—	43	6	no	—
192	C₄⋊C₄	—	43	6	no	—
193	C₄⋊C₄	—	43	2	no	—
194	C₄⋊C₄	—	43	1	no	—
195	C₄⋊C₄	—	43	3	no	—
196	C₄⋊C₄	—	43	6	no	—
197	C₄⋊C₄	—	43	6	no	—
198	C₄⋊C₄	C₂²×D₄₃	1	3	yes	—
199	C₄⋊C₄	C₂²×D₄₃	1	3	yes	—
200	C₄⋊C₄	C₂²×D₄₃	1	6	yes	—
201	C₂²⋊C₄	—	43	2	no	—
202	C₂²⋊C₄	—	43	6	no	—
203	C₂²⋊C₄	—	43	6	no	—
204	C₂²⋊C₄	—	43	6	no	—
205	C₂²⋊C₄	—	43	12	no	—
206	C₂²⋊C₄	—	43	6	no	—
207	C₄²	—	43	6	no	—
208	C₄²	C₂²×D₄₃	1	3	yes	—
209	C₄²	C₂²×D₄₃	1	3	yes	—
210	C₄²	C₂²×D₄₃	1	3	yes	—
211	C₄²	C₂²×D₄₃	1	3	yes	—
212	C₂³	—	43	6	no	—
213	C₂³	C₂³×D₄₃	1	1	yes	characteristic
214	Q₈	—	2	6	no	—
215	Q₈	—	86	2	no	—
216	Q₈	—	172	6	no	—
217	Q₈	#10621898	1	4	yes	complemented
218	D₄	—	86	12	no	—
219	D₄	—	86	12	no	—
220	D₄	—	86	6	no	—
221	D₄	—	172	6	no	—
222	C₂×C₄	—	2	6	no	—
223	C₂×C₄	—	2	6	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	43	2	no	—
226	C₂×C₄	—	43	6	no	—
227	C₂×C₄	—	43	2	no	—
228	C₂×C₄	—	86	12	no	—
229	C₂×C₄	—	172	12	no	—
230	C₂×C₄	C₂³×D₄₃	1	1	yes	characteristic
231	C₂×C₄	C₂³×D₄₃	1	3	yes	—
232	C₂×C₄	C₂³×D₄₃	1	3	yes	—
233	C₂×C₄	C₂³×D₄₃	1	3	yes	—
234	C₂×C₄	C₂³×D₄₃	1	3	yes	—
235	C₂×C₄	C₂³×D₄₃	1	1	yes	characteristic
236	C₂²	—	2	1	no	—
237	C₂²	—	2	1	no	—
238	C₂²	—	86	6	no	—
239	C₂²	—	86	6	no	—
240	C₂²	—	86	6	no	—
241	C₂²	#13063371	1	2	yes	—
242	C₂²	C₂⁴×D₄₃	1	1	yes	characteristic
243	C₄	—	2	3	no	—
244	C₄	—	2	3	no	—
245	C₄	—	2	3	no	—
246	C₄	—	2	1	no	—
247	C₄	—	86	2	no	—
248	C₄	—	86	6	no	—
249	C₄	—	86	2	no	—
250	C₄	C₂×#10621898	1	6	yes	—
251	C₄	#13063373	1	2	yes	—
252	C₂	—	2	2	no	—
253	C₂	—	172	6	no	—
254	C₂	C₂²×#10621898	1	1	yes	characteristic
255	C₂	C₂×#13063371	1	1	yes	characteristic
256	C₂	C₂×#13063373	1	1	yes	characteristic

	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	#3522
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. #14583634	C₂	semidirect
2. Q₈×Dic₈₆	C₂	semidirect
3. #14583609	C₂	semidirect
4. #14583571	C₂	semidirect
5. #14583560	C₂	semidirect
6. #14583550	C₂	semidirect
7. #14583521	C₂	semidirect
8. #14583449	C₂	semidirect
9. #14583433	C₂	semidirect
10. #14583381	C₂	semidirect
11. C₂×C₄×Q₈×C₄₃	C₂	semidirect
86. Q₈×C₄₃	#41	semidirect
139. C₂×C₄×Q₈	D₄₃	semidirect
140. C₄₃	#3522	semidirect
217. Q₈	#10621898	semidirect