Subgroups — C₄×#14582711 #25011853

Order 11008.

498 conjugacy classes of subgroups.

The subgroup lattice has 274 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×#14582711	C₂	1	2	yes	maximal
3	C₄×#13063265	C₂	1	1	yes	maximal, characteristic, complemented
4	#19132705	C₂	1	2	yes	maximal, complemented
5	C₄×#108×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₃₄₄	C₂²	1	1	yes	characteristic
7	C₂×C₄×D₁₇₂	—	2	1	no	—
8	C₂×#13063265	C₂²	1	2	yes	—
9	C₄×D₃₄₄	—	2	1	no	—
10	C₄×D₃₄₄	—	2	1	no	—
11	C₄×D₃₄₄	C₄	1	2	yes	—
12	C₄×D₃₄₄	C₄	1	2	yes	—
13	C₂×#13063259	C₂²	1	1	yes	characteristic
14	C₂×#13063247	—	2	1	no	—
15	#14582711	C₄	1	8	yes	direct product factor
16	#14582675	—	2	1	no	—
17	C₄×C₄₃⋊C₁₆	—	2	1	no	—
18	C₂×#108×C₄₃	C₂²	1	2	yes	—
19	C₄×M₁₆×C₄₃	—	2	1	no	—
20	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
21	C₄₃×#341	—	2	1	no	—
22	C₂²×D₁₇₂	—	2	1	no	—
23	C₂²×C₄×D₄₃	—	4	1	no	—
24	C₂×D₃₄₄	—	2	1	no	—
25	C₂×D₃₄₄	—	2	2	no	—
26	C₂×D₃₄₄	C₂×C₄	1	4	yes	—
27	C₂×D₃₄₄	C₂×C₄	1	2	yes	—
28	C₄×D₁₇₂	—	2	2	no	—
29	C₄×D₁₇₂	—	2	2	no	—
30	C₄×D₁₇₂	—	4	1	no	—
31	C₄×D₁₇₂	—	4	1	no	—
32	C₂×C₄⋊Dic₄₃	—	2	1	no	—
33	C₂×(C₂×D₄₃)⋊C₄	—	4	1	no	—
34	#13063265	—	2	2	no	—
35	#13063265	C₂×C₄	1	4	yes	complemented
36	C₂×C₄₃⋊C₁₆	—	2	2	no	—
37	#13063259	—	2	1	no	—
38	#13063259	C₂×C₄	1	2	yes	—
39	#13063247	—	2	2	no	—
40	#13063247	—	2	2	no	—
41	C₂×M₁₆×C₄₃	—	2	2	no	—
42	C₂²×C₈×C₄₃	D₄	1	1	yes	characteristic
43	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
44	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
45	#108×C₄₃	—	2	2	no	—
46	#108×C₄₃	C₂×C₄	1	4	yes	complemented
47	#97×C₄₃	—	4	1	no	—
48	C₄×C₈×C₄₃	—	4	1	no	—
49	C₄×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
50	C₄×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
51	C₄×C₈×C₄₃	D₄	1	1	yes	characteristic
52	C₄×C₈×C₄₃	D₄	1	1	yes	characteristic
53	C₂³×D₄₃	—	4	1	no	—
54	C₂²×C₄×C₄₃	SD₁₆	1	1	yes	characteristic
55	C₂²×C₄×C₄₃	SD₁₆	1	1	yes	characteristic
56	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
57	C₂²×Dic₄₃	—	4	1	no	—
58	C₂×D₁₇₂	—	2	2	no	—
59	C₂×D₁₇₂	—	2	4	no	—
60	C₂×D₁₇₂	—	4	1	no	—
61	C₂×D₁₇₂	—	4	2	no	—
62	C₂×C₄×D₄₃	—	4	2	no	—
63	C₂×C₄×D₄₃	—	4	2	no	—
64	C₂×C₄×D₄₃	—	8	4	no	—
65	M₁₆×C₄₃	—	2	4	no	—
66	M₁₆×C₄₃	—	4	2	no	—
67	C₂×C₈×C₄₃	—	2	2	no	—
68	C₂×C₈×C₄₃	—	4	2	no	—
69	C₂×C₈×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
70	C₂×C₈×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
