Subgroups — C₂×#13092188 #15673139

Order 2944.

388 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#67267	C₂	1	1	yes	maximal, characteristic
2	C₂×#10625097	C₂	1	1	yes	maximal, characteristic
3	#13092188	C₂	1	4	yes	maximal, direct product factor
4	C₂²×M₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic
5	C₂³×C₄×C₂₃	C₂²	1	1	yes	characteristic
6	C₂×M₁₆×C₂₃	—	2	1	no	—
7	C₂×M₁₆×C₂₃	—	2	1	no	—
8	C₂×M₁₆×C₂₃	—	2	1	no	—
9	C₂×M₁₆×C₂₃	—	2	1	no	—
10	C₂×M₁₆×C₂₃	C₄	1	2	yes	complemented
11	C₂×M₁₆×C₂₃	C₂²	1	2	yes	—
12	C₂²×C₈×C₂₃	C₄	1	2	yes	complemented
13	C₂×C₂²⋊Dic₂₃	C₄	1	2	yes	—
14	#10625097	—	2	1	no	—
15	#10625097	—	2	1	no	—
16	#10625097	C₄	1	2	yes	—
17	#10625097	C₂²	1	2	yes	—
18	C₂×C₄⋊Dic₂₃	—	2	1	no	—
19	C₂×C₄×Dic₂₃	—	2	1	no	—
20	C₂×#67267	—	2	1	no	—
21	C₂×#67267	—	2	1	no	—
22	C₂×#67267	—	2	1	no	—
23	C₂×#67267	C₄	1	2	yes	complemented
24	C₂×#67267	C₄	1	2	yes	complemented
25	C₂×#67267	C₂²	1	2	yes	—
26	C₂²×C₂₃⋊C₈	—	2	1	no	—
27	C₂⁴×C₂₃	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₄×C₂₃	D₄	1	1	yes	characteristic
34	C₂²×C₄×C₂₃	D₄	1	1	yes	characteristic
35	C₂²×C₄×C₂₃	D₄	1	1	yes	characteristic
36	C₂²×C₄×C₂₃	D₄	1	1	yes	characteristic
37	C₂²×C₄×C₂₃	D₄	1	1	yes	characteristic
38	C₂²×C₄×C₂₃	D₄	1	1	yes	characteristic
39	C₂²×C₄×C₂₃	Q₈	1	1	yes	characteristic
40	C₂²×C₄×C₂₃	Q₈	1	1	yes	characteristic
41	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
42	C₂²×Dic₂₃	—	2	2	no	—
43	M₁₆×C₂₃	—	2	2	no	—
44	M₁₆×C₂₃	—	2	2	no	—
45	M₁₆×C₂₃	—	2	2	no	—
46	M₁₆×C₂₃	—	2	2	no	—
47	C₂×C₈×C₂₃	—	2	2	no	—
48	C₂×C₈×C₂₃	—	2	2	no	—
49	C₂×C₈×C₂₃	C₂×C₄	1	4	yes	complemented
50	C₂²⋊Dic₂₃	—	2	2	no	—
51	C₂²⋊Dic₂₃	C₂×C₄	1	4	yes	—
52	C₄⋊Dic₂₃	—	2	2	no	—
53	C₄⋊Dic₂₃	—	2	2	no	—
54	C₄×Dic₂₃	—	2	2	no	—
55	C₄×Dic₂₃	—	2	2	no	—
56	#67267	—	2	2	no	—
57	#67267	—	2	2	no	—
58	#67267	—	2	2	no	—
59	#67267	C₂×C₄	1	4	yes	complemented
60	C₂×C₂₃⋊C₈	—	2	2	no	—
61	C₂×C₂₃⋊C₈	—	2	2	no	—
62	C₂×C₂₃⋊C₈	—	2	2	no	—
63	C₂³×C₂₃	—	4	2	no	—
64	C₂³×C₂₃	C₄²	1	1	yes	characteristic
65	C₂³×C₂₃	C₄²	1	1	yes	characteristic
66	C₂³×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
67	C₂³×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
68	C₂³×C₂₃	C₄⋊C₄	1	1	yes	characteristic
69	C₂³×C₂₃	C₄⋊C₄	1	1	yes	characteristic
70	C₂³×C₂₃	C₂²×C₄	1	1	yes	characteristic
71	C₂×C₄×C₂₃	C₄²	1	1	yes	characteristic
72	C₂×C₄×C₂₃	C₄²	1	1	yes	characteristic
73	C₂×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
74	C₂×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
76	C₂×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
