Subgroups — #27179652

Order 12928.

772 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13108074	C₂	1	1	yes	maximal, characteristic
2	C₂×#15692214	C₂	1	1	yes	maximal, characteristic, complemented
3	#20265187	C₂	1	2	yes	maximal, complemented
4	#20265183	C₂	1	2	yes	maximal, complemented
5	#20265179	C₂	1	4	yes	maximal, complemented
6	#20265174	C₂	1	4	yes	maximal, complemented
7	C₂²×M₁₆×C₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#13108074	—	2	1	no	—
9	C₂×#13108074	—	2	1	no	—
10	C₂×#13108074	—	2	4	no	—
11	C₂×#13108074	C₄	1	8	yes	complemented
12	C₂×#13108074	C₂²	1	2	yes	—
13	C₂×#13108074	C₂²	1	2	yes	—
14	C₂²×Dic₂₀₂	C₂²	1	1	yes	characteristic
15	C₂²×D₄₀₄	C₂²	1	1	yes	characteristic
16	C₂²×C₂²⋊D₁₀₁	—	2	1	no	—
17	C₂²×C₄×D₁₀₁	—	2	1	no	—
18	#15692214	—	2	1	no	—
19	#15692214	—	2	1	no	—
20	#15692214	C₂²	1	2	yes	—
21	#15692214	C₂²	1	2	yes	complemented
22	C₂×C₄⋊Dic₁₀₁	C₂²	1	2	yes	complemented
23	C₂×C₂²⋊Dic₁₀₁	—	2	1	no	—
24	C₂×C₄×Dic₁₀₁	—	2	1	no	—
25	#15692180	C₂²	1	8	yes	complemented
26	#15692177	C₂²	1	8	yes	complemented
27	C₂³×C₄×C₁₀₁	C₂²	1	1	yes	characteristic
28	C₂×M₁₆×C₁₀₁	—	2	4	no	—
29	C₂×M₁₆×C₁₀₁	C₂²	1	2	yes	—
30	C₂×M₁₆×C₁₀₁	C₂²	1	2	yes	complemented
31	C₂²×C₈×C₁₀₁	C₂²	1	2	yes	complemented
32	C₂³×D₁₀₁	—	2	1	no	—
33	#13108074	—	2	8	no	—
34	#13108074	—	2	8	no	—
35	#13108074	—	2	4	no	—
36	#13108074	—	2	4	no	—
37	#13108074	C₂×C₄	1	8	yes	—
38	#13108074	C₂×C₄	1	8	yes	complemented
39	C₂×Dic₂₀₂	—	2	2	no	—
40	C₂×Dic₂₀₂	—	2	1	no	—
41	C₂×Dic₂₀₂	C₂×C₄	1	4	yes	complemented
42	C₂×Dic₂₀₂	C₂³	1	2	yes	—
43	C₂×D₄₀₄	—	2	2	no	—
44	C₂×D₄₀₄	—	2	1	no	—
45	C₂×D₄₀₄	C₂×C₄	1	4	yes	complemented
46	C₂×D₄₀₄	C₂³	1	2	yes	—
47	C₂×C₂²⋊D₁₀₁	—	2	8	no	—
48	C₂×C₂²⋊D₁₀₁	—	2	2	no	—
49	C₂×C₂²⋊D₁₀₁	—	2	2	no	—
50	C₂²×Dic₁₀₁	—	2	1	no	—
51	C₂²×Dic₁₀₁	—	2	2	no	—
52	C₂×C₄×D₁₀₁	—	2	8	no	—
53	C₂×C₄×D₁₀₁	—	2	2	no	—
54	C₂×C₄×D₁₀₁	—	2	2	no	—
55	C₄⋊Dic₁₀₁	—	2	2	no	—
56	C₄⋊Dic₁₀₁	C₂³	1	4	yes	—
57	C₂²⋊Dic₁₀₁	—	2	2	no	—
58	C₂²⋊Dic₁₀₁	—	2	2	no	—
59	C₄×Dic₁₀₁	—	2	2	no	—
60	C₄×Dic₁₀₁	—	2	2	no	—
61	C₂⁴×C₁₀₁	D₄	1	1	yes	characteristic
62	C₂²×C₄×C₁₀₁	C₂×C₄	1	4	yes	—
63	C₂²×C₄×C₁₀₁	D₄	1	1	yes	characteristic
64	C₂²×C₄×C₁₀₁	D₄	1	1	yes	characteristic
65	C₂²×C₄×C₁₀₁	D₄	1	1	yes	characteristic
66	C₂²×C₄×C₁₀₁	D₄	1	4	yes	—
67	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
68	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