71	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
73	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
74	C₂×C₈×C₄₃	#41	1	2	yes	—
75	C₄²×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
76	C₄²×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
77	C₄²×C₄₃	D₈	1	1	yes	characteristic
78	C₄²×C₄₃	D₈	1	1	yes	characteristic
79	(C₂×D₄₃)⋊C₄	—	4	2	no	—
80	(C₂×D₄₃)⋊C₄	—	4	2	no	—
81	C₄⋊Dic₄₃	—	2	2	no	—
82	C₄⋊Dic₄₃	—	4	1	no	—
83	D₃₄₄	—	2	2	no	—
84	D₃₄₄	—	2	4	no	—
85	D₃₄₄	C₄²	1	4	yes	—
86	C₄₃⋊C₁₆	—	2	4	no	—
87	C₂³×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
88	C₂²×D₄₃	—	4	2	no	—
89	C₂²×D₄₃	—	4	4	no	—
90	C₂²×D₄₃	—	8	2	no	—
91	C₂²×D₄₃	—	8	2	no	—
92	C₂×C₄×C₄₃	—	2	2	no	—
93	C₂×C₄×C₄₃	—	4	1	no	—
94	C₂×C₄×C₄₃	#102	1	1	yes	characteristic
95	C₂×C₄×C₄₃	#102	1	1	yes	characteristic
96	C₂×C₄×C₄₃	#102	1	1	yes	characteristic
97	C₂×C₄×C₄₃	#102	1	1	yes	characteristic
98	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
99	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
100	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
101	C₂×C₄×C₄₃	#135	1	2	yes	—
102	C₂×Dic₄₃	—	4	2	no	—
103	C₂×Dic₄₃	—	8	2	no	—
104	D₁₇₂	—	2	4	no	—
105	D₁₇₂	—	4	2	no	—
106	D₁₇₂	—	4	4	no	—
107	C₄×D₄₃	—	8	4	no	—
108	C₄×D₄₃	—	8	4	no	—
109	C₈×C₄₃	—	2	1	no	—
110	C₈×C₄₃	—	2	1	no	—
111	C₈×C₄₃	—	4	4	no	—
112	C₈×C₄₃	C₂×C₄²	1	1	yes	characteristic
113	C₈×C₄₃	#117	1	1	yes	characteristic
114	C₈×C₄₃	C₄×D₄	1	2	yes	—
115	C₄×#361	—	43	1	no	maximal, Sylow
116	C₂²×C₄₃	—	2	1	no	—
117	C₂²×C₄₃	—	2	1	no	—
118	C₂²×C₄₃	C₂×#102	1	1	yes	characteristic
119	C₂²×C₄₃	C₄×SD₁₆	1	2	yes	—
120	C₂×D₄₃	—	4	4	no	—
121	C₂×D₄₃	—	8	8	no	—
122	C₂×D₄₃	—	8	2	no	—
123	C₂×D₄₃	—	8	2	no	—
124	C₄×C₄₃	—	2	1	no	—
125	C₄×C₄₃	—	2	1	no	—
126	C₄×C₄₃	—	2	1	no	—
127	C₄×C₄₃	#361	1	2	yes	complemented
128	C₄×C₄₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
129	C₄×C₄₃	#416	1	1	yes	characteristic
130	C₄×C₄₃	C₄×D₈	1	2	yes	—
131	Dic₄₃	—	8	2	no	—
132	C₂×C₄×D₈	—	43	1	no	—
133	C₂×#361	—	43	2	no	—
134	C₄×M₃₂	—	43	1	no	—
135	C₄×#108	D₄₃	1	1	yes	characteristic, complemented
136	#1387	—	43	2	no	—
137	C₂×C₄₃	—	2	2	no	—
138	C₂×C₄₃	C₄×#102	1	1	yes	characteristic
139	C₂×C₄₃	C₂×#361	1	1	yes	characteristic
140	C₂×C₄₃	#2148	1	1	yes	characteristic
141	D₄₃	—	8	4	no	—
142	C₂²×D₈	—	43	1	no	—
143	C₂×C₄×D₄	—	86	1	no	—
144	C₂×M₃₂	—	43	2	no	—
145	C₄×D₈	—	43	2	no	—
146	C₄×D₈	—	43	2	no	—
