Subgroups — C₂×#13093070 #15686856

Order 2944.

1632 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#67292	C₂	1	2	yes	maximal, complemented
2	#13093070	C₂	1	16	yes	maximal, direct product factor
3	C₂×#10625065	C₂	1	4	yes	maximal, complemented
4	C₂×#10625117	C₂	1	2	yes	maximal, complemented
5	C₂×#10625066	C₂	1	2	yes	maximal, complemented
6	C₂×C₄×D₉₂	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₄×D₉₂	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄⋊C₄×D₂₃	C₂	1	2	yes	maximal, complemented
9	C₂×C₂₃×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×Q₈×C₂₃	C₂²	1	2	yes	—
11	C₂×#67292	—	2	16	no	—
12	C₂×#67292	C₂²	1	16	yes	complemented
13	C₂²×D₉₂	C₂²	1	1	yes	characteristic
14	C₂²×D₉₂	C₂²	1	4	yes	—
15	C₂²×D₉₂	C₂²	1	1	yes	characteristic
16	C₂²×C₄×D₂₃	C₂²	1	2	yes	—
17	C₂²×C₄×D₂₃	C₂²	1	4	yes	—
18	C₂²×C₄×D₂₃	C₂²	1	2	yes	—
19	C₂₃×C₄⋊Q₈	C₂²	1	8	yes	complemented
20	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
21	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
22	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
23	#10625117	C₂²	1	16	yes	complemented
24	C₂×(C₂×D₂₃)⋊C₄	C₂²	1	2	yes	—
25	C₂×(C₂×D₂₃)⋊C₄	C₂²	1	2	yes	—
26	C₂×(C₂×D₂₃)⋊C₄	C₂²	1	4	yes	—
27	C₂×C₄⋊Dic₂₃	C₂²	1	1	yes	characteristic
28	C₂×C₄⋊Dic₂₃	C₂²	1	1	yes	characteristic
29	C₂×Dic₂₃⋊C₄	C₂²	1	4	yes	—
30	#10625066	C₂²	1	16	yes	complemented
31	#10625065	C₂²	1	32	yes	complemented
32	C₄⋊C₄×D₂₃	C₂²	1	16	yes	complemented
33	C₄×D₉₂	C₂²	1	8	yes	complemented
34	C₄×D₉₂	C₂²	1	8	yes	complemented
35	C₂³×D₂₃	C₂³	1	2	yes	—
36	C₂³×D₂₃	C₂³	1	2	yes	—
37	C₂×Q₈×C₂₃	—	2	8	no	—
38	C₂×Q₈×C₂₃	C₂³	1	8	yes	—
39	C₂²×C₄×C₂₃	C₂³	1	4	yes	—
40	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
41	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
42	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
43	C₂²×Dic₂₃	C₂³	1	2	yes	—
44	C₂²×Dic₂₃	C₂³	1	2	yes	—
45	#67292	—	2	64	no	—
46	C₂×D₉₂	—	2	16	no	—
47	C₂×D₉₂	—	2	4	no	—
48	C₂×D₉₂	D₄	1	8	yes	complemented
49	C₂×D₉₂	C₂³	1	4	yes	—
50	C₂×D₉₂	C₂³	1	16	yes	—
51	C₂×D₉₂	C₂³	1	4	yes	—
52	C₂×C₄×D₂₃	—	2	16	no	—
53	C₂×C₄×D₂₃	—	2	8	no	—
