Subgroups — C₂×#20265778 #27184488

Order 12928.

1260 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	#20265778	C₂	1	16	yes	maximal, direct product factor
2	C₂×#15692251	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#15692246	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#15692238	C₂	1	2	yes	maximal, complemented
5	C₂×#15692237	C₂	1	2	yes	maximal, complemented
6	C₂×#15692230	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#15692229	C₂	1	2	yes	maximal, complemented
8	C₂×#15692228	C₂	1	2	yes	maximal, complemented
9	C₂×#15692226	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₄×D₄₀₄	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×#15692212	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₄×D₄×C₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂²×D₄₀₄	C₂²	1	1	yes	characteristic
14	C₂²×D₄₀₄	C₂²	1	2	yes	—
15	C₂²×C₂²⋊D₁₀₁	C₂²	1	2	yes	—
16	C₂²×C₂²⋊D₁₀₁	C₂²	1	2	yes	—
17	C₂²×C₂²⋊D₁₀₁	C₂²	1	2	yes	—
18	C₂²×C₄×D₁₀₁	C₂²	1	1	yes	characteristic
19	C₂²×C₄×D₁₀₁	C₂²	1	2	yes	—
20	#15692251	C₂²	1	8	yes	complemented
21	#15692246	C₂²	1	8	yes	complemented
22	#15692238	C₂²	1	16	yes	complemented
23	#15692237	C₂²	1	16	yes	complemented
24	#15692230	C₂²	1	8	yes	complemented
25	#15692229	C₂²	1	16	yes	complemented
26	#15692228	C₂²	1	16	yes	complemented
27	#15692226	C₂²	1	8	yes	complemented
28	C₄×D₄₀₄	C₂²	1	8	yes	complemented
29	#15692212	C₂²	1	8	yes	complemented
30	C₂×C₄⋊Dic₁₀₁	C₂²	1	1	yes	characteristic
31	C₂×C₄⋊Dic₁₀₁	C₂²	1	2	yes	—
32	C₂×#13108053	C₂²	1	1	yes	characteristic
33	C₂×#13108053	C₂²	1	1	yes	characteristic
34	C₂×#13108053	C₂²	1	2	yes	—
35	C₂×C₂²⋊Dic₁₀₁	C₂²	1	2	yes	—
36	C₂×#13108051	C₂²	1	1	yes	characteristic
37	C₂×#13108051	C₂²	1	1	yes	characteristic
38	C₂×#13108051	C₂²	1	2	yes	—
39	C₂×#13108051	C₂²	1	2	yes	—
40	C₂×C₄×Dic₁₀₁	C₂²	1	1	yes	characteristic
41	C₂²×D₄×C₁₀₁	C₂²	1	1	yes	characteristic
42	C₂³×C₄×C₁₀₁	C₂²	1	2	yes	—
43	C₄×D₄×C₁₀₁	C₂²	1	8	yes	complemented
44	C₂×C₄⋊C₄×C₁₀₁	C₂²	1	1	yes	characteristic
45	C₂×C₂²⋊C₄×C₁₀₁	C₂²	1	2	yes	—
46	C₂×C₄²×C₁₀₁	C₂²	1	1	yes	characteristic
47	C₂³×D₁₀₁	C₂³	1	1	yes	characteristic
48	C₂³×D₁₀₁	C₂³	1	2	yes	—
49	C₂×D₄₀₄	—	2	4	no	—
50	C₂×D₄₀₄	—	4	4	no	—
51	C₂×D₄₀₄	C₂³	1	4	yes	—
52	C₂×D₄₀₄	C₂³	1	8	yes	—
53	C₂×C₂²⋊D₁₀₁	—	4	4	no	—
54	C₂×C₂²⋊D₁₀₁	—	4	4	no	—
55	C₂×C₂²⋊D₁₀₁	—	4	4	no	—
56	C₂×C₂²⋊D₁₀₁	C₂³	1	8	yes	—
57	C₂×C₂²⋊D₁₀₁	C₂³	1	8	yes	—
58	C₂×C₂²⋊D₁₀₁	C₂³	1	8	yes	—
59	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
