Subgroups — C₂×#20265656 #27184325

Order 12928.

1640 conjugacy classes of subgroups.

The subgroup lattice has 308 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	#20265656	C₂	1	16	yes	maximal, direct product factor
3	C₂×C₄×Dic₂₀₂	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#15692220	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×D₄×Dic₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₄×C₂²⋊D₁₀₁	C₂	1	2	yes	maximal, complemented
7	C₂×#15692215	C₂	1	2	yes	maximal, complemented
8	C₂×#15692212	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#15692211	C₂	1	2	yes	maximal, complemented
10	C₂²×#13108053	C₂	1	2	yes	maximal, complemented
11	C₂×C₄⋊C₄×D₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₄×D₄×C₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂×#13108073	C₄	1	16	yes	complemented
14	C₂×#13108073	C₂²	1	8	yes	—
15	C₂²×Dic₂₀₂	C₂²	1	1	yes	characteristic
16	C₂²×C₂²⋊D₁₀₁	C₂²	1	2	yes	—
17	C₂³×Dic₁₀₁	C₂²	1	2	yes	—
18	C₂³×Dic₁₀₁	C₂²	1	2	yes	—
19	C₂²×C₄×D₁₀₁	C₂²	1	2	yes	—
20	C₂²×C₄×D₁₀₁	C₂²	1	1	yes	characteristic
21	C₄×Dic₂₀₂	C₂²	1	8	yes	complemented
22	#15692220	C₂²	1	8	yes	complemented
23	D₄×Dic₁₀₁	C₂²	1	8	yes	complemented
24	C₄×C₂²⋊D₁₀₁	C₂²	1	16	yes	complemented
25	#15692215	C₂²	1	16	yes	complemented
26	#15692212	C₂²	1	8	yes	complemented
27	#15692211	C₂²	1	16	yes	complemented
28	C₂×C₄⋊Dic₁₀₁	C₂²	1	1	yes	characteristic
29	C₂×#13108053	C₂²	1	2	yes	—
30	C₂×#13108053	C₂²	1	1	yes	characteristic
31	C₂×#13108053	C₂²	1	2	yes	—
32	C₂×#13108053	C₂²	1	2	yes	—
33	C₂×#13108053	C₂²	1	1	yes	characteristic
34	C₂×#13108053	C₂²	1	16	yes	complemented
35	C₄⋊C₄×D₁₀₁	C₂²	1	8	yes	complemented
36	C₂×C₂²⋊Dic₁₀₁	C₂²	1	2	yes	—
37	C₂×#13108051	C₂²	1	1	yes	characteristic
38	C₂×#13108051	C₂²	1	1	yes	characteristic
39	C₂×C₄×Dic₁₀₁	C₂²	1	2	yes	—
40	C₂×C₄×Dic₁₀₁	C₂²	1	2	yes	—
41	C₂×C₄×Dic₁₀₁	C₂²	1	1	yes	characteristic
42	C₂²×D₄×C₁₀₁	C₂²	1	1	yes	characteristic
43	C₂³×C₄×C₁₀₁	C₂²	1	2	yes	—
44	C₄×D₄×C₁₀₁	C₂²	1	8	yes	complemented
45	C₂×C₄⋊C₄×C₁₀₁	C₂²	1	1	yes	characteristic
46	C₂×C₂²⋊C₄×C₁₀₁	C₂²	1	2	yes	—
47	C₂×C₄²×C₁₀₁	C₂²	1	1	yes	characteristic
48	C₂³×D₁₀₁	C₂³	1	1	yes	characteristic
49	#13108073	C₂×C₄	1	64	yes	complemented
50	C₂×Dic₂₀₂	C₂×C₄	1	8	yes	complemented
51	C₂×Dic₂₀₂	C₂³	1	4	yes	—
52	C₂×C₂²⋊D₁₀₁	C₂×C₄	1	16	yes	complemented