69	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
70	M₁₆×C₁₀₁	—	2	4	no	—
71	M₁₆×C₁₀₁	—	2	4	no	—
72	C₂×C₈×C₁₀₁	—	2	4	no	—
73	C₂×C₈×C₁₀₁	C₂³	1	4	yes	—
74	C₂³×C₁₀₁	—	4	2	no	—
75	C₂³×C₁₀₁	C₂²⋊C₄	1	4	yes	—
76	C₂³×C₁₀₁	C₂×D₄	1	1	yes	characteristic
77	C₂³×C₁₀₁	C₂×D₄	1	1	yes	characteristic
78	C₂³×C₁₀₁	C₂×D₄	1	1	yes	characteristic
79	C₂²×D₁₀₁	—	2	4	no	—
80	C₂²×D₁₀₁	—	2	2	no	—
81	C₂²×D₁₀₁	—	4	4	no	—
82	C₂×C₄×C₁₀₁	C₂²⋊C₄	1	4	yes	—
83	C₂×C₄×C₁₀₁	C₂²⋊C₄	1	4	yes	—
84	C₂×C₄×C₁₀₁	C₂²⋊C₄	1	4	yes	—
85	C₂×C₄×C₁₀₁	C₂²×C₄	1	2	yes	—
86	C₂×C₄×C₁₀₁	C₂²×C₄	1	2	yes	—
87	C₂×C₄×C₁₀₁	C₂²×C₄	1	2	yes	—
88	C₂×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
89	C₂×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
90	C₂×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
91	C₂×C₄×C₁₀₁	C₂×D₄	1	1	yes	characteristic
92	C₂×C₄×C₁₀₁	C₂×D₄	1	1	yes	characteristic
93	C₂×C₄×C₁₀₁	C₂×D₄	1	1	yes	characteristic
94	C₂×C₄×C₁₀₁	C₂⁴	1	1	yes	characteristic
95	C₂²⋊D₁₀₁	—	2	8	no	—
96	C₂²⋊D₁₀₁	—	2	8	no	—
97	C₂×Dic₁₀₁	—	2	4	no	—
98	C₂×Dic₁₀₁	—	2	2	no	—
99	C₂×Dic₁₀₁	—	2	4	no	—
100	C₂×Dic₁₀₁	—	4	4	no	—
101	D₄₀₄	—	2	4	no	—
102	D₄₀₄	—	2	2	no	—
103	D₄₀₄	C₂²×C₄	1	4	yes	—
104	C₄×D₁₀₁	—	2	8	no	—
105	C₄×D₁₀₁	—	2	8	no	—
106	Dic₂₀₂	—	2	4	no	—
107	Dic₂₀₂	—	2	2	no	—
108	Dic₂₀₂	C₂²×C₄	1	4	yes	—
109	C₈×C₁₀₁	—	2	4	no	—
110	C₂²×C₁₀₁	—	2	2	no	—
111	C₂²×C₁₀₁	—	2	2	no	—
112	C₂²×C₁₀₁	—	2	2	no	—
113	C₂²×C₁₀₁	—	4	4	no	—
114	C₂²×C₁₀₁	C₂×C₂²⋊C₄	1	2	yes	—
115	C₂²×C₁₀₁	C₂×C₂²⋊C₄	1	2	yes	—
116	C₂²×C₁₀₁	C₂×C₂²⋊C₄	1	2	yes	—
117	C₂²×C₁₀₁	C₂²×D₄	1	1	yes	characteristic
118	C₂×D₁₀₁	—	2	4	no	—
119	C₂×D₁₀₁	—	4	8	no	—
120	C₂×D₁₀₁	—	4	4	no	—
121	C₄×C₁₀₁	C₂×C₂²⋊C₄	1	2	yes	—
122	C₄×C₁₀₁	C₂×C₂²⋊C₄	1	2	yes	—
123	C₄×C₁₀₁	C₂×C₂²⋊C₄	1	2	yes	—
124	C₄×C₁₀₁	C₂³×C₄	1	1	yes	characteristic
125	C₄×C₁₀₁	C₂²×D₄	1	1	yes	characteristic
126	Dic₁₀₁	—	2	4	no	—
127	Dic₁₀₁	—	4	4	no	—
128	C₂×C₁₀₁	—	2	2	no	—
129	C₂×C₁₀₁	—	2	2	no	—
130	C₂×C₁₀₁	—	2	2	no	—
131	C₂×C₁₀₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
132	C₂×C₁₀₁	#575	1	2	yes	—
133	D₁₀₁	—	4	4	no	—
134	#2896	—	101	1	no	maximal, Sylow
135	C₁₀₁	#2896	1	1	yes	characteristic, complemented, Sylow
136	C₂²×#41	—	101	1	no	—
137	C₂²×M₁₆	D₁₀₁	1	1	yes	characteristic, complemented
138	C₂×#117	—	101	1	no	—
139	#419	—	101	2	no	—
140	#418	—	101	2	no	—
141	#417	—	101	4	no	—
142	#416	—	101	4	no	—
143	C₂×#41	—	101	8	no	—