60	C₂²×Dic₁₀₁	C₂³	1	1	yes	characteristic
61	C₂²×Dic₁₀₁	C₂³	1	1	yes	characteristic
62	C₂²×Dic₁₀₁	C₂³	1	1	yes	characteristic
63	C₂×C₄×D₁₀₁	—	2	4	no	—
64	C₂×C₄×D₁₀₁	—	4	4	no	—
65	C₂×C₄×D₁₀₁	C₂³	1	4	yes	—
66	C₂×C₄×D₁₀₁	C₂³	1	8	yes	—
67	C₄⋊Dic₁₀₁	C₂³	1	4	yes	—
68	C₄⋊Dic₁₀₁	C₂³	1	8	yes	—
69	#13108053	C₂³	1	4	yes	—
70	#13108053	C₂³	1	4	yes	—
71	#13108053	C₂³	1	8	yes	—
72	C₂²⋊Dic₁₀₁	C₂³	1	8	yes	—
73	#13108051	C₂³	1	4	yes	—
74	#13108051	C₂³	1	4	yes	—
75	#13108051	C₂³	1	8	yes	—
76	#13108051	C₂³	1	8	yes	—
77	C₄×Dic₁₀₁	C₂³	1	4	yes	—
78	C₂⁴×C₁₀₁	C₂³	1	2	yes	—
79	C₂×D₄×C₁₀₁	—	2	4	no	—
80	C₂×D₄×C₁₀₁	C₂³	1	4	yes	—
81	C₂²×C₄×C₁₀₁	—	4	4	no	—
82	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
83	C₂²×C₄×C₁₀₁	C₂³	1	2	yes	—
84	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
85	C₂²×C₄×C₁₀₁	C₂³	1	8	yes	—
86	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
87	C₄⋊C₄×C₁₀₁	C₂³	1	4	yes	—
88	C₂²⋊C₄×C₁₀₁	C₂³	1	8	yes	—
89	C₄²×C₁₀₁	C₂³	1	4	yes	—
90	C₂³×C₁₀₁	—	2	4	no	—
91	C₂³×C₁₀₁	—	2	4	no	—
92	C₂³×C₁₀₁	—	2	4	no	—
93	C₂³×C₁₀₁	C₂⁴	1	4	yes	—
94	C₂³×C₁₀₁	C₂⁴	1	1	yes	characteristic
95	C₂²×D₁₀₁	—	2	2	no	—
96	C₂²×D₁₀₁	—	2	4	no	—
97	C₂²×D₁₀₁	—	2	4	no	—
98	C₂²×D₁₀₁	—	2	2	no	—
99	C₂²×D₁₀₁	—	2	4	no	—
100	C₂²×D₁₀₁	—	2	2	no	—
101	C₂²×D₁₀₁	C₂⁴	1	2	yes	—
102	C₂²×D₁₀₁	C₂⁴	1	4	yes	—
103	D₄×C₁₀₁	—	2	8	no	—
104	C₂×C₄×C₁₀₁	—	2	2	no	—
105	C₂×C₄×C₁₀₁	—	2	4	no	—
106	C₂×C₄×C₁₀₁	—	2	2	no	—
107	C₂×C₄×C₁₀₁	—	4	8	no	—
108	C₂×C₄×C₁₀₁	#41	1	4	yes	—
109	C₂×C₄×C₁₀₁	C₂⁴	1	2	yes	—
110	C₂×C₄×C₁₀₁	C₂⁴	1	2	yes	—
111	C₂×C₄×C₁₀₁	C₂⁴	1	4	yes	—
112	C₂×C₄×C₁₀₁	C₂⁴	1	2	yes	—
113	C₂²⋊D₁₀₁	—	4	8	no	—
114	C₂²⋊D₁₀₁	—	4	8	no	—
115	C₂²⋊D₁₀₁	—	4	8	no	—
116	C₂×Dic₁₀₁	—	2	2	no	—
117	C₂×Dic₁₀₁	—	2	4	no	—
118	C₂×Dic₁₀₁	—	2	2	no	—
119	C₂×Dic₁₀₁	—	2	2	no	—
120	C₂×Dic₁₀₁	C₂⁴	1	2	yes	—
121	C₂×Dic₁₀₁	C₂⁴	1	2	yes	—
122	C₂×Dic₁₀₁	C₂⁴	1	2	yes	—
123	C₂×Dic₁₀₁	C₂⁴	1	4	yes	—
124	D₄₀₄	—	2	8	no	—
125	D₄₀₄	—	4	8	no	—
126	C₄×D₁₀₁	—	2	8	no	—
127	C₄×D₁₀₁	—	4	8	no	—
128	C₂²×C₁₀₁	—	2	4	no	—
129	C₂²×C₁₀₁	—	2	4	no	—
130	C₂²×C₁₀₁	—	2	4	no	—
131	C₂²×C₁₀₁	—	4	8	no	—
132	C₂²×C₁₀₁	C₂×#41	1	2	yes	—
133	C₂²×C₁₀₁	H(5,𝔽₂)	1	2	yes	—
134	C₂²×C₁₀₁	H(5,𝔽₂)	1	2	yes	—
135	C₂²×C₁₀₁	C₂⁵	1	1	yes	characteristic
136	C₂×D₁₀₁	—	2	2	no	—
137	C₂×D₁₀₁	—	2	4	no	—
138	C₂×D₁₀₁	—	2	4	no	—
139	C₂×D₁₀₁	—	2	2	no	—
140	C₂×D₁₀₁	—	2	4	no	—
141	C₂×D₁₀₁	—	2	2	no	—
142	C₂×D₁₀₁	—	4	4	no	—
143	C₂×D₁₀₁	—	4	8	no	—
144	C₄×C₁₀₁	—	2	2	no	—
145	C₄×C₁₀₁	—	2	4	no	—