53	C₂×C₂²⋊D₁₀₁	C₂³	1	8	yes	—
54	C₂²×Dic₁₀₁	—	4	4	no	—
55	C₂²×Dic₁₀₁	C₂×C₄	1	16	yes	complemented
56	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
57	C₂²×Dic₁₀₁	C₂³	1	1	yes	characteristic
58	C₂²×Dic₁₀₁	C₂³	1	1	yes	characteristic
59	C₂²×Dic₁₀₁	C₂³	1	8	yes	—
60	C₂²×Dic₁₀₁	C₂³	1	1	yes	characteristic
61	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
62	C₂²×Dic₁₀₁	C₂³	1	8	yes	—
63	C₂×C₄×D₁₀₁	—	4	4	no	—
64	C₂×C₄×D₁₀₁	C₂×C₄	1	8	yes	complemented
65	C₂×C₄×D₁₀₁	C₂³	1	8	yes	—
66	C₂×C₄×D₁₀₁	C₂³	1	4	yes	—
67	C₄⋊Dic₁₀₁	C₂³	1	4	yes	—
68	#13108053	C₂³	1	8	yes	—
69	#13108053	C₂³	1	4	yes	—
70	#13108053	C₂³	1	8	yes	—
71	#13108053	C₂³	1	8	yes	—
72	#13108053	C₂³	1	4	yes	—
73	C₂²⋊Dic₁₀₁	C₂³	1	8	yes	—
74	#13108051	C₂³	1	4	yes	—
75	#13108051	C₂³	1	4	yes	—
76	C₄×Dic₁₀₁	C₂³	1	8	yes	—
77	C₄×Dic₁₀₁	C₂³	1	8	yes	—
78	C₄×Dic₁₀₁	C₂³	1	4	yes	—
79	C₂⁴×C₁₀₁	C₂³	1	2	yes	—
80	C₂×D₄×C₁₀₁	C₂×C₄	1	8	yes	complemented
81	C₂×D₄×C₁₀₁	C₂³	1	4	yes	—
82	C₂²×C₄×C₁₀₁	—	4	4	no	—
83	C₂²×C₄×C₁₀₁	C₂³	1	2	yes	—
84	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
85	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
86	C₂²×C₄×C₁₀₁	C₂³	1	8	yes	—
87	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
88	C₄⋊C₄×C₁₀₁	C₂³	1	4	yes	—
89	C₂²⋊C₄×C₁₀₁	C₂³	1	8	yes	—
90	C₄²×C₁₀₁	C₂³	1	4	yes	—
91	C₂³×C₁₀₁	—	2	4	no	—
92	C₂³×C₁₀₁	—	2	4	no	—
93	C₂³×C₁₀₁	C₂²×C₄	1	8	yes	—
94	C₂³×C₁₀₁	C₂⁴	1	4	yes	—
95	C₂³×C₁₀₁	C₂⁴	1	1	yes	characteristic
96	C₂²×D₁₀₁	—	2	2	no	—
97	C₂²×D₁₀₁	—	2	2	no	—
98	C₂²×D₁₀₁	C₂²×C₄	1	4	yes	—
99	C₂²×D₁₀₁	C₂⁴	1	2	yes	—
100	D₄×C₁₀₁	C₂²×C₄	1	16	yes	complemented
101	C₂×C₄×C₁₀₁	—	2	2	no	—
102	C₂×C₄×C₁₀₁	—	2	4	no	—
103	C₂×C₄×C₁₀₁	—	2	2	no	—
104	C₂×C₄×C₁₀₁	—	4	8	no	—
105	C₂×C₄×C₁₀₁	C₂²×C₄	1	4	yes	—
106	C₂×C₄×C₁₀₁	C₂⁴	1	2	yes	—
107	C₂×C₄×C₁₀₁	C₂⁴	1	2	yes	—
108	C₂×C₄×C₁₀₁	C₂⁴	1	4	yes	—
109	C₂×C₄×C₁₀₁	C₂⁴	1	2	yes	—
110	C₂²⋊D₁₀₁	C₂²×C₄	1	32	yes	complemented
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	C₂²×C₄	1	4	yes	—
116	C₂×Dic₁₀₁	C₂²×C₄	1	8	yes	—
117	C₂×Dic₁₀₁	C₂²×C₄	1	32	yes	complemented
118	C₂×Dic₁₀₁	C₂⁴	1	2	yes	—
119	C₂×Dic₁₀₁	C₂⁴	1	2	yes	—
120	C₂×Dic₁₀₁	C₂⁴	1	4	yes	—
121	C₂×Dic₁₀₁	C₂⁴	1	4	yes	—
122	C₂×Dic₁₀₁	C₂⁴	1	2	yes	—
123	C₄×D₁₀₁	—	4	8	no	—
124	C₄×D₁₀₁	C₂²×C₄	1	16	yes	complemented
125	Dic₂₀₂	C₂²×C₄	1	16	yes	complemented
126	C₂²×C₁₀₁	—	2	16	no	—
