Subgroups — C₂×#20265794 #27184506

Order 12928.

1132 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20265794	C₂	1	16	yes	maximal, direct product factor
2	C₂×#15692254	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#15692248	C₂	1	2	yes	maximal, complemented
4	C₂×#15692242	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#15692242	C₂	1	2	yes	maximal, complemented
6	C₂×#15692236	C₂	1	2	yes	maximal, complemented
7	C₂×#15692234	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#15692232	C₂	1	2	yes	maximal, complemented
9	C₂×#15692220	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×D₄×Dic₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×#15692214	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#121×C₁₀₁	C₂	1	1	yes	maximal, characteristic
13	C₂²×Dic₂₀₂	C₂²	1	1	yes	characteristic
14	C₂²×Dic₂₀₂	C₂²	1	2	yes	—
15	C₂³×Dic₁₀₁	C₂²	1	2	yes	—
16	#15692254	C₂²	1	8	yes	complemented
17	#15692248	C₂²	1	16	yes	complemented
18	#15692242	C₂²	1	8	yes	complemented
19	#15692242	C₂²	1	16	yes	complemented
20	#15692236	C₂²	1	16	yes	complemented
21	#15692234	C₂²	1	8	yes	complemented
22	#15692232	C₂²	1	16	yes	complemented
23	#15692220	C₂²	1	8	yes	complemented
24	D₄×Dic₁₀₁	C₂²	1	8	yes	complemented
25	#15692214	C₂²	1	8	yes	complemented
26	C₂×C₄⋊Dic₁₀₁	C₂²	1	1	yes	characteristic
27	C₂×C₄⋊Dic₁₀₁	C₂²	1	1	yes	characteristic
28	C₂×C₄⋊Dic₁₀₁	C₂²	1	1	yes	characteristic
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	1	yes	characteristic
33	C₂×C₂²⋊Dic₁₀₁	C₂²	1	1	yes	characteristic
34	C₂×C₂²⋊Dic₁₀₁	C₂²	1	2	yes	—
35	C₂×C₂²⋊Dic₁₀₁	C₂²	1	2	yes	—
36	C₂×C₂²⋊Dic₁₀₁	C₂²	1	2	yes	—
37	C₂×C₂²⋊Dic₁₀₁	C₂²	1	2	yes	—
38	C₂×C₄×Dic₁₀₁	C₂²	1	1	yes	characteristic
39	C₂×C₄×Dic₁₀₁	C₂²	1	1	yes	characteristic
40	C₂×C₄×Dic₁₀₁	C₂²	1	2	yes	—
41	C₂²×D₄×C₁₀₁	C₂²	1	1	yes	characteristic
42	C₂²×D₄×C₁₀₁	C₂²	1	2	yes	—
43	C₂³×C₄×C₁₀₁	C₂²	1	1	yes	characteristic
44	#121×C₁₀₁	C₂²	1	8	yes	—
45	C₂×C₄⋊C₄×C₁₀₁	C₂²	1	1	yes	characteristic
46	C₂×C₂²⋊C₄×C₁₀₁	C₂²	1	2	yes	—
47	C₂×Dic₂₀₂	—	2	4	no	—
48	C₂×Dic₂₀₂	—	4	4	no	—
49	C₂×Dic₂₀₂	C₂³	1	4	yes	—
50	C₂×Dic₂₀₂	C₂³	1	8	yes	—
51	C₂²×Dic₁₀₁	—	4	4	no	—
52	C₂²×Dic₁₀₁	C₂³	1	1	yes	characteristic
53	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
