Subgroups — C₄×#10625080 #15682220

Order 2944.

484 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#10625080	C₂	1	1	yes	maximal, characteristic
2	C₄×#67288	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×Dic₄₆	C₂	1	1	yes	maximal, characteristic, complemented
4	#13092525	C₂	1	2	yes	maximal, complemented
5	#13092519	C₂	1	2	yes	maximal, complemented
6	C₄×Dic₉₂	C₂	1	2	yes	maximal, complemented
7	C₄×#67263	C₂	1	2	yes	maximal, complemented
8	#13092459	C₂	1	1	yes	maximal, characteristic, complemented
9	#13092427	C₂	1	1	yes	maximal, characteristic, complemented
10	#13092419	C₂	1	1	yes	maximal, characteristic
11	C₄×M₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#67288	C₂²	1	1	yes	characteristic
13	C₂²×Dic₄₆	C₂²	1	1	yes	characteristic
14	C₂×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
15	C₂×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
16	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
17	C₄×C₂²⋊D₂₃	—	2	1	no	—
18	C₄×C₂²⋊D₂₃	—	2	1	no	—
19	#10625097	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊Dic₂₃	C₂²	1	1	yes	characteristic
21	C₂×Dic₂₃⋊C₄	—	2	1	no	—
22	C₂×C₄×Dic₂₃	—	2	1	no	—
23	#10625080	C₄	1	8	yes	direct product factor
24	C₂×Dic₉₂	C₂²	1	2	yes	—
25	C₂×#67263	C₂²	1	2	yes	—
26	C₄×D₉₂	C₂²	1	1	yes	characteristic
27	C₄×D₉₂	C₂²	1	1	yes	characteristic
28	#10625043	—	2	1	no	—
29	C₄²×D₂₃	—	2	1	no	—
30	C₄×Dic₄₆	—	2	1	no	—
31	C₄×Dic₄₆	—	2	1	no	—
32	C₄×Dic₄₆	C₂²	1	2	yes	—
33	C₄×Dic₄₆	C₂²	1	2	yes	—
34	C₄×Dic₄₆	C₂²	1	1	yes	characteristic
35	C₄×Dic₄₆	C₂²	1	1	yes	characteristic
36	C₂₃×#97	C₂²	1	2	yes	—
37	C₄×C₈×C₂₃	C₂²	1	2	yes	—
38	#10625003	C₂²	1	2	yes	—
39	#10625000	C₂²	1	2	yes	—
40	#10624999	C₂²	1	2	yes	—
41	#10624998	C₂²	1	2	yes	—
42	#10624998	C₂²	1	2	yes	—
43	#10624998	C₂²	1	2	yes	—
44	C₂²×C₄×C₂₃	D₄	1	1	yes	characteristic
45	C₂²×C₄×C₂₃	D₄	1	1	yes	characteristic
46	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
47	C₂×C₂²⋊D₂₃	—	2	1	no	—
48	C₂²×Dic₂₃	—	2	1	no	—
49	C₂²×Dic₂₃	—	2	1	no	—
50	#67288	—	2	2	no	—
51	#67288	C₂×C₄	1	4	yes	—
52	C₂×D₉₂	C₂³	1	1	yes	characteristic
53	C₂×C₄×D₂₃	—	2	1	no	—
54	C₂×C₄×D₂₃	—	2	1	no	—
55	C₂×C₄×D₂₃	—	2	1	no	—
56	C₂×Dic₄₆	—	2	1	no	—
57	C₂×Dic₄₆	—	2	2	no	—
58	C₂×Dic₄₆	C₂×C₄	1	4	yes	complemented
59	C₂×Dic₄₆	C₂³	1	2	yes	—
60	C₂×Dic₄₆	C₂³	1	1	yes	characteristic
61	M₁₆×C₂₃	—	2	2	no	—
62	M₁₆×C₂₃	C₂×C₄	1	4	yes	complemented
63	C₂×C₈×C₂₃	C₂³	1	2	yes	—
64	C₂×C₈×C₂₃	C₂³	1	2	yes	—
65	C₄²×C₂₃	D₄	1	1	yes	characteristic
66	C₄²×C₂₃	D₄	1	1	yes	characteristic
67	C₄²×C₂₃	C₂³	1	1	yes	characteristic
68	C₄²×C₂₃	C₂³	1	1	yes	characteristic
69	C₂²⋊Dic₂₃	—	2	1	no	—
70	(C₂×D₂₃)⋊C₄	—	2	1	no	—
71	(C₂×D₂₃)⋊C₄	—	2	1	no	—
72	C₄⋊Dic₂₃	—	2	1	no	—
73	C₄⋊Dic₂₃	C₂³	1	2	yes	—
74	C₄⋊Dic₂₃	C₂³	1	1	yes	characteristic
75	C₄⋊Dic₂₃	C₂³	1	1	yes	characteristic
76	Dic₂₃⋊C₄	—	2	2	no	—
77	Dic₂₃⋊C₄	—	2	2	no	—
78	Dic₂₃⋊C₄	—	2	1	no	—
79	Dic₂₃⋊C₄	—	2	1	no	—
80	C₄×Dic₂₃	—	2	2	no	—
81	C₄×Dic₂₃	—	2	2	no	—
82	C₄×Dic₂₃	—	2	1	no	—
83	C₄×Dic₂₃	—	2	1	no	—
