Subgroups — C₂×#20265806 #27184531

Order 12928.

1080 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20265806	C₂	1	16	yes	maximal, direct product factor
2	C₂×#15692254	C₂	1	2	yes	maximal, complemented
3	C₂×#15692253	C₂	1	2	yes	maximal, complemented
4	C₂×#15692245	C₂	1	2	yes	maximal, complemented
5	C₂×#15692245	C₂	1	2	yes	maximal, complemented
6	C₂×#15692234	C₂	1	1	yes	maximal, characteristic
7	C₂×#15692232	C₂	1	2	yes	maximal
8	C₂×#15692211	C₂	1	2	yes	maximal
9	C₂²×#13108053	C₂	1	1	yes	maximal, characteristic
10	C₂×C₂²⋊Q₈×C₁₀₁	C₂	1	1	yes	maximal, characteristic
11	C₂²×Dic₂₀₂	C₂²	1	2	yes	—
12	C₂²×Dic₂₀₂	C₂²	1	1	yes	characteristic
13	C₂³×Dic₁₀₁	C₂²	1	1	yes	characteristic
14	C₂³×Dic₁₀₁	C₂²	1	1	yes	characteristic
15	#15692254	C₂²	1	16	yes	complemented
16	#15692253	C₂²	1	16	yes	complemented
17	#15692245	C₂²	1	16	yes	complemented
18	#15692245	C₂²	1	16	yes	complemented
19	#15692234	C₂²	1	8	yes	—
20	#15692232	C₂²	1	16	yes	—
21	#15692211	C₂²	1	16	yes	—
22	C₂×C₄⋊Dic₁₀₁	C₂²	1	2	yes	—
23	C₂×C₄⋊Dic₁₀₁	C₂²	1	1	yes	characteristic
24	C₂×#13108053	C₂²	1	8	yes	—
25	C₂×#13108053	C₂²	1	2	yes	—
26	C₂×#13108053	C₂²	1	2	yes	—
27	C₂×#13108053	C₂²	1	2	yes	—
28	C₂×#13108053	C₂²	1	2	yes	—
29	C₂×#13108053	C₂²	1	2	yes	—
30	C₂×#13108053	C₂²	1	2	yes	—
31	C₂×#13108053	C₂²	1	2	yes	—
32	C₂×C₂²⋊Dic₁₀₁	C₂²	1	2	yes	—
33	C₂×C₄×Dic₁₀₁	C₂²	1	2	yes	—
34	C₂×C₄×Dic₁₀₁	C₂²	1	2	yes	—
35	C₂²×Q₈×C₁₀₁	C₂²	1	1	yes	characteristic
36	C₂³×C₄×C₁₀₁	C₂²	1	1	yes	characteristic
37	C₂²⋊Q₈×C₁₀₁	C₂²	1	8	yes	—
38	C₂×C₄⋊C₄×C₁₀₁	C₂²	1	1	yes	characteristic
39	C₂×C₄⋊C₄×C₁₀₁	C₂²	1	2	yes	—
40	C₂×C₂²⋊C₄×C₁₀₁	C₂²	1	2	yes	—
41	C₂×Dic₂₀₂	—	4	4	no	—
42	C₂×Dic₂₀₂	—	4	2	no	—
43	C₂×Dic₂₀₂	C₂³	1	8	yes	—
44	C₂×Dic₂₀₂	C₂³	1	4	yes	—
45	C₂²×Dic₁₀₁	—	4	2	no	—
46	C₂²×Dic₁₀₁	Q₈	1	8	yes	complemented
47	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
48	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
49	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
50	C₂²×Dic₁₀₁	C₂³	1	4	yes	—
51	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
52	C₂²×Dic₁₀₁	C₂³	1	4	yes	—
53	C₄⋊Dic₁₀₁	C₂³	1	8	yes	—
54	C₄⋊Dic₁₀₁	C₂³	1	4	yes	—
55	#13108053	C₂³	1	8	yes	—
56	#13108053	C₂³	1	8	yes	—
