Subgroups — C₂×#20265911 #27184719

Order 12928.

1240 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	#20265911	C₂	1	16	yes	maximal, direct product factor
2	C₂×#15692251	C₂	1	1	yes	maximal, characteristic
3	C₂×#15692241	C₂	1	2	yes	maximal, complemented
4	C₂×#15692241	C₂	1	2	yes	maximal, complemented
5	C₂×#15692240	C₂	1	2	yes	maximal, complemented
6	C₂×#15692236	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#15692220	C₂	1	2	yes	maximal, complemented
8	C₂×C₄×C₂²⋊D₁₀₁	C₂	1	2	yes	maximal, complemented
9	C₂×#15692215	C₂	1	2	yes	maximal, complemented
10	C₂×#123×C₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂²×Dic₂₀₂	C₂²	1	2	yes	—
12	C₂²×D₄₀₄	C₂²	1	1	yes	characteristic
13	C₂²×C₂²⋊D₁₀₁	C₂²	1	2	yes	—
14	C₂²×C₂²⋊D₁₀₁	C₂²	1	2	yes	—
15	C₂³×Dic₁₀₁	C₂²	1	1	yes	characteristic
16	C₂²×C₄×D₁₀₁	C₂²	1	2	yes	—
17	#15692251	C₂²	1	8	yes	—
18	#15692241	C₂²	1	16	yes	complemented
19	#15692241	C₂²	1	16	yes	complemented
20	#15692240	C₂²	1	16	yes	complemented
21	#15692236	C₂²	1	8	yes	complemented
22	#15692220	C₂²	1	16	yes	complemented
23	C₄×C₂²⋊D₁₀₁	C₂²	1	16	yes	complemented
24	#15692215	C₂²	1	16	yes	complemented
25	C₂×#13108053	C₂²	1	2	yes	—
26	C₂×#13108053	C₂²	1	2	yes	—
27	C₂×C₂²⋊Dic₁₀₁	C₂²	1	2	yes	—
28	C₂×C₂²⋊Dic₁₀₁	C₂²	1	1	yes	characteristic
29	C₂×#13108051	C₂²	1	2	yes	—
30	C₂×#13108051	C₂²	1	2	yes	—
31	C₂×#13108051	C₂²	1	2	yes	—
32	C₂×C₄×Dic₁₀₁	C₂²	1	2	yes	—
33	C₂×C₄×Dic₁₀₁	C₂²	1	2	yes	—
34	C₂×C₄×Dic₁₀₁	C₂²	1	1	yes	characteristic
35	C₂²×D₄×C₁₀₁	C₂²	1	1	yes	characteristic
36	C₂³×C₄×C₁₀₁	C₂²	1	1	yes	characteristic
37	#123×C₁₀₁	C₂²	1	8	yes	complemented
38	C₂×C₄⋊C₄×C₁₀₁	C₂²	1	2	yes	—
39	C₂×C₂²⋊C₄×C₁₀₁	C₂²	1	2	yes	—
40	C₂×C₂²⋊C₄×C₁₀₁	C₂²	1	1	yes	characteristic
41	C₂³×D₁₀₁	C₂³	1	2	yes	—
42	C₂×Dic₂₀₂	—	2	8	no	—
43	C₂×Dic₂₀₂	C₂³	1	8	yes	—
44	C₂×D₄₀₄	—	4	2	no	—
45	C₂×D₄₀₄	C₂³	1	4	yes	—
46	C₂×C₂²⋊D₁₀₁	—	2	8	no	—
47	C₂×C₂²⋊D₁₀₁	—	2	8	no	—
48	C₂×C₂²⋊D₁₀₁	C₂³	1	8	yes	—
49	C₂×C₂²⋊D₁₀₁	C₂³	1	8	yes	—
50	C₂²×Dic₁₀₁	—	4	2	no	—
51	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
52	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
53	C₂²×Dic₁₀₁	C₂³	1	4	yes	—
54	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
55	C₂×C₄×D₁₀₁	—	4	4	no	—
56	C₂×C₄×D₁₀₁	C₂³	1	8	yes	—
57	#13108053	C₂³	1	8	yes	—
58	#13108053	C₂³	1	8	yes	—
59	C₂²⋊Dic₁₀₁	C₂³	1	8	yes	—
60	C₂²⋊Dic₁₀₁	C₂³	1	4	yes	—
61	#13108051	C₂³	1	8	yes	—
62	#13108051	C₂³	1	8	yes	—
63	#13108051	C₂³	1	8	yes	—
64	C₄×Dic₁₀₁	C₂³	1	8	yes	—
65	C₄×Dic₁₀₁	C₂³	1	8	yes	—
66	C₄×Dic₁₀₁	C₂³	1	4	yes	—
67	C₂⁴×C₁₀₁	C₂³	1	1	yes	characteristic
68	C₂⁴×C₁₀₁	C₂³	1	1	yes	characteristic
69	C₂×D₄×C₁₀₁	—	4	2	no	—
70	C₂×D₄×C₁₀₁	C₂³	1	4	yes	—
71	C₂²×C₄×C₁₀₁	—	4	2	no	—
72	C₂²×C₄×C₁₀₁	C₂³	1	2	yes	—
73	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
