Subgroups — C₂×#13092928 #15686639

Order 2944.

1880 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#67288	C₂	1	1	yes	maximal, characteristic, complemented
2	#13092928	C₂	1	16	yes	maximal, direct product factor
3	C₂×#10625100	C₂	1	2	yes	maximal, complemented
4	C₂×#10625095	C₂	1	2	yes	maximal, complemented
5	C₂×#10625101	C₂	1	2	yes	maximal, complemented
6	C₂×#10625124	C₂	1	2	yes	maximal, complemented
7	C₂×C₄×C₂²⋊D₂₃	C₂	1	4	yes	maximal, complemented
8	C₂×#10625097	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂⁴×C₄×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂⁵×C₂₃	C₂²	1	1	yes	characteristic
11	C₂³×C₄×C₂₃	—	2	8	no	—
12	C₂³×C₄×C₂₃	C₂²	1	2	yes	—
13	C₂³×C₄×C₂₃	C₂²	1	2	yes	—
14	C₂³×C₄×C₂₃	C₂²	1	1	yes	characteristic
15	C₂³×C₄×C₂₃	C₂²	1	1	yes	characteristic
16	C₂³×C₄×C₂₃	C₂²	1	8	yes	complemented
17	C₂²×C₂²⋊D₂₃	C₂²	1	4	yes	—
18	C₂²×C₂²⋊D₂₃	C₂²	1	2	yes	—
19	C₂×#67288	—	2	8	no	—
20	C₂×#67288	C₂²	1	8	yes	complemented
21	C₂²×D₉₂	C₂²	1	1	yes	characteristic, complemented
22	C₂²×C₄×D₂₃	C₂²	1	2	yes	complemented
23	C₂²×Dic₄₆	C₂²	1	1	yes	characteristic, complemented
24	#10625124	C₂²	1	16	yes	complemented
25	C₂×C₂²⋊Dic₂₃	C₂²	1	2	yes	—
26	C₂×C₂²⋊Dic₂₃	C₂²	1	4	yes	—
27	#10625101	C₂²	1	16	yes	complemented
28	#10625100	C₂²	1	16	yes	complemented
29	C₄×C₂²⋊D₂₃	C₂²	1	32	yes	complemented
30	C₂×(C₂×D₂₃)⋊C₄	C₂²	1	4	yes	complemented
31	#10625097	C₂²	1	8	yes	complemented
32	C₂×C₄⋊Dic₂₃	C₂²	1	2	yes	complemented
33	#10625095	C₂²	1	16	yes	complemented
34	C₂×Dic₂₃⋊C₄	C₂²	1	4	yes	complemented
35	C₂×C₄×Dic₂₃	C₂²	1	2	yes	complemented
36	C₂⁴×C₂₃	—	2	4	no	—
37	C₂⁴×C₂₃	—	2	8	no	—
38	C₂⁴×C₂₃	C₂³	1	4	yes	—
39	C₂⁴×C₂₃	C₂³	1	1	yes	characteristic
40	C₂⁴×C₂₃	C₂³	1	2	yes	—
41	C₂³×D₂₃	C₂³	1	2	yes	—
42	C₂²×C₄×C₂₃	—	2	4	no	—
43	C₂²×C₄×C₂₃	—	2	8	no	—
44	C₂²×C₄×C₂₃	—	2	8	no	—
45	C₂²×C₄×C₂₃	—	2	32	no	—
46	C₂²×C₄×C₂₃	D₄	1	8	yes	complemented
47	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