84	C₄×Dic₂₃	—	2	1	no	—
85	Dic₉₂	C₂×C₄	1	8	yes	complemented
86	#67263	C₂×C₄	1	8	yes	complemented
87	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
88	C₂²×D₂₃	—	2	1	no	—
89	C₂×C₄×C₂₃	—	2	2	no	—
90	C₂×C₄×C₂₃	—	4	1	no	—
91	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
92	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
93	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
94	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
95	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
96	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
97	C₂×C₄×C₂₃	#41	1	2	yes	—
98	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
99	C₂²⋊D₂₃	—	2	4	no	—
100	C₂×Dic₂₃	—	2	4	no	—
101	C₂×Dic₂₃	—	2	2	no	—
102	C₂×Dic₂₃	—	2	1	no	—
103	C₂×Dic₂₃	—	2	2	no	—
104	C₂×Dic₂₃	—	2	1	no	—
105	C₂×Dic₂₃	—	2	1	no	—
106	C₂×Dic₂₃	—	4	2	no	—
107	D₉₂	—	2	1	no	—
108	D₉₂	C₂²×C₄	1	2	yes	—
109	C₄×D₂₃	—	2	4	no	—
110	C₄×D₂₃	—	4	2	no	—
111	C₄×D₂₃	—	4	2	no	—
112	Dic₄₆	—	2	4	no	—
113	Dic₄₆	—	2	2	no	—
114	Dic₄₆	—	2	1	no	—
115	Dic₄₆	C₂²×C₄	1	4	yes	—
116	Dic₄₆	C₂²×C₄	1	2	yes	—
117	C₈×C₂₃	—	2	2	no	—
118	C₈×C₂₃	C₂²×C₄	1	4	yes	—
119	C₄×#137	—	23	1	no	maximal, Sylow
120	C₂²×C₂₃	—	2	1	no	—
121	C₂²×C₂₃	—	2	1	no	—
122	C₂²×C₂₃	C₄×D₄	1	2	yes	—
123	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
124	C₂×D₂₃	—	2	2	no	—
125	C₂×D₂₃	—	4	1	no	—
126	C₂×D₂₃	—	4	1	no	—
127	C₄×C₂₃	—	2	1	no	—
128	C₄×C₂₃	—	2	1	no	—
129	C₄×C₂₃	—	2	1	no	—
130	C₄×C₂₃	C₄×D₄	1	2	yes	—
131	C₄×C₂₃	#137	1	2	yes	complemented
132	C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
133	C₄×C₂₃	C₂×#41	1	1	yes	characteristic
134	Dic₂₃	—	2	4	no	—
135	Dic₂₃	—	2	2	no	—
136	Dic₂₃	—	4	1	no	—
137	Dic₂₃	—	4	1	no	—
138	Dic₂₃	—	4	2	no	—
139	C₂×#137	—	23	1	no	—
140	C₄×#41	—	23	1	no	—
141	C₂×C₄×Q₈	—	23	1	no	—
142	#441	—	23	2	no	—
143	#440	—	23	2	no	—
144	C₄×Q₁₆	—	23	2	no	—
145	C₄×SD₁₆	—	23	2	no	—
146	#428	—	23	1	no	—
147	#419	—	23	1	no	—
148	#417	—	23	1	no	—
149	C₄×M₁₆	D₂₃	1	1	yes	characteristic, complemented
150	C₂×C₂₃	—	2	2	no	—
151	C₂×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
152	C₂×C₂₃	C₂×#137	1	1	yes	characteristic
153	C₂×C₂₃	#575	1	1	yes	characteristic
154	D₂₃	—	4	2	no	—
155	C₂×#41	—	23	1	no	—
156	C₂²×Q₈	—	23	1	no	—
157	#137	—	23	8	no	—
158	C₂×Q₁₆	—	23	2	no	—
159	C₂×SD₁₆	—	23	2	no	—
160	C₂×M₁₆	C₂×D₂₃	1	1	yes	characteristic
161	C₂×M₁₆	C₂×D₂₃	1	1	yes	characteristic
162	C₄×Q₈	—	23	2	no	—
163	C₄×Q₈	—	23	2	no	—
164	C₄×Q₈	—	23	1	no	—
165	C₄×Q₈	—	23	1	no	—
166	C₄×Q₈	—	46	1	no	—
167	C₄×Q₈	—	46	1	no	—
168	C₄×D₄	—	23	1	no	—
169	C₄×D₄	—	23	1	no	—
170	C₄×D₄	—	46	1	no	—
171	C₄×D₄	—	46	1	no	—
172	#117	—	23	1	no	—
173	#117	—	46	1	no	—
174	C₂×C₄⋊C₄	—	23	1	no	—
175	C₂×C₄⋊C₄	—	46	1	no	—
176	C₂×C₄²	—	46	1	no	—
177	C₂×C₄²	—	46	1	no	—
178	C₂×C₄²	C₂×D₂₃	1	1	yes	characteristic
179	#107	—	23	2	no	—
180	#106	—	23	2	no	—
181	#103	—	23	2	no	—
182	#103	—	23	2	no	—
183	#103	—	23	2	no	—
184	#102	—	23	2	no	—
185	#97	C₂×D₂₃	1	2	yes	—
186	C₄×C₈	C₂×D₂₃	1	2	yes	—
