Subgroups — C₂×#20264943 #27170861

Order 12928.

680 conjugacy classes of subgroups.

The subgroup lattice has 298 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₂²⋊Dic₁₀₁	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₂²⋊Dic₁₀₁	C₂	1	1	yes	maximal, characteristic
3	#20264943	C₂	1	4	yes	maximal, direct product factor
4	C₂²×C₂²⋊C₄×C₁₀₁	C₂	1	1	yes	maximal, characteristic
5	C₂³×Dic₁₀₁	—	2	1	no	—
6	C₂³×Dic₁₀₁	C₄	1	2	yes	complemented
7	C₂×C₂²⋊Dic₁₀₁	—	2	1	no	—
8	C₂×C₂²⋊Dic₁₀₁	—	2	1	no	—
9	C₂×C₂²⋊Dic₁₀₁	—	2	1	no	—
10	C₂×C₂²⋊Dic₁₀₁	—	2	1	no	—
11	C₂×C₂²⋊Dic₁₀₁	—	2	2	no	—
12	C₂×C₂²⋊Dic₁₀₁	—	2	1	no	—
13	C₂×C₂²⋊Dic₁₀₁	C₄	1	2	yes	complemented
14	C₂×C₂²⋊Dic₁₀₁	C₄	1	4	yes	complemented
15	C₂×C₂²⋊Dic₁₀₁	C₂²	1	2	yes	—
16	C₂×C₂²⋊Dic₁₀₁	C₂²	1	2	yes	—
17	C₂⁵×C₁₀₁	C₂²	1	1	yes	characteristic
18	C₂³×C₄×C₁₀₁	—	2	1	no	—
19	C₂×C₂²⋊C₄×C₁₀₁	—	2	1	no	—
20	C₂×C₂²⋊C₄×C₁₀₁	—	2	2	no	—
21	C₂×C₂²⋊C₄×C₁₀₁	C₄	1	4	yes	complemented
22	C₂×C₂²⋊C₄×C₁₀₁	C₂²	1	2	yes	—
23	C₂²×Dic₁₀₁	—	2	4	no	—
24	C₂²×Dic₁₀₁	—	2	2	no	—
25	C₂²×Dic₁₀₁	—	2	2	no	—
26	C₂²×Dic₁₀₁	—	2	2	no	—
27	C₂²×Dic₁₀₁	—	4	4	no	—
28	C₂²×Dic₁₀₁	—	4	4	no	—
29	C₂²×Dic₁₀₁	C₂×C₄	1	4	yes	complemented
30	C₂²⋊Dic₁₀₁	—	2	2	no	—
31	C₂²⋊Dic₁₀₁	—	2	2	no	—
32	C₂²⋊Dic₁₀₁	—	2	4	no	—
33	C₂²⋊Dic₁₀₁	—	2	2	no	—
34	C₂²⋊Dic₁₀₁	—	2	2	no	—
35	C₂²⋊Dic₁₀₁	—	2	2	no	—
36	C₂²⋊Dic₁₀₁	C₂×C₄	1	4	yes	complemented
37	C₂⁴×C₁₀₁	—	4	4	no	—
38	C₂⁴×C₁₀₁	C₂×C₄	1	1	yes	characteristic
39	C₂⁴×C₁₀₁	C₂×C₄	1	2	yes	—
40	C₂⁴×C₁₀₁	C₂×C₄	1	1	yes	characteristic
41	C₂⁴×C₁₀₁	C₂×C₄	1	2	yes	—
42	C₂⁴×C₁₀₁	D₄	1	1	yes	characteristic
43	C₂⁴×C₁₀₁	D₄	1	1	yes	characteristic
44	C₂⁴×C₁₀₁	D₄	1	1	yes	characteristic
45	C₂⁴×C₁₀₁	D₄	1	1	yes	characteristic
46	C₂⁴×C₁₀₁	D₄	1	2	yes	—
47	C₂⁴×C₁₀₁	Q₈	1	2	yes	—
48	C₂⁴×C₁₀₁	C₂³	1	1	yes	characteristic
49	C₂²×C₄×C₁₀₁	—	2	4	no	—
50	C₂²×C₄×C₁₀₁	—	2	2	no	—
51	C₂²×C₄×C₁₀₁	—	4	4	no	—
52	C₂²⋊C₄×C₁₀₁	—	2	4	no	—
53	C₂²⋊C₄×C₁₀₁	—	2	2	no	—
54	C₂²⋊C₄×C₁₀₁	C₂×C₄	1	4	yes	complemented
55	C₂³×C₁₀₁	—	2	4	no	—