48	C₂²×C₄×C₂₃	C₂³	1	8	yes	—
49	C₂²×C₄×C₂₃	C₂³	1	8	yes	—
50	C₂²×C₄×C₂₃	C₂³	1	4	yes	—
51	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
52	C₂²×C₄×C₂₃	C₂³	1	4	yes	—
53	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
54	C₂×C₂²⋊D₂₃	—	2	8	no	—
55	C₂×C₂²⋊D₂₃	—	2	16	no	—
56	C₂×C₂²⋊D₂₃	C₂³	1	16	yes	—
57	C₂×C₂²⋊D₂₃	C₂³	1	8	yes	—
58	C₂²×Dic₂₃	C₂³	1	4	yes	—
59	C₂²×Dic₂₃	C₂³	1	2	yes	—
60	#67288	—	2	32	no	—
61	C₂×D₉₂	—	2	4	no	—
62	C₂×D₉₂	C₂³	1	4	yes	complemented
63	C₂×C₄×D₂₃	—	2	8	no	—
64	C₂×C₄×D₂₃	C₂³	1	8	yes	complemented
65	C₂×Dic₄₆	—	2	4	no	—
66	C₂×Dic₄₆	C₂³	1	4	yes	complemented
67	C₂²⋊Dic₂₃	C₂³	1	16	yes	—
68	C₂²⋊Dic₂₃	C₂³	1	8	yes	—
69	(C₂×D₂₃)⋊C₄	C₂³	1	16	yes	complemented
70	C₄⋊Dic₂₃	C₂³	1	8	yes	complemented
71	Dic₂₃⋊C₄	C₂³	1	16	yes	complemented
72	C₄×Dic₂₃	C₂³	1	8	yes	complemented
73	C₂³×C₂₃	—	2	32	no	—
74	C₂³×C₂₃	—	2	8	no	—
75	C₂³×C₂₃	—	2	4	no	—
76	C₂³×C₂₃	—	2	4	no	—
77	C₂³×C₂₃	—	2	4	no	—
78	C₂³×C₂₃	—	2	16	no	—
79	C₂³×C₂₃	C₂×D₄	1	4	yes	—
80	C₂³×C₂₃	#41	1	8	yes	complemented
81	C₂³×C₂₃	C₂⁴	1	2	yes	—
82	C₂³×C₂₃	C₂⁴	1	4	yes	—
83	C₂³×C₂₃	C₂⁴	1	1	yes	characteristic
84	C₂²×D₂₃	—	2	4	no	—
85	C₂²×D₂₃	—	2	8	no	—
86	C₂²×D₂₃	C₂⁴	1	4	yes	—
87	C₂×C₄×C₂₃	—	2	8	no	—
88	C₂×C₄×C₂₃	—	2	4	no	—
89	C₂×C₄×C₂₃	—	2	16	no	—
90	C₂×C₄×C₂₃	—	2	16	no	—
91	C₂×C₄×C₂₃	C₂×D₄	1	4	yes	—
92	C₂×C₄×C₂₃	C₂×D₄	1	4	yes	—
93	C₂×C₄×C₂₃	C₂×D₄	1	16	yes	complemented
94	C₂×C₄×C₂₃	C₂⁴	1	4	yes	—
95	C₂×C₄×C₂₃	C₂⁴	1	2	yes	—
96	C₂×C₄×C₂₃	C₂⁴	1	2	yes	—
97	C₂²⋊D₂₃	—	2	16	no	—
98	C₂²⋊D₂₃	—	2	32	no	—
99	C₂×Dic₂₃	—	2	4	no	—
100	C₂×Dic₂₃	—	2	8	no	—
101	C₂×Dic₂₃	C₂⁴	1	8	yes	—
102	C₂×Dic₂₃	C₂⁴	1	4	yes	—
103	D₉₂	—	2	8	no	—
104	C₄×D₂₃	—	2	16	no	—
105	Dic₄₆	—	2	8	no	—
106	C₂×#525	—	23	1	no	maximal, Sylow
107	C₂²×C₂₃	—	2	32	no	—
108	C₂²×C₂₃	—	2	8	no	—
109	C₂²×C₂₃	—	2	16	no	—
110	C₂²×C₂₃	—	2	4	no	—
111	C₂²×C₂₃	—	2	4	no	—
112	C₂²×C₂₃	—	2	4	no	—
113	C₂²×C₂₃	C₂²×D₄	1	4	yes	—
114	C₂²×C₂₃	C₂²×D₄	1	2	yes	—
115	C₂²×C₂₃	C₂×#41	1	4	yes	—
