Subgroups — C₂×#20265749 #27184447

Order 12928.

1512 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13108073	C₂	1	1	yes	maximal, characteristic
2	C₂³×Dic₂₀₂	C₂	1	1	yes	maximal, characteristic, complemented
3	#20265749	C₂	1	16	yes	maximal, direct product factor
4	C₂×#15692254	C₂	1	2	yes	maximal, complemented
5	C₂×#15692241	C₂	1	2	yes	maximal, complemented
6	C₂×#15692239	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#15692238	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#15692236	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#15692233	C₂	1	2	yes	maximal, complemented
10	C₂×#15692232	C₂	1	2	yes	maximal, complemented
11	C₂×#15692211	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#123×C₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂×#13108073	—	2	8	no	—
14	C₂×#13108073	C₂²	1	8	yes	—
15	C₂²×Dic₂₀₂	—	2	8	no	—
16	C₂²×Dic₂₀₂	C₂²	1	2	yes	—
17	C₂²×Dic₂₀₂	C₂²	1	2	yes	—
18	C₂²×Dic₂₀₂	C₂²	1	1	yes	characteristic
19	C₂²×Dic₂₀₂	C₂²	1	8	yes	complemented
20	C₂²×C₂²⋊D₁₀₁	C₂²	1	1	yes	characteristic
21	C₂²×C₂²⋊D₁₀₁	C₂²	1	1	yes	characteristic
22	C₂³×Dic₁₀₁	C₂²	1	1	yes	characteristic
23	C₂³×Dic₁₀₁	C₂²	1	1	yes	characteristic
24	C₂³×Dic₁₀₁	C₂²	1	1	yes	characteristic
25	C₂²×C₄×D₁₀₁	C₂²	1	1	yes	characteristic
26	#15692254	C₂²	1	16	yes	complemented
27	#15692241	C₂²	1	16	yes	complemented
28	#15692239	C₂²	1	8	yes	complemented
29	#15692238	C₂²	1	8	yes	complemented
30	#15692236	C₂²	1	8	yes	complemented
31	#15692233	C₂²	1	16	yes	complemented
32	#15692232	C₂²	1	16	yes	complemented
33	#15692211	C₂²	1	8	yes	complemented
34	C₂×C₄⋊Dic₁₀₁	C₂²	1	2	yes	—
35	C₂×#13108053	C₂²	1	2	yes	—
36	C₂×#13108053	C₂²	1	2	yes	—
37	C₂×#13108053	C₂²	1	2	yes	—
38	C₂×C₂²⋊Dic₁₀₁	C₂²	1	1	yes	characteristic
39	C₂×C₂²⋊Dic₁₀₁	C₂²	1	2	yes	—
40	C₂×#13108051	C₂²	1	2	yes	—
41	C₂×#13108051	C₂²	1	2	yes	—
42	C₂×C₄×Dic₁₀₁	C₂²	1	2	yes	—
43	C₂²×D₄×C₁₀₁	C₂²	1	1	yes	characteristic
44	C₂³×C₄×C₁₀₁	C₂²	1	1	yes	characteristic
45	#123×C₁₀₁	C₂²	1	8	yes	complemented
46	C₂×C₄⋊C₄×C₁₀₁	C₂²	1	2	yes	—
47	C₂×C₂²⋊C₄×C₁₀₁	C₂²	1	1	yes	characteristic
48	C₂×C₂²⋊C₄×C₁₀₁	C₂²	1	2	yes	—
49	C₂³×D₁₀₁	C₂³	1	1	yes	characteristic