56	C₂³×C₁₀₁	—	2	4	no	—
57	C₂³×C₁₀₁	—	2	4	no	—
58	C₂³×C₁₀₁	—	4	4	no	—
59	C₂³×C₁₀₁	—	4	2	no	—
60	C₂³×C₁₀₁	—	4	4	no	—
61	C₂³×C₁₀₁	—	4	2	no	—
62	C₂³×C₁₀₁	—	4	4	no	—
63	C₂³×C₁₀₁	—	4	2	no	—
64	C₂³×C₁₀₁	—	4	4	no	—
65	C₂³×C₁₀₁	—	4	2	no	—
66	C₂³×C₁₀₁	C₄²	1	2	yes	—
67	C₂³×C₁₀₁	C₄²	1	2	yes	—
68	C₂³×C₁₀₁	C₂²⋊C₄	1	1	yes	characteristic
69	C₂³×C₁₀₁	C₂²⋊C₄	1	1	yes	characteristic
70	C₂³×C₁₀₁	C₂²⋊C₄	1	2	yes	—
71	C₂³×C₁₀₁	C₂²⋊C₄	1	1	yes	characteristic
72	C₂³×C₁₀₁	C₂²⋊C₄	1	2	yes	—
73	C₂³×C₁₀₁	C₂²⋊C₄	1	1	yes	characteristic
74	C₂³×C₁₀₁	C₂²⋊C₄	1	2	yes	—
75	C₂³×C₁₀₁	C₂²⋊C₄	1	2	yes	—
76	C₂³×C₁₀₁	C₄⋊C₄	1	2	yes	—
77	C₂³×C₁₀₁	C₄⋊C₄	1	2	yes	—
78	C₂³×C₁₀₁	C₄⋊C₄	1	2	yes	—
79	C₂³×C₁₀₁	C₄⋊C₄	1	2	yes	—
80	C₂³×C₁₀₁	C₄⋊C₄	1	2	yes	—
81	C₂³×C₁₀₁	C₄⋊C₄	1	2	yes	—
82	C₂³×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
83	C₂³×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
84	C₂³×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
85	C₂³×C₁₀₁	C₂×D₄	1	1	yes	characteristic
86	C₂³×C₁₀₁	C₂×D₄	1	1	yes	characteristic
87	C₂³×C₁₀₁	C₂×D₄	1	1	yes	characteristic
88	C₂³×C₁₀₁	C₂×Q₈	1	1	yes	characteristic
89	C₂×C₄×C₁₀₁	—	2	4	no	—
90	C₂×C₄×C₁₀₁	—	4	8	no	—
91	C₂×C₄×C₁₀₁	—	4	4	no	—
92	C₂×Dic₁₀₁	—	2	4	no	—
93	C₂×Dic₁₀₁	—	2	4	no	—
94	C₂×Dic₁₀₁	—	4	8	no	—
95	C₂×Dic₁₀₁	—	4	4	no	—
96	C₂×Dic₁₀₁	—	4	4	no	—
97	C₂×Dic₁₀₁	—	4	4	no	—
98	C₂×Dic₁₀₁	—	4	4	no	—
99	C₂²×C₁₀₁	—	2	4	no	—
100	C₂²×C₁₀₁	—	2	4	no	—
101	C₂²×C₁₀₁	—	2	4	no	—
102	C₂²×C₁₀₁	—	2	4	no	—
103	C₂²×C₁₀₁	—	2	4	no	—
104	C₂²×C₁₀₁	—	2	4	no	—
105	C₂²×C₁₀₁	—	2	2	no	—
106	C₂²×C₁₀₁	—	2	2	no	—
107	C₂²×C₁₀₁	—	2	2	no	—
108	C₂²×C₁₀₁	—	2	2	no	—
109	C₂²×C₁₀₁	—	2	2	no	—
110	C₂²×C₁₀₁	—	2	2	no	—
111	C₂²×C₁₀₁	—	4	2	no	—
112	C₂²×C₁₀₁	—	4	2	no	—
113	C₂²×C₁₀₁	—	4	2	no	—
114	C₂²×C₁₀₁	—	4	2	no	—
115	C₂²×C₁₀₁	—	4	2	no	—
116	C₂²×C₁₀₁	—	4	2	no	—
117	C₂²×C₁₀₁	—	4	1	no	—
118	C₂²×C₁₀₁	—	4	1	no	—
119	C₂²×C₁₀₁	—	4	1	no	—
120	C₂²×C₁₀₁	—	4	1	no	—
121	C₂²×C₁₀₁	#95	1	2	yes	—
122	C₂²×C₁₀₁	#95	1	2	yes	—
123	C₂²×C₁₀₁	#95	1	2	yes	—
124	C₂²×C₁₀₁	#95	1	2	yes	—