116	C₂²×C₂₃	C₂×#41	1	8	yes	complemented
117	C₂²×C₂₃	C₂⁵	1	1	yes	characteristic
118	C₂×D₂₃	—	2	4	no	—
119	C₂×D₂₃	—	2	8	no	—
120	C₂×D₂₃	—	4	8	no	—
121	C₄×C₂₃	—	2	4	no	—
122	C₄×C₂₃	C₂²×D₄	1	4	yes	—
123	C₄×C₂₃	C₂²×D₄	1	4	yes	—
124	Dic₂₃	—	2	4	no	—
125	Dic₂₃	—	2	8	no	—
126	C₂²×#41	—	23	1	no	—
127	C₂⁴×C₄	D₂₃	1	1	yes	characteristic, complemented
128	#525	—	23	16	no	—
129	C₂×#123	—	23	2	no	—
130	C₂×C₂²⋊Q₈	—	23	2	no	—
131	C₂×#121	—	23	2	no	—
132	C₂×#120	—	23	2	no	—
133	C₂×C₄×D₄	—	23	4	no	—
134	C₂×#117	—	23	1	no	—
135	C₂×C₂₃	—	2	4	no	—
136	C₂×C₂₃	—	2	8	no	—
137	C₂×C₂₃	#525	1	4	yes	complemented
138	C₂×C₂₃	C₂³×D₄	1	1	yes	characteristic
139	C₂×C₂₃	C₂²×#41	1	2	yes	—
140	D₂₃	—	4	4	no	—
141	C₂⁵	C₂×D₂₃	1	1	yes	characteristic
142	C₂×#41	—	23	8	no	—
143	C₂×#41	—	46	8	no	—
144	C₂²×Q₈	—	23	1	no	—
145	C₂²×D₄	—	23	4	no	—
146	C₂²×D₄	—	23	2	no	—
147	C₂²×D₄	—	23	1	no	—
148	C₂³×C₄	—	2	8	no	—
149	C₂³×C₄	—	23	2	no	—
150	C₂³×C₄	C₂×D₂₃	1	2	yes	—
151	C₂³×C₄	C₂×D₂₃	1	1	yes	characteristic
152	C₂³×C₄	C₂×D₂₃	1	2	yes	—
153	C₂³×C₄	C₂×D₂₃	1	1	yes	characteristic
154	C₂³×C₄	C₂×D₂₃	1	8	yes	complemented
155	#123	—	23	16	no	—
156	C₂²⋊Q₈	—	23	16	no	—
157	#121	—	23	16	no	—
158	#120	—	23	16	no	—
159	C₄×D₄	—	23	32	no	—
160	#117	—	23	8	no	—
161	C₂×C₄⋊C₄	—	23	2	no	—
162	C₂×C₄⋊C₄	—	23	4	no	—
163	C₂×C₂²⋊C₄	—	23	2	no	—
164	C₂×C₂²⋊C₄	—	23	4	no	—
165	C₂×C₂²⋊C₄	—	23	4	no	—
166	C₂×C₄²	—	23	2	no	—
167	C₂₃	C₂×#525	1	1	yes	characteristic, complemented, Sylow
168	C₂⁴	—	2	8	no	—
169	C₂⁴	—	2	4	no	—
170	C₂⁴	—	23	2	no	—
171	C₂⁴	C₂²×D₂₃	1	4	yes	—
172	C₂⁴	C₂²×D₂₃	1	2	yes	—
173	C₂⁴	C₂²×D₂₃	1	1	yes	characteristic
174	#41	—	46	32	no	—
175	C₂×Q₈	—	23	4	no	—
176	C₂×Q₈	—	46	4	no	—
177	C₂×D₄	—	23	16	no	—
178	C₂×D₄	—	23	8	no	—
179	C₂×D₄	—	23	4	no	—
180	C₂×D₄	—	46	16	no	—
181	C₂×D₄	—	46	8	no	—
182	C₂×D₄	—	46	4	no	—
183	C₂²×C₄	—	2	4	no	—
184	C₂²×C₄	—	2	8	no	—
185	C₂²×C₄	—	2	8	no	—
186	C₂²×C₄	—	2	32	no	—
187	C₂²×C₄	—	23	4	no	—
188	C₂²×C₄	—	23	2	no	—
189	C₂²×C₄	—	23	8	no	—