146	C₄×C₁₀₁	—	2	2	no	—
147	C₄×C₁₀₁	C₂×#41	1	4	yes	—
148	Dic₁₀₁	—	2	2	no	—
149	Dic₁₀₁	—	2	4	no	—
150	Dic₁₀₁	—	2	2	no	—
151	Dic₁₀₁	—	2	2	no	—
152	C₂×C₁₀₁	—	4	4	no	—
153	C₂×C₁₀₁	#540	1	4	yes	complemented
154	C₂×C₁₀₁	C₂²×#41	1	1	yes	characteristic
155	C₂×C₁₀₁	C₂×H(5,𝔽₂)	1	1	yes	characteristic
156	C₂×C₁₀₁	C₂×H(5,𝔽₂)	1	1	yes	characteristic
157	D₁₀₁	—	4	2	no	—
158	D₁₀₁	—	4	4	no	—
159	C₂×#540	—	101	1	no	maximal, Sylow
160	C₁₀₁	C₂×#540	1	1	yes	characteristic, complemented, Sylow
161	#540	—	101	16	no	—
162	C₂×#127	—	101	1	no	—
163	C₂×#125	—	101	1	no	—
164	C₂×#123	—	101	2	no	—
165	C₂×#123	—	101	2	no	—
166	C₂×#121	—	101	1	no	—
167	C₂×#121	—	101	2	no	—
168	C₂×#121	—	101	1	no	—
169	C₂×#121	—	101	2	no	—
170	C₂×C₄×D₄	—	101	1	no	—
171	C₂×C₄×D₄	D₁₀₁	1	1	yes	characteristic, complemented
172	C₂×#117	—	101	1	no	—
173	C₂²×D₄	—	101	2	no	—
174	C₂²×D₄	—	101	2	no	—
175	C₂²×D₄	—	101	2	no	—
176	C₂²×D₄	—	101	2	no	—
177	C₂²×D₄	—	101	1	no	—
178	C₂²×D₄	C₂×D₁₀₁	1	1	yes	characteristic
179	C₂³×C₄	—	101	2	no	—
180	C₂³×C₄	—	101	1	no	—
181	C₂³×C₄	C₂×D₁₀₁	1	2	yes	—
182	#127	—	101	8	no	—
183	#125	—	101	8	no	—
184	#123	—	101	16	no	—
185	#123	—	101	16	no	—
186	#121	—	101	8	no	—
187	#121	—	101	16	no	—
188	#121	—	101	8	no	—
189	#121	—	101	16	no	—
190	C₄×D₄	—	101	8	no	—
191	C₄×D₄	C₂×D₁₀₁	1	8	yes	complemented
192	#117	—	101	8	no	—
193	C₂×C₄⋊C₄	—	101	1	no	—
194	C₂×C₄⋊C₄	—	101	1	no	—
195	C₂×C₄⋊C₄	—	101	1	no	—
196	C₂×C₄⋊C₄	—	101	2	no	—
197	C₂×C₄⋊C₄	—	101	2	no	—
198	C₂×C₄⋊C₄	C₂×D₁₀₁	1	1	yes	characteristic
199	C₂×C₂²⋊C₄	—	101	2	no	—
200	C₂×C₂²⋊C₄	—	101	1	no	—
201	C₂×C₂²⋊C₄	—	101	1	no	—
202	C₂×C₂²⋊C₄	—	101	2	no	—
203	C₂×C₂²⋊C₄	—	101	2	no	—
204	C₂×C₂²⋊C₄	C₂×D₁₀₁	1	2	yes	—
205	C₂×C₄²	—	101	1	no	—
206	C₂×C₄²	C₂×D₁₀₁	1	1	yes	characteristic
207	C₂⁴	—	101	1	no	—
208	C₂⁴	—	101	2	no	—
209	C₂⁴	C₂²×D₁₀₁	1	2	yes	—
210	C₂×D₄	—	2	4	no	—
211	C₂×D₄	—	101	8	no	—
212	C₂×D₄	—	101	8	no	—
213	C₂×D₄	—	101	8	no	—
214	C₂×D₄	—	101	8	no	—
215	C₂×D₄	—	101	4	no	—
216	C₂×D₄	—	404	4	no	—
217	C₂×D₄	—	404	4	no	—
218	C₂×D₄	—	404	4	no	—
219	C₂×D₄	—	404	4	no	—
220	C₂×D₄	—	202	4	no	—
221	C₂×D₄	C₂²×D₁₀₁	1	4	yes	—
222	C₂²×C₄	—	4	4	no	—
223	C₂²×C₄	—	101	1	no	—
224	C₂²×C₄	—	101	1	no	—
225	C₂²×C₄	—	101	1	no	—
226	C₂²×C₄	—	101	2	no	—
227	C₂²×C₄	—	101	8	no	—
228	C₂²×C₄	—	101	4	no	—
229	C₂²×C₄	—	404	4	no	—
230	C₂²×C₄	—	202	4	no	—
231	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
232	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