187	C₂₃	C₄×#137	1	1	yes	characteristic, complemented, Sylow
188	#41	—	23	4	no	—
189	#41	—	46	2	no	—
190	C₂×Q₈	—	23	4	no	—
191	C₂×Q₈	—	23	2	no	—
192	C₂×Q₈	—	23	1	no	—
193	C₂×Q₈	—	46	1	no	—
194	C₂×Q₈	—	46	2	no	—
195	C₂×D₄	—	23	1	no	—
196	C₂×D₄	—	46	1	no	—
197	C₂²×C₄	—	46	1	no	—
198	C₂²×C₄	—	46	1	no	—
199	C₂²×C₄	—	46	1	no	—
200	C₂²×C₄	—	46	1	no	—
201	C₂²×C₄	—	46	1	no	—
202	C₂²×C₄	D₉₂	1	1	yes	characteristic
203	C₂²×C₄	D₉₂	1	1	yes	characteristic
204	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
205	Q₁₆	—	23	8	no	—
206	SD₁₆	—	23	8	no	—
207	M₁₆	—	2	2	no	—
208	M₁₆	C₄×D₂₃	1	4	yes	complemented
209	C₂×C₈	C₂²×D₂₃	1	2	yes	—
210	C₂×C₈	C₂²×D₂₃	1	2	yes	—
211	C₄⋊C₄	—	23	2	no	—
212	C₄⋊C₄	—	23	1	no	—
213	C₄⋊C₄	—	23	1	no	—
214	C₄⋊C₄	—	46	1	no	—
215	C₄⋊C₄	—	46	2	no	—
216	C₄⋊C₄	—	46	2	no	—
217	C₄⋊C₄	—	46	1	no	—
218	C₄⋊C₄	—	46	1	no	—
219	C₂²⋊C₄	—	46	1	no	—
220	C₂²⋊C₄	—	46	1	no	—
221	C₂²⋊C₄	—	46	1	no	—
222	C₄²	—	46	1	no	—
223	C₄²	—	46	1	no	—
224	C₄²	—	46	2	no	—
225	C₄²	—	46	1	no	—
226	C₄²	—	46	2	no	—
227	C₄²	D₉₂	1	1	yes	characteristic
228	C₄²	D₉₂	1	1	yes	characteristic
229	C₄²	C₂²×D₂₃	1	1	yes	characteristic
230	C₄²	C₂²×D₂₃	1	1	yes	characteristic
231	C₂³	—	46	1	no	—
232	C₂³	C₂×D₉₂	1	1	yes	characteristic
233	Q₈	—	23	4	no	—
234	Q₈	—	23	2	no	—
235	Q₈	—	46	4	no	—
236	Q₈	—	46	2	no	—
237	Q₈	—	46	1	no	—
238	D₄	—	23	2	no	—
239	D₄	—	46	4	no	—
240	D₄	—	46	1	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	4	1	no	—
243	C₂×C₄	—	46	4	no	—
244	C₂×C₄	—	46	1	no	—
245	C₂×C₄	—	46	2	no	—
246	C₂×C₄	—	46	4	no	—
247	C₂×C₄	—	46	1	no	—
248	C₂×C₄	—	46	1	no	—
249	C₂×C₄	—	46	2	no	—
250	C₂×C₄	—	92	2	no	—
251	C₂×C₄	—	92	2	no	—
252	C₂×C₄	—	92	2	no	—
253	C₂×C₄	C₂×C₄×D₂₃	1	2	yes	—
254	C₂×C₄	C₂×D₉₂	1	1	yes	characteristic
255	C₂×C₄	C₂×D₉₂	1	1	yes	characteristic
256	C₂×C₄	C₂×D₉₂	1	1	yes	characteristic
257	C₂×C₄	C₂×D₉₂	1	1	yes	characteristic
258	C₂×C₄	C₂×D₉₂	1	1	yes	characteristic
259	C₂×C₄	#67288	1	2	yes	—
260	C₂×C₄	C₂³×D₂₃	1	1	yes	characteristic
261	C₈	—	2	2	no	—
262	C₈	C₂×C₄×D₂₃	1	4	yes	—
263	C₂²	—	2	1	no	—
264	C₂²	—	2	1	no	—
265	C₂²	—	46	2	no	—
266	C₂²	—	92	1	no	—
267	C₂²	—	92	1	no	—
268	C₂²	C₄×D₉₂	1	2	yes	—
269	C₂²	C₂²×D₉₂	1	1	yes	characteristic
270	C₄	—	2	1	no	—
271	C₄	—	2	1	no	—
272	C₄	—	2	1	no	—
273	C₄	—	46	2	no	—
274	C₄	—	46	4	no	—
275	C₄	—	92	1	no	—
276	C₄	—	92	1	no	—
277	C₄	—	92	2	no	—
278	C₄	C₄×D₉₂	1	2	yes	—
279	C₄	#10625080	1	2	yes	direct product factor
280	C₄	C₂²×C₄×D₂₃	1	1	yes	characteristic
281	C₄	C₂×#67288	1	1	yes	characteristic
282	C₂	—	2	2	no	—
283	C₂	—	92	2	no	—
284	C₂	C₂×C₄×D₉₂	1	1	yes	characteristic
285	C₂	C₂×#10625080	1	1	yes	characteristic
286	C₂	#13093217	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₉₂
commutator	C₄×C₂₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₂₃
Fitting	C₄×M₁₆×C₂₃	C₂
socle	C₂²×C₂₃	C₂²×D₄
prime radical	C₂×#67288	C₂²
cosocle	C₂×C₄×C₂₃	C₂⁴
hypercenter	C₄×M₁₆	D₂₃
hypocenter	C₂₃	C₄×#137
LCS	G ▷ C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×M₁₆

