Subgroups — C₂×#20265787 #27184498

Order 12928.

1004 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20265787	C₂	1	16	yes	maximal, direct product factor
2	C₂×#15692253	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#15692248	C₂	1	2	yes	maximal
4	C₂×#15692248	C₂	1	2	yes	maximal
5	C₂×#15692246	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#15692246	C₂	1	2	yes	maximal, complemented
7	C₂×#15692245	C₂	1	2	yes	maximal, complemented
8	C₂×#15692234	C₂	1	1	yes	maximal, characteristic
9	C₂×Q₈×Dic₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#15692220	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×#15692214	C₂	1	1	yes	maximal, characteristic
12	C₂×C₂²⋊Q₈×C₁₀₁	C₂	1	1	yes	maximal, characteristic
13	C₂²×Dic₂₀₂	C₂²	1	1	yes	characteristic
14	#15692253	C₂²	1	8	yes	complemented
15	#15692248	C₂²	1	16	yes	—
16	#15692248	C₂²	1	16	yes	—
17	#15692246	C₂²	1	8	yes	complemented
18	#15692246	C₂²	1	16	yes	complemented
19	#15692245	C₂²	1	16	yes	complemented
20	#15692234	C₂²	1	8	yes	—
21	Q₈×Dic₁₀₁	C₂²	1	8	yes	complemented
22	#15692220	C₂²	1	8	yes	complemented
23	#15692214	C₂²	1	8	yes	—
24	C₂×C₄⋊Dic₁₀₁	C₂²	1	1	yes	characteristic
25	C₂×C₄⋊Dic₁₀₁	C₂²	1	1	yes	characteristic
26	C₂×C₄⋊Dic₁₀₁	C₂²	1	1	yes	characteristic
27	C₂×C₄⋊Dic₁₀₁	C₂²	1	2	yes	—
28	C₂×C₄⋊Dic₁₀₁	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×#13108053	C₂²	1	2	yes	—
33	C₂×#13108053	C₂²	1	2	yes	—
34	C₂×C₂²⋊Dic₁₀₁	C₂²	1	1	yes	characteristic
35	C₂×C₂²⋊Dic₁₀₁	C₂²	1	1	yes	characteristic
36	C₂×C₂²⋊Dic₁₀₁	C₂²	1	2	yes	—
37	C₂×C₄×Dic₁₀₁	C₂²	1	1	yes	characteristic
38	C₂×C₄×Dic₁₀₁	C₂²	1	1	yes	characteristic
39	C₂×C₄×Dic₁₀₁	C₂²	1	2	yes	—
40	C₂×C₄×Dic₁₀₁	C₂²	1	2	yes	—
41	C₂²×Q₈×C₁₀₁	C₂²	1	1	yes	characteristic
42	C₂³×C₄×C₁₀₁	C₂²	1	1	yes	characteristic
43	C₂²⋊Q₈×C₁₀₁	C₂²	1	8	yes	—
44	C₂×C₄⋊C₄×C₁₀₁	C₂²	1	1	yes	characteristic
45	C₂×C₄⋊C₄×C₁₀₁	C₂²	1	2	yes	—
46	C₂×C₂²⋊C₄×C₁₀₁	C₂²	1	2	yes	—
47	C₂×Dic₂₀₂	—	2	4	no	—
48	C₂×Dic₂₀₂	C₂³	1	4	yes	—
49	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
50	C₂²×Dic₁₀₁	C₂³	1	1	yes	characteristic
51	C₂²×Dic₁₀₁	C₂³	1	1	yes	characteristic
52	C₂²×Dic₁₀₁	C₂³	1	1	yes	characteristic
53	C₂²×Dic₁₀₁	C₂³	1	1	yes	characteristic
54	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
55	C₄⋊Dic₁₀₁	C₂³	1	4	yes	—
