Subgroups — #27173808

Order 12928.

556 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15692231	C₂	1	1	yes	maximal, characteristic
2	C₂×#15692212	C₂	1	1	yes	maximal, characteristic
3	#20265084	C₂	1	1	yes	maximal, characteristic, complemented
4	#20265073	C₂	1	1	yes	maximal, characteristic, complemented
5	#20265072	C₂	1	2	yes	maximal, complemented
6	#20265068	C₂	1	2	yes	maximal
7	#20265067	C₂	1	2	yes	maximal
8	#20265061	C₂	1	2	yes	maximal, complemented
9	#20265034	C₂	1	1	yes	maximal, characteristic
10	C₄×C₄⋊Dic₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₄×Q₈×C₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×C₄×D₁₀₁	C₂²	1	1	yes	characteristic
13	#15692231	C₄	1	8	yes	complemented
14	#15692212	—	2	4	no	—
15	C₂×C₄⋊Dic₁₀₁	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊Dic₁₀₁	C₂²	1	1	yes	characteristic
17	C₂×#13108053	C₂²	1	1	yes	characteristic
18	C₂×#13108053	C₂²	1	1	yes	characteristic
19	C₂×#13108053	C₂²	1	2	yes	—
20	C₂×#13108053	C₂²	1	2	yes	—
21	C₂×#13108051	C₂²	1	1	yes	characteristic
22	C₂×#13108051	C₂²	1	1	yes	characteristic
23	C₂×#13108051	C₂²	1	2	yes	—
24	C₂×#13108051	C₂²	1	2	yes	—
25	C₂×C₄×Dic₁₀₁	C₂²	1	1	yes	characteristic
26	C₂×C₄×Dic₁₀₁	C₂²	1	2	yes	—
27	#15692162	C₂²	1	2	yes	—
28	#15692162	C₂²	1	2	yes	—
29	#15692162	C₂²	1	2	yes	—
30	#15692162	C₂²	1	2	yes	—
31	#15692162	C₂²	1	2	yes	—
32	C₂²×Q₈×C₁₀₁	C₂²	1	1	yes	characteristic
33	C₄×Q₈×C₁₀₁	—	2	4	no	—
34	C₂×C₄⋊C₄×C₁₀₁	C₂²	1	1	yes	characteristic
35	C₂×C₄⋊C₄×C₁₀₁	C₂²	1	2	yes	—
36	C₂×C₄²×C₁₀₁	C₂²	1	2	yes	—
37	C₂×C₄²×C₁₀₁	C₂²	1	1	yes	characteristic
38	C₂³×D₁₀₁	C₂³	1	1	yes	characteristic
39	C₂²×Dic₁₀₁	C₂³	1	1	yes	characteristic
40	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
41	C₂²×Dic₁₀₁	C₂³	1	1	yes	characteristic
42	C₂²×Dic₁₀₁	C₂³	1	1	yes	characteristic
43	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
44	C₂×C₄×D₁₀₁	—	2	2	no	—
45	C₂×C₄×D₁₀₁	—	2	2	no	—
46	C₂×C₄×D₁₀₁	C₂×C₄	1	4	yes	—
47	C₄⋊Dic₁₀₁	—	2	2	no	—
48	C₄⋊Dic₁₀₁	C₂×C₄	1	4	yes	—
49	#13108053	—	2	2	no	—
50	#13108053	—	2	2	no	—
51	#13108053	—	4	2	no	—
52	#13108053	C₂×C₄	1	8	yes	—
53	#13108051	—	2	2	no	—
54	#13108051	—	2	2	no	—
55	#13108051	—	4	2	no	—
56	#13108051	C₂×C₄	1	8	yes	—
57	C₄×Dic₁₀₁	—	2	2	no	—
58	C₄×Dic₁₀₁	—	4	2	no	—
59	C₂×Q₈×C₁₀₁	—	2	2	no	—
60	C₂×Q₈×C₁₀₁	—	2	2	no	—
61	C₂×Q₈×C₁₀₁	C₂×C₄	1	4	yes	—