Complements

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

N	H ≅ G/N	type
3. C₂×C₄×Dic₄₆	C₂	semidirect
4. #13092525	C₂	semidirect
5. #13092519	C₂	semidirect
6. C₄×Dic₉₂	C₂	semidirect
7. C₄×#67263	C₂	semidirect
8. #13092459	C₂	semidirect
9. #13092427	C₂	semidirect
11. C₄×M₁₆×C₂₃	C₂	semidirect
23. #10625080	C₄	direct (N × H)
58. C₂×Dic₄₆	C₂×C₄	semidirect
62. M₁₆×C₂₃	C₂×C₄	semidirect
85. Dic₉₂	C₂×C₄	semidirect
86. #67263	C₂×C₄	semidirect
131. C₄×C₂₃	#137	semidirect
149. C₄×M₁₆	D₂₃	semidirect
187. C₂₃	C₄×#137	semidirect
208. M₁₆	C₄×D₂₃	semidirect
279. C₄	#10625080	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×#10625080	2
C₄×#67288	2
C₂×C₄×Dic₄₆	2
#13092525	2
#13092519	2
C₄×Dic₉₂	2
C₄×#67263	2
#13092459	2
#13092427	2
#13092419	2
C₄×M₁₆×C₂₃	2
C₄×#137	23

Subgroups — C4×#10625080 #15682220

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#10625080 #15682220