Subgroups — C₂×#20266058 #27186839

Order 12928.

388 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13108083	C₂	1	1	yes	maximal, characteristic
2	C₂²×#13108083	C₂	1	1	yes	maximal, characteristic
3	#20266058	C₂	1	4	yes	maximal, direct product factor
4	C₂×#15692211	C₂	1	1	yes	maximal, characteristic
5	C₂×#13108083	—	2	1	no	—
6	C₂×#13108083	—	2	1	no	—
7	C₂×#13108083	—	2	1	no	—
8	C₂×#13108083	—	2	1	no	—
9	C₂×#13108083	—	2	1	no	—
10	C₂×#13108083	—	2	1	no	—
11	C₂×#13108083	—	2	1	no	—
12	C₂×#13108083	C₄	1	2	yes	complemented
13	C₂×#13108083	C₄	1	2	yes	complemented
14	C₂×#13108083	C₄	1	2	yes	complemented
15	C₂×#13108083	C₂²	1	2	yes	—
16	C₂×#13108083	C₂²	1	2	yes	—
17	C₂²×#11717581	—	2	1	no	—
18	C₂²×#11717581	C₄	1	2	yes	complemented
19	C₂³×Dic₁₀₁	C₂²	1	1	yes	characteristic
20	#15692211	—	2	1	no	—
21	#15692211	—	2	1	no	—
22	#15692211	C₄	1	2	yes	—
23	#15692211	C₂²	1	2	yes	—
24	C₂×#13108053	—	2	1	no	—
25	C₂×C₂²⋊Dic₁₀₁	C₄	1	1	yes	characteristic
26	C₂×C₄×Dic₁₀₁	—	2	1	no	—
27	C₂×C₂²⋊C₄×C₁₀₁	C₄	1	1	yes	characteristic
28	#13108083	—	2	2	no	—
29	#13108083	—	2	2	no	—
30	#13108083	—	2	2	no	—
31	#13108083	—	2	2	no	—
32	#13108083	—	2	2	no	—
33	#13108083	—	2	2	no	—
34	#13108083	—	2	2	no	—
35	#13108083	C₂×C₄	1	4	yes	complemented
36	C₂×#11717581	—	2	2	no	—
37	C₂×#11717581	—	2	2	no	—
38	C₂×#11717581	—	2	2	no	—
39	C₂×#11717581	—	2	2	no	—
40	C₂×#11717581	—	2	2	no	—
41	C₂×#11717581	C₂×C₄	1	4	yes	complemented
42	C₂²×Dic₁₀₁	—	2	1	no	—
43	C₂²×Dic₁₀₁	C₂×C₄	1	1	yes	characteristic
44	C₂²×Dic₁₀₁	C₂×C₄	1	1	yes	characteristic
45	C₂²×Dic₁₀₁	C₂×C₄	1	1	yes	characteristic
46	C₂²×Dic₁₀₁	C₂×C₄	1	1	yes	characteristic
47	C₂²×Dic₁₀₁	C₂×C₄	1	1	yes	characteristic
48	C₂²×Dic₁₀₁	D₄	1	1	yes	characteristic
49	C₂²×Dic₁₀₁	D₄	1	1	yes	characteristic
50	C₂²×Dic₁₀₁	D₄	1	1	yes	characteristic
51	C₂²×Dic₁₀₁	D₄	1	1	yes	characteristic
52	C₂²×Dic₁₀₁	D₄	1	1	yes	characteristic
53	C₂²×Dic₁₀₁	D₄	1	1	yes	characteristic
54	C₂²×Dic₁₀₁	Q₈	1	1	yes	characteristic
55	C₂²×Dic₁₀₁	Q₈	1	1	yes	characteristic
56	C₂²×Dic₁₀₁	C₂³	1	1	yes	characteristic
57	#13108053	—	2	2	no	—
58	#13108053	—	2	2	no	—
59	C₂²⋊Dic₁₀₁	—	2	1	no	—
60	C₂²⋊Dic₁₀₁	C₂×C₄	1	2	yes	—
61	C₄×Dic₁₀₁	—	2	2	no	—
62	C₄×Dic₁₀₁	—	2	2	no	—
63	C₂⁴×C₁₀₁	C₂×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₁₀₁	—	2	1	no	—
65	C₂²⋊C₄×C₁₀₁	—	2	1	no	—
66	C₂²⋊C₄×C₁₀₁	C₂×C₄	1	2	yes	—
67	C₂³×C₁₀₁	—	4	2	no	—
68	C₂³×C₁₀₁	C₄²	1	1	yes	characteristic
69	C₂³×C₁₀₁	C₄²	1	1	yes	characteristic
