Subgroups — #27173619

Order 12928.

936 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×#13108073	C₂	1	1	yes	maximal, characteristic
2	C₂×#15692213	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₄²×D₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
4	#20265083	C₂	1	1	yes	maximal, characteristic, complemented
5	#20265082	C₂	1	1	yes	maximal, characteristic, complemented
6	#20265080	C₂	1	1	yes	maximal, characteristic, complemented
7	#20265053	C₂	1	4	yes	maximal, complemented
8	C₂²⋊C₄×Dic₁₀₁	C₂	1	4	yes	maximal, complemented
9	#390×C₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#13108073	—	2	4	no	—
11	C₂×#13108073	—	2	4	no	—
12	C₂×#13108073	C₄	1	8	yes	complemented
13	C₂²×Dic₂₀₂	C₂²	1	1	yes	characteristic
14	C₂²×C₂²⋊D₁₀₁	C₂²	1	2	yes	—
15	C₂³×Dic₁₀₁	C₂²	1	2	yes	—
16	C₂²×C₄×D₁₀₁	C₂²	1	2	yes	—
17	C₂²×C₄×D₁₀₁	C₂²	1	1	yes	characteristic
18	#15692213	—	2	4	no	—
19	C₂×C₄⋊Dic₁₀₁	C₂²	1	1	yes	characteristic
20	C₂×C₂²⋊Dic₁₀₁	C₂²	1	4	yes	—
21	C₂×#13108051	C₂²	1	2	yes	—
22	C₂×C₄×Dic₁₀₁	C₂²	1	2	yes	—
23	C₂×C₄×Dic₁₀₁	C₂²	1	2	yes	—
24	C₂×C₄×Dic₁₀₁	C₂²	1	1	yes	characteristic
25	C₄²×D₁₀₁	—	2	4	no	—
26	#15692162	C₂²	1	4	yes	—
27	#15692162	C₂²	1	4	yes	—
28	C₂²×D₄×C₁₀₁	C₂²	1	1	yes	characteristic
29	C₂×C₄⋊C₄×C₁₀₁	C₂²	1	1	yes	characteristic
30	C₂×C₂²⋊C₄×C₁₀₁	C₂²	1	4	yes	—
31	C₂×C₄²×C₁₀₁	C₂²	1	1	yes	characteristic
32	C₂³×D₁₀₁	C₂³	1	1	yes	characteristic
33	#13108073	—	2	32	no	—
34	C₂×Dic₂₀₂	—	2	2	no	—
35	C₂×Dic₂₀₂	—	2	2	no	—
36	C₂×Dic₂₀₂	C₂×C₄	1	4	yes	complemented
37	C₂×C₂²⋊D₁₀₁	—	2	4	no	—
38	C₂×C₂²⋊D₁₀₁	—	2	4	no	—
39	C₂×C₂²⋊D₁₀₁	C₂×C₄	1	8	yes	—
40	C₂²×Dic₁₀₁	—	2	4	no	—
41	C₂²×Dic₁₀₁	—	2	4	no	—
42	C₂²×Dic₁₀₁	C₂×C₄	1	8	yes	complemented
43	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
44	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
45	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
46	C₂²×Dic₁₀₁	C₂³	1	1	yes	characteristic