54	C₂²×Dic₁₀₁	C₂³	1	1	yes	characteristic
55	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
56	C₂²×Dic₁₀₁	C₂³	1	1	yes	characteristic
57	C₂²×Dic₁₀₁	C₂³	1	1	yes	characteristic
58	C₂²×Dic₁₀₁	C₂³	1	8	yes	—
59	C₄⋊Dic₁₀₁	C₂³	1	4	yes	—
60	C₄⋊Dic₁₀₁	C₂³	1	4	yes	—
61	C₄⋊Dic₁₀₁	C₂³	1	4	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	C₂²⋊Dic₁₀₁	C₂³	1	8	yes	—
69	C₂²⋊Dic₁₀₁	C₂³	1	8	yes	—
70	C₂²⋊Dic₁₀₁	C₂³	1	8	yes	—
71	C₄×Dic₁₀₁	C₂³	1	4	yes	—
72	C₄×Dic₁₀₁	C₂³	1	4	yes	—
73	C₄×Dic₁₀₁	C₂³	1	8	yes	—
74	C₂⁴×C₁₀₁	C₂³	1	2	yes	—
75	C₂⁴×C₁₀₁	C₂³	1	1	yes	characteristic
76	C₂×D₄×C₁₀₁	—	2	4	no	—
77	C₂×D₄×C₁₀₁	—	4	4	no	—
78	C₂×D₄×C₁₀₁	C₂³	1	4	yes	—
79	C₂×D₄×C₁₀₁	C₂³	1	8	yes	—
80	C₂²×C₄×C₁₀₁	—	2	4	no	—
81	C₂²×C₄×C₁₀₁	C₂³	1	2	yes	—
82	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
83	C₂²×C₄×C₁₀₁	C₂³	1	4	yes	—
84	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
85	C₄⋊C₄×C₁₀₁	C₂³	1	4	yes	—
86	C₂²⋊C₄×C₁₀₁	C₂³	1	8	yes	—
87	C₂³×C₁₀₁	—	2	2	no	—
88	C₂³×C₁₀₁	—	2	4	no	—
89	C₂³×C₁₀₁	—	2	2	no	—
90	C₂³×C₁₀₁	—	2	4	no	—
91	C₂³×C₁₀₁	—	2	4	no	—
92	C₂³×C₁₀₁	—	2	2	no	—
93	C₂³×C₁₀₁	C₂⁴	1	4	yes	—
94	C₂³×C₁₀₁	C₂⁴	1	2	yes	—
95	C₂³×C₁₀₁	C₂⁴	1	1	yes	characteristic
96	D₄×C₁₀₁	—	2	8	no	—
97	D₄×C₁₀₁	—	4	8	no	—
98	C₂×C₄×C₁₀₁	—	2	2	no	—
99	C₂×C₄×C₁₀₁	—	2	4	no	—
100	C₂×C₄×C₁₀₁	—	2	8	no	—
101	C₂×C₄×C₁₀₁	#41	1	4	yes	—
102	C₂×C₄×C₁₀₁	C₂⁴	1	2	yes	—
103	C₂×C₄×C₁₀₁	C₂⁴	1	4	yes	—
104	C₂×C₄×C₁₀₁	C₂⁴	1	2	yes	—
105	C₂×Dic₁₀₁	—	2	4	no	—
106	C₂×Dic₁₀₁	—	2	2	no	—
107	C₂×Dic₁₀₁	—	2	4	no	—
108	C₂×Dic₁₀₁	—	2	2	no	—
109	C₂×Dic₁₀₁	—	2	2	no	—
110	C₂×Dic₁₀₁	—	2	2	no	—
111	C₂×Dic₁₀₁	—	4	8	no	—
112	C₂×Dic₁₀₁	C₂⁴	1	2	yes	—
113	C₂×Dic₁₀₁	C₂⁴	1	2	yes	—
114	C₂×Dic₁₀₁	C₂⁴	1	4	yes	—
115	C₂×Dic₁₀₁	C₂⁴	1	2	yes	—
116	C₂×Dic₁₀₁	C₂⁴	1	2	yes	—
117	C₂×Dic₁₀₁	C₂⁴	1	4	yes	—
118	Dic₂₀₂	—	2	8	no	—
119	Dic₂₀₂	—	4	8	no	—
120	C₂²×C₁₀₁	—	2	2	no	—
121	C₂²×C₁₀₁	—	2	4	no	—
122	C₂²×C₁₀₁	—	2	4	no	—
123	C₂²×C₁₀₁	—	2	2	no	—
124	C₂²×C₁₀₁	—	2	4	no	—
125	C₂²×C₁₀₁	—	2	2	no	—
126	C₂²×C₁₀₁	—	4	4	no	—
127	C₂²×C₁₀₁	—	4	8	no	—
128	C₂²×C₁₀₁	C₂×#41	1	2	yes	—
129	C₂²×C₁₀₁	H(5,𝔽₂)	1	2	yes	—
130	C₂²×C₁₀₁	2₋¹⁺⁴	1	2	yes	—
131	C₂²×C₁₀₁	C₂⁵	1	1	yes	characteristic
132	C₄×C₁₀₁	—	2	2	no	—
133	C₄×C₁₀₁	—	2	4	no	—
134	C₄×C₁₀₁	C₂×#41	1	4	yes	—
135	Dic₁₀₁	—	2	4	no	—
136	Dic₁₀₁	—	2	2	no	—
