Subgroups — C₄×#14582701 #25011802

Order 11008.

534 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	C₂×C₄×D₃₄₄	C₂	1	1	yes	maximal, characteristic
2	C₂×#14582701	C₂	1	2	yes	maximal
3	C₄×#13063259	C₂	1	1	yes	maximal, characteristic, complemented
4	#19132687	C₂	1	2	yes	maximal, complemented
5	C₂×C₄×C₁₆×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₃₄₄	C₂²	1	1	yes	characteristic
7	C₂×C₄×D₁₇₂	—	2	1	no	—
8	C₄×D₃₄₄	—	2	2	no	—
9	C₄×D₃₄₄	C₄	1	4	yes	complemented
10	C₂×#13063259	C₂²	1	1	yes	characteristic
11	C₂×#13063259	C₂²	1	2	yes	—
12	C₂×#13063247	—	2	1	no	—
13	C₄×C₄⋊Dic₄₃	—	2	1	no	—
14	#14582701	C₄	1	8	yes	direct product factor
15	#14582662	—	2	1	no	—
16	C₂²×C₁₆×C₄₃	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
18	C₄×C₁₆×C₄₃	—	2	2	no	—
19	C₂²×D₁₇₂	—	2	1	no	—
20	C₂²×C₄×D₄₃	—	4	1	no	—
21	C₂×D₃₄₄	—	2	1	no	—
22	C₂×D₃₄₄	—	2	2	no	—
23	C₂×D₃₄₄	C₂×C₄	1	4	yes	—
24	C₂×D₃₄₄	C₂×C₄	1	2	yes	—
25	C₄×D₁₇₂	—	2	4	no	—
26	C₄×D₁₇₂	—	4	2	no	—
27	C₂×C₄⋊Dic₄₃	—	2	1	no	—
28	C₂×C₄⋊Dic₄₃	—	2	2	no	—
29	C₂×(C₂×D₄₃)⋊C₄	—	4	1	no	—
30	C₂×C₄×Dic₄₃	—	4	1	no	—
31	#13063259	—	2	1	no	—
32	#13063259	—	2	2	no	—
33	#13063259	C₂×C₄	1	2	yes	—
34	#13063259	C₂×C₄	1	4	yes	complemented
35	#13063247	—	2	4	no	—
36	C₄₃⋊#95	—	4	1	no	—
37	C₂²×C₈×C₄₃	D₄	1	1	yes	characteristic
38	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
39	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
40	C₂×C₁₆×C₄₃	—	2	2	no	—
41	C₂×C₁₆×C₄₃	—	2	2	no	—
42	C₂×C₁₆×C₄₃	C₂×C₄	1	4	yes	complemented
43	C₄×C₈×C₄₃	C₂×C₄	1	2	yes	—
44	C₄×C₈×C₄₃	D₄	1	2	yes	—
45	C₂³×D₄₃	—	4	1	no	—
46	C₂²×C₄×C₄₃	D₈	1	1	yes	characteristic
47	C₂²×C₄×C₄₃	D₈	1	1	yes	characteristic
48	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
49	C₂²×Dic₄₃	—	4	2	no	—
50	C₂²×Dic₄₃	—	4	1	no	—
51	C₂×D₁₇₂	—	2	2	no	—
52	C₂×D₁₇₂	—	2	4	no	—
53	C₂×D₁₇₂	—	4	1	no	—
54	C₂×D₁₇₂	—	4	2	no	—
55	C₂×C₄×D₄₃	—	4	4	no	—
56	C₂×C₄×D₄₃	—	8	2	no	—
57	C₂×C₄×D₄₃	—	8	2	no	—
58	C₂×C₈×C₄₃	—	2	2	no	—
59	C₂×C₈×C₄₃	C₂²⋊C₄	1	2	yes	—
60	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
62	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
63	C₂×C₈×C₄₃	#41	1	2	yes	—
64	C₄²×C₄₃	C₂²⋊C₄	1	2	yes	—
65	C₄²×C₄₃	SD₁₆	1	2	yes	—