80	C₂×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
81	C₂×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
82	C₂×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
83	C₂×C₄×C₂₃	C₄⋊C₄	1	1	yes	characteristic
84	C₂×C₄×C₂₃	C₄⋊C₄	1	1	yes	characteristic
85	C₂×C₄×C₂₃	C₄⋊C₄	1	1	yes	characteristic
86	C₂×C₄×C₂₃	C₄⋊C₄	1	1	yes	characteristic
87	C₂×C₄×C₂₃	C₄⋊C₄	1	1	yes	characteristic
88	C₂×C₄×C₂₃	C₄⋊C₄	1	1	yes	characteristic
89	C₂×C₄×C₂₃	C₄⋊C₄	1	1	yes	characteristic
90	C₂×C₄×C₂₃	C₄⋊C₄	1	1	yes	characteristic
91	C₂×C₄×C₂₃	C₄⋊C₄	1	1	yes	characteristic
92	C₂×C₄×C₂₃	C₄⋊C₄	1	1	yes	characteristic
93	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₄×C₂₃	C₂²×C₄	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	C₂×D₄	1	1	yes	characteristic
98	C₂×C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
99	C₂×Dic₂₃	—	2	4	no	—
100	C₂×Dic₂₃	—	4	4	no	—
101	C₈×C₂₃	—	2	4	no	—
102	C₂₃⋊C₈	—	2	4	no	—
103	C₂×#344	—	23	1	no	maximal, Sylow
104	C₂²×C₂₃	—	2	2	no	—
105	C₂²×C₂₃	—	2	2	no	—
106	C₂²×C₂₃	—	2	2	no	—
107	C₂²×C₂₃	—	4	1	no	—
108	C₂²×C₂₃	—	4	1	no	—
109	C₂²×C₂₃	—	4	1	no	—
110	C₂²×C₂₃	—	4	1	no	—
111	C₂²×C₂₃	#95	1	1	yes	characteristic
112	C₂²×C₂₃	#95	1	1	yes	characteristic
113	C₂²×C₂₃	#95	1	1	yes	characteristic
114	C₂²×C₂₃	#95	1	1	yes	characteristic
115	C₂²×C₂₃	C₂×C₄²	1	1	yes	characteristic
116	C₂²×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
117	C₂²×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
118	C₄×C₂₃	#95	1	1	yes	characteristic
119	C₄×C₂₃	#95	1	1	yes	characteristic
120	C₄×C₂₃	#95	1	1	yes	characteristic
121	C₄×C₂₃	#95	1	1	yes	characteristic
122	C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
123	C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
124	C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
125	C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
126	Dic₂₃	—	4	4	no	—
127	C₂²×M₁₆	—	23	1	no	—
128	C₂²×M₁₆	D₂₃	1	1	yes	characteristic
129	C₂×#117	—	23	1	no	—
130	#344	—	23	4	no	—
131	C₂×C₂₃	—	2	1	no	—
132	C₂×C₂₃	—	2	1	no	—
133	C₂×C₂₃	—	2	1	no	—
134	C₂×C₂₃	—	2	1	no	—
135	C₂×C₂₃	—	2	1	no	—
136	C₂×C₂₃	—	2	1	no	—
137	C₂×C₂₃	#344	1	2	yes	complemented
138	C₂×C₂₃	C₂×#95	1	1	yes	characteristic
139	C₂³×C₄	C₂×D₂₃	1	1	yes	characteristic
140	C₂×M₁₆	—	2	1	no	—
141	C₂×M₁₆	—	2	1	no	—
142	C₂×M₁₆	—	2	1	no	—
143	C₂×M₁₆	—	2	1	no	—
144	C₂×M₁₆	—	23	2	no	—
145	C₂×M₁₆	—	23	2	no	—
146	C₂×M₁₆	—	23	2	no	—
147	C₂×M₁₆	—	46	1	no	—
148	C₂×M₁₆	—	46	1	no	—
149	C₂×M₁₆	—	46	1	no	—
150	C₂×M₁₆	Dic₂₃	1	2	yes	complemented
151	C₂×M₁₆	C₂×D₂₃	1	2	yes	—