137	Dic₁₀₁	—	2	4	no	—
138	Dic₁₀₁	—	2	2	no	—
139	Dic₁₀₁	—	2	2	no	—
140	Dic₁₀₁	—	2	2	no	—
141	C₂×C₁₀₁	—	4	2	no	—
142	C₂×C₁₀₁	—	4	4	no	—
143	C₂×C₁₀₁	#542	1	4	yes	complemented
144	C₂×C₁₀₁	C₂²×#41	1	1	yes	characteristic
145	C₂×C₁₀₁	C₂×H(5,𝔽₂)	1	1	yes	characteristic
146	C₂×C₁₀₁	C₂×2₋¹⁺⁴	1	1	yes	characteristic
147	C₂×#542	—	101	1	no	maximal, Sylow
148	C₁₀₁	C₂×#542	1	1	yes	characteristic, complemented, Sylow
149	#542	—	101	16	no	—
150	C₂×C₄⋊Q₈	—	101	1	no	—
151	C₂×#126	—	101	2	no	—
152	C₂×#124	—	101	1	no	—
153	C₂×#124	—	101	2	no	—
154	C₂×#123	—	101	2	no	—
155	C₂×C₂²⋊Q₈	—	101	1	no	—
156	C₂×C₂²⋊Q₈	—	101	2	no	—
157	C₂×#121	D₁₀₁	1	1	yes	characteristic
158	C₂×C₄×Q₈	—	101	1	no	—
159	C₂×C₄×D₄	—	101	1	no	—
160	C₂×#117	—	101	1	no	—
161	C₂²×Q₈	—	101	2	no	—
162	C₂²×Q₈	—	101	1	no	—
163	C₂²×D₄	C₂×D₁₀₁	1	1	yes	characteristic
164	C₂²×D₄	C₂×D₁₀₁	1	2	yes	—
165	C₂³×C₄	—	101	2	no	—
166	C₂³×C₄	C₂×D₁₀₁	1	1	yes	characteristic
167	C₄⋊Q₈	—	101	8	no	—
168	#126	—	101	16	no	—
169	#124	—	101	8	no	—
170	#124	—	101	16	no	—
171	#123	—	101	16	no	—
172	C₂²⋊Q₈	—	101	8	no	—
173	C₂²⋊Q₈	—	101	16	no	—
174	#121	C₂×D₁₀₁	1	8	yes	—
175	C₄×Q₈	—	101	8	no	—
176	C₄×D₄	—	101	8	no	—
177	#117	—	101	8	no	—
178	C₂×C₄⋊C₄	—	101	1	no	—
179	C₂×C₄⋊C₄	—	101	1	no	—
180	C₂×C₄⋊C₄	—	101	1	no	—
181	C₂×C₄⋊C₄	—	101	2	no	—
182	C₂×C₄⋊C₄	—	101	2	no	—
183	C₂×C₄⋊C₄	—	101	2	no	—
184	C₂×C₄⋊C₄	C₂×D₁₀₁	1	1	yes	characteristic
185	C₂×C₂²⋊C₄	—	101	1	no	—
186	C₂×C₂²⋊C₄	—	101	1	no	—
187	C₂×C₂²⋊C₄	—	101	2	no	—
188	C₂×C₂²⋊C₄	—	101	2	no	—
189	C₂×C₂²⋊C₄	—	101	2	no	—
190	C₂×C₂²⋊C₄	—	101	2	no	—
191	C₂×C₂²⋊C₄	C₂×D₁₀₁	1	2	yes	—
192	C₂×C₄²	—	101	1	no	—
193	C₂×C₄²	—	101	1	no	—
194	C₂×C₄²	—	101	2	no	—
195	C₂⁴	C₂²×D₁₀₁	1	1	yes	characteristic
196	C₂⁴	C₂²×D₁₀₁	1	2	yes	—
197	C₂×Q₈	—	101	8	no	—
198	C₂×Q₈	—	101	4	no	—
199	C₂×Q₈	—	404	4	no	—
200	C₂×Q₈	—	202	4	no	—
201	C₂×D₄	—	2	4	no	—
202	C₂×D₄	—	4	4	no	—
203	C₂×D₄	C₂²×D₁₀₁	1	4	yes	—
204	C₂×D₄	C₂²×D₁₀₁	1	8	yes	—
205	C₂²×C₄	—	2	4	no	—
206	C₂²×C₄	—	101	8	no	—
207	C₂²×C₄	—	101	1	no	—
208	C₂²×C₄	—	101	1	no	—
209	C₂²×C₄	—	101	2	no	—
210	C₂²×C₄	—	101	1	no	—
211	C₂²×C₄	—	101	1	no	—
212	C₂²×C₄	—	101	2	no	—
213	C₂²×C₄	—	404	4	no	—
214	C₂²×C₄	C₂²×D₁₀₁	1	4	yes	—
215	C₂²×C₄	C₂²×D₁₀₁	1	2	yes	—
216	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
