Subgroups — C₂×#20265909 #27184717

Order 12928.

1240 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20265909	C₂	1	16	yes	maximal, direct product factor
2	C₂×#15692251	C₂	1	1	yes	maximal, characteristic
3	C₂×#15692244	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#15692241	C₂	1	2	yes	maximal, complemented
5	C₂×#15692239	C₂	1	2	yes	maximal, complemented
6	C₂×#15692237	C₂	1	2	yes	maximal, complemented
7	C₂×Q₈×Dic₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄×Dic₂₀₂	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₄×D₄₀₄	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×D₄×Dic₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×#15692215	C₂	1	2	yes	maximal, complemented
12	C₂×#124×C₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂²×Dic₂₀₂	C₂²	1	1	yes	characteristic
14	C₂²×D₄₀₄	C₂²	1	1	yes	characteristic
15	C₂²×C₂²⋊D₁₀₁	C₂²	1	2	yes	—
16	C₂²×C₂²⋊D₁₀₁	C₂²	1	2	yes	—
17	C₂³×Dic₁₀₁	C₂²	1	2	yes	—
18	C₂²×C₄×D₁₀₁	C₂²	1	2	yes	—
19	#15692251	C₂²	1	8	yes	—
20	#15692244	C₂²	1	8	yes	complemented
21	#15692241	C₂²	1	16	yes	complemented
22	#15692239	C₂²	1	16	yes	complemented
23	#15692237	C₂²	1	16	yes	complemented
24	Q₈×Dic₁₀₁	C₂²	1	8	yes	complemented
25	C₄×Dic₂₀₂	C₂²	1	8	yes	complemented
26	C₄×D₄₀₄	C₂²	1	8	yes	complemented
27	D₄×Dic₁₀₁	C₂²	1	8	yes	complemented
28	#15692215	C₂²	1	16	yes	complemented
29	C₂×C₄⋊Dic₁₀₁	C₂²	1	1	yes	characteristic
30	C₂×C₄⋊Dic₁₀₁	C₂²	1	1	yes	characteristic
31	C₂×C₄⋊Dic₁₀₁	C₂²	1	2	yes	—
32	C₂×#13108053	C₂²	1	1	yes	characteristic
33	C₂×#13108053	C₂²	1	1	yes	characteristic
34	C₂×C₂²⋊Dic₁₀₁	C₂²	1	2	yes	—
35	C₂×#13108051	C₂²	1	2	yes	—
36	C₂×#13108051	C₂²	1	2	yes	—
37	C₂×#13108051	C₂²	1	2	yes	—
38	C₂×C₄×Dic₁₀₁	C₂²	1	2	yes	—
39	C₂×C₄×Dic₁₀₁	C₂²	1	1	yes	characteristic
40	C₂×C₄×Dic₁₀₁	C₂²	1	1	yes	characteristic
41	C₂²×Q₈×C₁₀₁	C₂²	1	1	yes	characteristic
42	C₂²×D₄×C₁₀₁	C₂²	1	1	yes	characteristic
43	#124×C₁₀₁	C₂²	1	8	yes	complemented
44	C₂×C₂²⋊C₄×C₁₀₁	C₂²	1	2	yes	—
45	C₂×C₂²⋊C₄×C₁₀₁	C₂²	1	2	yes	—
46	C₂×C₄²×C₁₀₁	C₂²	1	1	yes	characteristic
47	C₂³×D₁₀₁	C₂³	1	2	yes	—
48	C₂×Dic₂₀₂	—	2	4	no	—
49	C₂×Dic₂₀₂	C₂³	1	4	yes	—
50	C₂×D₄₀₄	—	2	4	no	—
51	C₂×D₄₀₄	C₂³	1	4	yes	—
52	C₂×C₂²⋊D₁₀₁	—	2	8	no	—
53	C₂×C₂²⋊D₁₀₁	—	4	4	no	—
54	C₂×C₂²⋊D₁₀₁	C₂³	1	8	yes	—
55	C₂×C₂²⋊D₁₀₁	C₂³	1	8	yes	—
56	C₂²×Dic₁₀₁	—	4	4	no	—
57	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