66	(C₂×D₄₃)⋊C₄	—	4	4	no	—
67	C₄⋊Dic₄₃	—	2	2	no	—
68	C₄⋊Dic₄₃	—	2	4	no	—
69	C₄⋊Dic₄₃	—	4	1	no	—
70	C₄⋊Dic₄₃	—	4	2	no	—
71	C₄×Dic₄₃	—	8	2	no	—
72	D₃₄₄	—	2	2	no	—
73	D₃₄₄	—	2	4	no	—
74	D₃₄₄	C₄²	1	4	yes	complemented
75	C₁₆×C₄₃	—	2	4	no	—
76	C₂³×C₄₃	C₂×D₈	1	1	yes	characteristic
77	C₂²×D₄₃	—	4	2	no	—
78	C₂²×D₄₃	—	4	4	no	—
79	C₂²×D₄₃	—	8	1	no	—
80	C₂²×D₄₃	—	8	1	no	—
81	C₂²×D₄₃	—	8	2	no	—
82	C₂×C₄×C₄₃	—	2	2	no	—
83	C₂×C₄×C₄₃	#102	1	2	yes	—
84	C₂×C₄×C₄₃	#102	1	2	yes	—
85	C₂×C₄×C₄₃	D₁₆	1	2	yes	—
86	C₂×C₄×C₄₃	SD₃₂	1	2	yes	—
87	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
88	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
89	C₂×C₄×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
90	C₂×C₄×C₄₃	#135	1	2	yes	—
91	C₂×Dic₄₃	—	4	2	no	—
92	C₂×Dic₄₃	—	4	4	no	—
93	C₂×Dic₄₃	—	8	2	no	—
94	C₂×Dic₄₃	—	8	1	no	—
95	C₂×Dic₄₃	—	8	1	no	—
96	C₂×Dic₄₃	—	8	2	no	—
97	D₁₇₂	—	2	4	no	—
98	D₁₇₂	—	4	2	no	—
99	D₁₇₂	—	4	4	no	—
100	C₄×D₄₃	—	8	8	no	—
101	C₈×C₄₃	—	2	2	no	—
102	C₈×C₄₃	C₂×C₄²	1	1	yes	characteristic
103	C₈×C₄₃	#117	1	1	yes	characteristic
104	C₈×C₄₃	C₄×D₄	1	2	yes	—
105	C₄×#357	—	43	1	no	maximal, Sylow
106	C₂²×C₄₃	C₂×#102	1	1	yes	characteristic
107	C₂²×C₄₃	C₄×D₈	1	2	yes	—
108	C₂²×C₄₃	C₂×D₁₆	1	1	yes	characteristic
109	C₂²×C₄₃	C₂×SD₃₂	1	1	yes	characteristic
110	C₂²×C₄₃	#508	1	2	yes	—
111	C₂×D₄₃	—	4	4	no	—
112	C₂×D₄₃	—	8	2	no	—
113	C₂×D₄₃	—	8	2	no	—
114	C₂×D₄₃	—	8	4	no	—
115	C₂×D₄₃	—	8	4	no	—
116	C₄×C₄₃	—	2	2	no	—
117	C₄×C₄₃	#357	1	4	yes	complemented
118	C₄×C₄₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
119	C₄×C₄₃	#416	1	1	yes	characteristic
120	C₄×C₄₃	C₄×SD₁₆	1	2	yes	—
121	Dic₄₃	—	8	2	no	—
122	Dic₄₃	—	8	4	no	—
123	C₂×C₄×D₈	—	43	1	no	—
124	C₂×#357	—	43	2	no	—
125	C₂×C₄×C₁₆	D₄₃	1	1	yes	characteristic, complemented
126	C₄×#107	—	43	1	no	—
127	#1380	—	43	2	no	—
128	C₂×C₄₃	C₄×#102	1	1	yes	characteristic
129	C₂×C₄₃	C₂×#357	1	1	yes	characteristic
130	C₂×C₄₃	#2138	1	1	yes	characteristic
131	C₂×C₄₃	C₄×D₁₆	1	2	yes	—
132	C₂×C₄₃	C₄×SD₃₂	1	2	yes	—
133	D₄₃	—	8	4	no	—
134	C₂²×D₈	—	43	1	no	—
135	C₂×C₄×D₄	—	86	1	no	—
136	C₂²×C₁₆	C₂×D₄₃	1	2	yes	—
137	C₄×D₈	—	43	4	no	—
138	C₄×D₈	—	86	2	no	—
139	C₂×#107	—	43	2	no	—