54	C₂×C₄×D₂₃	—	2	8	no	—
55	C₂×C₄×D₂₃	C₂³	1	8	yes	—
56	C₂×C₄×D₂₃	C₂³	1	8	yes	—
57	C₂×C₄×D₂₃	C₂³	1	16	yes	—
58	C₄⋊C₄×C₂₃	C₂³	1	8	yes	—
59	C₄⋊C₄×C₂₃	C₂³	1	8	yes	—
60	C₄²×C₂₃	C₂³	1	4	yes	—
61	(C₂×D₂₃)⋊C₄	C₂³	1	8	yes	—
62	(C₂×D₂₃)⋊C₄	C₂³	1	8	yes	—
63	(C₂×D₂₃)⋊C₄	C₂³	1	16	yes	—
64	C₄⋊Dic₂₃	C₂³	1	4	yes	—
65	C₄⋊Dic₂₃	C₂³	1	4	yes	—
66	Dic₂₃⋊C₄	C₂³	1	16	yes	—
67	C₂³×C₂₃	C₂⁴	1	1	yes	characteristic
68	C₂²×D₂₃	—	2	4	no	—
69	C₂²×D₂₃	—	2	4	no	—
70	C₂²×D₂₃	—	2	4	no	—
71	C₂²×D₂₃	—	2	4	no	—
72	C₂²×D₂₃	—	2	4	no	—
73	C₂²×D₂₃	C₂×D₄	1	8	yes	—
74	C₂²×D₂₃	C₂⁴	1	4	yes	—
75	C₂²×D₂₃	C₂⁴	1	4	yes	—
76	Q₈×C₂₃	—	2	16	no	—
77	C₂×C₄×C₂₃	—	2	8	no	—
78	C₂×C₄×C₂₃	—	2	2	no	—
79	C₂×C₄×C₂₃	C₂×D₄	1	4	yes	—
80	C₂×C₄×C₂₃	#41	1	4	yes	complemented
81	C₂×C₄×C₂₃	C₂⁴	1	8	yes	—
82	C₂×C₄×C₂₃	C₂⁴	1	2	yes	—
83	C₂×C₄×C₂₃	C₂⁴	1	2	yes	—
84	C₂×C₄×C₂₃	C₂⁴	1	2	yes	—
85	C₂×Dic₂₃	—	2	4	no	—
86	C₂×Dic₂₃	—	2	4	no	—
87	C₂×Dic₂₃	C₂⁴	1	4	yes	—
88	C₂×Dic₂₃	C₂⁴	1	4	yes	—
89	D₉₂	—	2	32	no	—
90	D₉₂	—	2	8	no	—
91	D₉₂	C₂×D₄	1	16	yes	complemented
92	C₄×D₂₃	—	2	32	no	—
93	C₄×D₂₃	—	2	16	no	—
94	C₄×D₂₃	—	2	16	no	—
95	C₂×#547	—	23	1	no	maximal, Sylow
96	C₂²×C₂₃	C₂²×D₄	1	2	yes	—
97	C₂²×C₂₃	C₂×#41	1	2	yes	—
98	C₂²×C₂₃	2₋¹⁺⁴	1	2	yes	—
99	C₂²×C₂₃	C₂⁵	1	1	yes	characteristic
100	C₂×D₂₃	—	2	16	no	—
101	C₂×D₂₃	—	2	4	no	—
102	C₂×D₂₃	—	2	4	no	—
103	C₂×D₂₃	—	2	4	no	—
104	C₂×D₂₃	—	2	4	no	—
105	C₂×D₂₃	—	2	4	no	—
106	C₂×D₂₃	—	4	8	no	—
107	C₂×D₂₃	C₂²×D₄	1	8	yes	—
108	C₄×C₂₃	—	2	8	no	—
109	C₄×C₂₃	—	2	2	no	—
110	C₄×C₂₃	C₂²×D₄	1	4	yes	—
111	C₄×C₂₃	C₂×#41	1	4	yes	complemented
112	Dic₂₃	—	2	4	no	—
113	Dic₂₃	—	2	4	no	—
114	C₂²×#41	—	23	2	no	—
115	#547	—	23	16	no	—
116	C₂×C₄⋊Q₈	D₂₃	1	1	yes	characteristic, complemented
117	C₂×#123	—	23	4	no	—
118	C₂×C₂²⋊Q₈	—	23	2	no	—
119	C₂×#121	—	23	2	no	—
120	C₂×C₄×D₄	—	23	1	no	—
121	C₂×C₄×D₄	—	23	1	no	—
122	C₂²×C₄⋊C₄	—	23	2	no	—
123	C₂×C₂₃	#547	1	4	yes	complemented
124	C₂×C₂₃	C₂³×D₄	1	1	yes	characteristic
125	C₂×C₂₃	C₂²×#41	1	1	yes	characteristic
126	C₂×C₂₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
127	D₂₃	—	2	8	no	—
128	D₂₃	—	4	4	no	—
129	C₂×#41	—	23	16	no	—
130	C₂×#41	—	46	16	no	—