50	#13108073	—	2	32	no	—
51	C₂×Dic₂₀₂	—	2	8	no	—
52	C₂×Dic₂₀₂	—	2	8	no	—
53	C₂×Dic₂₀₂	—	2	32	no	—
54	C₂×Dic₂₀₂	—	2	4	no	—
55	C₂×Dic₂₀₂	C₂³	1	8	yes	—
56	C₂×Dic₂₀₂	C₂³	1	8	yes	—
57	C₂×Dic₂₀₂	C₂³	1	4	yes	—
58	C₂×C₂²⋊D₁₀₁	—	2	4	no	—
59	C₂×C₂²⋊D₁₀₁	—	2	4	no	—
60	C₂×C₂²⋊D₁₀₁	C₂³	1	4	yes	—
61	C₂×C₂²⋊D₁₀₁	C₂³	1	4	yes	—
62	C₂²×Dic₁₀₁	—	2	4	no	—
63	C₂²×Dic₁₀₁	—	2	4	no	—
64	C₂²×Dic₁₀₁	D₄	1	8	yes	—
65	C₂²×Dic₁₀₁	C₂³	1	1	yes	characteristic
66	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
67	C₂²×Dic₁₀₁	C₂³	1	4	yes	—
68	C₂²×Dic₁₀₁	C₂³	1	4	yes	—
69	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
70	C₂²×Dic₁₀₁	C₂³	1	1	yes	characteristic
71	C₂²×Dic₁₀₁	C₂³	1	1	yes	characteristic
72	C₂²×Dic₁₀₁	C₂³	1	4	yes	—
73	C₂×C₄×D₁₀₁	—	2	4	no	—
74	C₂×C₄×D₁₀₁	C₂³	1	4	yes	—
75	C₄⋊Dic₁₀₁	C₂³	1	8	yes	—
76	#13108053	C₂³	1	8	yes	—
77	#13108053	C₂³	1	8	yes	—
78	#13108053	C₂³	1	8	yes	—
79	C₂²⋊Dic₁₀₁	C₂³	1	4	yes	—
80	C₂²⋊Dic₁₀₁	C₂³	1	8	yes	—
81	#13108051	C₂³	1	8	yes	—
82	#13108051	C₂³	1	8	yes	—
83	C₄×Dic₁₀₁	C₂³	1	8	yes	—
84	C₂⁴×C₁₀₁	C₂³	1	1	yes	characteristic
85	C₂⁴×C₁₀₁	C₂³	1	1	yes	characteristic
86	C₂×D₄×C₁₀₁	—	2	4	no	—
87	C₂×D₄×C₁₀₁	C₂³	1	4	yes	—
88	C₂²×C₄×C₁₀₁	—	2	4	no	—
89	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
90	C₂²×C₄×C₁₀₁	C₂³	1	2	yes	—
91	C₂²×C₄×C₁₀₁	C₂³	1	4	yes	—
92	C₂²×C₄×C₁₀₁	C₂³	1	2	yes	—
93	C₄⋊C₄×C₁₀₁	C₂³	1	8	yes	—
94	C₂²⋊C₄×C₁₀₁	C₂³	1	4	yes	—
95	C₂²⋊C₄×C₁₀₁	C₂³	1	8	yes	—
96	C₂³×C₁₀₁	—	2	4	no	—
97	C₂³×C₁₀₁	—	2	2	no	—
98	C₂³×C₁₀₁	—	2	4	no	—
99	C₂³×C₁₀₁	C₂×D₄	1	4	yes	—
100	C₂³×C₁₀₁	C₂⁴	1	2	yes	—
101	C₂³×C₁₀₁	C₂⁴	1	2	yes	—
102	C₂³×C₁₀₁	C₂⁴	1	1	yes	characteristic
103	C₂²×D₁₀₁	—	2	2	no	—
104	C₂²×D₁₀₁	—	2	4	no	—
105	C₂²×D₁₀₁	C₂⁴	1	2	yes	—
106	D₄×C₁₀₁	—	2	8	no	—
107	C₂×C₄×C₁₀₁	—	2	8	no	—
108	C₂×C₄×C₁₀₁	—	2	4	no	—
109	C₂×C₄×C₁₀₁	—	2	2	no	—
110	C₂×C₄×C₁₀₁	—	2	4	no	—
111	C₂×C₄×C₁₀₁	C₂⁴	1	4	yes	—
112	C₂×C₄×C₁₀₁	C₂⁴	1	4	yes	—
113	C₂×C₄×C₁₀₁	C₂⁴	1	2	yes	—
114	C₂²⋊D₁₀₁	—	2	8	no	—
115	C₂²⋊D₁₀₁	—	2	8	no	—
116	C₂×Dic₁₀₁	—	2	2	no	—
117	C₂×Dic₁₀₁	—	2	8	no	—
118	C₂×Dic₁₀₁	—	2	4	no	—
119	C₂×Dic₁₀₁	—	2	4	no	—
120	C₂×Dic₁₀₁	—	2	8	no	—
121	C₂×Dic₁₀₁	C₂×D₄	1	4	yes	—
122	C₂×Dic₁₀₁	C₂×D₄	1	4	yes	—