147	C₄×D₈	—	86	1	no	—
148	C₄×D₈	—	86	1	no	—
149	C₂×#108	C₂×D₄₃	1	2	yes	—
150	C₂×#107	—	43	1	no	—
151	C₂×#102	—	86	1	no	—
152	C₄×M₁₆	—	2	1	no	—
153	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
154	#361	—	43	8	no	—
155	#346	—	86	1	no	—
156	C₄×C₁₆	—	86	1	no	—
157	#341	—	2	1	no	—
158	C₄₃	C₄×#361	1	1	yes	characteristic, complemented, Sylow
159	C₂²×D₄	—	86	1	no	—
160	C₂³×C₄	—	172	1	no	—
161	C₂×D₈	—	43	4	no	—
162	C₂×D₈	—	43	2	no	—
163	C₂×D₈	—	86	1	no	—
164	C₂×D₈	—	86	2	no	—
165	C₂×M₁₆	—	2	2	no	—
166	C₂²×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
167	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
168	C₄×D₄	—	86	2	no	—
169	C₄×D₄	—	86	2	no	—
170	C₄×D₄	—	172	1	no	—
171	C₄×D₄	—	172	1	no	—
172	C₂×C₄⋊C₄	—	86	1	no	—
173	C₂×C₂²⋊C₄	—	172	1	no	—
174	C₂×C₄²	C₂²⋊D₄₃	1	1	yes	characteristic
175	M₃₂	—	43	4	no	—
176	M₃₂	—	86	2	no	—
177	C₂×C₁₆	—	86	2	no	—
178	#108	—	2	2	no	—
179	#108	C₄×D₄₃	1	4	yes	complemented
180	#107	—	43	2	no	—
181	#107	—	86	1	no	—
182	#102	—	86	2	no	—
183	#102	—	86	2	no	—
184	#97	—	4	1	no	—
185	C₄×C₈	—	4	1	no	—
186	C₄×C₈	C₄×D₄₃	1	1	yes	characteristic
187	C₄×C₈	C₄×D₄₃	1	1	yes	characteristic
188	C₄×C₈	D₁₇₂	1	1	yes	characteristic
189	C₄×C₈	D₁₇₂	1	1	yes	characteristic
190	C₂⁴	—	172	1	no	—
191	C₂×D₄	—	86	2	no	—
192	C₂×D₄	—	86	4	no	—
193	C₂×D₄	—	172	1	no	—
194	C₂×D₄	—	172	2	no	—
195	C₂²×C₄	—	344	4	no	—
196	C₂²×C₄	—	172	1	no	—
197	C₂²×C₄	—	172	2	no	—
198	C₂²×C₄	—	172	2	no	—
199	C₂²×C₄	#10621884	1	1	yes	characteristic
200	C₂²×C₄	#10621884	1	1	yes	characteristic
201	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
202	D₈	—	43	4	no	—
203	D₈	—	86	2	no	—
204	D₈	—	86	4	no	—
205	M₁₆	—	2	4	no	—
206	M₁₆	—	4	2	no	—
207	C₂×C₈	—	2	2	no	—
208	C₂×C₈	—	4	2	no	—
209	C₂×C₈	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
210	C₂×C₈	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
211	C₂×C₈	C₂×C₄×D₄₃	1	1	yes	characteristic
212	C₂×C₈	C₂×C₄×D₄₃	1	2	yes	—
213	C₂×C₈	C₂×D₁₇₂	1	1	yes	characteristic
214	C₂×C₈	#10621898	1	2	yes	—
215	C₄⋊C₄	—	86	2	no	—
216	C₄⋊C₄	—	172	1	no	—
217	C₂²⋊C₄	—	172	2	no	—
218	C₂²⋊C₄	—	172	2	no	—
219	C₄²	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
220	C₄²	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
221	C₄²	#10621882	1	1	yes	characteristic
222	C₄²	#10621882	1	1	yes	characteristic
223	C₁₆	—	86	4	no	—
224	C₂³	—	344	2	no	—
225	C₂³	—	344	2	no	—
226	C₂³	—	172	2	no	—