233	C₂²×C₄	C₂²×D₁₀₁	1	2	yes	—
234	C₂²×C₄	C₂²×D₁₀₁	1	8	yes	—
235	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
236	C₄⋊C₄	—	101	4	no	—
237	C₄⋊C₄	—	101	4	no	—
238	C₄⋊C₄	—	101	8	no	—
239	C₄⋊C₄	—	101	4	no	—
240	C₄⋊C₄	—	101	8	no	—
241	C₄⋊C₄	C₂²×D₁₀₁	1	4	yes	—
242	C₂²⋊C₄	—	101	8	no	—
243	C₂²⋊C₄	—	101	4	no	—
244	C₂²⋊C₄	—	101	4	no	—
245	C₂²⋊C₄	—	101	8	no	—
246	C₂²⋊C₄	—	101	8	no	—
247	C₂²⋊C₄	C₂²×D₁₀₁	1	8	yes	—
248	C₄²	—	101	4	no	—
249	C₄²	C₂²×D₁₀₁	1	4	yes	—
250	C₂³	—	2	4	no	—
251	C₂³	—	2	4	no	—
252	C₂³	—	2	4	no	—
253	C₂³	—	101	2	no	—
254	C₂³	—	101	4	no	—
255	C₂³	—	202	2	no	—
256	C₂³	—	202	4	no	—
257	C₂³	—	202	2	no	—
258	C₂³	—	202	4	no	—
259	C₂³	—	202	4	no	—
260	C₂³	—	202	2	no	—
261	C₂³	C₂³×D₁₀₁	1	4	yes	—
262	C₂³	C₂³×D₁₀₁	1	1	yes	characteristic
263	D₄	—	2	8	no	—
264	D₄	—	404	8	no	—
265	D₄	—	404	8	no	—
266	D₄	—	404	8	no	—
267	D₄	—	404	8	no	—
268	D₄	—	202	8	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	2	4	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	4	8	no	—
273	C₂×C₄	—	101	2	no	—
274	C₂×C₄	—	101	2	no	—
275	C₂×C₄	—	101	2	no	—
276	C₂×C₄	—	101	4	no	—
277	C₂×C₄	—	404	8	no	—
278	C₂×C₄	—	202	2	no	—
279	C₂×C₄	—	202	2	no	—
280	C₂×C₄	—	202	4	no	—
281	C₂×C₄	—	202	2	no	—
282	C₂×C₄	—	202	8	no	—
283	C₂×C₄	#13108074	1	4	yes	—
284	C₂×C₄	C₂³×D₁₀₁	1	2	yes	—
285	C₂×C₄	C₂³×D₁₀₁	1	2	yes	—
286	C₂×C₄	C₂³×D₁₀₁	1	4	yes	—
287	C₂×C₄	C₂³×D₁₀₁	1	2	yes	—
288	C₂²	—	2	4	no	—
289	C₂²	—	2	4	no	—
290	C₂²	—	2	4	no	—
291	C₂²	—	4	8	no	—
292	C₂²	—	404	4	no	—
293	C₂²	—	404	8	no	—
294	C₂²	—	202	2	no	—
295	C₂²	—	202	4	no	—
296	C₂²	—	202	2	no	—
297	C₂²	—	202	4	no	—
298	C₂²	—	202	4	no	—
299	C₂²	—	202	2	no	—
300	C₂²	C₂×#13108074	1	2	yes	—
301	C₂²	#15692300	1	2	yes	—
302	C₂²	#15692301	1	2	yes	—
303	C₂²	C₂⁴×D₁₀₁	1	1	yes	characteristic
304	C₄	—	2	2	no	—
305	C₄	—	2	4	no	—
306	C₄	—	2	2	no	—
307	C₄	—	202	2	no	—
308	C₄	—	202	2	no	—
309	C₄	—	202	4	no	—
310	C₄	—	202	2	no	—
311	C₄	C₂×#13108074	1	4	yes	—
312	C₂	—	4	4	no	—
313	C₂	—	404	2	no	—
314	C₂	—	404	4	no	—
315	C₂	#20265778	1	4	yes	direct product factor
316	C₂	C₂²×#13108074	1	1	yes	characteristic
317	C₂	C₂×#15692300	1	1	yes	characteristic
318	C₂	C₂×#15692301	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂³×D₁₀₁
commutator, cosocle	C₂²×C₁₀₁	C₂⁵
Frattini	C₂²	C₂⁴×D₁₀₁
Fitting	C₂×C₄×D₄×C₁₀₁	C₂
socle	C₂³×C₁₀₁	C₂⁴
hypercenter	C₂×C₄×D₄	D₁₀₁
hypocenter	C₁₀₁	C₂×#540
LCS	G ▷ C₂²×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×D₄

