Subgroups — C₂×#20265048 #27173541

Order 12928.

1232 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₄×D₁₀₁	C₂	1	1	yes	maximal, characteristic
2	C₂²×#13108053	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×#13108051	C₂	1	1	yes	maximal, characteristic
4	C₂²×#13108051	C₂	1	1	yes	maximal, characteristic
5	#20265048	C₂	1	8	yes	maximal, direct product factor
6	C₂×#15692162	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#15692162	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₄⋊C₄×C₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂⁴×D₁₀₁	C₂²	1	1	yes	characteristic
10	C₂³×Dic₁₀₁	C₂²	1	1	yes	characteristic
11	C₂³×Dic₁₀₁	C₂²	1	1	yes	characteristic
12	C₂³×Dic₁₀₁	C₂²	1	1	yes	characteristic
13	C₂²×C₄×D₁₀₁	—	2	4	no	—
14	C₂²×C₄×D₁₀₁	—	2	4	no	—
15	C₂²×C₄×D₁₀₁	—	2	4	no	—
16	C₂²×C₄×D₁₀₁	C₂²	1	4	yes	—
17	C₂×#13108053	—	2	4	no	—
18	C₂×#13108053	C₂²	1	4	yes	complemented
19	C₂×#13108051	—	2	4	no	—
20	C₂×#13108051	C₄	1	8	yes	complemented
21	C₂×#13108051	C₂²	1	4	yes	—
22	C₂×#13108051	C₂²	1	4	yes	—
23	#15692162	C₂²	1	4	yes	complemented
24	#15692162	C₂²	1	4	yes	complemented
25	C₂³×C₄×C₁₀₁	C₂²	1	1	yes	characteristic
26	C₂³×C₄×C₁₀₁	C₂²	1	1	yes	characteristic
27	C₂³×C₄×C₁₀₁	C₂²	1	1	yes	characteristic
28	C₂×C₄⋊C₄×C₁₀₁	—	2	4	no	—
29	C₂×C₄⋊C₄×C₁₀₁	C₂²	1	4	yes	complemented
30	C₂³×D₁₀₁	—	2	2	no	—
31	C₂³×D₁₀₁	—	2	2	no	—
32	C₂³×D₁₀₁	—	2	2	no	—
33	C₂³×D₁₀₁	—	2	2	no	—
34	C₂³×D₁₀₁	C₂×C₄	1	4	yes	—
35	C₂³×D₁₀₁	D₄	1	4	yes	—
36	C₂³×D₁₀₁	Q₈	1	4	yes	—
37	C₂³×D₁₀₁	C₂³	1	2	yes	—
38	C₂²×Dic₁₀₁	—	2	2	no	—
39	C₂²×Dic₁₀₁	—	2	2	no	—
40	C₂²×Dic₁₀₁	—	2	2	no	—
41	C₂²×Dic₁₀₁	—	2	2	no	—
42	C₂²×Dic₁₀₁	—	2	2	no	—
43	C₂²×Dic₁₀₁	—	2	2	no	—
44	C₂²×Dic₁₀₁	—	2	2	no	—
45	C₂²×Dic₁₀₁	C₂×C₄	1	4	yes	complemented
46	C₂²×Dic₁₀₁	D₄	1	4	yes	—
47	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
48	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
49	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
50	C₂×C₄×D₁₀₁	—	2	32	no	—
51	C₂×C₄×D₁₀₁	—	2	8	no	—
52	C₂×C₄×D₁₀₁	—	2	8	no	—
53	C₂×C₄×D₁₀₁	—	2	8	no	—
54	#13108053	—	2	8	no	—
55	#13108051	—	2	8	no	—
56	#13108051	C₂×C₄	1	16	yes	complemented