190	C₂²×C₄	—	46	8	no	—
191	C₂²×C₄	C₂²⋊D₂₃	1	8	yes	complemented
192	C₂²×C₄	C₂²×D₂₃	1	2	yes	—
193	C₂²×C₄	C₂²×D₂₃	1	8	yes	—
194	C₂²×C₄	C₂²×D₂₃	1	8	yes	—
195	C₂²×C₄	C₂²×D₂₃	1	4	yes	—
196	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
197	C₂²×C₄	C₂²×D₂₃	1	4	yes	—
198	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
199	C₄⋊C₄	—	23	8	no	—
200	C₄⋊C₄	—	23	16	no	—
201	C₂²⋊C₄	—	23	8	no	—
202	C₂²⋊C₄	—	23	16	no	—
203	C₂²⋊C₄	—	23	16	no	—
204	C₄²	—	23	8	no	—
205	C₂³	—	2	32	no	—
206	C₂³	—	2	4	no	—
207	C₂³	—	2	4	no	—
208	C₂³	—	2	16	no	—
209	C₂³	—	2	8	no	—
210	C₂³	—	2	4	no	—
211	C₂³	—	23	4	no	—
212	C₂³	—	46	8	no	—
213	C₂³	—	46	4	no	—
214	C₂³	#67288	1	8	yes	complemented
215	C₂³	C₂×C₂²⋊D₂₃	1	4	yes	—
216	C₂³	C₂³×D₂₃	1	4	yes	—
217	C₂³	C₂³×D₂₃	1	2	yes	—
218	C₂³	C₂³×D₂₃	1	1	yes	characteristic
219	Q₈	—	46	8	no	—
220	D₄	—	46	32	no	—
221	D₄	—	46	16	no	—
222	D₄	—	46	8	no	—
223	C₂×C₄	—	2	8	no	—
224	C₂×C₄	—	2	4	no	—
225	C₂×C₄	—	2	16	no	—
226	C₂×C₄	—	2	16	no	—
227	C₂×C₄	—	23	8	no	—
228	C₂×C₄	—	23	4	no	—
229	C₂×C₄	—	46	8	no	—
230	C₂×C₄	—	46	4	no	—
231	C₂×C₄	—	46	16	no	—
232	C₂×C₄	C₂×C₂²⋊D₂₃	1	4	yes	—
233	C₂×C₄	C₂×C₂²⋊D₂₃	1	4	yes	—
234	C₂×C₄	C₂×C₂²⋊D₂₃	1	16	yes	complemented
235	C₂×C₄	C₂³×D₂₃	1	4	yes	—
236	C₂×C₄	C₂³×D₂₃	1	2	yes	—
237	C₂×C₄	C₂³×D₂₃	1	2	yes	—
238	C₂²	—	2	32	no	—
239	C₂²	—	2	16	no	—
240	C₂²	—	2	8	no	—
241	C₂²	—	2	4	no	—
242	C₂²	—	2	4	no	—
243	C₂²	—	2	4	no	—
244	C₂²	—	46	4	no	—
245	C₂²	—	46	8	no	—
246	C₂²	—	92	8	no	—
247	C₂²	C₂×#67288	1	4	yes	—
248	C₂²	C₂×#67288	1	8	yes	complemented
249	C₂²	C₂²×C₂²⋊D₂₃	1	4	yes	—
250	C₂²	C₂²×C₂²⋊D₂₃	1	2	yes	—
251	C₂²	C₂⁴×D₂₃	1	1	yes	characteristic
252	C₄	—	2	4	no	—
253	C₄	—	46	4	no	—
254	C₄	—	46	8	no	—
255	C₄	C₂²×C₂²⋊D₂₃	1	4	yes	—
256	C₄	C₂²×C₂²⋊D₂₃	1	4	yes	—
257	C₂	—	2	8	no	—
258	C₂	—	2	4	no	—
259	C₂	—	92	4	no	—
260	C₂	#13092928	1	4	yes	direct product factor
261	C₂	C₂³×C₂²⋊D₂₃	1	1	yes	characteristic
262	C₂	C₂²×#67288	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂²×D₂₃