57	#13108053	C₂³	1	8	yes	—
58	#13108053	C₂³	1	8	yes	—
59	#13108053	C₂³	1	8	yes	—
60	#13108053	C₂³	1	8	yes	—
61	#13108053	C₂³	1	8	yes	—
62	C₂²⋊Dic₁₀₁	C₂³	1	8	yes	—
63	C₄×Dic₁₀₁	C₂³	1	8	yes	—
64	C₄×Dic₁₀₁	C₂³	1	8	yes	—
65	C₂⁴×C₁₀₁	C₂³	1	1	yes	characteristic
66	C₂×Q₈×C₁₀₁	—	4	2	no	—
67	C₂×Q₈×C₁₀₁	C₂³	1	4	yes	—
68	C₂²×C₄×C₁₀₁	—	4	2	no	—
69	C₂²×C₄×C₁₀₁	C₂³	1	2	yes	—
70	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
71	C₂²×C₄×C₁₀₁	C₂³	1	2	yes	—
72	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
73	C₂²×C₄×C₁₀₁	C₂³	1	4	yes	—
74	C₄⋊C₄×C₁₀₁	C₂³	1	4	yes	—
75	C₄⋊C₄×C₁₀₁	C₂³	1	8	yes	—
76	C₂²⋊C₄×C₁₀₁	C₂³	1	8	yes	—
77	C₂³×C₁₀₁	—	2	2	no	—
78	C₂³×C₁₀₁	—	2	2	no	—
79	C₂³×C₁₀₁	C₂×Q₈	1	4	yes	—
80	C₂³×C₁₀₁	C₂⁴	1	2	yes	—
81	C₂³×C₁₀₁	C₂⁴	1	1	yes	characteristic
82	Q₈×C₁₀₁	—	4	4	no	—
83	C₂×C₄×C₁₀₁	—	2	2	no	—
84	C₂×C₄×C₁₀₁	—	2	2	no	—
85	C₂×C₄×C₁₀₁	—	2	4	no	—
86	C₂×C₄×C₁₀₁	—	2	4	no	—
87	C₂×C₄×C₁₀₁	—	4	4	no	—
88	C₂×C₄×C₁₀₁	C₂⁴	1	4	yes	—
89	C₂×C₄×C₁₀₁	C₂⁴	1	4	yes	—
90	C₂×C₄×C₁₀₁	C₂⁴	1	2	yes	—
91	C₂×C₄×C₁₀₁	C₂⁴	1	2	yes	—
92	C₂×Dic₁₀₁	—	2	4	no	—
93	C₂×Dic₁₀₁	—	2	4	no	—
94	C₂×Dic₁₀₁	—	2	4	no	—
95	C₂×Dic₁₀₁	—	4	4	no	—
96	C₂×Dic₁₀₁	C₂×Q₈	1	8	yes	—
97	C₂×Dic₁₀₁	C₂×Q₈	1	16	yes	complemented
98	C₂×Dic₁₀₁	C₂⁴	1	4	yes	—
99	C₂×Dic₁₀₁	C₂⁴	1	4	yes	—
100	C₂×Dic₁₀₁	C₂⁴	1	4	yes	—
101	C₂×Dic₁₀₁	C₂⁴	1	4	yes	—
102	Dic₂₀₂	—	4	4	no	—
103	Dic₂₀₂	—	4	8	no	—
104	C₂²×C₁₀₁	—	2	8	no	—
105	C₂²×C₁₀₁	—	2	2	no	—
106	C₂²×C₁₀₁	—	2	2	no	—
107	C₂²×C₁₀₁	C₂²×Q₈	1	4	yes	—
108	C₂²×C₁₀₁	C₂²×Q₈	1	2	yes	—
109	C₂²×C₁₀₁	H(5,𝔽₂)	1	2	yes	—
110	C₂²×C₁₀₁	2₋¹⁺⁴	1	2	yes	—
111	C₂²×C₁₀₁	C₂⁵	1	1	yes	characteristic
112	C₄×C₁₀₁	—	2	2	no	—
113	C₄×C₁₀₁	—	2	4	no	—
114	C₄×C₁₀₁	—	2	2	no	—
115	C₄×C₁₀₁	—	2	4	no	—
116	Dic₁₀₁	—	2	4	no	—
117	Dic₁₀₁	—	2	4	no	—
118	Dic₁₀₁	—	2	4	no	—
119	Dic₁₀₁	C₂²×Q₈	1	8	yes	—
120	C₂×C₁₀₁	—	2	4	no	—
121	C₂×C₁₀₁	#544	1	4	yes	complemented
122	C₂×C₁₀₁	C₂³×Q₈	1	1	yes	characteristic
123	C₂×C₁₀₁	C₂×H(5,𝔽₂)	1	1	yes	characteristic
124	C₂×C₁₀₁	C₂×2₋¹⁺⁴	1	1	yes	characteristic
125	C₂×#544	—	101	1	no	maximal, Sylow
126	C₁₀₁	C₂×#544	1	1	yes	characteristic, complemented, Sylow
127	#544	—	101	16	no	—
128	C₂×C₄⋊Q₈	—	101	2	no	—
129	C₂×C₄⋊Q₈	—	101	2	no	—
130	C₂×#125	—	101	2	no	—