57	C₂⁴×C₁₀₁	C₂³	1	1	yes	characteristic
58	C₂²×C₄×C₁₀₁	—	2	2	no	—
59	C₂²×C₄×C₁₀₁	—	2	2	no	—
60	C₂²×C₄×C₁₀₁	—	2	2	no	—
61	C₂²×C₄×C₁₀₁	—	2	2	no	—
62	C₂²×C₄×C₁₀₁	—	2	2	no	—
63	C₂²×C₄×C₁₀₁	—	2	2	no	—
64	C₂²×C₄×C₁₀₁	—	2	2	no	—
65	C₂²×C₄×C₁₀₁	C₂×C₄	1	4	yes	complemented
66	C₂²×C₄×C₁₀₁	D₄	1	4	yes	—
67	C₂²×C₄×C₁₀₁	C₂³	1	2	yes	—
68	C₂²×C₄×C₁₀₁	C₂³	1	2	yes	—
69	C₂²×C₄×C₁₀₁	C₂³	1	2	yes	—
70	C₄⋊C₄×C₁₀₁	—	2	8	no	—
71	C₂³×C₁₀₁	C₂²×C₄	1	2	yes	—
72	C₂³×C₁₀₁	C₂×D₄	1	2	yes	—
73	C₂³×C₁₀₁	C₂×D₄	1	2	yes	—
74	C₂³×C₁₀₁	C₂×D₄	1	2	yes	—
75	C₂³×C₁₀₁	C₂×Q₈	1	2	yes	—
76	C₂³×C₁₀₁	#41	1	2	yes	—
77	C₂³×C₁₀₁	#41	1	2	yes	—
78	C₂³×C₁₀₁	C₂⁴	1	1	yes	characteristic
79	C₂²×D₁₀₁	—	2	8	no	—
80	C₂²×D₁₀₁	—	2	8	no	—
81	C₂²×D₁₀₁	—	2	8	no	—
82	C₂²×D₁₀₁	—	2	8	no	—
83	C₂²×D₁₀₁	—	2	8	no	—
84	C₂²×D₁₀₁	—	2	8	no	—
85	C₂²×D₁₀₁	—	2	2	no	—
86	C₂²×D₁₀₁	—	2	2	no	—
87	C₂²×D₁₀₁	—	2	2	no	—
88	C₂²×D₁₀₁	—	2	2	no	—
89	C₂²×D₁₀₁	C₄⋊C₄	1	16	yes	complemented
90	C₂²×D₁₀₁	C₂²×C₄	1	4	yes	—
91	C₂²×D₁₀₁	C₂×D₄	1	4	yes	—
92	C₂²×D₁₀₁	C₂×Q₈	1	4	yes	—
93	C₂×C₄×C₁₀₁	—	2	8	no	—
94	C₂×C₄×C₁₀₁	—	2	8	no	—
95	C₂×C₄×C₁₀₁	—	2	2	no	—
96	C₂×C₄×C₁₀₁	—	2	2	no	—
97	C₂×C₄×C₁₀₁	—	2	2	no	—
98	C₂×C₄×C₁₀₁	—	2	2	no	—
99	C₂×C₄×C₁₀₁	—	2	2	no	—
100	C₂×C₄×C₁₀₁	—	2	2	no	—
101	C₂×C₄×C₁₀₁	—	2	2	no	—
102	C₂×C₄×C₁₀₁	—	4	4	no	—
103	C₂×C₄×C₁₀₁	C₂²×C₄	1	4	yes	complemented
104	C₂×C₄×C₁₀₁	C₂×D₄	1	4	yes	—
105	C₂×Dic₁₀₁	—	2	8	no	—
106	C₂×Dic₁₀₁	—	2	8	no	—
107	C₂×Dic₁₀₁	—	2	2	no	—
108	C₂×Dic₁₀₁	—	2	2	no	—
109	C₂×Dic₁₀₁	—	2	2	no	—
110	C₂×Dic₁₀₁	—	2	2	no	—
111	C₂×Dic₁₀₁	—	2	2	no	—
112	C₂×Dic₁₀₁	—	2	2	no	—
113	C₂×Dic₁₀₁	—	2	2	no	—
114	C₂×Dic₁₀₁	—	4	4	no	—
115	C₂×Dic₁₀₁	C₂²×C₄	1	4	yes	complemented
116	C₂×Dic₁₀₁	C₂×D₄	1	4	yes	—
117	C₄×D₁₀₁	—	2	32	no	—
118	C₂²×C₁₀₁	C₂×C₄⋊C₄	1	4	yes	—
119	C₂²×C₁₀₁	C₄×D₄	1	4	yes	—
120	C₂²×C₁₀₁	C₄×D₄	1	4	yes	—
121	C₂²×C₁₀₁	#120	1	4	yes	—
122	C₂²×C₁₀₁	C₂²⋊Q₈	1	4	yes	—
123	C₂²×C₁₀₁	C₂²⋊Q₈	1	4	yes	—
124	C₂²×C₁₀₁	#123	1	4	yes	—
125	C₂²×C₁₀₁	C₂³×C₄	1	1	yes	characteristic
126	C₂²×C₁₀₁	C₂²×D₄	1	1	yes	characteristic
127	C₂²×C₁₀₁	C₂²×D₄	1	1	yes	characteristic
128	C₂²×C₁₀₁	C₂²×D₄	1	1	yes	characteristic
129	C₂²×C₁₀₁	C₂²×Q₈	1	1	yes	characteristic