127	C₂²×C₁₀₁	—	2	4	no	—
128	C₂²×C₁₀₁	—	2	4	no	—
129	C₂²×C₁₀₁	C₂³×C₄	1	8	yes	—
130	C₂²×C₁₀₁	C₂³×C₄	1	2	yes	—
131	C₂²×C₁₀₁	H(5,𝔽₂)	1	2	yes	—
132	C₂²×C₁₀₁	2₋¹⁺⁴	1	2	yes	—
133	C₂²×C₁₀₁	C₂⁵	1	1	yes	characteristic
134	C₂×D₁₀₁	—	2	8	no	—
135	C₂×D₁₀₁	—	2	2	no	—
136	C₂×D₁₀₁	—	2	2	no	—
137	C₂×D₁₀₁	C₂³×C₄	1	4	yes	—
138	C₄×C₁₀₁	—	2	4	no	—
139	C₄×C₁₀₁	—	2	2	no	—
140	C₄×C₁₀₁	—	2	2	no	—
141	C₄×C₁₀₁	C₂³×C₄	1	4	yes	—
142	Dic₁₀₁	—	2	4	no	—
143	Dic₁₀₁	—	2	2	no	—
144	Dic₁₀₁	—	2	2	no	—
145	Dic₁₀₁	C₂³×C₄	1	8	yes	—
146	Dic₁₀₁	C₂³×C₄	1	4	yes	—
147	C₂×C₁₀₁	—	2	8	no	—
148	C₂×C₁₀₁	#520	1	4	yes	complemented
149	C₂×C₁₀₁	C₂⁴×C₄	1	1	yes	characteristic
150	C₂×C₁₀₁	C₂×H(5,𝔽₂)	1	1	yes	characteristic
151	C₂×C₁₀₁	C₂×2₋¹⁺⁴	1	1	yes	characteristic
152	D₁₀₁	—	2	4	no	—
153	C₂×#520	—	101	1	no	maximal, Sylow
154	C₁₀₁	C₂×#520	1	1	yes	characteristic, complemented, Sylow
155	C₂²×#41	—	101	1	no	—
156	#520	—	101	16	no	—
157	C₂×C₄×Q₈	—	101	1	no	—
158	C₂×C₄×Q₈	—	101	1	no	—
159	C₂×C₄×D₄	—	101	1	no	—
160	C₂×C₄×D₄	—	101	2	no	—
161	C₂×C₄×D₄	—	101	2	no	—
162	C₂×C₄×D₄	D₁₀₁	1	1	yes	characteristic, complemented
163	C₂×#117	—	101	2	no	—
164	C₂×#117	—	101	1	no	—
165	C₂²×C₄⋊C₄	—	101	2	no	—
166	C₂²×C₄⋊C₄	—	101	1	no	—
167	C₂×#41	—	101	16	no	—
168	C₂×#41	—	101	8	no	—
169	C₂²×Q₈	—	101	1	no	—
170	C₂²×D₄	—	101	2	no	—
171	C₂²×D₄	C₂×D₁₀₁	1	1	yes	characteristic
172	C₂³×C₄	—	101	2	no	—
173	C₂³×C₄	—	101	1	no	—
174	C₂³×C₄	—	101	2	no	—
175	C₂³×C₄	—	101	2	no	—
176	C₂³×C₄	C₂×D₁₀₁	1	2	yes	—
177	C₄×Q₈	—	101	8	no	—
178	C₄×Q₈	—	101	8	no	—
179	C₄×D₄	—	101	8	no	—
180	C₄×D₄	—	101	16	no	—
181	C₄×D₄	—	101	16	no	—
182	C₄×D₄	C₂×D₁₀₁	1	8	yes	complemented
183	#117	—	101	16	no	—
184	#117	—	101	8	no	—
185	C₂×C₄⋊C₄	—	101	2	no	—
186	C₂×C₄⋊C₄	—	101	1	no	—
187	C₂×C₄⋊C₄	—	101	2	no	—
188	C₂×C₄⋊C₄	—	101	2	no	—
189	C₂×C₄⋊C₄	—	101	1	no	—
190	C₂×C₄⋊C₄	—	101	16	no	—
191	C₂×C₄⋊C₄	—	101	1	no	—
192	C₂×C₄⋊C₄	—	101	8	no	—
193	C₂×C₄⋊C₄	C₂×D₁₀₁	1	1	yes	characteristic
194	C₂×C₂²⋊C₄	—	101	2	no	—
195	C₂×C₂²⋊C₄	—	101	1	no	—
196	C₂×C₂²⋊C₄	—	101	1	no	—
197	C₂×C₂²⋊C₄	C₂×D₁₀₁	1	2	yes	—
198	C₂×C₄²	—	101	2	no	—
199	C₂×C₄²	—	101	2	no	—
200	C₂×C₄²	—	101	1	no	—
201	C₂×C₄²	C₂×D₁₀₁	1	1	yes	characteristic
202	C₂⁴	—	101	1	no	—
203	C₂⁴	C₂²×D₁₀₁	1	2	yes	—
204	#41	—	101	64	no	—
205	C₂×Q₈	—	101	8	no	—
206	C₂×Q₈	—	101	4	no	—
207	C₂×D₄	—	101	16	no	—