131	C₂×#125	—	101	2	no	—
132	C₂×C₂²⋊Q₈	—	101	1	no	—
133	C₂×C₂²⋊Q₈	—	101	2	no	—
134	C₂×C₂²⋊Q₈	D₁₀₁	1	1	yes	characteristic
135	C₂×#117	—	101	2	no	—
136	C₂²×C₄⋊C₄	—	101	1	no	—
137	C₂²×Q₈	—	101	1	no	—
138	C₂²×Q₈	—	101	2	no	—
139	C₂²×Q₈	C₂×D₁₀₁	1	1	yes	characteristic
140	C₂³×C₄	—	101	1	no	—
141	C₂³×C₄	—	101	1	no	—
142	C₂³×C₄	C₂×D₁₀₁	1	1	yes	characteristic
143	C₄⋊Q₈	—	101	16	no	—
144	C₄⋊Q₈	—	101	16	no	—
145	#125	—	101	16	no	—
146	#125	—	101	16	no	—
147	C₂²⋊Q₈	—	101	8	no	—
148	C₂²⋊Q₈	—	101	16	no	—
149	C₂²⋊Q₈	C₂×D₁₀₁	1	8	yes	—
150	#117	—	101	16	no	—
151	C₂×C₄⋊C₄	—	101	2	no	—
152	C₂×C₄⋊C₄	—	101	2	no	—
153	C₂×C₄⋊C₄	—	101	8	no	—
154	C₂×C₄⋊C₄	—	101	2	no	—
155	C₂×C₄⋊C₄	—	101	2	no	—
156	C₂×C₄⋊C₄	—	101	2	no	—
157	C₂×C₄⋊C₄	—	101	1	no	—
158	C₂×C₄⋊C₄	—	101	2	no	—
159	C₂×C₄⋊C₄	—	101	2	no	—
160	C₂×C₄⋊C₄	—	101	2	no	—
161	C₂×C₄⋊C₄	C₂×D₁₀₁	1	1	yes	characteristic
162	C₂×C₄⋊C₄	C₂×D₁₀₁	1	2	yes	—
163	C₂×C₂²⋊C₄	—	101	2	no	—
164	C₂×C₂²⋊C₄	C₂×D₁₀₁	1	2	yes	—
165	C₂×C₄²	—	101	2	no	—
166	C₂×C₄²	—	101	2	no	—
167	C₂⁴	C₂²×D₁₀₁	1	1	yes	characteristic
168	C₂×Q₈	—	4	2	no	—
169	C₂×Q₈	—	101	4	no	—
170	C₂×Q₈	—	101	8	no	—
171	C₂×Q₈	—	404	2	no	—
172	C₂×Q₈	—	404	4	no	—
173	C₂×Q₈	C₂²×D₁₀₁	1	4	yes	—
174	C₂²×C₄	—	4	2	no	—
175	C₂²×C₄	—	101	8	no	—
176	C₂²×C₄	—	101	4	no	—
177	C₂²×C₄	—	101	2	no	—
178	C₂²×C₄	—	101	4	no	—
179	C₂²×C₄	—	101	2	no	—
180	C₂²×C₄	—	101	2	no	—
181	C₂²×C₄	—	101	2	no	—
182	C₂²×C₄	—	404	2	no	—
183	C₂²×C₄	C₂²×D₁₀₁	1	2	yes	—
184	C₂²×C₄	C₂²×D₁₀₁	1	2	yes	—
185	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
186	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
187	C₂²×C₄	C₂²×D₁₀₁	1	4	yes	—
188	C₄⋊C₄	—	101	8	no	—
189	C₄⋊C₄	—	101	8	no	—
190	C₄⋊C₄	—	101	8	no	—
191	C₄⋊C₄	—	101	8	no	—
192	C₄⋊C₄	—	101	8	no	—
193	C₄⋊C₄	—	101	8	no	—
194	C₄⋊C₄	—	101	8	no	—
195	C₄⋊C₄	—	101	4	no	—
196	C₄⋊C₄	—	101	8	no	—
197	C₄⋊C₄	C₂²×D₁₀₁	1	4	yes	—
198	C₄⋊C₄	C₂²×D₁₀₁	1	8	yes	—
199	C₂²⋊C₄	—	101	8	no	—
200	C₂²⋊C₄	C₂²×D₁₀₁	1	8	yes	—
201	C₄²	—	101	8	no	—
202	C₄²	—	101	8	no	—
203	C₂³	—	2	2	no	—
204	C₂³	—	2	2	no	—
205	C₂³	Q₈×D₁₀₁	1	4	yes	—
206	C₂³	C₂³×D₁₀₁	1	2	yes	—
207	C₂³	C₂³×D₁₀₁	1	1	yes	characteristic
208	Q₈	—	4	4	no	—
209	Q₈	—	404	4	no	—