123	C₂×Dic₁₀₁	C₂×D₄	1	16	yes	—
124	C₂×Dic₁₀₁	C₂⁴	1	4	yes	—
125	C₂×Dic₁₀₁	C₂⁴	1	4	yes	—
126	C₂×Dic₁₀₁	C₂⁴	1	2	yes	—
127	C₂×Dic₁₀₁	C₂⁴	1	2	yes	—
128	C₂×Dic₁₀₁	C₂⁴	1	2	yes	—
129	C₄×D₁₀₁	—	2	8	no	—
130	Dic₂₀₂	—	2	16	no	—
131	Dic₂₀₂	—	2	16	no	—
132	Dic₂₀₂	—	2	8	no	—
133	C₂²×C₁₀₁	—	2	8	no	—
134	C₂²×C₁₀₁	—	2	4	no	—
135	C₂²×C₁₀₁	—	2	2	no	—
136	C₂²×C₁₀₁	—	2	4	no	—
137	C₂²×C₁₀₁	—	4	4	no	—
138	C₂²×C₁₀₁	C₂²×D₄	1	4	yes	—
139	C₂²×C₁₀₁	C₂²×D₄	1	2	yes	—
140	C₂²×C₁₀₁	2₋¹⁺⁴	1	4	yes	—
141	C₂²×C₁₀₁	C₂⁵	1	1	yes	characteristic
142	C₂×D₁₀₁	—	2	2	no	—
143	C₂×D₁₀₁	—	2	4	no	—
144	C₂×D₁₀₁	—	4	4	no	—
145	C₄×C₁₀₁	—	2	4	no	—
146	C₄×C₁₀₁	—	2	2	no	—
147	C₄×C₁₀₁	—	2	4	no	—
148	Dic₁₀₁	—	2	4	no	—
149	Dic₁₀₁	—	2	2	no	—
150	Dic₁₀₁	—	2	4	no	—
151	Dic₁₀₁	C₂²×D₄	1	4	yes	—
152	Dic₁₀₁	C₂²×D₄	1	4	yes	—
153	C₂×C₁₀₁	—	2	4	no	—
154	C₂×C₁₀₁	—	4	2	no	—
155	C₂×C₁₀₁	#536	1	4	yes	complemented
156	C₂×C₁₀₁	C₂³×D₄	1	1	yes	characteristic
157	C₂×C₁₀₁	C₂×2₋¹⁺⁴	1	2	yes	—
158	D₁₀₁	—	4	2	no	—
159	C₂×#536	—	101	1	no	maximal, Sylow
160	C₁₀₁	C₂×#536	1	1	yes	characteristic, complemented, Sylow
161	C₂²×#41	—	101	1	no	—
162	C₂³×Q₈	—	101	1	no	—
163	#536	—	101	16	no	—
164	C₂×C₄⋊Q₈	—	101	2	no	—
165	C₂×#124	—	101	2	no	—
166	C₂×#123	—	101	1	no	—
167	C₂×#123	—	101	1	no	—
168	C₂×#123	—	101	1	no	—
169	C₂×#123	D₁₀₁	1	1	yes	characteristic, complemented
170	C₂×C₂²⋊Q₈	—	101	2	no	—
171	C₂×C₂²⋊Q₈	—	101	2	no	—
172	C₂×#117	—	101	1	no	—
173	C₂×#41	—	101	8	no	—
174	C₂×#41	—	202	8	no	—
175	C₂²×Q₈	—	101	2	no	—
176	C₂²×Q₈	—	101	2	no	—
177	C₂²×Q₈	—	101	1	no	—
178	C₂²×Q₈	—	101	8	no	—
179	C₂²×Q₈	—	202	8	no	—
180	C₂²×D₄	—	101	1	no	—
181	C₂²×D₄	—	101	1	no	—
182	C₂²×D₄	C₂×D₁₀₁	1	1	yes	characteristic
183	C₂³×C₄	—	101	1	no	—
184	C₂³×C₄	—	101	1	no	—
185	C₂³×C₄	—	101	1	no	—
186	C₂³×C₄	—	101	1	no	—
187	C₂³×C₄	C₂×D₁₀₁	1	1	yes	characteristic
188	C₄⋊Q₈	—	101	16	no	—
189	#124	—	101	16	no	—
190	#123	—	101	8	no	—
191	#123	—	101	8	no	—
192	#123	—	101	8	no	—
193	#123	C₂×D₁₀₁	1	8	yes	complemented
194	C₂²⋊Q₈	—	101	16	no	—
195	C₂²⋊Q₈	—	101	16	no	—
196	#117	—	101	8	no	—
197	C₂×C₄⋊C₄	—	101	2	no	—
198	C₂×C₄⋊C₄	—	101	2	no	—
199	C₂×C₄⋊C₄	—	101	2	no	—
200	C₂×C₄⋊C₄	—	101	2	no	—