217	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
218	C₄⋊C₄	—	101	8	no	—
219	C₄⋊C₄	—	101	8	no	—
220	C₄⋊C₄	—	101	8	no	—
221	C₄⋊C₄	—	101	4	no	—
222	C₄⋊C₄	—	101	4	no	—
223	C₄⋊C₄	—	101	4	no	—
224	C₄⋊C₄	C₂²×D₁₀₁	1	4	yes	—
225	C₂²⋊C₄	—	101	4	no	—
226	C₂²⋊C₄	—	101	4	no	—
227	C₂²⋊C₄	—	101	8	no	—
228	C₂²⋊C₄	—	101	8	no	—
229	C₂²⋊C₄	—	101	8	no	—
230	C₂²⋊C₄	—	101	8	no	—
231	C₂²⋊C₄	C₂²×D₁₀₁	1	8	yes	—
232	C₄²	—	101	4	no	—
233	C₄²	—	101	4	no	—
234	C₄²	—	101	8	no	—
235	C₂³	—	2	2	no	—
236	C₂³	—	2	4	no	—
237	C₂³	—	2	4	no	—
238	C₂³	—	2	2	no	—
239	C₂³	—	2	2	no	—
240	C₂³	—	2	4	no	—
241	C₂³	C₂³×D₁₀₁	1	2	yes	—
242	C₂³	C₂³×D₁₀₁	1	4	yes	—
243	C₂³	C₂³×D₁₀₁	1	1	yes	characteristic
244	Q₈	—	404	8	no	—
245	Q₈	—	202	8	no	—
246	D₄	—	2	8	no	—
247	D₄	—	4	8	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	2	4	no	—
250	C₂×C₄	—	2	8	no	—
251	C₂×C₄	—	101	2	no	—
252	C₂×C₄	—	101	2	no	—
253	C₂×C₄	—	101	4	no	—
254	C₂×C₄	—	101	2	no	—
255	C₂×C₄	—	101	2	no	—
256	C₂×C₄	—	101	4	no	—
257	C₂×C₄	—	404	8	no	—
258	C₂×C₄	—	202	4	no	—
259	C₂×C₄	—	202	2	no	—
260	C₂×C₄	—	202	4	no	—
261	C₂×C₄	—	202	2	no	—
262	C₂×C₄	—	202	2	no	—
263	C₂×C₄	—	202	2	no	—
264	C₂×C₄	#13108073	1	4	yes	—
265	C₂×C₄	C₂³×D₁₀₁	1	4	yes	—
266	C₂×C₄	C₂³×D₁₀₁	1	2	yes	—
267	C₂×C₄	C₂³×D₁₀₁	1	2	yes	—
268	C₂²	—	2	2	no	—
269	C₂²	—	2	4	no	—
270	C₂²	—	2	4	no	—
271	C₂²	—	2	2	no	—
272	C₂²	—	2	2	no	—
273	C₂²	—	2	4	no	—
274	C₂²	—	4	4	no	—
275	C₂²	—	4	8	no	—
276	C₂²	C₂×#13108073	1	2	yes	—
277	C₂²	#15692300	1	2	yes	—
278	C₂²	#15692302	1	2	yes	—
279	C₂²	C₂⁴×D₁₀₁	1	1	yes	characteristic
280	C₄	—	2	2	no	—
281	C₄	—	2	4	no	—
282	C₄	—	202	4	no	—
283	C₄	—	202	2	no	—
284	C₄	—	202	4	no	—
285	C₄	—	202	2	no	—
286	C₄	—	202	2	no	—
287	C₄	—	202	2	no	—
288	C₄	C₂×#13108073	1	4	yes	—
289	C₂	—	4	2	no	—
290	C₂	—	4	4	no	—
291	C₂	#20265794	1	4	yes	direct product factor
292	C₂	C₂²×#13108073	1	1	yes	characteristic
293	C₂	C₂×#15692300	1	1	yes	characteristic
294	C₂	C₂×#15692302	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂³×D₁₀₁
commutator, cosocle	C₂²×C₁₀₁	C₂⁵
Frattini	C₂²	C₂⁴×D₁₀₁
Fitting, prime radical	C₂×#121×C₁₀₁	C₂
socle	C₂³×C₁₀₁	C₂⁴
hypercenter	C₂×#121	D₁₀₁
hypocenter	C₁₀₁	C₂×#542
LCS	G ▷ C₂²×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂³ ◁ C₂×#121