62	C₂²×C₄×C₁₀₁	C₂³	1	2	yes	—
63	C₂²×C₄×C₁₀₁	C₂³	1	2	yes	—
64	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
65	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
66	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
67	C₄⋊C₄×C₁₀₁	—	2	4	no	—
68	C₄⋊C₄×C₁₀₁	—	2	2	no	—
69	C₄²×C₁₀₁	—	2	4	no	—
70	C₄²×C₁₀₁	D₄	1	4	yes	—
71	C₂³×C₁₀₁	C₂⁴	1	1	yes	characteristic
72	C₂²×D₁₀₁	—	2	1	no	—
73	C₂²×D₁₀₁	—	2	1	no	—
74	C₂²×D₁₀₁	—	2	1	no	—
75	C₂²×D₁₀₁	—	2	1	no	—
76	C₂²×D₁₀₁	—	2	1	no	—
77	C₂²×D₁₀₁	—	2	1	no	—
78	C₂²×D₁₀₁	C₂²×C₄	1	2	yes	—
79	Q₈×C₁₀₁	—	2	8	no	—
80	C₂×C₄×C₁₀₁	—	2	2	no	—
81	C₂×C₄×C₁₀₁	—	2	2	no	—
82	C₂×C₄×C₁₀₁	—	2	1	no	—
83	C₂×C₄×C₁₀₁	—	2	1	no	—
84	C₂×C₄×C₁₀₁	—	2	1	no	—
85	C₂×C₄×C₁₀₁	—	2	1	no	—
86	C₂×C₄×C₁₀₁	—	2	2	no	—
87	C₂×C₄×C₁₀₁	—	2	2	no	—
88	C₂×C₄×C₁₀₁	—	2	2	no	—
89	C₂×C₄×C₁₀₁	C₂²×C₄	1	4	yes	—
90	C₂×C₄×C₁₀₁	C₂²×C₄	1	2	yes	—
91	C₂×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
92	C₂×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
93	C₂×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
94	C₂×C₄×C₁₀₁	#41	1	2	yes	—
95	C₂×Dic₁₀₁	—	2	1	no	—
96	C₂×Dic₁₀₁	—	2	1	no	—
97	C₂×Dic₁₀₁	—	2	2	no	—
98	C₂×Dic₁₀₁	—	2	2	no	—
99	C₂×Dic₁₀₁	—	2	1	no	—
100	C₂×Dic₁₀₁	—	2	1	no	—
101	C₂×Dic₁₀₁	—	2	2	no	—
102	C₂×Dic₁₀₁	—	2	2	no	—
103	C₂×Dic₁₀₁	—	2	1	no	—
104	C₂×Dic₁₀₁	—	2	1	no	—
105	C₂×Dic₁₀₁	—	2	1	no	—
106	C₂×Dic₁₀₁	—	2	1	no	—
107	C₂×Dic₁₀₁	—	2	2	no	—
108	C₂×Dic₁₀₁	C₂²×C₄	1	4	yes	—
109	C₂×Dic₁₀₁	C₂²×C₄	1	2	yes	—
110	C₄×D₁₀₁	—	2	8	no	—
111	C₂²×C₁₀₁	C₂³×C₄	1	1	yes	characteristic
112	C₂²×C₁₀₁	C₂²×D₄	1	1	yes	characteristic
113	C₂²×C₁₀₁	C₂×#41	1	1	yes	characteristic
114	C₂²×C₁₀₁	H(5,𝔽₂)	1	1	yes	characteristic
115	C₂²×C₁₀₁	2₋¹⁺⁴	1	1	yes	characteristic
116	C₂²×C₁₀₁	2₋¹⁺⁴	1	1	yes	characteristic
117	C₂²×C₁₀₁	2₋¹⁺⁴	1	1	yes	characteristic
118	C₂×D₁₀₁	—	2	2	no	—
119	C₂×D₁₀₁	—	2	2	no	—
120	C₂×D₁₀₁	—	2	2	no	—
121	C₂×D₁₀₁	—	4	1	no	—
122	C₂×D₁₀₁	—	4	1	no	—
123	C₂×D₁₀₁	—	4	1	no	—
124	C₂×D₁₀₁	—	4	1	no	—
125	C₄×C₁₀₁	—	2	4	no	—
126	C₄×C₁₀₁	—	2	2	no	—
127	C₄×C₁₀₁	—	2	2	no	—
128	C₄×C₁₀₁	—	4	2	no	—
129	C₄×C₁₀₁	C₄×D₄	1	4	yes	—
130	Dic₁₀₁	—	2	2	no	—
131	Dic₁₀₁	—	2	4	no	—
132	Dic₁₀₁	—	4	1	no	—
133	Dic₁₀₁	—	4	1	no	—
134	Dic₁₀₁	—	4	2	no	—
135	C₂×C₁₀₁	C₂×C₄×D₄	1	1	yes	characteristic
136	C₂×C₁₀₁	#519	1	1	yes	characteristic
137	C₂×C₁₀₁	#520	1	1	yes	characteristic
138	C₂×C₁₀₁	#536	1	1	yes	characteristic