140	C₂×#107	—	43	1	no	—
141	C₂×#102	—	86	1	no	—
142	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
143	C₄×C₄⋊C₄	—	86	1	no	—
144	#357	—	43	8	no	—
145	C₄×C₁₆	—	2	2	no	—
146	#340	—	86	1	no	—
147	C₄₃	C₄×#357	1	1	yes	characteristic, complemented, Sylow
148	C₂²×D₄	—	86	1	no	—
149	C₂³×C₄	—	172	1	no	—
150	C₂×D₈	—	43	4	no	—
151	C₂×D₈	—	43	2	no	—
152	C₂×D₈	—	86	2	no	—
153	C₂×D₈	—	86	1	no	—
154	C₂²×C₈	D₁₇₂	1	1	yes	characteristic
155	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
156	C₄×D₄	—	86	4	no	—
157	C₄×D₄	—	172	2	no	—
158	C₂×C₄⋊C₄	—	86	1	no	—
159	C₂×C₄⋊C₄	—	86	2	no	—
160	C₂×C₂²⋊C₄	—	172	1	no	—
161	C₂×C₄²	—	172	1	no	—
162	C₂×C₄²	D₁₇₂	1	1	yes	characteristic
163	C₂×C₁₆	—	2	2	no	—
164	C₂×C₁₆	—	2	2	no	—
165	C₂×C₁₆	C₄×D₄₃	1	4	yes	complemented
166	#107	—	43	2	no	—
167	#107	—	43	4	no	—
168	#107	—	86	2	no	—
169	#107	—	86	1	no	—
170	#102	—	86	4	no	—
171	C₄×C₈	C₄×D₄₃	1	2	yes	—
172	C₄×C₈	C₂²⋊D₄₃	1	2	yes	—
173	#95	—	172	1	no	—
174	C₂⁴	—	172	1	no	—
175	C₂×D₄	—	86	2	no	—
176	C₂×D₄	—	86	4	no	—
177	C₂×D₄	—	172	1	no	—
178	C₂×D₄	—	172	2	no	—
179	C₂²×C₄	—	344	2	no	—
180	C₂²×C₄	—	344	2	no	—
181	C₂²×C₄	—	172	2	no	—
182	C₂²×C₄	—	172	1	no	—
183	C₂²×C₄	—	172	4	no	—
184	C₂²×C₄	D₃₄₄	1	1	yes	characteristic
185	C₂²×C₄	D₃₄₄	1	1	yes	characteristic
186	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
187	D₈	—	43	4	no	—
188	D₈	—	86	2	no	—
189	D₈	—	86	4	no	—
190	C₂×C₈	—	2	2	no	—
191	C₂×C₈	(C₂×D₄₃)⋊C₄	1	2	yes	—
192	C₂×C₈	C₂×C₄×D₄₃	1	1	yes	characteristic
193	C₂×C₈	C₂×C₄×D₄₃	1	2	yes	—
194	C₂×C₈	#10621898	1	2	yes	—
195	C₂×C₈	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
196	C₄⋊C₄	—	86	2	no	—
197	C₄⋊C₄	—	86	4	no	—
198	C₄⋊C₄	—	172	1	no	—
199	C₄⋊C₄	—	172	2	no	—
200	C₂²⋊C₄	—	172	4	no	—
201	C₄²	—	344	2	no	—
202	C₄²	#10621873	1	2	yes	—
203	C₄²	(C₂×D₄₃)⋊C₄	1	2	yes	—
204	C₁₆	—	2	4	no	—
205	C₂³	—	344	1	no	—
206	C₂³	—	344	1	no	—
207	C₂³	—	344	2	no	—
208	C₂³	—	172	2	no	—
209	C₂³	—	172	4	no	—
210	C₂³	C₂×D₃₄₄	1	1	yes	characteristic
211	D₄	—	86	4	no	—
212	D₄	—	172	2	no	—
213	D₄	—	172	4	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	344	2	no	—
216	C₂×C₄	—	344	1	no	—
217	C₂×C₄	—	344	1	no	—
218	C₂×C₄	—	344	2	no	—