56	C₄⋊Dic₁₀₁	C₂³	1	4	yes	—
57	C₄⋊Dic₁₀₁	C₂³	1	4	yes	—
58	C₄⋊Dic₁₀₁	C₂³	1	8	yes	—
59	C₄⋊Dic₁₀₁	C₂³	1	8	yes	—
60	#13108053	C₂³	1	8	yes	—
61	#13108053	C₂³	1	8	yes	—
62	#13108053	C₂³	1	8	yes	—
63	#13108053	C₂³	1	8	yes	—
64	#13108053	C₂³	1	8	yes	—
65	C₂²⋊Dic₁₀₁	C₂³	1	4	yes	—
66	C₂²⋊Dic₁₀₁	C₂³	1	4	yes	—
67	C₂²⋊Dic₁₀₁	C₂³	1	8	yes	—
68	C₄×Dic₁₀₁	C₂³	1	4	yes	—
69	C₄×Dic₁₀₁	C₂³	1	4	yes	—
70	C₄×Dic₁₀₁	C₂³	1	8	yes	—
71	C₄×Dic₁₀₁	C₂³	1	8	yes	—
72	C₂⁴×C₁₀₁	C₂³	1	1	yes	characteristic
73	C₂×Q₈×C₁₀₁	—	2	4	no	—
74	C₂×Q₈×C₁₀₁	C₂³	1	4	yes	—
75	C₂²×C₄×C₁₀₁	—	2	4	no	—
76	C₂²×C₄×C₁₀₁	C₂³	1	2	yes	—
77	C₂²×C₄×C₁₀₁	C₂³	1	2	yes	—
78	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
79	C₂²×C₄×C₁₀₁	C₂³	1	4	yes	—
80	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
81	C₄⋊C₄×C₁₀₁	C₂³	1	4	yes	—
82	C₄⋊C₄×C₁₀₁	C₂³	1	8	yes	—
83	C₂²⋊C₄×C₁₀₁	C₂³	1	8	yes	—
84	C₂³×C₁₀₁	—	2	2	no	—
85	C₂³×C₁₀₁	—	2	2	no	—
86	C₂³×C₁₀₁	—	2	2	no	—
87	C₂³×C₁₀₁	C₂⁴	1	2	yes	—
88	C₂³×C₁₀₁	C₂⁴	1	1	yes	characteristic
89	Q₈×C₁₀₁	—	2	8	no	—
90	C₂×C₄×C₁₀₁	—	2	2	no	—
91	C₂×C₄×C₁₀₁	—	2	4	no	—
92	C₂×C₄×C₁₀₁	—	2	4	no	—
93	C₂×C₄×C₁₀₁	—	2	8	no	—
94	C₂×C₄×C₁₀₁	#41	1	4	yes	—
95	C₂×C₄×C₁₀₁	C₂⁴	1	4	yes	—
96	C₂×C₄×C₁₀₁	C₂⁴	1	4	yes	—
97	C₂×C₄×C₁₀₁	C₂⁴	1	2	yes	—
98	C₂×C₄×C₁₀₁	C₂⁴	1	2	yes	—
99	C₂×Dic₁₀₁	—	2	4	no	—
100	C₂×Dic₁₀₁	—	2	2	no	—
101	C₂×Dic₁₀₁	—	2	2	no	—
102	C₂×Dic₁₀₁	—	2	2	no	—
103	C₂×Dic₁₀₁	—	2	2	no	—
104	C₂×Dic₁₀₁	—	2	4	no	—
105	C₂×Dic₁₀₁	C₂⁴	1	2	yes	—
106	C₂×Dic₁₀₁	C₂⁴	1	2	yes	—
107	C₂×Dic₁₀₁	C₂⁴	1	2	yes	—
108	C₂×Dic₁₀₁	C₂⁴	1	4	yes	—
109	C₂×Dic₁₀₁	C₂⁴	1	2	yes	—
110	C₂×Dic₁₀₁	C₂⁴	1	4	yes	—
111	Dic₂₀₂	—	2	8	no	—
112	C₂²×C₁₀₁	—	2	2	no	—
113	C₂²×C₁₀₁	—	2	2	no	—
114	C₂²×C₁₀₁	—	2	2	no	—
115	C₂²×C₁₀₁	—	4	4	no	—
116	C₂²×C₁₀₁	C₂×#41	1	2	yes	—
117	C₂²×C₁₀₁	2₋¹⁺⁴	1	2	yes	—
118	C₂²×C₁₀₁	2₋¹⁺⁴	1	2	yes	—
119	C₂²×C₁₀₁	C₂⁵	1	1	yes	characteristic
120	C₄×C₁₀₁	—	2	2	no	—
121	C₄×C₁₀₁	—	2	4	no	—
122	C₄×C₁₀₁	—	2	4	no	—
123	C₄×C₁₀₁	C₂×#41	1	4	yes	—
124	Dic₁₀₁	—	2	4	no	—
125	Dic₁₀₁	—	2	2	no	—
126	Dic₁₀₁	—	2	2	no	—
127	Dic₁₀₁	—	2	2	no	—
128	Dic₁₀₁	—	2	2	no	—
129	Dic₁₀₁	—	2	4	no	—
130	C₂×C₁₀₁	—	4	2	no	—
131	C₂×C₁₀₁	#541	1	4	yes	complemented
132	C₂×C₁₀₁	C₂²×#41	1	1	yes	characteristic
133	C₂×C₁₀₁	C₂×2₋¹⁺⁴	1	1	yes	characteristic
134	C₂×C₁₀₁	C₂×2₋¹⁺⁴	1	1	yes	characteristic
135	C₂×#541	—	101	1	no	maximal, Sylow
