Subgroups — C₂×#13093132 #15686937

Order 2944.

1240 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13093132	C₂	1	16	yes	maximal, direct product factor
2	C₂×#10625040	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#10625056	C₂	1	4	yes	maximal, complemented
4	C₂×#10625101	C₂	1	2	yes	maximal, complemented
5	C₂×#10625109	C₂	1	2	yes	maximal, complemented
6	C₂×C₄×D₉₂	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#10625097	C₂	1	2	yes	maximal, complemented
8	C₂×#10625063	C₂	1	2	yes	maximal, complemented
9	C₂×C₄×D₄×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×D₄×C₂₃	C₂²	1	1	yes	characteristic
11	C₂³×C₄×C₂₃	C₂²	1	2	yes	—
12	C₂²×C₂²⋊D₂₃	C₂²	1	4	yes	—
13	C₂²×D₉₂	C₂²	1	1	yes	characteristic
14	C₂²×C₄×D₂₃	C₂²	1	2	yes	—
15	C₂²×Dic₄₆	C₂²	1	2	yes	—
16	C₄×D₄×C₂₃	C₂²	1	8	yes	complemented
17	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
18	C₂×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
19	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
20	C₂×C₂²⋊Dic₂₃	C₂²	1	4	yes	—
21	#10625109	C₂²	1	16	yes	complemented
22	#10625101	C₂²	1	16	yes	complemented
23	C₂×(C₂×D₂₃)⋊C₄	C₂²	1	4	yes	—
24	C₂×(C₂×D₂₃)⋊C₄	C₂²	1	2	yes	—
25	#10625097	C₂²	1	16	yes	complemented
26	C₂×C₄⋊Dic₂₃	C₂²	1	2	yes	—
27	C₂×C₄⋊Dic₂₃	C₂²	1	1	yes	characteristic
28	C₂×C₄⋊Dic₂₃	C₂²	1	2	yes	—
29	C₂×C₄×Dic₂₃	C₂²	1	4	yes	—
30	#10625063	C₂²	1	16	yes	complemented
31	#10625056	C₂²	1	32	yes	complemented
32	C₄×D₉₂	C₂²	1	8	yes	complemented
33	#10625040	C₂²	1	8	yes	complemented
34	C₂⁴×C₂₃	C₂³	1	2	yes	—
35	C₂³×D₂₃	C₂³	1	2	yes	—
36	C₂×D₄×C₂₃	—	2	4	no	—
37	C₂×D₄×C₂₃	C₂³	1	4	yes	—
38	C₂²×C₄×C₂₃	—	2	8	no	—
39	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
40	C₂²×C₄×C₂₃	C₂³	1	8	yes	—
41	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
42	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
43	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
44	C₂×C₂²⋊D₂₃	—	4	8	no	—
45	C₂×C₂²⋊D₂₃	C₂³	1	16	yes	—
46	C₂²×Dic₂₃	C₂³	1	2	yes	—
47	C₂²×Dic₂₃	C₂³	1	4	yes	—
48	C₂×D₉₂	—	2	4	no	—
49	C₂×D₉₂	C₂³	1	4	yes	—
50	C₂×C₄×D₂₃	—	2	8	no	—
51	C₂×C₄×D₂₃	C₂³	1	8	yes	—
52	C₂×Dic₄₆	—	4	4	no	—
53	C₂×Dic₄₆	C₂³	1	8	yes	—
54	C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
55	C₂²⋊C₄×C₂₃	C₂³	1	8	yes	—
56	C₄²×C₂₃	C₂³	1	4	yes	—
57	C₂²⋊Dic₂₃	C₂³	1	16	yes	—
58	(C₂×D₂₃)⋊C₄	C₂³	1	8	yes	—
59	(C₂×D₂₃)⋊C₄	C₂³	1	16	yes	—
60	C₄⋊Dic₂₃	C₂³	1	8	yes	—