70	C₂³×C₁₀₁	C₂²⋊C₄	1	1	yes	characteristic
71	C₂³×C₁₀₁	C₂²⋊C₄	1	1	yes	characteristic
72	C₂³×C₁₀₁	C₄⋊C₄	1	1	yes	characteristic
73	C₂³×C₁₀₁	C₄⋊C₄	1	1	yes	characteristic
74	C₂³×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₄×C₁₀₁	—	2	2	no	—
76	C₂×C₄×C₁₀₁	—	4	2	no	—
77	C₂×Dic₁₀₁	—	2	2	no	—
78	C₂×Dic₁₀₁	—	4	2	no	—
79	C₂×Dic₁₀₁	C₄²	1	1	yes	characteristic
80	C₂×Dic₁₀₁	C₄²	1	1	yes	characteristic
81	C₂×Dic₁₀₁	C₂²⋊C₄	1	1	yes	characteristic
82	C₂×Dic₁₀₁	C₂²⋊C₄	1	1	yes	characteristic
83	C₂×Dic₁₀₁	C₂²⋊C₄	1	1	yes	characteristic
84	C₂×Dic₁₀₁	C₂²⋊C₄	1	1	yes	characteristic
85	C₂×Dic₁₀₁	C₂²⋊C₄	1	1	yes	characteristic
86	C₂×Dic₁₀₁	C₂²⋊C₄	1	1	yes	characteristic
87	C₂×Dic₁₀₁	C₂²⋊C₄	1	1	yes	characteristic
88	C₂×Dic₁₀₁	C₂²⋊C₄	1	1	yes	characteristic
89	C₂×Dic₁₀₁	C₂²⋊C₄	1	1	yes	characteristic
90	C₂×Dic₁₀₁	C₂²⋊C₄	1	1	yes	characteristic
91	C₂×Dic₁₀₁	C₄⋊C₄	1	1	yes	characteristic
92	C₂×Dic₁₀₁	C₄⋊C₄	1	1	yes	characteristic
93	C₂×Dic₁₀₁	C₄⋊C₄	1	1	yes	characteristic
94	C₂×Dic₁₀₁	C₄⋊C₄	1	1	yes	characteristic
95	C₂×Dic₁₀₁	C₄⋊C₄	1	1	yes	characteristic
96	C₂×Dic₁₀₁	C₄⋊C₄	1	1	yes	characteristic
97	C₂×Dic₁₀₁	C₄⋊C₄	1	1	yes	characteristic
98	C₂×Dic₁₀₁	C₄⋊C₄	1	1	yes	characteristic
99	C₂×Dic₁₀₁	C₄⋊C₄	1	1	yes	characteristic
100	C₂×Dic₁₀₁	C₄⋊C₄	1	1	yes	characteristic
101	C₂×Dic₁₀₁	C₂²×C₄	1	1	yes	characteristic
102	C₂×Dic₁₀₁	C₂²×C₄	1	1	yes	characteristic
103	C₂×Dic₁₀₁	C₂×D₄	1	1	yes	characteristic
104	C₂×Dic₁₀₁	C₂×D₄	1	1	yes	characteristic
105	C₂×Dic₁₀₁	C₂×D₄	1	1	yes	characteristic
106	C₂×Dic₁₀₁	C₂×Q₈	1	1	yes	characteristic
107	#11717581	—	2	4	no	—
108	#11717581	—	2	4	no	—
109	C₂²×C₁₀₁	—	2	2	no	—
110	C₂²×C₁₀₁	—	2	2	no	—
111	C₂²×C₁₀₁	—	2	2	no	—
112	C₂²×C₁₀₁	—	4	1	no	—
113	C₂²×C₁₀₁	—	4	1	no	—
114	C₂²×C₁₀₁	—	4	1	no	—
115	C₂²×C₁₀₁	—	4	1	no	—
116	C₂²×C₁₀₁	#95	1	1	yes	characteristic
117	C₂²×C₁₀₁	#95	1	1	yes	characteristic
118	C₂²×C₁₀₁	#95	1	1	yes	characteristic
119	C₂²×C₁₀₁	#95	1	1	yes	characteristic
120	C₂²×C₁₀₁	C₂×C₄²	1	1	yes	characteristic
121	C₂²×C₁₀₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
122	C₂²×C₁₀₁	C₂×C₄⋊C₄	1	1	yes	characteristic
123	C₄×C₁₀₁	—	4	2	no	—
124	Dic₁₀₁	—	4	2	no	—
125	Dic₁₀₁	#95	1	1	yes	characteristic
126	Dic₁₀₁	#95	1	1	yes	characteristic
127	Dic₁₀₁	#95	1	1	yes	characteristic
128	Dic₁₀₁	#95	1	1	yes	characteristic
129	Dic₁₀₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
130	Dic₁₀₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
131	Dic₁₀₁	C₂×C₄⋊C₄	1	1	yes	characteristic