125	C₂²×C₁₀₁	#99	1	4	yes	—
126	C₂²×C₁₀₁	C₂×C₄²	1	1	yes	characteristic
127	C₂²×C₁₀₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
128	C₂²×C₁₀₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
129	C₂²×C₁₀₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
130	C₂²×C₁₀₁	C₂×C₄⋊C₄	1	1	yes	characteristic
131	C₂²×C₁₀₁	C₂×C₄⋊C₄	1	1	yes	characteristic
132	C₂²×C₁₀₁	C₂×C₄⋊C₄	1	1	yes	characteristic
133	C₄×C₁₀₁	—	4	4	no	—
134	Dic₁₀₁	—	4	4	no	—
135	Dic₁₀₁	—	4	4	no	—
136	C₂×C₁₀₁	—	2	1	no	—
137	C₂×C₁₀₁	—	2	1	no	—
138	C₂×C₁₀₁	—	2	2	no	—
139	C₂×C₁₀₁	—	2	2	no	—
140	C₂×C₁₀₁	—	2	2	no	—
141	C₂×C₁₀₁	—	2	1	no	—
142	C₂×C₁₀₁	—	2	1	no	—
143	C₂×C₁₀₁	—	2	1	no	—
144	C₂×C₁₀₁	—	2	1	no	—
145	C₂×C₁₀₁	#342	1	4	yes	complemented
146	C₂×C₁₀₁	C₂×#95	1	1	yes	characteristic
147	C₂×C₁₀₁	C₂×#99	1	2	yes	—
148	C₂×#342	—	101	1	no	maximal, Sylow
149	C₁₀₁	C₂×#342	1	1	yes	characteristic, complemented, Sylow
150	C₂²×C₂²⋊C₄	—	101	1	no	—
151	C₂²×C₂²⋊C₄	—	101	1	no	—
152	C₂²×C₂²⋊C₄	D₁₀₁	1	1	yes	characteristic
153	#342	—	101	4	no	—
154	C₂⁵	C₂×D₁₀₁	1	1	yes	characteristic
155	C₂³×C₄	—	2	1	no	—
156	C₂³×C₄	—	101	2	no	—
157	C₂³×C₄	—	202	1	no	—
158	C₂×C₂²⋊C₄	—	2	1	no	—
159	C₂×C₂²⋊C₄	—	2	2	no	—
160	C₂×C₂²⋊C₄	—	101	4	no	—
161	C₂×C₂²⋊C₄	—	101	2	no	—
162	C₂×C₂²⋊C₄	—	101	2	no	—
163	C₂×C₂²⋊C₄	—	101	2	no	—
164	C₂×C₂²⋊C₄	—	202	1	no	—
165	C₂×C₂²⋊C₄	—	202	1	no	—
166	C₂×C₂²⋊C₄	—	202	1	no	—
167	C₂×C₂²⋊C₄	—	202	2	no	—
168	C₂×C₂²⋊C₄	—	202	1	no	—
169	C₂×C₂²⋊C₄	—	202	1	no	—
170	C₂×C₂²⋊C₄	Dic₁₀₁	1	4	yes	complemented
171	C₂×C₂²⋊C₄	C₂×D₁₀₁	1	2	yes	—
172	C₂⁴	—	4	4	no	—
173	C₂⁴	Dic₂₀₂	1	2	yes	—
174	C₂⁴	C₄×D₁₀₁	1	1	yes	characteristic
175	C₂⁴	C₄×D₁₀₁	1	2	yes	—
176	C₂⁴	C₄×D₁₀₁	1	1	yes	characteristic
177	C₂⁴	D₄₀₄	1	1	yes	characteristic
178	C₂⁴	D₄₀₄	1	1	yes	characteristic
179	C₂⁴	C₂×Dic₁₀₁	1	2	yes	—
180	C₂⁴	C₂²⋊D₁₀₁	1	1	yes	characteristic
181	C₂⁴	C₂²⋊D₁₀₁	1	1	yes	characteristic
182	C₂⁴	C₂²⋊D₁₀₁	1	2	yes	—
183	C₂⁴	C₂²×D₁₀₁	1	1	yes	characteristic
184	C₂²×C₄	—	2	4	no	—