61	C₄⋊Dic₂₃	C₂³	1	8	yes	—
62	C₄⋊Dic₂₃	C₂³	1	4	yes	—
63	C₄×Dic₂₃	C₂³	1	16	yes	—
64	C₂³×C₂₃	—	2	4	no	—
65	C₂³×C₂₃	—	2	4	no	—
66	C₂³×C₂₃	—	2	4	no	—
67	C₂³×C₂₃	C₂⁴	1	4	yes	—
68	C₂³×C₂₃	C₂⁴	1	1	yes	characteristic
69	C₂²×D₂₃	—	2	4	no	—
70	C₂²×D₂₃	—	2	4	no	—
71	C₂²×D₂₃	—	2	4	no	—
72	C₂²×D₂₃	C₂⁴	1	4	yes	—
73	D₄×C₂₃	—	2	8	no	—
74	C₂×C₄×C₂₃	—	2	4	no	—
75	C₂×C₄×C₂₃	—	2	2	no	—
76	C₂×C₄×C₂₃	—	2	16	no	—
77	C₂×C₄×C₂₃	#41	1	4	yes	—
78	C₂×C₄×C₂₃	#41	1	4	yes	—
79	C₂×C₄×C₂₃	C₂⁴	1	4	yes	—
80	C₂×C₄×C₂₃	C₂⁴	1	2	yes	—
81	C₂×C₄×C₂₃	C₂⁴	1	2	yes	—
82	C₂×C₄×C₂₃	C₂⁴	1	2	yes	—
83	C₂²⋊D₂₃	—	4	16	no	—
84	C₂×Dic₂₃	—	2	4	no	—
85	C₂×Dic₂₃	—	2	8	no	—
86	C₂×Dic₂₃	C₂⁴	1	4	yes	—
87	C₂×Dic₂₃	C₂⁴	1	8	yes	—
88	D₉₂	—	2	8	no	—
89	C₄×D₂₃	—	2	16	no	—
90	Dic₄₆	—	4	8	no	—
91	C₂×#555	—	23	1	no	maximal, Sylow
92	C₂²×C₂₃	—	2	4	no	—
93	C₂²×C₂₃	—	2	4	no	—
94	C₂²×C₂₃	—	2	4	no	—
95	C₂²×C₂₃	—	4	8	no	—
96	C₂²×C₂₃	C₂×#41	1	2	yes	—
97	C₂²×C₂₃	C₂×#41	1	2	yes	—
98	C₂²×C₂₃	H(5,𝔽₂)	1	2	yes	—
99	C₂²×C₂₃	C₂⁵	1	1	yes	characteristic
100	C₂×D₂₃	—	2	4	no	—
101	C₂×D₂₃	—	2	4	no	—
102	C₂×D₂₃	—	2	4	no	—
103	C₂×D₂₃	—	4	8	no	—
104	C₄×C₂₃	—	2	4	no	—
105	C₄×C₂₃	—	2	2	no	—
106	C₄×C₂₃	C₂×#41	1	4	yes	—
107	C₄×C₂₃	C₂×#41	1	4	yes	—
108	Dic₂₃	—	2	4	no	—
109	Dic₂₃	—	2	8	no	—
110	#555	—	23	16	no	—
111	C₂×C₄⋊Q₈	—	23	1	no	—
112	C₂×#124	—	23	4	no	—
113	C₂×#121	—	23	2	no	—
114	C₂×#121	—	23	2	no	—
115	C₂×C₄×D₄	—	23	1	no	—
116	C₂×C₄×D₄	D₂₃	1	1	yes	characteristic, complemented
117	C₂×#117	—	23	2	no	—
118	C₂×#117	—	23	2	no	—
119	C₂×C₂₃	—	4	4	no	—
120	C₂×C₂₃	#555	1	4	yes	complemented
121	C₂×C₂₃	C₂²×#41	1	1	yes	characteristic
122	C₂×C₂₃	C₂²×#41	1	1	yes	characteristic
123	C₂×C₂₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
124	D₂₃	—	4	4	no	—
125	C₂²×Q₈	—	23	2	no	—
126	C₂²×D₄	—	23	4	no	—
127	C₂²×D₄	—	23	1	no	—
128	C₂²×D₄	C₂×D₂₃	1	1	yes	characteristic
129	C₂³×C₄	—	23	2	no	—
130	C₂³×C₄	C₂×D₂₃	1	2	yes	—
131	C₄⋊Q₈	—	23	8	no	—
132	#124	—	23	32	no	—
133	#121	—	23	16	no	—
134	#121	—	23	16	no	—
135	C₄×D₄	—	23	8	no	—
136	C₄×D₄	C₂×D₂₃	1	8	yes	complemented
137	#117	—	23	16	no	—
138	#117	—	23	16	no	—
139	C₂×C₄⋊C₄	—	23	1	no	—
140	C₂×C₄⋊C₄	—	23	2	no	—
141	C₂×C₄⋊C₄	—	23	2	no	—
142	C₂×C₄⋊C₄	C₂×D₂₃	1	1	yes	characteristic
143	C₂×C₂²⋊C₄	—	23	4	no	—
144	C₂×C₂²⋊C₄	—	23	4	no	—
145	C₂×C₂²⋊C₄	—	23	2	no	—
146	C₂×C₂²⋊C₄	C₂×D₂₃	1	2	yes	—
147	C₂×C₄²	—	23	4	no	—
148	C₂×C₄²	C₂×D₂₃	1	1	yes	characteristic