Complements

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

N	H ≅ G/N	type
1. #20265794	C₂	direct (N × H)
2. C₂×#15692254	C₂	semidirect
3. C₂×#15692248	C₂	semidirect
4. C₂×#15692242	C₂	semidirect
5. C₂×#15692242	C₂	semidirect
6. C₂×#15692236	C₂	semidirect
7. C₂×#15692234	C₂	semidirect
8. C₂×#15692232	C₂	semidirect
9. C₂×#15692220	C₂	semidirect
10. C₂×D₄×Dic₁₀₁	C₂	semidirect
11. C₂×#15692214	C₂	semidirect
16. #15692254	C₂²	semidirect
17. #15692248	C₂²	semidirect
18. #15692242	C₂²	semidirect
19. #15692242	C₂²	semidirect
20. #15692236	C₂²	semidirect
21. #15692234	C₂²	semidirect
22. #15692232	C₂²	semidirect
23. #15692220	C₂²	semidirect
24. D₄×Dic₁₀₁	C₂²	semidirect
25. #15692214	C₂²	semidirect
143. C₂×C₁₀₁	#542	semidirect
148. C₁₀₁	C₂×#542	semidirect
291. C₂	#20265794	direct (N × H)

Maximal subgroups

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

subgroup	index
#20265794	2
C₂×#15692254	2
C₂×#15692248	2
C₂×#15692242	2
C₂×#15692236	2
C₂×#15692234	2
C₂×#15692232	2
C₂×#15692220	2
C₂×D₄×Dic₁₀₁	2
C₂×#15692214	2
C₂×#121×C₁₀₁	2
C₂×#542	101

Subgroups — C2×#20265794 #27184506

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20265794 #27184506