152	C₂²×C₈	—	46	1	no	—
153	C₂²×C₈	Dic₂₃	1	2	yes	complemented
154	#117	—	23	2	no	—
155	#117	—	23	2	no	—
156	#117	—	46	1	no	—
157	#117	—	46	1	no	—
158	C₂×C₄⋊C₄	—	46	1	no	—
159	C₂×C₂²⋊C₄	—	23	2	no	—
160	C₂×C₄²	—	46	1	no	—
161	C₂₃	C₂×#344	1	1	yes	characteristic, complemented, Sylow
162	C₂⁴	C₄×D₂₃	1	1	yes	characteristic
163	C₂²×C₄	—	46	2	no	—
164	C₂²×C₄	Dic₄₆	1	1	yes	characteristic
165	C₂²×C₄	Dic₄₆	1	1	yes	characteristic
166	C₂²×C₄	C₄×D₂₃	1	1	yes	characteristic
167	C₂²×C₄	C₄×D₂₃	1	1	yes	characteristic
168	C₂²×C₄	C₄×D₂₃	1	1	yes	characteristic
169	C₂²×C₄	D₉₂	1	1	yes	characteristic
170	C₂²×C₄	D₉₂	1	1	yes	characteristic
171	C₂²×C₄	C₂×Dic₂₃	1	1	yes	characteristic
172	C₂²×C₄	C₂×Dic₂₃	1	1	yes	characteristic
173	C₂²×C₄	C₂²⋊D₂₃	1	1	yes	characteristic
174	C₂²×C₄	C₂²⋊D₂₃	1	1	yes	characteristic
175	C₂²×C₄	C₂²⋊D₂₃	1	1	yes	characteristic
176	C₂²×C₄	C₂²⋊D₂₃	1	1	yes	characteristic
177	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
178	M₁₆	—	2	2	no	—
179	M₁₆	—	2	2	no	—
180	M₁₆	—	2	2	no	—
181	M₁₆	—	2	2	no	—
182	M₁₆	—	23	4	no	—
183	M₁₆	—	46	2	no	—
184	M₁₆	—	46	2	no	—
185	M₁₆	—	46	2	no	—
186	C₂×C₈	—	2	2	no	—
187	C₂×C₈	—	2	2	no	—
188	C₂×C₈	—	46	2	no	—
189	C₂×C₈	—	46	2	no	—
190	C₂×C₈	—	46	2	no	—
191	C₂×C₈	C₂×Dic₂₃	1	4	yes	complemented
192	C₄⋊C₄	—	46	2	no	—
193	C₄⋊C₄	—	46	2	no	—
194	C₂²⋊C₄	—	23	4	no	—
195	C₂²⋊C₄	—	46	2	no	—
196	C₄²	—	46	2	no	—
197	C₄²	—	46	2	no	—
198	C₂³	—	4	2	no	—
199	C₂³	C₄×Dic₂₃	1	1	yes	characteristic
200	C₂³	C₄×Dic₂₃	1	1	yes	characteristic
201	C₂³	Dic₂₃⋊C₄	1	1	yes	characteristic
202	C₂³	Dic₂₃⋊C₄	1	1	yes	characteristic
203	C₂³	(C₂×D₂₃)⋊C₄	1	1	yes	characteristic
204	C₂³	(C₂×D₂₃)⋊C₄	1	1	yes	characteristic
205	C₂³	C₂×C₄×D₂₃	1	1	yes	characteristic
206	C₂×C₄	—	46	4	no	—
207	C₂×C₄	—	92	4	no	—
208	C₂×C₄	C₄×Dic₂₃	1	1	yes	characteristic
209	C₂×C₄	C₄×Dic₂₃	1	1	yes	characteristic
210	C₂×C₄	Dic₂₃⋊C₄	1	1	yes	characteristic
211	C₂×C₄	Dic₂₃⋊C₄	1	1	yes	characteristic
212	C₂×C₄	Dic₂₃⋊C₄	1	1	yes	characteristic
213	C₂×C₄	Dic₂₃⋊C₄	1	1	yes	characteristic
214	C₂×C₄	Dic₂₃⋊C₄	1	1	yes	characteristic
215	C₂×C₄	Dic₂₃⋊C₄	1	1	yes	characteristic
216	C₂×C₄	C₄⋊Dic₂₃	1	1	yes	characteristic
217	C₂×C₄	C₄⋊Dic₂₃	1	1	yes	characteristic
218	C₂×C₄	C₄⋊Dic₂₃	1	1	yes	characteristic
219	C₂×C₄	C₄⋊Dic₂₃	1	1	yes	characteristic
220	C₂×C₄	(C₂×D₂₃)⋊C₄	1	1	yes	characteristic
221	C₂×C₄	(C₂×D₂₃)⋊C₄	1	1	yes	characteristic
222	C₂×C₄	(C₂×D₂₃)⋊C₄	1	1	yes	characteristic
223	C₂×C₄	(C₂×D₂₃)⋊C₄	1	1	yes	characteristic
224	C₂×C₄	(C₂×D₂₃)⋊C₄	1	1	yes	characteristic
225	C₂×C₄	(C₂×D₂₃)⋊C₄	1	1	yes	characteristic