132	Dic₁₀₁	C₂×C₄⋊C₄	1	1	yes	characteristic
133	C₂×C₁₀₁	—	2	1	no	—
134	C₂×C₁₀₁	—	2	1	no	—
135	C₂×C₁₀₁	—	2	1	no	—
136	C₂×C₁₀₁	—	2	1	no	—
137	C₂×C₁₀₁	—	2	1	no	—
138	C₂×C₁₀₁	—	2	1	no	—
139	C₂×C₁₀₁	#344	1	2	yes	complemented
140	C₂×C₁₀₁	C₂×#95	1	1	yes	characteristic
141	C₂×#344	—	101	1	no	maximal, Sylow
142	C₁₀₁	C₂×#344	1	1	yes	characteristic, complemented, Sylow
143	C₂²×M₁₆	—	101	1	no	—
144	C₂²×M₁₆	—	101	1	no	—
145	C₂×#117	—	101	1	no	—
146	#344	—	101	4	no	—
147	C₂³×C₄	—	101	1	no	—
148	C₂×M₁₆	—	101	2	no	—
149	C₂×M₁₆	—	101	2	no	—
150	C₂×M₁₆	—	101	2	no	—
151	C₂×M₁₆	—	101	2	no	—
152	C₂×M₁₆	—	101	2	no	—
153	C₂×M₁₆	—	202	1	no	—
154	C₂×M₁₆	—	202	1	no	—
155	C₂×M₁₆	—	202	1	no	—
156	C₂×M₁₆	—	202	1	no	—
157	C₂×M₁₆	—	202	1	no	—
158	C₂×M₁₆	—	202	1	no	—
159	C₂×M₁₆	—	202	1	no	—
160	C₂²×C₈	—	101	2	no	—
161	C₂²×C₈	—	202	1	no	—
162	#117	—	101	2	no	—
163	#117	—	101	2	no	—
164	#117	—	202	1	no	—
165	#117	—	202	1	no	—
166	C₂×C₄⋊C₄	—	202	1	no	—
167	C₂×C₂²⋊C₄	—	101	1	no	—
168	C₂×C₂²⋊C₄	#87938	1	1	yes	characteristic
169	C₂×C₄²	—	202	1	no	—
170	C₂⁴	C₂×#87938	1	1	yes	characteristic
171	C₂²×C₄	—	2	1	no	—
172	C₂²×C₄	—	101	1	no	—
173	C₂²×C₄	—	101	1	no	—
174	C₂²×C₄	—	101	1	no	—
175	C₂²×C₄	—	101	1	no	—
176	C₂²×C₄	—	101	1	no	—
177	C₂²×C₄	—	101	1	no	—
178	C₂²×C₄	—	101	1	no	—
179	C₂²×C₄	—	101	1	no	—
180	C₂²×C₄	—	101	1	no	—
181	C₂²×C₄	—	101	1	no	—
182	C₂²×C₄	—	101	1	no	—
183	C₂²×C₄	—	101	1	no	—
184	C₂²×C₄	—	101	1	no	—
185	C₂²×C₄	—	101	1	no	—
186	C₂²×C₄	—	202	1	no	—
187	M₁₆	—	101	4	no	—
188	M₁₆	—	202	2	no	—
189	M₁₆	—	202	2	no	—
190	M₁₆	—	202	2	no	—
191	M₁₆	—	202	2	no	—
192	M₁₆	—	202	2	no	—
193	M₁₆	—	202	2	no	—
194	M₁₆	—	202	2	no	—
195	C₂×C₈	—	101	4	no	—
196	C₂×C₈	—	202	2	no	—
197	C₂×C₈	—	202	2	no	—
198	C₂×C₈	—	202	2	no	—
199	C₂×C₈	—	202	2	no	—
200	C₂×C₈	—	202	2	no	—
201	C₄⋊C₄	—	202	2	no	—
202	C₄⋊C₄	—	202	2	no	—
203	C₂²⋊C₄	—	2	1	no	—
204	C₂²⋊C₄	—	101	2	no	—
205	C₂²⋊C₄	—	202	1	no	—
206	C₂²⋊C₄	C₂×#87938	1	2	yes	—
207	C₄²	—	202	2	no	—
208	C₄²	—	202	2	no	—
209	C₂³	—	4	2	no	—
210	C₂³	C₄×#87938	1	1	yes	characteristic
211	C₂³	C₄×#87938	1	1	yes	characteristic
212	C₂³	C₂²⋊#87938	1	1	yes	characteristic
213	C₂³	C₂²⋊#87938	1	1	yes	characteristic
214	C₂³	C₄⋊#87938	1	1	yes	characteristic
215	C₂³	C₄⋊#87938	1	1	yes	characteristic
216	C₂³	C₂²×#87938	1	1	yes	characteristic
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	4	2	no	—
219	C₂×C₄	—	101	1	no	—
220	C₂×C₄	—	101	1	no	—
221	C₂×C₄	—	101	1	no	—