149	C₂₃	C₂×#555	1	1	yes	characteristic, complemented, Sylow
150	C₂⁴	—	23	2	no	—
151	C₂⁴	C₂²×D₂₃	1	2	yes	—
152	C₂×Q₈	—	23	8	no	—
153	C₂×Q₈	—	92	4	no	—
154	C₂×D₄	—	2	4	no	—
155	C₂×D₄	—	23	16	no	—
156	C₂×D₄	—	23	4	no	—
157	C₂×D₄	—	46	4	no	—
158	C₂×D₄	—	92	8	no	—
159	C₂×D₄	C₂²×D₂₃	1	4	yes	—
160	C₂²×C₄	—	2	8	no	—
161	C₂²×C₄	—	23	2	no	—
162	C₂²×C₄	—	23	4	no	—
163	C₂²×C₄	—	23	8	no	—
164	C₂²×C₄	—	46	8	no	—
165	C₂²×C₄	C₂²×D₂₃	1	8	yes	—
166	C₂²×C₄	C₂²×D₂₃	1	2	yes	—
167	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
168	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
169	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
170	C₄⋊C₄	—	23	4	no	—
171	C₄⋊C₄	—	23	8	no	—
172	C₄⋊C₄	—	23	8	no	—
173	C₄⋊C₄	C₂²×D₂₃	1	4	yes	—
174	C₂²⋊C₄	—	23	16	no	—
175	C₂²⋊C₄	—	23	16	no	—
176	C₂²⋊C₄	—	23	8	no	—
177	C₂²⋊C₄	C₂²×D₂₃	1	8	yes	—
178	C₄²	—	23	16	no	—
179	C₄²	C₂²×D₂₃	1	4	yes	—
180	C₂³	—	2	4	no	—
181	C₂³	—	2	4	no	—
182	C₂³	—	2	4	no	—
183	C₂³	—	23	4	no	—
184	C₂³	—	46	4	no	—
185	C₂³	—	46	4	no	—
186	C₂³	—	46	4	no	—
187	C₂³	C₂³×D₂₃	1	4	yes	—
188	C₂³	C₂³×D₂₃	1	1	yes	characteristic
189	Q₈	—	92	8	no	—
190	D₄	—	2	8	no	—
191	D₄	—	46	8	no	—
192	D₄	—	92	16	no	—
193	C₂×C₄	—	2	4	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	2	16	no	—
196	C₂×C₄	—	23	4	no	—
197	C₂×C₄	—	23	8	no	—
198	C₂×C₄	—	46	4	no	—
199	C₂×C₄	—	46	8	no	—
200	C₂×C₄	—	46	16	no	—
201	C₂×C₄	#67288	1	4	yes	—
202	C₂×C₄	#67290	1	4	yes	—
203	C₂×C₄	C₂³×D₂₃	1	4	yes	—
204	C₂×C₄	C₂³×D₂₃	1	2	yes	—
205	C₂×C₄	C₂³×D₂₃	1	2	yes	—
206	C₂×C₄	C₂³×D₂₃	1	2	yes	—
207	C₂²	—	2	4	no	—
208	C₂²	—	2	4	no	—
209	C₂²	—	2	4	no	—
210	C₂²	—	4	8	no	—
211	C₂²	—	46	4	no	—
212	C₂²	—	46	4	no	—
213	C₂²	—	46	4	no	—
214	C₂²	—	92	8	no	—
215	C₂²	C₂×#67288	1	2	yes	—
216	C₂²	C₂×#67290	1	2	yes	—
217	C₂²	#10625160	1	2	yes	—
218	C₂²	C₂⁴×D₂₃	1	1	yes	characteristic
219	C₄	—	2	4	no	—
220	C₄	—	2	2	no	—
221	C₄	—	46	4	no	—
222	C₄	—	46	8	no	—
223	C₄	C₂×#67288	1	4	yes	—
224	C₄	C₂×#67290	1	4	yes	—
225	C₂	—	4	4	no	—
226	C₂	—	92	4	no	—
227	C₂	#13093132	1	4	yes	direct product factor
228	C₂	C₂²×#67290	1	1	yes	characteristic
229	C₂	C₂²×#67288	1	1	yes	characteristic
230	C₂	C₂×#10625160	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂³×D₂₃
commutator, cosocle	C₂²×C₂₃	C₂⁵
Frattini	C₂²	C₂⁴×D₂₃
Fitting	C₂×C₄×D₄×C₂₃	C₂
socle	C₂³×C₂₃	C₂⁴
hypercenter	C₂×C₄×D₄	D₂₃
hypocenter	C₂₃	C₂×#555
LCS	G ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×D₄