130	C₂²×C₁₀₁	C₂×#41	1	1	yes	characteristic
131	C₂²×C₁₀₁	C₂×#41	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	—	2	4	no	—
136	C₂×D₁₀₁	—	2	4	no	—
137	C₂×D₁₀₁	—	2	4	no	—
138	C₂×D₁₀₁	—	2	32	no	—
139	C₂×D₁₀₁	C₂×C₄⋊C₄	1	8	yes	complemented
140	C₄×C₁₀₁	—	2	4	no	—
141	C₄×C₁₀₁	—	2	4	no	—
142	C₄×C₁₀₁	—	4	2	no	—
143	Dic₁₀₁	—	2	4	no	—
144	Dic₁₀₁	—	2	4	no	—
145	Dic₁₀₁	—	4	2	no	—
146	C₂×C₁₀₁	#385	1	8	yes	complemented
147	C₂×C₁₀₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
148	C₂×C₁₀₁	C₂×C₄×D₄	1	1	yes	characteristic
149	C₂×C₁₀₁	C₂×C₄×D₄	1	1	yes	characteristic
150	C₂×C₁₀₁	C₂×#120	1	1	yes	characteristic
151	C₂×C₁₀₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
152	C₂×C₁₀₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
153	C₂×C₁₀₁	C₂×#123	1	1	yes	characteristic
154	D₁₀₁	—	2	8	no	—
155	C₂×#385	—	101	1	no	maximal, Sylow
156	C₁₀₁	C₂×#385	1	1	yes	characteristic, complemented, Sylow
157	C₂⁴×C₄	—	101	1	no	—
158	C₂²×C₄⋊C₄	—	101	1	no	—
159	C₂²×C₄⋊C₄	D₁₀₁	1	1	yes	characteristic, complemented
160	C₂²×C₂²⋊C₄	—	101	1	no	—
161	C₂²×C₂²⋊C₄	—	101	1	no	—
162	#385	—	101	8	no	—
163	C₂×#95	—	101	1	no	—
164	C₂×#95	—	101	1	no	—
165	C₂⁵	—	101	1	no	—
166	C₂³×C₄	—	101	1	no	—
167	C₂³×C₄	—	101	1	no	—
168	C₂³×C₄	—	101	1	no	—
169	C₂³×C₄	—	101	4	no	—
170	C₂³×C₄	—	202	4	no	—
171	C₂³×C₄	—	202	4	no	—
172	C₂³×C₄	—	202	4	no	—
173	C₂³×C₄	C₂×D₁₀₁	1	1	yes	characteristic
174	C₂³×C₄	C₂×D₁₀₁	1	1	yes	characteristic
175	C₂³×C₄	C₂×D₁₀₁	1	1	yes	characteristic
176	C₂×C₄⋊C₄	—	2	4	no	—
177	C₂×C₄⋊C₄	—	101	4	no	—
178	C₂×C₄⋊C₄	—	202	4	no	—
179	C₂×C₄⋊C₄	C₂×D₁₀₁	1	4	yes	complemented
180	C₂×C₂²⋊C₄	—	101	8	no	—
181	C₂×C₂²⋊C₄	—	101	4	no	—
182	C₂×C₂²⋊C₄	—	101	4	no	—
183	C₂×C₂²⋊C₄	—	202	4	no	—
184	#95	—	101	4	no	—
185	#95	—	101	4	no	—
186	C₂⁴	—	101	4	no	—
187	C₂⁴	—	101	4	no	—
188	C₂⁴	—	101	4	no	—
189	C₂⁴	—	101	2	no	—
190	C₂⁴	—	202	2	no	—
191	C₂⁴	—	202	2	no	—
192	C₂⁴	—	202	2	no	—
193	C₂⁴	—	202	2	no	—
194	C₂⁴	C₂²×D₁₀₁	1	1	yes	characteristic
195	C₂²×C₄	—	2	2	no	—
196	C₂²×C₄	—	2	2	no	—
197	C₂²×C₄	—	2	2	no	—
198	C₂²×C₄	—	2	2	no	—
199	C₂²×C₄	—	2	2	no	—
200	C₂²×C₄	—	2	2	no	—
201	C₂²×C₄	—	2	2	no	—