201	C₂×C₄⋊C₄	C₂×D₁₀₁	1	2	yes	—
202	C₂×C₂²⋊C₄	—	101	1	no	—
203	C₂×C₂²⋊C₄	—	101	2	no	—
204	C₂×C₂²⋊C₄	—	101	2	no	—
205	C₂×C₂²⋊C₄	—	101	2	no	—
206	C₂×C₂²⋊C₄	C₂×D₁₀₁	1	1	yes	characteristic
207	C₂×C₂²⋊C₄	C₂×D₁₀₁	1	2	yes	—
208	C₂×C₄²	—	101	2	no	—
209	C₂⁴	—	101	1	no	—
210	C₂⁴	C₂²×D₁₀₁	1	1	yes	characteristic
211	C₂⁴	C₂²×D₁₀₁	1	1	yes	characteristic
212	#41	—	202	32	no	—
213	C₂×Q₈	—	101	8	no	—
214	C₂×Q₈	—	101	8	no	—
215	C₂×Q₈	—	101	4	no	—
216	C₂×Q₈	—	202	8	no	—
217	C₂×Q₈	—	202	8	no	—
218	C₂×Q₈	—	202	32	no	—
219	C₂×Q₈	—	202	4	no	—
220	C₂×D₄	—	2	4	no	—
221	C₂×D₄	—	101	4	no	—
222	C₂×D₄	—	101	4	no	—
223	C₂×D₄	—	202	4	no	—
224	C₂×D₄	—	202	4	no	—
225	C₂×D₄	C₂²×D₁₀₁	1	4	yes	—
226	C₂²×C₄	—	2	4	no	—
227	C₂²×C₄	—	101	8	no	—
228	C₂²×C₄	—	101	4	no	—
229	C₂²×C₄	—	101	1	no	—
230	C₂²×C₄	—	101	1	no	—
231	C₂²×C₄	—	101	4	no	—
232	C₂²×C₄	—	101	4	no	—
233	C₂²×C₄	—	101	2	no	—
234	C₂²×C₄	—	101	2	no	—
235	C₂²×C₄	—	101	1	no	—
236	C₂²×C₄	—	101	4	no	—
237	C₂²×C₄	—	202	4	no	—
238	C₂²×C₄	—	202	4	no	—
239	C₂²×C₄	—	202	4	no	—
240	C₂²×C₄	C₂²×D₁₀₁	1	4	yes	—
241	C₂²×C₄	C₂²×D₁₀₁	1	2	yes	—
242	C₂²×C₄	C₂²×D₁₀₁	1	2	yes	—
243	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
244	C₄⋊C₄	—	101	8	no	—
245	C₄⋊C₄	—	101	8	no	—
246	C₄⋊C₄	—	101	8	no	—
247	C₄⋊C₄	—	101	8	no	—
248	C₄⋊C₄	C₂²×D₁₀₁	1	8	yes	—
249	C₂²⋊C₄	—	101	4	no	—
250	C₂²⋊C₄	—	101	8	no	—
251	C₂²⋊C₄	—	101	8	no	—
252	C₂²⋊C₄	—	101	8	no	—
253	C₂²⋊C₄	C₂²×D₁₀₁	1	4	yes	—
254	C₂²⋊C₄	C₂²×D₁₀₁	1	8	yes	—
255	C₄²	—	101	8	no	—
256	C₂³	—	2	4	no	—
257	C₂³	—	2	2	no	—
258	C₂³	—	2	4	no	—
259	C₂³	—	101	2	no	—
260	C₂³	—	202	2	no	—
261	C₂³	—	202	4	no	—
262	C₂³	D₄×D₁₀₁	1	4	yes	—
263	C₂³	C₂³×D₁₀₁	1	2	yes	—
264	C₂³	C₂³×D₁₀₁	1	2	yes	—
265	C₂³	C₂³×D₁₀₁	1	1	yes	characteristic
266	Q₈	—	202	16	no	—
267	Q₈	—	202	16	no	—
268	Q₈	—	202	8	no	—
269	D₄	—	2	8	no	—
270	D₄	—	202	8	no	—
271	D₄	—	202	8	no	—
272	C₂×C₄	—	2	8	no	—
273	C₂×C₄	—	2	4	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	2	4	no	—
276	C₂×C₄	—	101	4	no	—
277	C₂×C₄	—	101	4	no	—
278	C₂×C₄	—	101	16	no	—
279	C₂×C₄	—	101	2	no	—
280	C₂×C₄	—	101	2	no	—
281	C₂×C₄	—	101	4	no	—
282	C₂×C₄	—	101	4	no	—
283	C₂×C₄	—	101	2	no	—
284	C₂×C₄	—	202	8	no	—