131	C₂²×Q₈	C₂×D₂₃	1	2	yes	—
132	C₂²×D₄	—	23	1	no	—
133	C₂²×D₄	—	23	1	no	—
134	C₂²×D₄	—	23	4	no	—
135	C₂³×C₄	—	23	2	no	—
136	C₂³×C₄	—	23	4	no	—
137	C₂³×C₄	—	23	2	no	—
138	C₄⋊Q₈	C₂×D₂₃	1	8	yes	complemented
139	#123	—	23	32	no	—
140	C₂²⋊Q₈	—	23	16	no	—
141	#121	—	23	16	no	—
142	C₄×D₄	—	23	8	no	—
143	C₄×D₄	—	23	8	no	—
144	C₂×C₄⋊C₄	—	23	1	no	—
145	C₂×C₄⋊C₄	—	23	4	no	—
146	C₂×C₄⋊C₄	—	23	1	no	—
147	C₂×C₄⋊C₄	—	23	16	no	—
148	C₂×C₄⋊C₄	C₂×D₂₃	1	2	yes	—
149	C₂×C₄⋊C₄	C₂×D₂₃	1	2	yes	—
150	C₂×C₂²⋊C₄	—	23	2	no	—
151	C₂×C₂²⋊C₄	—	23	2	no	—
152	C₂×C₂²⋊C₄	—	23	4	no	—
153	C₂×C₄²	C₂×D₂₃	1	1	yes	characteristic
154	C₂₃	C₂×#547	1	1	yes	characteristic, complemented, Sylow
155	C₂⁴	—	23	2	no	—
156	C₂⁴	—	23	2	no	—
157	#41	—	46	64	no	—
158	C₂×Q₈	—	2	8	no	—
159	C₂×Q₈	C₂²×D₂₃	1	8	yes	—
160	C₂×D₄	—	23	8	no	—
161	C₂×D₄	—	23	4	no	—
162	C₂×D₄	—	23	4	no	—
163	C₂×D₄	—	23	16	no	—
164	C₂×D₄	—	46	16	no	—
165	C₂×D₄	—	46	4	no	—
166	C₂²×C₄	—	23	2	no	—
167	C₂²×C₄	—	23	2	no	—
168	C₂²×C₄	—	23	8	no	—
169	C₂²×C₄	—	23	16	no	—
170	C₂²×C₄	—	23	8	no	—
171	C₂²×C₄	—	46	16	no	—
172	C₂²×C₄	—	46	8	no	—
173	C₂²×C₄	—	46	8	no	—
174	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
175	C₂²×C₄	C₂²×D₂₃	1	4	yes	—
176	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
177	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
178	C₄⋊C₄	—	23	4	no	—
179	C₄⋊C₄	—	23	16	no	—
180	C₄⋊C₄	—	23	4	no	—
181	C₄⋊C₄	C₂²×D₂₃	1	8	yes	—
182	C₄⋊C₄	C₂²×D₂₃	1	8	yes	—
183	C₂²⋊C₄	—	23	8	no	—
184	C₂²⋊C₄	—	23	8	no	—
185	C₂²⋊C₄	—	23	16	no	—
186	C₄²	C₂²×D₂₃	1	4	yes	—
187	C₂³	—	23	8	no	—
188	C₂³	—	23	4	no	—
189	C₂³	—	23	4	no	—
190	C₂³	—	46	4	no	—
191	C₂³	—	46	4	no	—
192	C₂³	—	46	4	no	—
193	C₂³	—	46	4	no	—
194	C₂³	—	46	4	no	—
195	C₂³	C₂³×D₂₃	1	1	yes	characteristic
196	Q₈	—	2	16	no	—
197	D₄	—	23	16	no	—
198	D₄	—	46	32	no	—
199	D₄	—	46	8	no	—
200	C₂×C₄	—	2	8	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	23	4	no	—
203	C₂×C₄	—	23	4	no	—
204	C₂×C₄	—	46	32	no	—
205	C₂×C₄	—	46	4	no	—
206	C₂×C₄	—	46	4	no	—
207	C₂×C₄	—	46	16	no	—
208	C₂×C₄	—	46	16	no	—
209	C₂×C₄	D₄×D₂₃	1	4	yes	—
210	C₂×C₄	#67292	1	4	yes	complemented