185	C₂²×C₄	—	2	2	no	—
186	C₂²×C₄	—	4	4	no	—
187	C₂²×C₄	—	101	4	no	—
188	C₂²×C₄	—	404	4	no	—
189	C₂²×C₄	—	404	4	no	—
190	C₂²×C₄	—	202	4	no	—
191	C₂²×C₄	—	202	2	no	—
192	C₂²×C₄	—	202	2	no	—
193	C₂²×C₄	—	202	2	no	—
194	C₂²⋊C₄	—	2	4	no	—
195	C₂²⋊C₄	—	2	2	no	—
196	C₂²⋊C₄	—	101	4	no	—
197	C₂²⋊C₄	—	202	4	no	—
198	C₂²⋊C₄	—	202	2	no	—
199	C₂²⋊C₄	—	202	2	no	—
200	C₂²⋊C₄	—	202	2	no	—
201	C₂²⋊C₄	—	202	2	no	—
202	C₂²⋊C₄	—	202	2	no	—
203	C₂²⋊C₄	C₂×Dic₁₀₁	1	4	yes	complemented
204	C₂³	—	2	4	no	—
205	C₂³	—	2	4	no	—
206	C₂³	—	2	4	no	—
207	C₂³	—	4	4	no	—
208	C₂³	—	4	2	no	—
209	C₂³	—	4	4	no	—
210	C₂³	—	4	2	no	—
211	C₂³	—	4	4	no	—
212	C₂³	—	4	2	no	—
213	C₂³	—	4	4	no	—
214	C₂³	—	4	2	no	—
215	C₂³	C₄×Dic₁₀₁	1	2	yes	—
216	C₂³	C₄×Dic₁₀₁	1	2	yes	—
217	C₂³	#13108051	1	1	yes	characteristic
218	C₂³	#13108051	1	1	yes	characteristic
219	C₂³	#13108051	1	1	yes	characteristic
220	C₂³	#13108051	1	2	yes	—
221	C₂³	#13108051	1	2	yes	—
222	C₂³	#13108051	1	1	yes	characteristic
223	C₂³	C₂²⋊Dic₁₀₁	1	2	yes	—
224	C₂³	C₂²⋊Dic₁₀₁	1	2	yes	—
225	C₂³	#13108053	1	2	yes	—
226	C₂³	#13108053	1	2	yes	—
227	C₂³	#13108053	1	2	yes	—
228	C₂³	#13108053	1	2	yes	—
229	C₂³	C₄⋊Dic₁₀₁	1	2	yes	—
230	C₂³	C₄⋊Dic₁₀₁	1	2	yes	—
231	C₂³	C₂×C₄×D₁₀₁	1	1	yes	characteristic
232	C₂³	C₂×C₄×D₁₀₁	1	1	yes	characteristic
233	C₂³	C₂²×Dic₁₀₁	1	1	yes	characteristic
234	C₂³	C₂×C₂²⋊D₁₀₁	1	1	yes	characteristic
235	C₂³	C₂×C₂²⋊D₁₀₁	1	1	yes	characteristic
236	C₂³	C₂×D₄₀₄	1	1	yes	characteristic
237	C₂³	C₂×Dic₂₀₂	1	1	yes	characteristic
238	C₂×C₄	—	2	4	no	—
239	C₂×C₄	—	4	4	no	—
240	C₂×C₄	—	4	8	no	—
241	C₂×C₄	—	404	4	no	—
242	C₂×C₄	—	404	8	no	—
243	C₂×C₄	—	404	4	no	—
244	C₂×C₄	—	404	4	no	—
245	C₂×C₄	—	404	4	no	—
246	C₂×C₄	—	202	4	no	—
247	C₂×C₄	—	202	4	no	—
248	C₂²	—	2	2	no	—
249	C₂²	—	2	2	no	—
250	C₂²	—	2	2	no	—
251	C₂²	—	2	2	no	—
252	C₂²	—	2	2	no	—
253	C₂²	—	2	2	no	—
254	C₂²	—	2	4	no	—
255	C₂²	—	2	4	no	—
256	C₂²	—	2	4	no	—
257	C₂²	—	2	4	no	—
258	C₂²	—	2	4	no	—
259	C₂²	—	2	4	no	—
260	C₂²	—	4	2	no	—