285	C₂×C₄	—	202	8	no	—
286	C₂×C₄	—	202	2	no	—
287	C₂×C₄	—	202	8	no	—
288	C₂×C₄	—	202	4	no	—
289	C₂×C₄	—	202	4	no	—
290	C₂×C₄	C₂³×D₁₀₁	1	4	yes	—
291	C₂×C₄	C₂³×D₁₀₁	1	4	yes	—
292	C₂×C₄	C₂³×D₁₀₁	1	2	yes	—
293	C₂²	—	2	8	no	—
294	C₂²	—	2	4	no	—
295	C₂²	—	2	2	no	—
296	C₂²	—	2	4	no	—
297	C₂²	—	4	4	no	—
298	C₂²	—	404	4	no	—
299	C₂²	—	202	2	no	—
300	C₂²	—	202	4	no	—
301	C₂²	C₂×D₄×D₁₀₁	1	4	yes	—
302	C₂²	C₂×D₄×D₁₀₁	1	2	yes	—
303	C₂²	#15692302	1	4	yes	—
304	C₂²	C₂⁴×D₁₀₁	1	1	yes	characteristic
305	C₄	—	2	4	no	—
306	C₄	—	2	2	no	—
307	C₄	—	2	4	no	—
308	C₄	—	101	4	no	—
309	C₄	—	101	4	no	—
310	C₄	—	202	2	no	—
311	C₄	—	202	4	no	—
312	C₄	—	202	4	no	—
313	C₂	—	2	4	no	—
314	C₂	—	4	2	no	—
315	C₂	—	404	2	no	—
316	C₂	#20265749	1	4	yes	direct product factor
317	C₂	C₂²×D₄×D₁₀₁	1	1	yes	characteristic
318	C₂	C₂×#15692302	1	2	yes	—

	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	C₂²×#13108073	C₂
hypercenter	C₂×#123	D₁₀₁
hypocenter	C₁₀₁	C₂×#536
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
2. C₂³×Dic₂₀₂	C₂	semidirect
3. #20265749	C₂	direct (N × H)
4. C₂×#15692254	C₂	semidirect
5. C₂×#15692241	C₂	semidirect
6. C₂×#15692239	C₂	semidirect
7. C₂×#15692238	C₂	semidirect
8. C₂×#15692236	C₂	semidirect
9. C₂×#15692233	C₂	semidirect
10. C₂×#15692232	C₂	semidirect
11. C₂×#15692211	C₂	semidirect
12. C₂×#123×C₁₀₁	C₂	semidirect
19. C₂²×Dic₂₀₂	C₂²	semidirect
26. #15692254	C₂²	semidirect
27. #15692241	C₂²	semidirect
28. #15692239	C₂²	semidirect
29. #15692238	C₂²	semidirect
30. #15692236	C₂²	semidirect
31. #15692233	C₂²	semidirect
32. #15692232	C₂²	semidirect
33. #15692211	C₂²	semidirect
45. #123×C₁₀₁	C₂²	semidirect
155. C₂×C₁₀₁	#536	semidirect
160. C₁₀₁	C₂×#536	semidirect
169. C₂×#123	D₁₀₁	semidirect
193. #123	C₂×D₁₀₁	semidirect
316. C₂	#20265749	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×#13108073	2
C₂³×Dic₂₀₂	2
#20265749	2
C₂×#15692254	2
C₂×#15692241	2
C₂×#15692239	2
C₂×#15692238	2
C₂×#15692236	2
C₂×#15692233	2
C₂×#15692232	2
C₂×#15692211	2
C₂×#123×C₁₀₁	2
C₂×#536	101

Subgroups — C2×#20265749 #27184447

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20265749 #27184447