58	C₂²×Dic₁₀₁	C₂³	1	1	yes	characteristic
59	C₂²×Dic₁₀₁	C₂³	1	1	yes	characteristic
60	C₂²×Dic₁₀₁	C₂³	1	1	yes	characteristic
61	C₂²×Dic₁₀₁	C₂³	1	8	yes	—
62	C₂²×Dic₁₀₁	C₂³	1	1	yes	characteristic
63	C₂×C₄×D₁₀₁	—	4	4	no	—
64	C₂×C₄×D₁₀₁	C₂³	1	8	yes	—
65	C₄⋊Dic₁₀₁	C₂³	1	4	yes	—
66	C₄⋊Dic₁₀₁	C₂³	1	4	yes	—
67	C₄⋊Dic₁₀₁	C₂³	1	8	yes	—
68	#13108053	C₂³	1	4	yes	—
69	#13108053	C₂³	1	4	yes	—
70	C₂²⋊Dic₁₀₁	C₂³	1	8	yes	—
71	#13108051	C₂³	1	8	yes	—
72	#13108051	C₂³	1	8	yes	—
73	#13108051	C₂³	1	8	yes	—
74	C₄×Dic₁₀₁	C₂³	1	8	yes	—
75	C₄×Dic₁₀₁	C₂³	1	4	yes	—
76	C₄×Dic₁₀₁	C₂³	1	4	yes	—
77	C₂⁴×C₁₀₁	C₂³	1	2	yes	—
78	C₂×Q₈×C₁₀₁	—	2	4	no	—
79	C₂×Q₈×C₁₀₁	C₂³	1	4	yes	—
80	C₂×D₄×C₁₀₁	—	2	4	no	—
81	C₂×D₄×C₁₀₁	C₂³	1	4	yes	—
82	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
83	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
84	C₂²×C₄×C₁₀₁	C₂³	1	2	yes	—
85	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
86	C₂²⋊C₄×C₁₀₁	C₂³	1	8	yes	—
87	C₂²⋊C₄×C₁₀₁	C₂³	1	8	yes	—
88	C₄²×C₁₀₁	C₂³	1	4	yes	—
89	C₂³×C₁₀₁	—	2	4	no	—
90	C₂³×C₁₀₁	—	2	4	no	—
91	C₂³×C₁₀₁	—	2	4	no	—
92	C₂³×C₁₀₁	C₂⁴	1	4	yes	—
93	C₂³×C₁₀₁	C₂⁴	1	1	yes	characteristic
94	C₂²×D₁₀₁	—	2	4	no	—
95	C₂²×D₁₀₁	—	2	4	no	—
96	C₂²×D₁₀₁	—	2	4	no	—
97	C₂²×D₁₀₁	C₂⁴	1	4	yes	—
98	Q₈×C₁₀₁	—	2	8	no	—
99	D₄×C₁₀₁	—	2	8	no	—
100	C₂×C₄×C₁₀₁	—	2	4	no	—
101	C₂×C₄×C₁₀₁	—	2	2	no	—
102	C₂×C₄×C₁₀₁	—	2	2	no	—
103	C₂×C₄×C₁₀₁	#41	1	4	yes	—
104	C₂×C₄×C₁₀₁	C₂⁴	1	2	yes	—
105	C₂×C₄×C₁₀₁	C₂⁴	1	2	yes	—
106	C₂×C₄×C₁₀₁	C₂⁴	1	4	yes	—
107	C₂×C₄×C₁₀₁	C₂⁴	1	2	yes	—
108	C₂²⋊D₁₀₁	—	2	16	no	—
109	C₂²⋊D₁₀₁	—	4	8	no	—
110	C₂×Dic₁₀₁	—	2	2	no	—
111	C₂×Dic₁₀₁	—	2	4	no	—
112	C₂×Dic₁₀₁	—	2	2	no	—
113	C₂×Dic₁₀₁	—	2	2	no	—
114	C₂×Dic₁₀₁	—	4	8	no	—
115	C₂×Dic₁₀₁	#41	1	4	yes	—
116	C₂×Dic₁₀₁	C₂⁴	1	2	yes	—
117	C₂×Dic₁₀₁	C₂⁴	1	2	yes	—
118	C₂×Dic₁₀₁	C₂⁴	1	4	yes	—
119	C₂×Dic₁₀₁	C₂⁴	1	2	yes	—
120	C₂×Dic₁₀₁	C₂⁴	1	2	yes	—
121	D₄₀₄	—	2	8	no	—
122	C₄×D₁₀₁	—	4	8	no	—
123	Dic₂₀₂	—	2	8	no	—
124	C₂²×C₁₀₁	—	2	4	no	—
125	C₂²×C₁₀₁	—	2	4	no	—
126	C₂²×C₁₀₁	—	2	4	no	—
127	C₂²×C₁₀₁	—	4	8	no	—
128	C₂²×C₁₀₁	C₂×#41	1	2	yes	—
129	C₂²×C₁₀₁	C₂×#41	1	2	yes	—
130	C₂²×C₁₀₁	H(5,𝔽₂)	1	2	yes	—
131	C₂²×C₁₀₁	C₂⁵	1	1	yes	characteristic
132	C₂×D₁₀₁	—	2	4	no	—
133	C₂×D₁₀₁	—	2	4	no	—
134	C₂×D₁₀₁	—	2	4	no	—
135	C₂×D₁₀₁	—	4	8	no	—
136	C₄×C₁₀₁	—	2	2	no	—
137	C₄×C₁₀₁	—	2	4	no	—
138	C₄×C₁₀₁	—	2	2	no	—
139	C₄×C₁₀₁	C₂×#41	1	4	yes	—