227	C₂³	—	172	4	no	—
228	C₂³	C₂×#10621884	1	1	yes	characteristic
229	D₄	—	86	4	no	—
230	D₄	—	172	2	no	—
231	D₄	—	172	4	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	4	1	no	—
234	C₂×C₄	—	344	4	no	—
235	C₂×C₄	—	344	2	no	—
236	C₂×C₄	—	344	4	no	—
237	C₂×C₄	—	172	2	no	—
238	C₂×C₄	#13063245	1	1	yes	characteristic
239	C₂×C₄	#13063245	1	1	yes	characteristic
240	C₂×C₄	#13063245	1	1	yes	characteristic
241	C₂×C₄	#13063245	1	1	yes	characteristic
242	C₂×C₄	C₂×(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
243	C₂×C₄	C₄×C₂²⋊D₄₃	1	2	yes	—
244	C₂×C₄	C₂×#10621882	1	1	yes	characteristic
245	C₂×C₄	#13063345	1	2	yes	—
246	C₈	—	2	1	no	—
247	C₈	—	2	1	no	—
248	C₈	—	4	4	no	—
249	C₈	C₄²×D₄₃	1	1	yes	characteristic
250	C₈	#13063282	1	1	yes	characteristic
251	C₈	C₄×D₁₇₂	1	2	yes	—
252	C₂²	—	2	1	no	—
253	C₂²	—	2	1	no	—
254	C₂²	—	344	8	no	—
255	C₂²	—	344	2	no	—
256	C₂²	—	344	2	no	—
257	C₂²	—	172	4	no	—
258	C₂²	C₂×#13063245	1	1	yes	characteristic
259	C₂²	C₄×#10621884	1	2	yes	—
260	C₄	—	2	1	no	—
261	C₄	—	2	1	no	—
262	C₄	—	2	1	no	—
263	C₄	—	344	2	no	—
264	C₄	#14582711	1	2	yes	direct product factor
265	C₄	C₄×(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
266	C₄	#14582893	1	1	yes	characteristic
267	C₄	C₄×#10621882	1	2	yes	—
268	C₂	—	2	2	no	—
269	C₂	—	344	4	no	—
270	C₂	C₄×#13063245	1	1	yes	characteristic
271	C₂	C₂×#14582711	1	1	yes	characteristic
272	C₂	#19136342	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#13063245
commutator	C₈×C₄₃	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₄₃
Fitting	C₄×#108×C₄₃	C₂
socle	C₂²×C₄₃	C₂×#102
prime radical	C₂²×D₃₄₄	C₂²
cosocle	C₂²×C₈×C₄₃	C₂³
hypercenter	C₄×#108	D₄₃
hypocenter	C₄₃	C₄×#361
LCS	G ▷ C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×#108

Complements

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

N	H ≅ G/N	type
3. C₄×#13063265	C₂	semidirect
4. #19132705	C₂	semidirect
5. C₄×#108×C₄₃	C₂	semidirect
15. #14582711	C₄	direct (N × H)
35. #13063265	C₂×C₄	semidirect
46. #108×C₄₃	C₂×C₄	semidirect
127. C₄×C₄₃	#361	semidirect
135. C₄×#108	D₄₃	semidirect
158. C₄₃	C₄×#361	semidirect
179. #108	C₄×D₄₃	semidirect
264. C₄	#14582711	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×#14582711	2
C₄×#13063265	2
#19132705	2
C₄×#108×C₄₃	2
C₄×#361	43

Subgroups — C4×#14582711 #25011853

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#14582711 #25011853