Complements

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

N	H ≅ G/N	type
1. #13093132	C₂	direct (N × H)
2. C₂×#10625040	C₂	semidirect
3. C₂×#10625056	C₂	semidirect
4. C₂×#10625101	C₂	semidirect
5. C₂×#10625109	C₂	semidirect
6. C₂×C₄×D₉₂	C₂	semidirect
7. C₂×#10625097	C₂	semidirect
8. C₂×#10625063	C₂	semidirect
9. C₂×C₄×D₄×C₂₃	C₂	semidirect
16. C₄×D₄×C₂₃	C₂²	semidirect
21. #10625109	C₂²	semidirect
22. #10625101	C₂²	semidirect
25. #10625097	C₂²	semidirect
30. #10625063	C₂²	semidirect
31. #10625056	C₂²	semidirect
32. C₄×D₉₂	C₂²	semidirect
33. #10625040	C₂²	semidirect
116. C₂×C₄×D₄	D₂₃	semidirect
120. C₂×C₂₃	#555	semidirect
136. C₄×D₄	C₂×D₂₃	semidirect
149. C₂₃	C₂×#555	semidirect
227. C₂	#13093132	direct (N × H)

Maximal subgroups

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

subgroup	index
#13093132	2
C₂×#10625040	2
C₂×#10625056	2
C₂×#10625101	2
C₂×#10625109	2
C₂×C₄×D₉₂	2
C₂×#10625097	2
C₂×#10625063	2
C₂×C₄×D₄×C₂₃	2
C₂×#555	23

Subgroups — C2×#13093132 #15686937

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13093132 #15686937