136	C₁₀₁	C₂×#541	1	1	yes	characteristic, complemented, Sylow
137	#541	—	101	16	no	—
138	C₂×C₄⋊Q₈	—	101	1	no	—
139	C₂×#126	—	101	2	no	—
140	C₂×#126	—	101	2	no	—
141	C₂×#125	—	101	2	no	—
142	C₂×#125	—	101	1	no	—
143	C₂×#125	—	101	2	no	—
144	C₂×C₂²⋊Q₈	—	101	1	no	—
145	C₂×C₂²⋊Q₈	D₁₀₁	1	1	yes	characteristic
146	C₂×C₄×Q₈	—	101	1	no	—
147	C₂×C₄×Q₈	—	101	1	no	—
148	C₂×#117	—	101	1	no	—
149	C₂²×Q₈	—	101	1	no	—
150	C₂²×Q₈	C₂×D₁₀₁	1	1	yes	characteristic
151	C₂³×C₄	C₂×D₁₀₁	1	1	yes	characteristic
152	C₄⋊Q₈	—	101	8	no	—
153	#126	—	101	16	no	—
154	#126	—	101	16	no	—
155	#125	—	101	16	no	—
156	#125	—	101	8	no	—
157	#125	—	101	16	no	—
158	C₂²⋊Q₈	—	101	8	no	—
159	C₂²⋊Q₈	C₂×D₁₀₁	1	8	yes	—
160	C₄×Q₈	—	101	8	no	—
161	C₄×Q₈	—	101	8	no	—
162	#117	—	101	8	no	—
163	C₂×C₄⋊C₄	—	101	1	no	—
164	C₂×C₄⋊C₄	—	101	1	no	—
165	C₂×C₄⋊C₄	—	101	1	no	—
166	C₂×C₄⋊C₄	—	101	2	no	—
167	C₂×C₄⋊C₄	—	101	2	no	—
168	C₂×C₄⋊C₄	—	101	2	no	—
169	C₂×C₄⋊C₄	—	101	2	no	—
170	C₂×C₄⋊C₄	—	101	2	no	—
171	C₂×C₄⋊C₄	—	101	2	no	—
172	C₂×C₄⋊C₄	—	101	2	no	—
173	C₂×C₄⋊C₄	C₂×D₁₀₁	1	1	yes	characteristic
174	C₂×C₄⋊C₄	C₂×D₁₀₁	1	2	yes	—
175	C₂×C₂²⋊C₄	—	101	1	no	—
176	C₂×C₂²⋊C₄	—	101	1	no	—
177	C₂×C₂²⋊C₄	—	101	2	no	—
178	C₂×C₂²⋊C₄	C₂×D₁₀₁	1	2	yes	—
179	C₂×C₄²	—	101	1	no	—
180	C₂×C₄²	—	101	1	no	—
181	C₂×C₄²	—	101	2	no	—
182	C₂×C₄²	—	101	2	no	—
183	C₂⁴	C₂²×D₁₀₁	1	1	yes	characteristic
184	C₂×Q₈	—	2	4	no	—
185	C₂×Q₈	—	101	4	no	—
186	C₂×Q₈	—	202	4	no	—
187	C₂×Q₈	C₂²×D₁₀₁	1	4	yes	—
188	C₂²×C₄	—	2	4	no	—
189	C₂²×C₄	—	101	1	no	—
190	C₂²×C₄	—	101	1	no	—
191	C₂²×C₄	—	101	1	no	—
192	C₂²×C₄	—	101	1	no	—
193	C₂²×C₄	—	101	2	no	—
194	C₂²×C₄	—	101	2	no	—
195	C₂²×C₄	C₂²×D₁₀₁	1	4	yes	—
196	C₂²×C₄	C₂²×D₁₀₁	1	2	yes	—
197	C₂²×C₄	C₂²×D₁₀₁	1	2	yes	—
198	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
199	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
200	C₄⋊C₄	—	101	8	no	—
201	C₄⋊C₄	—	101	8	no	—
202	C₄⋊C₄	—	101	8	no	—
203	C₄⋊C₄	—	101	8	no	—
204	C₄⋊C₄	—	101	8	no	—
205	C₄⋊C₄	—	101	4	no	—
206	C₄⋊C₄	—	101	4	no	—
207	C₄⋊C₄	—	101	4	no	—
208	C₄⋊C₄	—	101	8	no	—
209	C₄⋊C₄	—	101	8	no	—
210	C₄⋊C₄	C₂²×D₁₀₁	1	4	yes	—
211	C₄⋊C₄	C₂²×D₁₀₁	1	8	yes	—
212	C₂²⋊C₄	—	101	4	no	—
213	C₂²⋊C₄	—	101	4	no	—
214	C₂²⋊C₄	—	101	8	no	—
215	C₂²⋊C₄	C₂²×D₁₀₁	1	8	yes	—
216	C₄²	—	101	4	no	—
217	C₄²	—	101	4	no	—