Complements

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

N	H ≅ G/N	type
1. #20265778	C₂	direct (N × H)
2. C₂×#15692251	C₂	semidirect
3. C₂×#15692246	C₂	semidirect
4. C₂×#15692238	C₂	semidirect
5. C₂×#15692237	C₂	semidirect
6. C₂×#15692230	C₂	semidirect
7. C₂×#15692229	C₂	semidirect
8. C₂×#15692228	C₂	semidirect
9. C₂×#15692226	C₂	semidirect
10. C₂×C₄×D₄₀₄	C₂	semidirect
11. C₂×#15692212	C₂	semidirect
12. C₂×C₄×D₄×C₁₀₁	C₂	semidirect
20. #15692251	C₂²	semidirect
21. #15692246	C₂²	semidirect
22. #15692238	C₂²	semidirect
23. #15692237	C₂²	semidirect
24. #15692230	C₂²	semidirect
25. #15692229	C₂²	semidirect
26. #15692228	C₂²	semidirect
27. #15692226	C₂²	semidirect
28. C₄×D₄₀₄	C₂²	semidirect
29. #15692212	C₂²	semidirect
43. C₄×D₄×C₁₀₁	C₂²	semidirect
153. C₂×C₁₀₁	#540	semidirect
160. C₁₀₁	C₂×#540	semidirect
171. C₂×C₄×D₄	D₁₀₁	semidirect
191. C₄×D₄	C₂×D₁₀₁	semidirect
315. C₂	#20265778	direct (N × H)

Maximal subgroups

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

subgroup	index
#20265778	2
C₂×#15692251	2
C₂×#15692246	2
C₂×#15692238	2
C₂×#15692237	2
C₂×#15692230	2
C₂×#15692229	2
C₂×#15692228	2
C₂×#15692226	2
C₂×C₄×D₄₀₄	2
C₂×#15692212	2
C₂×C₄×D₄×C₁₀₁	2
C₂×#540	101

Subgroups — C2×#20265778 #27184488

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20265778 #27184488