222	C₂×C₄	—	101	1	no	—
223	C₂×C₄	—	101	1	no	—
224	C₂×C₄	—	101	1	no	—
225	C₂×C₄	—	101	1	no	—
226	C₂×C₄	—	101	1	no	—
227	C₂×C₄	—	101	1	no	—
228	C₂×C₄	—	101	1	no	—
229	C₂×C₄	—	101	1	no	—
230	C₂×C₄	—	101	1	no	—
231	C₂×C₄	—	101	1	no	—
232	C₂×C₄	—	101	1	no	—
233	C₂×C₄	—	101	1	no	—
234	C₂×C₄	—	101	1	no	—
235	C₂×C₄	—	101	1	no	—
236	C₂×C₄	—	101	1	no	—
237	C₂×C₄	—	101	1	no	—
238	C₂×C₄	—	101	1	no	—
239	C₂×C₄	—	101	1	no	—
240	C₂×C₄	—	101	1	no	—
241	C₂×C₄	—	101	1	no	—
242	C₂×C₄	—	101	1	no	—
243	C₂×C₄	—	101	1	no	—
244	C₂×C₄	—	101	1	no	—
245	C₂×C₄	—	101	1	no	—
246	C₂×C₄	—	101	1	no	—
247	C₂×C₄	—	404	2	no	—
248	C₂×C₄	—	202	2	no	—
249	C₈	—	202	4	no	—
250	C₈	—	202	4	no	—
251	C₂²	—	2	2	no	—
252	C₂²	—	2	2	no	—
253	C₂²	—	2	2	no	—
254	C₂²	—	4	1	no	—
255	C₂²	—	4	1	no	—
256	C₂²	—	4	1	no	—
257	C₂²	—	4	1	no	—
258	C₂²	(C₂×#87938)⋊C₄	1	1	yes	characteristic
259	C₂²	(C₂×#87938)⋊C₄	1	1	yes	characteristic
260	C₂²	(C₂×#87938)⋊C₄	1	1	yes	characteristic
261	C₂²	(C₂×#87938)⋊C₄	1	1	yes	characteristic
262	C₂²	C₂×C₄×#87938	1	1	yes	characteristic
263	C₂²	C₂×C₂²⋊#87938	1	1	yes	characteristic
264	C₂²	C₂×C₄⋊#87938	1	1	yes	characteristic
265	C₄	—	4	2	no	—
266	C₄	—	101	1	no	—
267	C₄	—	101	1	no	—
268	C₄	—	101	1	no	—
269	C₄	—	101	1	no	—
270	C₄	—	101	1	no	—
271	C₄	—	101	1	no	—
272	C₄	—	101	1	no	—
273	C₄	—	101	1	no	—
274	C₄	—	404	2	no	—
275	C₂	—	2	1	no	—
276	C₂	—	2	1	no	—
277	C₂	—	2	1	no	—
278	C₂	—	2	1	no	—
279	C₂	—	2	1	no	—
280	C₂	—	2	1	no	—
281	C₂	#20266058	1	2	yes	direct product factor
282	C₂	C₂×(C₂×#87938)⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	(C₂×#87938)⋊C₄
commutator	C₂²×C₁₀₁	C₂×C₄²
Frattini	C₂³	C₂²×#87938
Fitting	C₂×C₂²⋊C₄×C₁₀₁	C₄
socle	C₂²×C₁₀₁	#95
prime radical	C₂⁴×C₁₀₁	C₂×C₄
cosocle	C₂²×Dic₁₀₁	C₂³
hypercenter	C₂×C₂²⋊C₄	#87938
hypocenter	C₁₀₁	C₂×#344
LCS	G ▷ C₂²×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂×C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
3. #20266058	C₂	direct (N × H)
12. C₂×#13108083	C₄	semidirect
13. C₂×#13108083	C₄	semidirect
14. C₂×#13108083	C₄	semidirect
18. C₂²×#11717581	C₄	semidirect
35. #13108083	C₂×C₄	semidirect
41. C₂×#11717581	C₂×C₄	semidirect
139. C₂×C₁₀₁	#344	semidirect
142. C₁₀₁	C₂×#344	semidirect
281. C₂	#20266058	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×#13108083	2
#20266058	2
C₂×#15692211	2
C₂×#344	101

Subgroups — C2×#20266058 #27186839

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20266058 #27186839