144	C₂×#41	—	101	2	no	—
145	C₂×#41	—	101	2	no	—
146	C₂×#41	—	202	1	no	—
147	C₂×#41	—	202	1	no	—
148	C₂×#41	—	202	4	no	—
149	C₂²×Q₈	—	101	1	no	—
150	C₂²×D₄	—	101	1	no	—
151	C₂²×D₄	—	202	1	no	—
152	C₂³×C₄	—	202	1	no	—
153	C₂³×C₄	C₂×D₁₀₁	1	1	yes	characteristic
154	C₂×M₁₆	—	2	4	no	—
155	C₂×M₁₆	C₂×D₁₀₁	1	2	yes	—
156	C₂×M₁₆	C₂×D₁₀₁	1	2	yes	complemented
157	C₂²×C₈	C₂×D₁₀₁	1	2	yes	complemented
158	#117	—	101	2	no	—
159	#117	—	101	2	no	—
160	#117	—	202	1	no	—
161	#117	—	202	1	no	—
162	C₂×C₄⋊C₄	—	101	2	no	—
163	C₂×C₂²⋊C₄	—	202	1	no	—
164	C₂×C₄²	—	202	1	no	—
165	#103	—	101	8	no	—
166	#102	—	101	8	no	—
167	C₂⁴	—	202	1	no	—
168	C₂⁴	D₄₀₄	1	1	yes	characteristic
169	#41	—	101	8	no	—
170	#41	—	101	8	no	—
171	#41	—	202	8	no	—
172	#41	—	202	8	no	—
173	#41	—	202	4	no	—
174	#41	—	202	4	no	—
175	C₂×Q₈	—	101	4	no	—
176	C₂×Q₈	—	101	2	no	—
177	C₂×Q₈	—	202	2	no	—
178	C₂×Q₈	—	202	1	no	—
179	C₂×D₄	—	101	4	no	—
180	C₂×D₄	—	101	2	no	—
181	C₂×D₄	—	202	8	no	—
182	C₂×D₄	—	202	2	no	—
183	C₂×D₄	—	202	2	no	—
184	C₂×D₄	—	202	2	no	—
185	C₂×D₄	—	202	1	no	—
186	C₂²×C₄	—	202	2	no	—
187	C₂²×C₄	—	202	1	no	—
188	C₂²×C₄	—	202	8	no	—
189	C₂²×C₄	—	202	2	no	—
190	C₂²×C₄	—	202	2	no	—
191	C₂²×C₄	C₄×D₁₀₁	1	4	yes	—
192	C₂²×C₄	D₄₀₄	1	1	yes	characteristic
193	C₂²×C₄	D₄₀₄	1	1	yes	characteristic
194	C₂²×C₄	D₄₀₄	1	1	yes	characteristic
195	C₂²×C₄	C₂²⋊D₁₀₁	1	4	yes	—
196	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
197	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
198	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
199	M₁₆	—	2	4	no	—
200	M₁₆	—	2	4	no	—
201	C₂×C₈	—	2	4	no	—
202	C₂×C₈	C₂²×D₁₀₁	1	4	yes	—
203	C₄⋊C₄	—	101	4	no	—
204	C₄⋊C₄	—	202	2	no	—
205	C₂²⋊C₄	—	202	2	no	—
206	C₂²⋊C₄	—	202	2	no	—
207	C₄²	—	202	2	no	—
208	C₄²	—	202	2	no	—
209	C₂³	—	4	2	no	—
210	C₂³	—	404	4	no	—
211	C₂³	—	202	2	no	—
212	C₂³	—	202	4	no	—
213	C₂³	#13108051	1	4	yes	—
214	C₂³	C₂×D₄₀₄	1	1	yes	characteristic
215	C₂³	C₂×D₄₀₄	1	1	yes	characteristic
216	C₂³	C₂×D₄₀₄	1	1	yes	characteristic
217	Q₈	—	101	4	no	—
218	Q₈	—	202	2	no	—
219	Q₈	—	202	4	no	—
220	D₄	—	101	4	no	—
221	D₄	—	202	8	no	—
222	D₄	—	202	8	no	—
223	D₄	—	202	2	no	—
224	D₄	—	202	4	no	—
225	C₂×C₄	—	404	4	no	—
226	C₂×C₄	—	202	4	no	—
227	C₂×C₄	—	202	2	no	—
228	C₂×C₄	—	202	4	no	—
229	C₂×C₄	—	202	8	no	—
230	C₂×C₄	—	202	8	no	—
231	C₂×C₄	#13108051	1	4	yes	—