139	C₂×C₁₀₁	#537	1	1	yes	characteristic
140	C₂×C₁₀₁	#541	1	1	yes	characteristic
141	C₂×C₁₀₁	#542	1	1	yes	characteristic
142	D₁₀₁	—	4	2	no	—
143	#2323	—	101	1	no	maximal, Sylow
144	C₁₀₁	#2323	1	1	yes	characteristic, complemented, Sylow
145	C₂×C₂²⋊Q₈	—	101	1	no	—
146	C₂×C₄×Q₈	D₁₀₁	1	1	yes	characteristic, complemented
147	C₂×#117	—	101	1	no	—
148	#391	—	101	1	no	—
149	#389	—	101	2	no	—
150	#389	—	101	1	no	—
151	#388	—	101	2	no	—
152	#388	—	101	2	no	—
153	#387	—	101	2	no	—
154	#380	—	101	1	no	—
155	C₄×C₄⋊C₄	—	101	1	no	—
156	C₂²×Q₈	C₂×D₁₀₁	1	1	yes	characteristic
157	C₂³×C₄	—	101	1	no	—
158	C₂²⋊Q₈	—	101	8	no	—
159	C₄×Q₈	—	2	4	no	—
160	#117	—	202	4	no	—
161	C₂×C₄⋊C₄	—	101	1	no	—
162	C₂×C₄⋊C₄	—	101	2	no	—
163	C₂×C₄⋊C₄	—	101	2	no	—
164	C₂×C₄⋊C₄	—	101	1	no	—
165	C₂×C₄⋊C₄	—	101	1	no	—
166	C₂×C₄⋊C₄	—	101	1	no	—
167	C₂×C₄⋊C₄	C₂×D₁₀₁	1	1	yes	characteristic
168	C₂×C₄⋊C₄	C₂×D₁₀₁	1	2	yes	—
169	C₂×C₂²⋊C₄	—	101	2	no	—
170	C₂×C₂²⋊C₄	—	101	2	no	—
171	C₂×C₂²⋊C₄	—	101	1	no	—
172	C₂×C₂²⋊C₄	—	101	1	no	—
173	C₂×C₄²	—	101	2	no	—
174	C₂×C₄²	—	101	1	no	—
175	C₂×C₄²	C₂×D₁₀₁	1	2	yes	—
176	C₂×C₄²	C₂×D₁₀₁	1	1	yes	characteristic
177	#95	—	101	2	no	—
178	#95	—	101	2	no	—
179	#95	—	101	2	no	—
180	#95	—	101	2	no	—
181	#95	—	101	2	no	—
182	C₂⁴	—	101	1	no	—
183	C₂×Q₈	—	2	2	no	—
184	C₂×Q₈	—	2	2	no	—
185	C₂×Q₈	C₄×D₁₀₁	1	4	yes	—
186	C₂²×C₄	—	101	2	no	—
187	C₂²×C₄	—	101	1	no	—
188	C₂²×C₄	—	101	4	no	—
189	C₂²×C₄	—	101	1	no	—
190	C₂²×C₄	—	101	2	no	—
191	C₂²×C₄	—	101	1	no	—
192	C₂²×C₄	—	202	2	no	—
193	C₂²×C₄	—	202	2	no	—
194	C₂²×C₄	C₂²×D₁₀₁	1	2	yes	—
195	C₂²×C₄	C₂²×D₁₀₁	1	2	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	C₄⋊C₄	—	2	4	no	—
200	C₄⋊C₄	—	2	2	no	—
201	C₄⋊C₄	—	101	8	no	—
202	C₄⋊C₄	—	101	4	no	—
203	C₄⋊C₄	—	404	2	no	—
204	C₄⋊C₄	—	202	2	no	—
205	C₄⋊C₄	—	202	2	no	—
206	C₄⋊C₄	—	202	2	no	—
207	C₂²⋊C₄	—	101	8	no	—
208	C₂²⋊C₄	—	404	2	no	—
209	C₂²⋊C₄	—	202	2	no	—
210	C₂²⋊C₄	—	202	2	no	—
211	C₄²	—	2	4	no	—
212	C₄²	—	404	2	no	—
213	C₄²	—	202	2	no	—
214	C₄²	C₂²⋊D₁₀₁	1	4	yes	—
215	C₂³	—	101	2	no	—
216	C₂³	—	202	1	no	—
217	C₂³	—	202	1	no	—
218	C₂³	—	202	1	no	—
219	C₂³	—	202	1	no	—
220	C₂³	—	202	1	no	—
221	C₂³	—	202	1	no	—
222	C₂³	C₂³×D₁₀₁	1	1	yes	characteristic
223	Q₈	—	2	8	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	101	4	no	—
234	C₂×C₄	—	101	2	no	—
235	C₂×C₄	—	202	1	no	—
236	C₂×C₄	—	202	1	no	—