202	C₂²×C₄	—	101	4	no	—
203	C₂²×C₄	—	101	4	no	—
204	C₂²×C₄	—	101	2	no	—
205	C₂²×C₄	—	101	2	no	—
206	C₂²×C₄	—	101	2	no	—
207	C₂²×C₄	—	202	8	no	—
208	C₂²×C₄	—	202	8	no	—
209	C₂²×C₄	—	202	32	no	—
210	C₂²×C₄	—	202	2	no	—
211	C₂²×C₄	—	202	2	no	—
212	C₂²×C₄	—	202	2	no	—
213	C₂²×C₄	—	202	2	no	—
214	C₂²×C₄	—	202	2	no	—
215	C₂²×C₄	—	202	2	no	—
216	C₂²×C₄	—	202	2	no	—
217	C₂²×C₄	—	202	8	no	—
218	C₂²×C₄	C₄×D₁₀₁	1	4	yes	complemented
219	C₂²×C₄	C₂²⋊D₁₀₁	1	4	yes	—
220	C₂²×C₄	C₂²×D₁₀₁	1	2	yes	—
221	C₂²×C₄	C₂²×D₁₀₁	1	2	yes	—
222	C₂²×C₄	C₂²×D₁₀₁	1	2	yes	—
223	C₄⋊C₄	—	2	8	no	—
224	C₄⋊C₄	—	202	8	no	—
225	C₂²⋊C₄	—	101	16	no	—
226	C₂²⋊C₄	—	202	8	no	—
227	C₂³	—	101	4	no	—
228	C₂³	—	101	4	no	—
229	C₂³	—	101	4	no	—
230	C₂³	—	101	16	no	—
231	C₂³	—	202	2	no	—
232	C₂³	—	202	2	no	—
233	C₂³	—	202	2	no	—
234	C₂³	—	202	2	no	—
235	C₂³	—	202	8	no	—
236	C₂³	—	202	8	no	—
237	C₂³	—	202	8	no	—
238	C₂³	—	202	8	no	—
239	C₂³	—	202	8	no	—
240	C₂³	—	202	8	no	—
241	C₂³	C₂×C₄×D₁₀₁	1	2	yes	—
242	C₂³	D₄×D₁₀₁	1	2	yes	—
243	C₂³	D₄×D₁₀₁	1	2	yes	—
244	C₂³	C₂×C₂²⋊D₁₀₁	1	2	yes	—
245	C₂³	Q₈×D₁₀₁	1	2	yes	—
246	C₂³	#13108074	1	2	yes	—
247	C₂³	#13108075	1	2	yes	—
248	C₂³	C₂³×D₁₀₁	1	1	yes	characteristic
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	2	8	no	—
254	C₂×C₄	—	2	8	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	4	4	no	—
259	C₂×C₄	—	101	4	no	—
260	C₂×C₄	—	101	4	no	—
261	C₂×C₄	—	404	4	no	—
262	C₂×C₄	—	202	32	no	—
263	C₂×C₄	—	202	2	no	—
264	C₂×C₄	—	202	2	no	—
265	C₂×C₄	—	202	2	no	—
266	C₂×C₄	—	202	2	no	—
267	C₂×C₄	—	202	2	no	—
268	C₂×C₄	—	202	2	no	—
269	C₂×C₄	—	202	2	no	—
270	C₂×C₄	—	202	8	no	—
271	C₂×C₄	—	202	8	no	—
272	C₂×C₄	C₂×C₄×D₁₀₁	1	4	yes	complemented
273	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	4	yes	—
274	C₂²	—	101	8	no	—
275	C₂²	—	202	32	no	—
276	C₂²	—	202	4	no	—
277	C₂²	—	202	4	no	—
278	C₂²	—	202	4	no	—
279	C₂²	—	202	4	no	—
280	C₂²	—	202	4	no	—
281	C₂²	—	202	4	no	—
282	C₂²	C₄⋊C₄×D₁₀₁	1	4	yes	—
283	C₂²	C₄×C₂²⋊D₁₀₁	1	4	yes	—
284	C₂²	#15692217	1	4	yes	—
285	C₂²	#15692223	1	4	yes	—
286	C₂²	#15692231	1	4	yes	—
287	C₂²	#15692233	1	4	yes	—
288	C₂²	#15692240	1	4	yes	—