211	C₂×C₄	C₂³×D₂₃	1	2	yes	—
212	C₂×C₄	C₂³×D₂₃	1	8	yes	—
213	C₂×C₄	C₂³×D₂₃	1	2	yes	—
214	C₂×C₄	C₂³×D₂₃	1	2	yes	—
215	C₂²	—	23	8	no	—
216	C₂²	—	46	4	no	—
217	C₂²	—	46	4	no	—
218	C₂²	—	46	16	no	—
219	C₂²	—	46	4	no	—
220	C₂²	—	46	4	no	—
221	C₂²	—	46	4	no	—
222	C₂²	—	92	8	no	—
223	C₂²	C₂×D₄×D₂₃	1	2	yes	—
224	C₂²	C₂×#67292	1	2	yes	—
225	C₂²	#10625158	1	2	yes	—
226	C₂²	C₂⁴×D₂₃	1	1	yes	characteristic
227	C₄	—	2	8	no	—
228	C₄	—	2	2	no	—
229	C₄	—	46	4	no	—
230	C₄	—	46	4	no	—
231	C₄	C₂×D₄×D₂₃	1	4	yes	—
232	C₄	C₂×#67292	1	4	yes	complemented
233	C₂	—	46	8	no	—
234	C₂	—	92	4	no	—
235	C₂	#13093070	1	4	yes	direct product factor
236	C₂	C₂²×D₄×D₂₃	1	1	yes	characteristic
237	C₂	C₂²×#67292	1	1	yes	characteristic
238	C₂	C₂×#10625158	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂³×D₂₃
commutator, cosocle	C₂²×C₂₃	C₂⁵
Frattini	C₂²	C₂⁴×D₂₃
Fitting	C₂×C₂₃×C₄⋊Q₈	C₂
socle	C₂³×C₂₃	C₂⁴
hypercenter	C₂×C₄⋊Q₈	D₂₃
hypocenter	C₂₃	C₂×#547
LCS	G ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
1. C₂²×#67292	C₂	semidirect
2. #13093070	C₂	direct (N × H)
3. C₂×#10625065	C₂	semidirect
4. C₂×#10625117	C₂	semidirect
5. C₂×#10625066	C₂	semidirect
6. C₂×C₄×D₉₂	C₂	semidirect
7. C₂×C₄×D₉₂	C₂	semidirect
8. C₂×C₄⋊C₄×D₂₃	C₂	semidirect
9. C₂×C₂₃×C₄⋊Q₈	C₂	semidirect
12. C₂×#67292	C₂²	semidirect
19. C₂₃×C₄⋊Q₈	C₂²	semidirect
23. #10625117	C₂²	semidirect
30. #10625066	C₂²	semidirect
31. #10625065	C₂²	semidirect
32. C₄⋊C₄×D₂₃	C₂²	semidirect
33. C₄×D₉₂	C₂²	semidirect
34. C₄×D₉₂	C₂²	semidirect
48. C₂×D₉₂	D₄	semidirect
80. C₂×C₄×C₂₃	#41	semidirect
91. D₉₂	C₂×D₄	semidirect
111. C₄×C₂₃	C₂×#41	semidirect
116. C₂×C₄⋊Q₈	D₂₃	semidirect
123. C₂×C₂₃	#547	semidirect
138. C₄⋊Q₈	C₂×D₂₃	semidirect
154. C₂₃	C₂×#547	semidirect
210. C₂×C₄	#67292	semidirect
232. C₄	C₂×#67292	semidirect
235. C₂	#13093070	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×#67292	2
#13093070	2
C₂×#10625065	2
C₂×#10625117	2
C₂×#10625066	2
C₂×C₄×D₉₂	2
C₂×C₄⋊C₄×D₂₃	2
C₂×C₂₃×C₄⋊Q₈	2
C₂×#547	23

Subgroups — C2×#13093070 #15686856

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13093070 #15686856