74	C₂²×C₄×C₁₀₁	C₂³	1	2	yes	—
75	C₂²×C₄×C₁₀₁	C₂³	1	4	yes	—
76	C₄⋊C₄×C₁₀₁	C₂³	1	8	yes	—
77	C₂²⋊C₄×C₁₀₁	C₂³	1	8	yes	—
78	C₂²⋊C₄×C₁₀₁	C₂³	1	4	yes	—
79	C₂³×C₁₀₁	—	2	4	no	—
80	C₂³×C₁₀₁	—	2	4	no	—
81	C₂³×C₁₀₁	—	2	2	no	—
82	C₂³×C₁₀₁	—	2	2	no	—
83	C₂³×C₁₀₁	C₂⁴	1	2	yes	—
84	C₂³×C₁₀₁	C₂⁴	1	2	yes	—
85	C₂³×C₁₀₁	C₂⁴	1	1	yes	characteristic
86	C₂²×D₁₀₁	—	2	4	no	—
87	C₂²×D₁₀₁	—	2	4	no	—
88	C₂²×D₁₀₁	—	2	4	no	—
89	C₂²×D₁₀₁	C₂⁴	1	4	yes	—
90	D₄×C₁₀₁	—	4	4	no	—
91	C₂×C₄×C₁₀₁	—	2	4	no	—
92	C₂×C₄×C₁₀₁	—	2	4	no	—
93	C₂×C₄×C₁₀₁	—	2	2	no	—
94	C₂×C₄×C₁₀₁	—	4	4	no	—
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	C₂²⋊D₁₀₁	—	2	16	no	—
99	C₂²⋊D₁₀₁	—	2	16	no	—
100	C₂×Dic₁₀₁	—	2	4	no	—
101	C₂×Dic₁₀₁	—	2	4	no	—
102	C₂×Dic₁₀₁	—	4	4	no	—
103	C₂×Dic₁₀₁	#41	1	8	yes	—
104	C₂×Dic₁₀₁	C₂⁴	1	4	yes	—
105	C₂×Dic₁₀₁	C₂⁴	1	4	yes	—
106	C₂×Dic₁₀₁	C₂⁴	1	4	yes	—
107	D₄₀₄	—	4	4	no	—
108	C₄×D₁₀₁	—	4	8	no	—
109	Dic₂₀₂	—	2	16	no	—
110	C₂²×C₁₀₁	—	2	4	no	—
111	C₂²×C₁₀₁	—	2	4	no	—
112	C₂²×C₁₀₁	—	2	2	no	—
113	C₂²×C₁₀₁	—	2	2	no	—
114	C₂²×C₁₀₁	—	4	4	no	—
115	C₂²×C₁₀₁	—	4	4	no	—
116	C₂²×C₁₀₁	C₂×#41	1	4	yes	—
117	C₂²×C₁₀₁	H(5,𝔽₂)	1	2	yes	—
118	C₂²×C₁₀₁	C₂⁵	1	1	yes	characteristic
119	C₂×D₁₀₁	—	2	4	no	—
120	C₂×D₁₀₁	—	2	4	no	—
121	C₂×D₁₀₁	—	2	4	no	—
122	C₂×D₁₀₁	—	4	8	no	—
123	C₄×C₁₀₁	—	2	4	no	—
124	C₄×C₁₀₁	—	2	4	no	—
125	C₄×C₁₀₁	—	2	2	no	—
126	Dic₁₀₁	—	2	4	no	—
127	Dic₁₀₁	—	2	4	no	—
128	Dic₁₀₁	C₂×#41	1	8	yes	—
129	C₂×C₁₀₁	—	4	2	no	—
130	C₂×C₁₀₁	—	4	2	no	—
131	C₂×C₁₀₁	#559	1	4	yes	complemented
132	C₂×C₁₀₁	C₂²×#41	1	2	yes	—
133	C₂×C₁₀₁	C₂×H(5,𝔽₂)	1	1	yes	characteristic
134	D₁₀₁	—	4	4	no	—
135	C₂×#559	—	101	1	no	maximal, Sylow
136	C₁₀₁	C₂×#559	1	1	yes	characteristic, complemented, Sylow
137	#559	—	101	16	no	—
138	C₂×#127	—	101	1	no	—
139	C₂×#124	—	101	2	no	—
140	C₂×#124	—	101	2	no	—
141	C₂×#123	—	101	1	no	—
142	C₂×#123	—	101	2	no	—
143	C₂×#123	D₁₀₁	1	1	yes	characteristic, complemented
144	C₂×C₄×Q₈	—	101	2	no	—
145	C₂×C₄×D₄	—	101	2	no	—
146	C₂×C₄×D₄	—	101	2	no	—
147	C₂²×Q₈	—	101	2	no	—
148	C₂²×D₄	—	101	2	no	—
149	C₂²×D₄	—	101	2	no	—
150	C₂²×D₄	—	101	1	no	—
151	C₂²×D₄	C₂×D₁₀₁	1	1	yes	characteristic
152	C₂³×C₄	—	101	1	no	—
153	C₂³×C₄	—	101	2	no	—
154	C₂³×C₄	C₂×D₁₀₁	1	1	yes	characteristic
155	#127	—	101	8	no	—
156	#124	—	101	16	no	—
157	#124	—	101	16	no	—
158	#123	—	101	8	no	—
159	#123	—	101	16	no	—
160	#123	C₂×D₁₀₁	1	8	yes	complemented
161	C₄×Q₈	—	101	16	no	—
162	C₄×D₄	—	101	16	no	—
163	C₄×D₄	—	101	16	no	—
164	C₂×C₄⋊C₄	—	101	2	no	—
165	C₂×C₄⋊C₄	—	101	2	no	—
166	C₂×C₄⋊C₄	C₂×D₁₀₁	1	2	yes	—
167	C₂×C₂²⋊C₄	—	101	2	no	—
168	C₂×C₂²⋊C₄	—	101	1	no	—
169	C₂×C₂²⋊C₄	—	101	2	no	—
170	C₂×C₂²⋊C₄	—	101	2	no	—
171	C₂×C₂²⋊C₄	—	101	2	no	—
172	C₂×C₂²⋊C₄	C₂×D₁₀₁	1	2	yes	—
173	C₂×C₂²⋊C₄	C₂×D₁₀₁	1	1	yes	characteristic