140	Dic₁₀₁	—	2	2	no	—
141	Dic₁₀₁	—	2	4	no	—
142	Dic₁₀₁	—	2	2	no	—
143	Dic₁₀₁	—	2	2	no	—
144	Dic₁₀₁	C₂×#41	1	4	yes	—
145	C₂×C₁₀₁	—	4	4	no	—
146	C₂×C₁₀₁	#559	1	4	yes	complemented
147	C₂×C₁₀₁	C₂²×#41	1	1	yes	characteristic
148	C₂×C₁₀₁	C₂²×#41	1	1	yes	characteristic
149	C₂×C₁₀₁	C₂×H(5,𝔽₂)	1	1	yes	characteristic
150	D₁₀₁	—	4	4	no	—
151	C₂×#559	—	101	1	no	maximal, Sylow
152	C₁₀₁	C₂×#559	1	1	yes	characteristic, complemented, Sylow
153	#559	—	101	16	no	—
154	C₂×#127	—	101	1	no	—
155	C₂×#124	—	101	1	no	—
156	C₂×#124	—	101	2	no	—
157	C₂×#124	D₁₀₁	1	1	yes	characteristic, complemented
158	C₂×#123	—	101	2	no	—
159	C₂×#123	—	101	2	no	—
160	C₂×C₄×Q₈	—	101	1	no	—
161	C₂×C₄×Q₈	—	101	1	no	—
162	C₂×C₄×D₄	—	101	1	no	—
163	C₂×C₄×D₄	—	101	2	no	—
164	C₂×C₄×D₄	—	101	1	no	—
165	C₂²×Q₈	—	101	1	no	—
166	C₂²×Q₈	C₂×D₁₀₁	1	1	yes	characteristic
167	C₂²×D₄	—	101	2	no	—
168	C₂²×D₄	—	101	2	no	—
169	C₂²×D₄	—	101	1	no	—
170	C₂²×D₄	C₂×D₁₀₁	1	1	yes	characteristic
171	C₂³×C₄	—	101	2	no	—
172	C₂³×C₄	—	101	2	no	—
173	#127	—	101	8	no	—
174	#124	—	101	8	no	—
175	#124	—	101	16	no	—
176	#124	C₂×D₁₀₁	1	8	yes	complemented
177	#123	—	101	16	no	—
178	#123	—	101	16	no	—
179	C₄×Q₈	—	101	8	no	—
180	C₄×Q₈	—	101	8	no	—
181	C₄×D₄	—	101	8	no	—
182	C₄×D₄	—	101	16	no	—
183	C₄×D₄	—	101	8	no	—
184	C₂×C₄⋊C₄	—	101	1	no	—
185	C₂×C₄⋊C₄	—	101	1	no	—
186	C₂×C₄⋊C₄	—	101	1	no	—
187	C₂×C₄⋊C₄	—	101	2	no	—
188	C₂×C₄⋊C₄	—	101	1	no	—
189	C₂×C₂²⋊C₄	—	101	2	no	—
190	C₂×C₂²⋊C₄	—	101	2	no	—
191	C₂×C₂²⋊C₄	—	101	2	no	—
192	C₂×C₂²⋊C₄	—	101	2	no	—
193	C₂×C₂²⋊C₄	C₂×D₁₀₁	1	2	yes	—
194	C₂×C₂²⋊C₄	C₂×D₁₀₁	1	2	yes	—
195	C₂×C₄²	—	101	2	no	—
196	C₂×C₄²	—	101	1	no	—
197	C₂×C₄²	—	101	1	no	—
198	C₂×C₄²	C₂×D₁₀₁	1	1	yes	characteristic
199	C₂⁴	—	101	2	no	—
200	C₂⁴	C₂²×D₁₀₁	1	2	yes	—
201	C₂×Q₈	—	2	4	no	—
202	C₂×Q₈	—	101	4	no	—
203	C₂×Q₈	—	202	4	no	—
204	C₂×Q₈	C₂²×D₁₀₁	1	4	yes	—
205	C₂×D₄	—	2	4	no	—
206	C₂×D₄	—	101	8	no	—
207	C₂×D₄	—	101	8	no	—
208	C₂×D₄	—	101	4	no	—
209	C₂×D₄	—	404	4	no	—
210	C₂×D₄	—	202	8	no	—
211	C₂×D₄	—	202	4	no	—
212	C₂×D₄	C₂²×D₁₀₁	1	4	yes	—
213	C₂²×C₄	—	101	1	no	—
214	C₂²×C₄	—	101	8	no	—
215	C₂²×C₄	—	101	1	no	—
216	C₂²×C₄	—	101	2	no	—
217	C₂²×C₄	—	101	1	no	—
218	C₂²×C₄	—	101	1	no	—
219	C₂²×C₄	—	101	8	no	—
220	C₂²×C₄	—	404	4	no	—
221	C₂²×C₄	—	404	4	no	—
222	C₂²×C₄	C₂²×D₁₀₁	1	2	yes	—
223	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