237	C₂×C₄	—	202	2	no	—
238	C₂×C₄	—	202	2	no	—
239	C₂×C₄	—	202	1	no	—
240	C₂×C₄	—	202	1	no	—
241	C₂×C₄	—	202	1	no	—
242	C₂×C₄	—	202	1	no	—
243	C₂×C₄	—	202	2	no	—
244	C₂×C₄	—	202	1	no	—
245	C₂×C₄	—	202	1	no	—
246	C₂×C₄	—	202	2	no	—
247	C₂×C₄	—	202	2	no	—
248	C₂×C₄	—	202	8	no	—
249	C₂×C₄	C₂×C₄×D₁₀₁	1	4	yes	—
250	C₂×C₄	C₂×C₄×D₁₀₁	1	2	yes	—
251	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	2	yes	—
252	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	2	yes	—
253	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	2	yes	—
254	C₂×C₄	#13108074	1	2	yes	—
255	C₂²	—	404	1	no	—
256	C₂²	—	404	1	no	—
257	C₂²	—	404	1	no	—
258	C₂²	—	404	1	no	—
259	C₂²	—	202	2	no	—
260	C₂²	—	202	2	no	—
261	C₂²	—	202	2	no	—
262	C₂²	C₂²×C₄×D₁₀₁	1	1	yes	characteristic
263	C₂²	C₂²×C₂²⋊D₁₀₁	1	1	yes	characteristic
264	C₂²	C₂×#13108074	1	1	yes	characteristic
265	C₂²	#15692301	1	1	yes	characteristic
266	C₂²	#15692302	1	1	yes	characteristic
267	C₂²	#15692303	1	1	yes	characteristic
268	C₂²	#15692303	1	1	yes	characteristic
269	C₄	—	2	2	no	—
270	C₄	—	2	2	no	—
271	C₄	—	2	4	no	—
272	C₄	—	4	2	no	—
273	C₄	—	404	1	no	—
274	C₄	—	404	1	no	—
275	C₄	—	404	2	no	—
276	C₄	—	202	2	no	—
277	C₄	—	202	4	no	—
278	C₄	C₄×C₂²⋊D₁₀₁	1	4	yes	—
279	C₂	—	404	2	no	—
280	C₂	C₂×C₄×C₂²⋊D₁₀₁	1	1	yes	characteristic
281	C₂	#20265653	1	1	yes	characteristic
282	C₂	#20265659	1	1	yes	characteristic
283	C₂	#20265751	1	1	yes	characteristic
284	C₂	#20265755	1	1	yes	characteristic
285	C₂	#20265785	1	1	yes	characteristic
286	C₂	#20265799	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₀₁
commutator	C₂²×C₁₀₁	C₂³×C₄
Fitting	C₂×C₄×Q₈×C₁₀₁	C₂
socle, cosocle	C₂³×C₁₀₁	C₂⁴
prime radical	C₂³×D₁₀₁	C₂³
hypercenter	C₂×C₄×Q₈	D₁₀₁
hypocenter	C₁₀₁	#2323
LCS	G ▷ C₂²×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×Q₈

Complements

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

N	H ≅ G/N	type
3. #20265084	C₂	semidirect
4. #20265073	C₂	semidirect
5. #20265072	C₂	semidirect
8. #20265061	C₂	semidirect
10. C₄×C₄⋊Dic₁₀₁	C₂	semidirect
11. C₂×C₄×Q₈×C₁₀₁	C₂	semidirect
13. #15692231	C₄	semidirect
144. C₁₀₁	#2323	semidirect
146. C₂×C₄×Q₈	D₁₀₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15692231	2
C₂×#15692212	2
#20265084	2
#20265073	2
#20265072	2
#20265068	2
#20265067	2
#20265061	2
#20265034	2
C₄×C₄⋊Dic₁₀₁	2
C₂×C₄×Q₈×C₁₀₁	2
#2323	101