219	C₂×C₄	—	344	8	no	—
220	C₂×C₄	—	172	2	no	—
221	C₂×C₄	—	172	4	no	—
222	C₂×C₄	#13063247	1	2	yes	—
223	C₂×C₄	#13063247	1	2	yes	—
224	C₂×C₄	D₆₈₈	1	2	yes	—
225	C₂×C₄	#13063270	1	2	yes	—
226	C₂×C₄	C₂×(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
227	C₂×C₄	C₄×D₁₇₂	1	2	yes	—
228	C₂×C₄	C₂×#10621873	1	1	yes	characteristic
229	C₂×C₄	#13063347	1	2	yes	—
230	C₈	—	2	2	no	—
231	C₈	C₄²×D₄₃	1	1	yes	characteristic
232	C₈	#13063282	1	1	yes	characteristic
233	C₈	C₄×C₂²⋊D₄₃	1	2	yes	—
234	C₂²	—	344	2	no	—
235	C₂²	—	344	2	no	—
236	C₂²	—	344	4	no	—
237	C₂²	—	344	4	no	—
238	C₂²	—	172	4	no	—
239	C₂²	C₂×#13063247	1	1	yes	characteristic
240	C₂²	C₄×D₃₄₄	1	2	yes	—
241	C₂²	C₂×D₆₈₈	1	1	yes	characteristic
242	C₂²	C₂×#13063270	1	1	yes	characteristic
243	C₂²	#14583331	1	2	yes	—
244	C₄	—	2	2	no	—
245	C₄	—	344	2	no	—
246	C₄	—	344	4	no	—
247	C₄	#14582701	1	4	yes	direct product factor
248	C₄	C₄×(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
249	C₄	#14582896	1	1	yes	characteristic
250	C₄	C₄×#10621873	1	2	yes	—
251	C₂	—	344	4	no	—
252	C₂	C₄×#13063247	1	1	yes	characteristic
253	C₂	C₂×#14582701	1	1	yes	characteristic
254	C₂	#19136303	1	1	yes	characteristic
255	C₂	C₄×D₆₈₈	1	2	yes	—
256	C₂	C₄×#13063270	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	D₃₄₄
commutator	C₈×C₄₃	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₄₃
Fitting	C₂×C₄×C₁₆×C₄₃	C₂
socle	C₂³×C₄₃	C₂×D₈
prime radical	C₂²×D₃₄₄	C₂²
cosocle	C₂²×C₈×C₄₃	C₂³
hypercenter	C₂×C₄×C₁₆	D₄₃
hypocenter	C₄₃	C₄×#357
LCS	G ▷ C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
3. C₄×#13063259	C₂	semidirect
4. #19132687	C₂	semidirect
5. C₂×C₄×C₁₆×C₄₃	C₂	semidirect
9. C₄×D₃₄₄	C₄	semidirect
14. #14582701	C₄	direct (N × H)
34. #13063259	C₂×C₄	semidirect
42. C₂×C₁₆×C₄₃	C₂×C₄	semidirect
74. D₃₄₄	C₄²	semidirect
117. C₄×C₄₃	#357	semidirect
125. C₂×C₄×C₁₆	D₄₃	semidirect
147. C₄₃	C₄×#357	semidirect
165. C₂×C₁₆	C₄×D₄₃	semidirect
247. C₄	#14582701	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₄×D₃₄₄	2
C₂×#14582701	2
C₄×#13063259	2
#19132687	2
C₂×C₄×C₁₆×C₄₃	2
C₄×#357	43

Subgroups — C4×#14582701 #25011802

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#14582701 #25011802