218	C₄²	—	101	8	no	—
219	C₄²	—	101	8	no	—
220	C₂³	—	2	2	no	—
221	C₂³	—	2	2	no	—
222	C₂³	—	2	2	no	—
223	C₂³	C₂³×D₁₀₁	1	2	yes	—
224	C₂³	C₂³×D₁₀₁	1	1	yes	characteristic
225	Q₈	—	2	8	no	—
226	Q₈	—	202	8	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	4	no	—
229	C₂×C₄	—	2	4	no	—
230	C₂×C₄	—	2	8	no	—
231	C₂×C₄	—	101	2	no	—
232	C₂×C₄	—	101	2	no	—
233	C₂×C₄	—	101	2	no	—
234	C₂×C₄	—	101	2	no	—
235	C₂×C₄	—	101	4	no	—
236	C₂×C₄	—	101	4	no	—
237	C₂×C₄	—	202	4	no	—
238	C₂×C₄	—	202	2	no	—
239	C₂×C₄	—	202	2	no	—
240	C₂×C₄	—	202	2	no	—
241	C₂×C₄	—	202	2	no	—
242	C₂×C₄	—	202	4	no	—
243	C₂×C₄	#13108073	1	4	yes	—
244	C₂×C₄	C₂³×D₁₀₁	1	4	yes	—
245	C₂×C₄	C₂³×D₁₀₁	1	4	yes	—
246	C₂×C₄	C₂³×D₁₀₁	1	2	yes	—
247	C₂×C₄	C₂³×D₁₀₁	1	2	yes	—
248	C₂²	—	2	2	no	—
249	C₂²	—	2	2	no	—
250	C₂²	—	2	2	no	—
251	C₂²	—	4	4	no	—
252	C₂²	C₂×#13108073	1	2	yes	—
253	C₂²	#15692302	1	2	yes	—
254	C₂²	#15692303	1	2	yes	—
255	C₂²	C₂⁴×D₁₀₁	1	1	yes	characteristic
256	C₄	—	2	2	no	—
257	C₄	—	2	4	no	—
258	C₄	—	2	4	no	—
259	C₄	—	202	4	no	—
260	C₄	—	202	2	no	—
261	C₄	—	202	2	no	—
262	C₄	—	202	2	no	—
263	C₄	—	202	2	no	—
264	C₄	—	202	4	no	—
265	C₄	C₂×#13108073	1	4	yes	—
266	C₂	—	4	2	no	—
267	C₂	#20265787	1	4	yes	direct product factor
268	C₂	C₂²×#13108073	1	1	yes	characteristic
269	C₂	C₂×#15692302	1	1	yes	characteristic
270	C₂	C₂×#15692303	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×#541
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. #20265787	C₂	direct (N × H)
2. C₂×#15692253	C₂	semidirect
5. C₂×#15692246	C₂	semidirect
6. C₂×#15692246	C₂	semidirect
7. C₂×#15692245	C₂	semidirect
9. C₂×Q₈×Dic₁₀₁	C₂	semidirect
10. C₂×#15692220	C₂	semidirect
14. #15692253	C₂²	semidirect
17. #15692246	C₂²	semidirect
18. #15692246	C₂²	semidirect
19. #15692245	C₂²	semidirect
21. Q₈×Dic₁₀₁	C₂²	semidirect
22. #15692220	C₂²	semidirect
131. C₂×C₁₀₁	#541	semidirect
136. C₁₀₁	C₂×#541	semidirect
267. C₂	#20265787	direct (N × H)

Maximal subgroups

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

subgroup	index
#20265787	2
C₂×#15692253	2
C₂×#15692248	2
C₂×#15692246	2
C₂×#15692245	2
C₂×#15692234	2
C₂×Q₈×Dic₁₀₁	2
C₂×#15692220	2
C₂×#15692214	2
C₂×C₂²⋊Q₈×C₁₀₁	2
C₂×#541	101

Subgroups — C2×#20265787 #27184498

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20265787 #27184498