261	C₂²	—	4	2	no	—
262	C₂²	—	4	2	no	—
263	C₂²	—	4	2	no	—
264	C₂²	—	4	2	no	—
265	C₂²	—	4	2	no	—
266	C₂²	—	4	1	no	—
267	C₂²	—	4	1	no	—
268	C₂²	—	4	1	no	—
269	C₂²	—	4	1	no	—
270	C₂²	#15692162	1	2	yes	—
271	C₂²	#15692162	1	2	yes	—
272	C₂²	#15692162	1	2	yes	—
273	C₂²	#15692162	1	2	yes	—
274	C₂²	#15692169	1	4	yes	—
275	C₂²	C₂×C₄×Dic₁₀₁	1	1	yes	characteristic
276	C₂²	C₂×#13108051	1	1	yes	characteristic
277	C₂²	C₂×#13108051	1	1	yes	characteristic
278	C₂²	C₂×C₂²⋊Dic₁₀₁	1	1	yes	characteristic
279	C₂²	C₂×#13108053	1	1	yes	characteristic
280	C₂²	C₂×#13108053	1	1	yes	characteristic
281	C₂²	C₂×C₄⋊Dic₁₀₁	1	1	yes	characteristic
282	C₄	—	4	4	no	—
283	C₄	—	404	4	no	—
284	C₄	—	404	4	no	—
285	C₂	—	2	1	no	—
286	C₂	—	2	1	no	—
287	C₂	—	2	2	no	—
288	C₂	—	2	2	no	—
289	C₂	—	2	2	no	—
290	C₂	—	2	1	no	—
291	C₂	—	2	1	no	—
292	C₂	—	2	1	no	—
293	C₂	—	2	1	no	—
294	C₂	#20264943	1	4	yes	direct product factor
295	C₂	C₂×#15692162	1	1	yes	characteristic
296	C₂	C₂×#15692169	1	2	yes	—

	subgroup	G/H
center	C₂³	#13108051
commutator	C₂²×C₁₀₁	C₂×C₄²
Frattini	C₂⁴	C₂²×D₁₀₁
Fitting	C₂²×C₂²⋊C₄×C₁₀₁	C₂
socle	C₂³×C₁₀₁	C₂²⋊C₄
prime radical	C₂⁵×C₁₀₁	C₂²
cosocle	C₂⁴×C₁₀₁	C₂³
hypercenter	C₂²×C₂²⋊C₄	D₁₀₁
hypocenter	C₁₀₁	C₂×#342
LCS	G ▷ C₂²×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂³ ◁ C₂⁵ ◁ C₂²×C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
3. #20264943	C₂	direct (N × H)
6. C₂³×Dic₁₀₁	C₄	semidirect
13. C₂×C₂²⋊Dic₁₀₁	C₄	semidirect
14. C₂×C₂²⋊Dic₁₀₁	C₄	semidirect
21. C₂×C₂²⋊C₄×C₁₀₁	C₄	semidirect
29. C₂²×Dic₁₀₁	C₂×C₄	semidirect
36. C₂²⋊Dic₁₀₁	C₂×C₄	semidirect
54. C₂²⋊C₄×C₁₀₁	C₂×C₄	semidirect
145. C₂×C₁₀₁	#342	semidirect
149. C₁₀₁	C₂×#342	semidirect
170. C₂×C₂²⋊C₄	Dic₁₀₁	semidirect
203. C₂²⋊C₄	C₂×Dic₁₀₁	semidirect
294. C₂	#20264943	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₂²⋊Dic₁₀₁	2
#20264943	2
C₂²×C₂²⋊C₄×C₁₀₁	2
C₂×#342	101

Subgroups — C2×#20264943 #27170861

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20264943 #27170861