224	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
225	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
226	C₄⋊C₄	—	101	4	no	—
227	C₄⋊C₄	—	101	4	no	—
228	C₄⋊C₄	—	101	4	no	—
229	C₄⋊C₄	—	101	8	no	—
230	C₄⋊C₄	—	101	4	no	—
231	C₂²⋊C₄	—	101	8	no	—
232	C₂²⋊C₄	—	101	8	no	—
233	C₂²⋊C₄	—	101	8	no	—
234	C₂²⋊C₄	—	101	8	no	—
235	C₂²⋊C₄	C₂²×D₁₀₁	1	8	yes	—
236	C₂²⋊C₄	C₂²×D₁₀₁	1	8	yes	—
237	C₄²	—	101	8	no	—
238	C₄²	—	101	4	no	—
239	C₄²	—	101	4	no	—
240	C₄²	C₂²×D₁₀₁	1	4	yes	—
241	C₂³	—	2	4	no	—
242	C₂³	—	2	4	no	—
243	C₂³	—	2	4	no	—
244	C₂³	—	101	4	no	—
245	C₂³	—	202	4	no	—
246	C₂³	—	202	4	no	—
247	C₂³	—	202	4	no	—
248	C₂³	C₂³×D₁₀₁	1	4	yes	—
249	C₂³	C₂³×D₁₀₁	1	1	yes	characteristic
250	Q₈	—	2	8	no	—
251	Q₈	—	202	8	no	—
252	D₄	—	2	8	no	—
253	D₄	—	404	8	no	—
254	D₄	—	202	16	no	—
255	D₄	—	202	8	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	4	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	101	4	no	—
260	C₂×C₄	—	101	2	no	—
261	C₂×C₄	—	101	2	no	—
262	C₂×C₄	—	101	4	no	—
263	C₂×C₄	—	101	2	no	—
264	C₂×C₄	—	101	2	no	—
265	C₂×C₄	—	404	8	no	—
266	C₂×C₄	—	404	8	no	—
267	C₂×C₄	—	202	2	no	—
268	C₂×C₄	—	202	4	no	—
269	C₂×C₄	—	202	2	no	—
270	C₂×C₄	—	202	2	no	—
271	C₂×C₄	#13108073	1	4	yes	—
272	C₂×C₄	C₂³×D₁₀₁	1	4	yes	—
273	C₂×C₄	C₂³×D₁₀₁	1	2	yes	—
274	C₂×C₄	C₂³×D₁₀₁	1	2	yes	—
275	C₂×C₄	C₂³×D₁₀₁	1	2	yes	—
276	C₂²	—	2	4	no	—
277	C₂²	—	2	4	no	—
278	C₂²	—	2	4	no	—
279	C₂²	—	4	8	no	—
280	C₂²	—	404	8	no	—
281	C₂²	—	202	4	no	—
282	C₂²	—	202	4	no	—
283	C₂²	—	202	4	no	—
284	C₂²	#41×D₁₀₁	1	2	yes	—
285	C₂²	C₂×#13108073	1	2	yes	—
286	C₂²	#15692301	1	2	yes	—
287	C₂²	C₂⁴×D₁₀₁	1	1	yes	characteristic
288	C₄	—	2	2	no	—
289	C₄	—	2	4	no	—
290	C₄	—	2	2	no	—
291	C₄	—	101	4	no	—
292	C₄	—	202	2	no	—
293	C₄	—	202	4	no	—
294	C₄	—	202	2	no	—
295	C₄	—	202	2	no	—
296	C₄	C₂×#13108073	1	4	yes	—
297	C₂	—	4	4	no	—
298	C₂	—	404	4	no	—
299	C₂	#20265909	1	4	yes	direct product factor
300	C₂	C₂×#41×D₁₀₁	1	1	yes	characteristic
301	C₂	C₂²×#13108073	1	1	yes	characteristic
302	C₂	C₂×#15692301	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂³×D₁₀₁
commutator, cosocle	C₂²×C₁₀₁	C₂⁵
Frattini	C₂²	C₂⁴×D₁₀₁
Fitting	C₂×#124×C₁₀₁	C₂
socle	C₂³×C₁₀₁	C₂⁴
prime radical	C₂×#15692251	C₂
hypercenter	C₂×#124	D₁₀₁
hypocenter	C₁₀₁	C₂×#559
LCS	G ▷ C₂²×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂³ ◁ C₂×#124