289	C₂²	C₂²×C₄×D₁₀₁	1	1	yes	characteristic
290	C₂²	C₂×D₄×D₁₀₁	1	1	yes	characteristic
291	C₂²	C₂×D₄×D₁₀₁	1	1	yes	characteristic
292	C₂²	C₂²×C₂²⋊D₁₀₁	1	1	yes	characteristic
293	C₂²	C₂×Q₈×D₁₀₁	1	1	yes	characteristic
294	C₂²	C₂×#13108074	1	1	yes	characteristic
295	C₂²	C₂×#13108075	1	1	yes	characteristic
296	C₄	—	2	4	no	—
297	C₄	—	2	4	no	—
298	C₄	—	4	2	no	—
299	C₄	—	404	2	no	—
300	C₄	—	202	4	no	—
301	C₄	—	202	4	no	—
302	C₂	—	202	8	no	—
303	C₂	#20265048	1	8	yes	direct product factor
304	C₂	C₂×C₄⋊C₄×D₁₀₁	1	1	yes	characteristic
305	C₂	C₂×C₄×C₂²⋊D₁₀₁	1	1	yes	characteristic
306	C₂	C₂×#15692217	1	1	yes	characteristic
307	C₂	C₂×#15692223	1	1	yes	characteristic
308	C₂	C₂×#15692231	1	1	yes	characteristic
309	C₂	C₂×#15692233	1	1	yes	characteristic
310	C₂	C₂×#15692240	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂²×D₁₀₁
commutator	C₂²×C₁₀₁	C₂³×C₄
Frattini	C₂³	C₂³×D₁₀₁
Fitting	C₂²×C₄⋊C₄×C₁₀₁	C₂
socle	C₂⁴×C₁₀₁	C₂³
prime radical	C₂⁴×D₁₀₁	C₂²
cosocle	C₂³×C₁₀₁	C₂⁴
hypercenter	C₂²×C₄⋊C₄	D₁₀₁
hypocenter	C₁₀₁	C₂×#385
LCS	G ▷ C₂²×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
2. C₂²×#13108053	C₂	semidirect
5. #20265048	C₂	direct (N × H)
6. C₂×#15692162	C₂	semidirect
7. C₂×#15692162	C₂	semidirect
8. C₂²×C₄⋊C₄×C₁₀₁	C₂	semidirect
18. C₂×#13108053	C₂²	semidirect
20. C₂×#13108051	C₄	semidirect
23. #15692162	C₂²	semidirect
24. #15692162	C₂²	semidirect
29. C₂×C₄⋊C₄×C₁₀₁	C₂²	semidirect
45. C₂²×Dic₁₀₁	C₂×C₄	semidirect
56. #13108051	C₂×C₄	semidirect
65. C₂²×C₄×C₁₀₁	C₂×C₄	semidirect
89. C₂²×D₁₀₁	C₄⋊C₄	semidirect
103. C₂×C₄×C₁₀₁	C₂²×C₄	semidirect
115. C₂×Dic₁₀₁	C₂²×C₄	semidirect
139. C₂×D₁₀₁	C₂×C₄⋊C₄	semidirect
146. C₂×C₁₀₁	#385	semidirect
156. C₁₀₁	C₂×#385	semidirect
159. C₂²×C₄⋊C₄	D₁₀₁	semidirect
179. C₂×C₄⋊C₄	C₂×D₁₀₁	semidirect
218. C₂²×C₄	C₄×D₁₀₁	semidirect
272. C₂×C₄	C₂×C₄×D₁₀₁	semidirect
303. C₂	#20265048	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₄×D₁₀₁	2
C₂²×#13108053	2
C₂²×#13108051	2
#20265048	2
C₂×#15692162	2
C₂²×C₄⋊C₄×C₁₀₁	2
C₂×#385	101

Subgroups — C2×#20265048 #27173541

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20265048 #27173541