210	Q₈	—	404	8	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	2	4	no	—
214	C₂×C₄	—	2	4	no	—
215	C₂×C₄	—	4	4	no	—
216	C₂×C₄	—	101	8	no	—
217	C₂×C₄	—	101	16	no	—
218	C₂×C₄	—	101	4	no	—
219	C₂×C₄	—	101	4	no	—
220	C₂×C₄	—	101	4	no	—
221	C₂×C₄	—	101	4	no	—
222	C₂×C₄	—	404	4	no	—
223	C₂×C₄	—	202	4	no	—
224	C₂×C₄	—	202	4	no	—
225	C₂×C₄	—	202	4	no	—
226	C₂×C₄	C₂³×D₁₀₁	1	4	yes	—
227	C₂×C₄	C₂³×D₁₀₁	1	4	yes	—
228	C₂×C₄	C₂³×D₁₀₁	1	2	yes	—
229	C₂×C₄	C₂³×D₁₀₁	1	2	yes	—
230	C₂²	—	2	2	no	—
231	C₂²	—	2	2	no	—
232	C₂²	—	2	8	no	—
233	C₂²	C₂×Q₈×D₁₀₁	1	4	yes	—
234	C₂²	C₂×Q₈×D₁₀₁	1	2	yes	—
235	C₂²	#15692300	1	2	yes	—
236	C₂²	#15692302	1	2	yes	—
237	C₂²	C₂⁴×D₁₀₁	1	1	yes	characteristic
238	C₄	—	2	2	no	—
239	C₄	—	2	2	no	—
240	C₄	—	2	4	no	—
241	C₄	—	2	4	no	—
242	C₄	—	101	8	no	—
243	C₄	—	202	4	no	—
244	C₄	—	202	4	no	—
245	C₄	—	202	4	no	—
246	C₂	—	2	4	no	—
247	C₂	#20265806	1	4	yes	direct product factor
248	C₂	C₂²×Q₈×D₁₀₁	1	1	yes	characteristic
249	C₂	C₂×#15692300	1	1	yes	characteristic
250	C₂	C₂×#15692302	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂³×D₁₀₁
commutator, cosocle	C₂²×C₁₀₁	C₂⁵
Frattini	C₂²	C₂⁴×D₁₀₁
Fitting	C₂×C₂²⋊Q₈×C₁₀₁	C₂
socle	C₂³×C₁₀₁	C₂⁴
prime radical	C₂⁴×C₁₀₁	C₂³
hypercenter	C₂×C₂²⋊Q₈	D₁₀₁
hypocenter	C₁₀₁	C₂×#544
LCS	G ▷ C₂²×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₂²⋊Q₈

Complements

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

N	H ≅ G/N	type
1. #20265806	C₂	direct (N × H)
2. C₂×#15692254	C₂	semidirect
3. C₂×#15692253	C₂	semidirect
4. C₂×#15692245	C₂	semidirect
5. C₂×#15692245	C₂	semidirect
15. #15692254	C₂²	semidirect
16. #15692253	C₂²	semidirect
17. #15692245	C₂²	semidirect
18. #15692245	C₂²	semidirect
46. C₂²×Dic₁₀₁	Q₈	semidirect
97. C₂×Dic₁₀₁	C₂×Q₈	semidirect
121. C₂×C₁₀₁	#544	semidirect
126. C₁₀₁	C₂×#544	semidirect
247. C₂	#20265806	direct (N × H)

Maximal subgroups

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

subgroup	index
#20265806	2
C₂×#15692254	2
C₂×#15692253	2
C₂×#15692245	2
C₂×#15692234	2
C₂×#15692232	2
C₂×#15692211	2
C₂²×#13108053	2
C₂×C₂²⋊Q₈×C₁₀₁	2
C₂×#544	101

Subgroups — C2×#20265806 #27184531

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20265806 #27184531