174	C₂×C₄²	—	101	2	no	—
175	C₂×C₄²	—	101	2	no	—
176	C₂×C₄²	—	101	1	no	—
177	C₂⁴	—	101	2	no	—
178	C₂⁴	C₂²×D₁₀₁	1	1	yes	characteristic
179	C₂⁴	C₂²×D₁₀₁	1	1	yes	characteristic
180	C₂×Q₈	—	101	8	no	—
181	C₂×Q₈	—	202	8	no	—
182	C₂×D₄	—	4	2	no	—
183	C₂×D₄	—	101	8	no	—
184	C₂×D₄	—	101	8	no	—
185	C₂×D₄	—	101	4	no	—
186	C₂×D₄	—	404	2	no	—
187	C₂×D₄	—	202	8	no	—
188	C₂×D₄	—	202	8	no	—
189	C₂×D₄	C₂²×D₁₀₁	1	4	yes	—
190	C₂²×C₄	—	4	2	no	—
191	C₂²×C₄	—	101	2	no	—
192	C₂²×C₄	—	101	4	no	—
193	C₂²×C₄	—	101	2	no	—
194	C₂²×C₄	—	101	2	no	—
195	C₂²×C₄	—	101	8	no	—
196	C₂²×C₄	—	404	4	no	—
197	C₂²×C₄	—	404	2	no	—
198	C₂²×C₄	C₂²×D₁₀₁	1	2	yes	—
199	C₂²×C₄	C₂²×D₁₀₁	1	2	yes	—
200	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
201	C₂²×C₄	C₂²×D₁₀₁	1	4	yes	—
202	C₄⋊C₄	—	101	8	no	—
203	C₄⋊C₄	—	101	8	no	—
204	C₄⋊C₄	C₂²×D₁₀₁	1	8	yes	—
205	C₂²⋊C₄	—	101	8	no	—
206	C₂²⋊C₄	—	101	4	no	—
207	C₂²⋊C₄	—	101	8	no	—
208	C₂²⋊C₄	—	101	8	no	—
209	C₂²⋊C₄	—	101	8	no	—
210	C₂²⋊C₄	C₂²×D₁₀₁	1	8	yes	—
211	C₂²⋊C₄	C₂²×D₁₀₁	1	4	yes	—
212	C₄²	—	101	8	no	—
213	C₄²	—	101	8	no	—
214	C₄²	—	101	4	no	—
215	C₂³	—	2	4	no	—
216	C₂³	—	2	4	no	—
217	C₂³	—	2	2	no	—
218	C₂³	—	2	2	no	—
219	C₂³	—	101	4	no	—
220	C₂³	—	202	4	no	—
221	C₂³	—	202	4	no	—
222	C₂³	—	202	4	no	—
223	C₂³	C₂³×D₁₀₁	1	2	yes	—
224	C₂³	C₂³×D₁₀₁	1	2	yes	—
225	C₂³	C₂³×D₁₀₁	1	1	yes	characteristic
226	Q₈	—	202	16	no	—
227	D₄	—	4	4	no	—
228	D₄	—	404	4	no	—
229	D₄	—	202	16	no	—
230	D₄	—	202	16	no	—
231	C₂×C₄	—	2	4	no	—
232	C₂×C₄	—	2	4	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	4	4	no	—
235	C₂×C₄	—	101	8	no	—
236	C₂×C₄	—	101	4	no	—
237	C₂×C₄	—	101	4	no	—
238	C₂×C₄	—	101	4	no	—
239	C₂×C₄	—	404	8	no	—
240	C₂×C₄	—	404	4	no	—
241	C₂×C₄	—	202	4	no	—
242	C₂×C₄	—	202	4	no	—
243	C₂×C₄	C₂³×D₁₀₁	1	4	yes	—
244	C₂×C₄	C₂³×D₁₀₁	1	4	yes	—
245	C₂×C₄	C₂³×D₁₀₁	1	2	yes	—
246	C₂²	—	2	4	no	—
247	C₂²	—	2	4	no	—
248	C₂²	—	2	2	no	—
249	C₂²	—	2	2	no	—
250	C₂²	—	4	4	no	—
251	C₂²	—	4	4	no	—
252	C₂²	—	404	8	no	—
253	C₂²	—	202	4	no	—
254	C₂²	—	202	4	no	—
255	C₂²	—	202	4	no	—
256	C₂²	#41×D₁₀₁	1	4	yes	—
257	C₂²	#15692300	1	2	yes	—
258	C₂²	C₂⁴×D₁₀₁	1	1	yes	characteristic
259	C₄	—	2	4	no	—
260	C₄	—	2	4	no	—
261	C₄	—	2	2	no	—
262	C₄	—	101	8	no	—
263	C₄	—	202	4	no	—
264	C₄	—	202	4	no	—
265	C₂	—	4	2	no	—
266	C₂	—	4	2	no	—
267	C₂	—	404	4	no	—
268	C₂	#20265911	1	4	yes	direct product factor
269	C₂	C₂×#41×D₁₀₁	1	2	yes	—
270	C₂	C₂×#15692300	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂³×D₁₀₁
commutator, cosocle	C₂²×C₁₀₁	C₂⁵
Frattini	C₂²	C₂⁴×D₁₀₁
Fitting	C₂×#123×C₁₀₁	C₂
socle	C₂³×C₁₀₁	C₂⁴
prime radical	C₂×#15692251	C₂
hypercenter	C₂×#123	D₁₀₁
hypocenter	C₁₀₁	C₂×#559
LCS	G ▷ C₂²×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂³ ◁ C₂×#123