226	C₂×C₄	C₂²⋊Dic₂₃	1	1	yes	characteristic
227	C₂×C₄	C₂²⋊Dic₂₃	1	1	yes	characteristic
228	C₂×C₄	C₂²⋊Dic₂₃	1	1	yes	characteristic
229	C₂×C₄	C₂²⋊Dic₂₃	1	1	yes	characteristic
230	C₂×C₄	C₂×Dic₄₆	1	1	yes	characteristic
231	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
232	C₂×C₄	C₂×D₉₂	1	1	yes	characteristic
233	C₂×C₄	C₂²×Dic₂₃	1	1	yes	characteristic
234	C₂×C₄	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
235	C₂×C₄	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
236	C₈	—	2	4	no	—
237	C₈	—	46	4	no	—
238	C₂²	—	2	2	no	—
239	C₂²	—	2	2	no	—
240	C₂²	—	2	2	no	—
241	C₂²	—	4	1	no	—
242	C₂²	—	4	1	no	—
243	C₂²	—	4	1	no	—
244	C₂²	—	4	1	no	—
245	C₂²	C₂₃⋊#95	1	1	yes	characteristic
246	C₂²	C₂₃⋊#95	1	1	yes	characteristic
247	C₂²	C₂₃⋊#95	1	1	yes	characteristic
248	C₂²	C₂₃⋊#95	1	1	yes	characteristic
249	C₂²	C₂×C₄×Dic₂₃	1	1	yes	characteristic
250	C₂²	C₂×Dic₂₃⋊C₄	1	1	yes	characteristic
251	C₂²	C₂×(C₂×D₂₃)⋊C₄	1	1	yes	characteristic
252	C₄	—	92	4	no	—
253	C₄	C₂₃⋊#95	1	1	yes	characteristic
254	C₄	C₂₃⋊#95	1	1	yes	characteristic
255	C₄	C₂₃⋊#95	1	1	yes	characteristic
256	C₄	C₂₃⋊#95	1	1	yes	characteristic
257	C₄	C₂×Dic₂₃⋊C₄	1	1	yes	characteristic
258	C₄	C₂×C₄⋊Dic₂₃	1	1	yes	characteristic
259	C₄	C₂×(C₂×D₂₃)⋊C₄	1	1	yes	characteristic
260	C₄	C₂×C₂²⋊Dic₂₃	1	1	yes	characteristic
261	C₂	—	2	1	no	—
262	C₂	—	2	1	no	—
263	C₂	—	2	1	no	—
264	C₂	—	2	1	no	—
265	C₂	—	2	1	no	—
266	C₂	—	2	1	no	—
267	C₂	#13092188	1	2	yes	direct product factor
268	C₂	C₂×C₂₃⋊#95	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²⋊Dic₂₃
commutator	C₂²×C₂₃	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₂₃
Fitting	C₂²×M₁₆×C₂₃	C₂
socle	C₂²×C₂₃	#95
prime radical	C₂⁴×C₂₃	C₂×C₄
cosocle	C₂²×C₄×C₂₃	C₂³
hypercenter	C₂²×M₁₆	D₂₃
hypocenter	C₂₃	C₂×#344
LCS	G ▷ C₂²×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×M₁₆

Complements

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

N	H ≅ G/N	type
3. #13092188	C₂	direct (N × H)
10. C₂×M₁₆×C₂₃	C₄	semidirect
12. C₂²×C₈×C₂₃	C₄	semidirect
23. C₂×#67267	C₄	semidirect
24. C₂×#67267	C₄	semidirect
49. C₂×C₈×C₂₃	C₂×C₄	semidirect
59. #67267	C₂×C₄	semidirect
137. C₂×C₂₃	#344	semidirect
150. C₂×M₁₆	Dic₂₃	semidirect
153. C₂²×C₈	Dic₂₃	semidirect
161. C₂₃	C₂×#344	semidirect
191. C₂×C₈	C₂×Dic₂₃	semidirect
267. C₂	#13092188	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#67267	2
C₂×#10625097	2
#13092188	2
C₂²×M₁₆×C₂₃	2
C₂×#344	23

Subgroups — C2×#13092188 #15673139

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13092188 #15673139