232	C₂×C₄	#13108051	1	4	yes	—
233	C₂×C₄	#13108051	1	4	yes	—
234	C₂×C₄	C₂×C₄×D₁₀₁	1	2	yes	—
235	C₂×C₄	C₂×C₄×D₁₀₁	1	2	yes	—
236	C₂×C₄	C₂×C₄×D₁₀₁	1	2	yes	—
237	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	2	yes	—
238	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	2	yes	—
239	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	2	yes	—
240	C₂×C₄	C₂×D₄₀₄	1	1	yes	characteristic
241	C₂×C₄	C₂×D₄₀₄	1	1	yes	characteristic
242	C₂×C₄	C₂×D₄₀₄	1	1	yes	characteristic
243	C₂×C₄	C₂³×D₁₀₁	1	1	yes	characteristic
244	C₈	—	2	4	no	—
245	C₂²	—	2	2	no	—
246	C₂²	—	2	2	no	—
247	C₂²	—	2	2	no	—
248	C₂²	—	4	4	no	—
249	C₂²	—	404	8	no	—
250	C₂²	—	404	4	no	—
251	C₂²	—	202	4	no	—
252	C₂²	C₂×#13108051	1	2	yes	—
253	C₂²	C₂×#13108051	1	2	yes	—
254	C₂²	C₂×#13108051	1	2	yes	—
255	C₂²	C₂²×D₄₀₄	1	1	yes	characteristic
256	C₄	—	404	4	no	—
257	C₄	—	202	4	no	—
258	C₄	C₂×#13108051	1	2	yes	—
259	C₄	C₂×#13108051	1	2	yes	—
260	C₄	C₂×#13108051	1	2	yes	—
261	C₄	C₂²×C₄×D₁₀₁	1	1	yes	characteristic
262	C₄	C₂²×C₂²⋊D₁₀₁	1	1	yes	characteristic
263	C₂	—	2	2	no	—
264	C₂	—	2	2	no	—
265	C₂	—	2	2	no	—
266	C₂	—	404	4	no	—
267	C₂	C₂²×#13108051	1	1	yes	characteristic
268	C₂	#20265977	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₄₀₄
commutator	C₄×C₁₀₁	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₀₁
Fitting	C₂²×M₁₆×C₁₀₁	C₂
socle	C₂²×C₁₀₁	C₂²×D₄
prime radical	C₂²×#13108074	C₂
cosocle	C₂×C₄×C₁₀₁	C₂⁴
hypercenter	C₂²×M₁₆	D₁₀₁
hypocenter	C₁₀₁	#2896
LCS	G ▷ C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×M₁₆

Complements

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

N	H ≅ G/N	type
2. C₂×#15692214	C₂	semidirect
3. #20265187	C₂	semidirect
4. #20265183	C₂	semidirect
5. #20265179	C₂	semidirect
6. #20265174	C₂	semidirect
7. C₂²×M₁₆×C₁₀₁	C₂	semidirect
11. C₂×#13108074	C₄	semidirect
21. #15692214	C₂²	semidirect
22. C₂×C₄⋊Dic₁₀₁	C₂²	semidirect
25. #15692180	C₂²	semidirect
26. #15692177	C₂²	semidirect
30. C₂×M₁₆×C₁₀₁	C₂²	semidirect
31. C₂²×C₈×C₁₀₁	C₂²	semidirect
38. #13108074	C₂×C₄	semidirect
41. C₂×Dic₂₀₂	C₂×C₄	semidirect
45. C₂×D₄₀₄	C₂×C₄	semidirect
135. C₁₀₁	#2896	semidirect
137. C₂²×M₁₆	D₁₀₁	semidirect
156. C₂×M₁₆	C₂×D₁₀₁	semidirect
157. C₂²×C₈	C₂×D₁₀₁	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#13108074	2
C₂×#15692214	2
#20265187	2
#20265183	2
#20265179	2
#20265174	2
C₂²×M₁₆×C₁₀₁	2
#2896	101