Complements

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

N	H ≅ G/N	type
1. #20265911	C₂	direct (N × H)
3. C₂×#15692241	C₂	semidirect
4. C₂×#15692241	C₂	semidirect
5. C₂×#15692240	C₂	semidirect
6. C₂×#15692236	C₂	semidirect
7. C₂×#15692220	C₂	semidirect
8. C₂×C₄×C₂²⋊D₁₀₁	C₂	semidirect
9. C₂×#15692215	C₂	semidirect
10. C₂×#123×C₁₀₁	C₂	semidirect
18. #15692241	C₂²	semidirect
19. #15692241	C₂²	semidirect
20. #15692240	C₂²	semidirect
21. #15692236	C₂²	semidirect
22. #15692220	C₂²	semidirect
23. C₄×C₂²⋊D₁₀₁	C₂²	semidirect
24. #15692215	C₂²	semidirect
37. #123×C₁₀₁	C₂²	semidirect
131. C₂×C₁₀₁	#559	semidirect
136. C₁₀₁	C₂×#559	semidirect
143. C₂×#123	D₁₀₁	semidirect
160. #123	C₂×D₁₀₁	semidirect
268. C₂	#20265911	direct (N × H)

Maximal subgroups

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

subgroup	index
#20265911	2
C₂×#15692251	2
C₂×#15692241	2
C₂×#15692240	2
C₂×#15692236	2
C₂×#15692220	2
C₂×C₄×C₂²⋊D₁₀₁	2
C₂×#15692215	2
C₂×#123×C₁₀₁	2
C₂×#559	101

Subgroups — C2×#20265911 #27184719

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20265911 #27184719