Complements

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

N	H ≅ G/N	type
1. #20265909	C₂	direct (N × H)
3. C₂×#15692244	C₂	semidirect
4. C₂×#15692241	C₂	semidirect
5. C₂×#15692239	C₂	semidirect
6. C₂×#15692237	C₂	semidirect
7. C₂×Q₈×Dic₁₀₁	C₂	semidirect
8. C₂×C₄×Dic₂₀₂	C₂	semidirect
9. C₂×C₄×D₄₀₄	C₂	semidirect
10. C₂×D₄×Dic₁₀₁	C₂	semidirect
11. C₂×#15692215	C₂	semidirect
12. C₂×#124×C₁₀₁	C₂	semidirect
20. #15692244	C₂²	semidirect
21. #15692241	C₂²	semidirect
22. #15692239	C₂²	semidirect
23. #15692237	C₂²	semidirect
24. Q₈×Dic₁₀₁	C₂²	semidirect
25. C₄×Dic₂₀₂	C₂²	semidirect
26. C₄×D₄₀₄	C₂²	semidirect
27. D₄×Dic₁₀₁	C₂²	semidirect
28. #15692215	C₂²	semidirect
43. #124×C₁₀₁	C₂²	semidirect
146. C₂×C₁₀₁	#559	semidirect
152. C₁₀₁	C₂×#559	semidirect
157. C₂×#124	D₁₀₁	semidirect
176. #124	C₂×D₁₀₁	semidirect
299. C₂	#20265909	direct (N × H)

Maximal subgroups

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

subgroup	index
#20265909	2
C₂×#15692251	2
C₂×#15692244	2
C₂×#15692241	2
C₂×#15692239	2
C₂×#15692237	2
C₂×Q₈×Dic₁₀₁	2
C₂×C₄×Dic₂₀₂	2
C₂×C₄×D₄₀₄	2
C₂×D₄×Dic₁₀₁	2
C₂×#15692215	2
C₂×#124×C₁₀₁	2
C₂×#559	101

Subgroups — C2×#20265909 #27184717

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20265909 #27184717