208	C₂×D₄	—	101	8	no	—
209	C₂×D₄	C₄×D₁₀₁	1	8	yes	complemented
210	C₂×D₄	C₂²×D₁₀₁	1	4	yes	—
211	C₂²×C₄	—	4	4	no	—
212	C₂²×C₄	—	101	16	no	—
213	C₂²×C₄	—	101	8	no	—
214	C₂²×C₄	—	101	2	no	—
215	C₂²×C₄	—	101	1	no	—
216	C₂²×C₄	—	101	8	no	—
217	C₂²×C₄	—	101	4	no	—
218	C₂²×C₄	—	101	8	no	—
219	C₂²×C₄	—	101	1	no	—
220	C₂²×C₄	—	101	2	no	—
221	C₂²×C₄	—	101	1	no	—
222	C₂²×C₄	—	101	8	no	—
223	C₂²×C₄	—	404	4	no	—
224	C₂²×C₄	—	404	4	no	—
225	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
226	C₂²×C₄	C₂²×D₁₀₁	1	2	yes	—
227	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
228	C₂²×C₄	C₂²×D₁₀₁	1	8	yes	—
229	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
230	C₄⋊C₄	—	101	8	no	—
231	C₄⋊C₄	—	101	4	no	—
232	C₄⋊C₄	—	101	8	no	—
233	C₄⋊C₄	—	101	8	no	—
234	C₄⋊C₄	—	101	4	no	—
235	C₄⋊C₄	—	101	4	no	—
236	C₄⋊C₄	C₂²×D₁₀₁	1	4	yes	—
237	C₂²⋊C₄	—	101	8	no	—
238	C₂²⋊C₄	—	101	4	no	—
239	C₂²⋊C₄	—	101	4	no	—
240	C₂²⋊C₄	C₂²×D₁₀₁	1	8	yes	—
241	C₄²	—	101	8	no	—
242	C₄²	—	101	8	no	—
243	C₄²	—	101	4	no	—
244	C₄²	C₂²×D₁₀₁	1	4	yes	—
245	C₂³	—	2	4	no	—
246	C₂³	—	2	4	no	—
247	C₂³	—	101	4	no	—
248	C₂³	—	101	2	no	—
249	C₂³	—	202	2	no	—
250	C₂³	—	202	2	no	—
251	C₂³	C₂×C₄×D₁₀₁	1	8	yes	—
252	C₂³	C₂³×D₁₀₁	1	4	yes	—
253	C₂³	C₂³×D₁₀₁	1	1	yes	characteristic
254	Q₈	—	101	16	no	—
255	D₄	—	101	32	no	—
256	D₄	C₂×C₄×D₁₀₁	1	16	yes	complemented
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	2	4	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	4	8	no	—
261	C₂×C₄	—	101	4	no	—
262	C₂×C₄	—	101	8	no	—
263	C₂×C₄	—	101	32	no	—
264	C₂×C₄	—	101	4	no	—
265	C₂×C₄	—	101	2	no	—
266	C₂×C₄	—	101	16	no	—
267	C₂×C₄	—	101	2	no	—
268	C₂×C₄	—	101	4	no	—
269	C₂×C₄	—	101	2	no	—
270	C₂×C₄	—	404	8	no	—
271	C₂×C₄	—	404	8	no	—
272	C₂×C₄	—	202	4	no	—
273	C₂×C₄	—	202	2	no	—
274	C₂×C₄	—	202	2	no	—
275	C₂×C₄	C₂×C₄×D₁₀₁	1	4	yes	—
276	C₂×C₄	C₂³×D₁₀₁	1	2	yes	—
277	C₂×C₄	C₂³×D₁₀₁	1	4	yes	—
278	C₂×C₄	C₂³×D₁₀₁	1	2	yes	—
279	C₂×C₄	C₂³×D₁₀₁	1	2	yes	—
280	C₂²	—	2	4	no	—
281	C₂²	—	2	4	no	—
282	C₂²	—	2	16	no	—
283	C₂²	—	101	4	no	—
284	C₂²	—	202	2	no	—
285	C₂²	—	202	2	no	—
286	C₂²	—	202	8	no	—
287	C₂²	C₂²×C₄×D₁₀₁	1	8	yes	—
288	C₂²	C₂²×C₄×D₁₀₁	1	2	yes	—
289	C₂²	#15692300	1	2	yes	—
290	C₂²	#15692302	1	2	yes	—
291	C₂²	C₂⁴×D₁₀₁	1	1	yes	characteristic
292	C₄	—	2	2	no	—
293	C₄	—	2	4	no	—
294	C₄	—	2	2	no	—