commutator, cosocle	C₂²×C₂₃	C₂⁵
Frattini	C₂²	C₂⁴×D₂₃
Fitting	C₂⁴×C₄×C₂₃	C₂
socle	C₂³×C₂₃	C₂⁴
hypercenter	C₂⁴×C₄	D₂₃
hypocenter	C₂₃	C₂×#525
LCS	G ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂⁴×C₄

Complements

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

N	H ≅ G/N	type
1. C₂²×#67288	C₂	semidirect
2. #13092928	C₂	direct (N × H)
3. C₂×#10625100	C₂	semidirect
4. C₂×#10625095	C₂	semidirect
5. C₂×#10625101	C₂	semidirect
6. C₂×#10625124	C₂	semidirect
7. C₂×C₄×C₂²⋊D₂₃	C₂	semidirect
8. C₂×#10625097	C₂	semidirect
9. C₂⁴×C₄×C₂₃	C₂	semidirect
16. C₂³×C₄×C₂₃	C₂²	semidirect
20. C₂×#67288	C₂²	semidirect
21. C₂²×D₉₂	C₂²	semidirect
22. C₂²×C₄×D₂₃	C₂²	semidirect
23. C₂²×Dic₄₆	C₂²	semidirect
24. #10625124	C₂²	semidirect
27. #10625101	C₂²	semidirect
28. #10625100	C₂²	semidirect
29. C₄×C₂²⋊D₂₃	C₂²	semidirect
30. C₂×(C₂×D₂₃)⋊C₄	C₂²	semidirect
31. #10625097	C₂²	semidirect
32. C₂×C₄⋊Dic₂₃	C₂²	semidirect
33. #10625095	C₂²	semidirect
34. C₂×Dic₂₃⋊C₄	C₂²	semidirect
35. C₂×C₄×Dic₂₃	C₂²	semidirect
46. C₂²×C₄×C₂₃	D₄	semidirect
62. C₂×D₉₂	C₂³	semidirect
64. C₂×C₄×D₂₃	C₂³	semidirect
66. C₂×Dic₄₆	C₂³	semidirect
69. (C₂×D₂₃)⋊C₄	C₂³	semidirect
70. C₄⋊Dic₂₃	C₂³	semidirect
71. Dic₂₃⋊C₄	C₂³	semidirect
72. C₄×Dic₂₃	C₂³	semidirect
80. C₂³×C₂₃	#41	semidirect
93. C₂×C₄×C₂₃	C₂×D₄	semidirect
116. C₂²×C₂₃	C₂×#41	semidirect
127. C₂⁴×C₄	D₂₃	semidirect
137. C₂×C₂₃	#525	semidirect
154. C₂³×C₄	C₂×D₂₃	semidirect
167. C₂₃	C₂×#525	semidirect
191. C₂²×C₄	C₂²⋊D₂₃	semidirect
214. C₂³	#67288	semidirect
234. C₂×C₄	C₂×C₂²⋊D₂₃	semidirect
248. C₂²	C₂×#67288	semidirect
260. C₂	#13092928	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×#67288	2
#13092928	2
C₂×#10625100	2
C₂×#10625095	2
C₂×#10625101	2
C₂×#10625124	2
C₂×C₄×C₂²⋊D₂₃	2
C₂×#10625097	2
C₂⁴×C₄×C₂₃	2
C₂×#525	23

Subgroups — C2×#13092928 #15686639

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13092928 #15686639