295	C₄	—	101	4	no	—
296	C₄	—	101	8	no	—
297	C₄	—	202	4	no	—
298	C₄	—	202	2	no	—
299	C₄	—	202	2	no	—
300	C₄	C₂²×C₄×D₁₀₁	1	4	yes	—
301	C₂	—	2	8	no	—
302	C₂	—	202	4	no	—
303	C₂	#20265656	1	4	yes	direct product factor
304	C₂	C₂³×C₄×D₁₀₁	1	1	yes	characteristic
305	C₂	C₂×#15692300	1	1	yes	characteristic
306	C₂	C₂×#15692302	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂³×D₁₀₁
commutator	C₂×C₁₀₁	C₂⁴×C₄
Frattini	C₂²	C₂⁴×D₁₀₁
Fitting	C₂×C₄×D₄×C₁₀₁	C₂
socle	C₂³×C₁₀₁	C₂⁴
prime radical	C₂²×#13108073	C₂
cosocle	C₂²×C₁₀₁	C₂⁵
hypercenter	C₂×C₄×D₄	D₁₀₁
hypocenter	C₁₀₁	C₂×#520
LCS	G ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×D₄

Complements

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

N	H ≅ G/N	type
2. #20265656	C₂	direct (N × H)
3. C₂×C₄×Dic₂₀₂	C₂	semidirect
4. C₂×#15692220	C₂	semidirect
5. C₂×D₄×Dic₁₀₁	C₂	semidirect
6. C₂×C₄×C₂²⋊D₁₀₁	C₂	semidirect
7. C₂×#15692215	C₂	semidirect
8. C₂×#15692212	C₂	semidirect
9. C₂×#15692211	C₂	semidirect
10. C₂²×#13108053	C₂	semidirect
11. C₂×C₄⋊C₄×D₁₀₁	C₂	semidirect
12. C₂×C₄×D₄×C₁₀₁	C₂	semidirect
13. C₂×#13108073	C₄	semidirect
21. C₄×Dic₂₀₂	C₂²	semidirect
22. #15692220	C₂²	semidirect
23. D₄×Dic₁₀₁	C₂²	semidirect
24. C₄×C₂²⋊D₁₀₁	C₂²	semidirect
25. #15692215	C₂²	semidirect
26. #15692212	C₂²	semidirect
27. #15692211	C₂²	semidirect
34. C₂×#13108053	C₂²	semidirect
35. C₄⋊C₄×D₁₀₁	C₂²	semidirect
44. C₄×D₄×C₁₀₁	C₂²	semidirect
49. #13108073	C₂×C₄	semidirect
50. C₂×Dic₂₀₂	C₂×C₄	semidirect
52. C₂×C₂²⋊D₁₀₁	C₂×C₄	semidirect
55. C₂²×Dic₁₀₁	C₂×C₄	semidirect
64. C₂×C₄×D₁₀₁	C₂×C₄	semidirect
80. C₂×D₄×C₁₀₁	C₂×C₄	semidirect
100. D₄×C₁₀₁	C₂²×C₄	semidirect
110. C₂²⋊D₁₀₁	C₂²×C₄	semidirect
117. C₂×Dic₁₀₁	C₂²×C₄	semidirect
124. C₄×D₁₀₁	C₂²×C₄	semidirect
125. Dic₂₀₂	C₂²×C₄	semidirect
148. C₂×C₁₀₁	#520	semidirect
154. C₁₀₁	C₂×#520	semidirect
162. C₂×C₄×D₄	D₁₀₁	semidirect
182. C₄×D₄	C₂×D₁₀₁	semidirect
209. C₂×D₄	C₄×D₁₀₁	semidirect
256. D₄	C₂×C₄×D₁₀₁	semidirect
303. C₂	#20265656	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
#20265656	2
C₂×C₄×Dic₂₀₂	2
C₂×#15692220	2
C₂×D₄×Dic₁₀₁	2
C₂×C₄×C₂²⋊D₁₀₁	2
C₂×#15692215	2
C₂×#15692212	2
C₂×#15692211	2
C₂²×#13108053	2
C₂×C₄⋊C₄×D₁₀₁	2
C₂×C₄×D₄×C₁₀₁	2
C₂×#520	101

Subgroups — C2×#20265656 #27184325

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20265656 #27184325