47	C₂×C₄×D₁₀₁	—	2	4	no	—
48	C₂×C₄×D₁₀₁	—	2	4	no	—
49	C₂×C₄×D₁₀₁	—	2	4	no	—
50	C₂×C₄×D₁₀₁	C₂×C₄	1	4	yes	—
51	C₂×C₄×D₁₀₁	D₄	1	4	yes	—
52	C₂×C₄×D₁₀₁	D₄	1	4	yes	—
53	C₄⋊Dic₁₀₁	—	2	2	no	—
54	C₂²⋊Dic₁₀₁	—	2	8	no	—
55	#13108051	—	2	4	no	—
56	C₄×Dic₁₀₁	—	2	2	no	—
57	C₄×Dic₁₀₁	—	2	4	no	—
58	C₄×Dic₁₀₁	—	2	4	no	—
59	C₂⁴×C₁₀₁	C₂³	1	2	yes	—
60	C₂×D₄×C₁₀₁	—	2	2	no	—
61	C₂×D₄×C₁₀₁	—	2	2	no	—
62	C₂×D₄×C₁₀₁	C₂×C₄	1	4	yes	complemented
63	C₂²×C₄×C₁₀₁	C₂³	1	2	yes	—
64	C₂²×C₄×C₁₀₁	C₂³	1	2	yes	—
65	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
66	C₄⋊C₄×C₁₀₁	—	2	2	no	—
67	C₂²⋊C₄×C₁₀₁	—	2	8	no	—
68	C₄²×C₁₀₁	—	2	2	no	—
69	C₂³×C₁₀₁	—	2	2	no	—
70	C₂³×C₁₀₁	—	2	2	no	—
71	C₂³×C₁₀₁	—	2	4	no	—
72	C₂³×C₁₀₁	—	2	4	no	—
73	C₂³×C₁₀₁	C₂²×C₄	1	4	yes	—
74	C₂³×C₁₀₁	C₂⁴	1	1	yes	characteristic
75	C₂²×D₁₀₁	—	2	2	no	—
76	C₂²×D₁₀₁	—	2	2	no	—
77	C₂²×D₁₀₁	C₂²×C₄	1	2	yes	—
78	C₂²×D₁₀₁	C₂×D₄	1	2	yes	—
79	C₂²×D₁₀₁	C₂×D₄	1	2	yes	—
80	D₄×C₁₀₁	—	2	8	no	—
81	C₂×C₄×C₁₀₁	—	2	2	no	—
82	C₂×C₄×C₁₀₁	—	2	2	no	—
83	C₂×C₄×C₁₀₁	—	2	2	no	—
84	C₂×C₄×C₁₀₁	—	2	4	no	—
85	C₂×C₄×C₁₀₁	C₂²×C₄	1	2	yes	—
86	C₂×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
87	C₂×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
88	C₂×C₄×C₁₀₁	#41	1	4	yes	complemented
89	C₂²⋊D₁₀₁	—	2	16	no	—
90	C₂×Dic₁₀₁	—	2	2	no	—
91	C₂×Dic₁₀₁	—	2	2	no	—
92	C₂×Dic₁₀₁	—	2	2	no	—
93	C₂×Dic₁₀₁	—	2	2	no	—
94	C₂×Dic₁₀₁	—	2	16	no	—
95	C₂×Dic₁₀₁	—	2	2	no	—
96	C₂×Dic₁₀₁	—	2	4	no	—
97	C₂×Dic₁₀₁	C₂²×C₄	1	4	yes	—
98	C₂×Dic₁₀₁	C₂²×C₄	1	2	yes	—
99	C₂×Dic₁₀₁	C₂×D₄	1	2	yes	—
100	C₂×Dic₁₀₁	C₂×D₄	1	2	yes	—
101	C₂×Dic₁₀₁	#41	1	4	yes	complemented
102	C₄×D₁₀₁	—	2	16	no	—
103	C₄×D₁₀₁	C₂²⋊C₄	1	16	yes	complemented
104	Dic₂₀₂	—	2	8	no	—
105	C₂²×C₁₀₁	—	2	4	no	—
106	C₂²×C₁₀₁	—	2	8	no	—
107	C₂²×C₁₀₁	—	4	4	no	—
108	C₂²×C₁₀₁	—	4	2	no	—
109	C₂²×C₁₀₁	—	4	2	no	—
110	C₂²×C₁₀₁	C₂³×C₄	1	1	yes	characteristic
111	C₂²×C₁₀₁	C₂²×D₄	1	1	yes	characteristic
112	C₂²×C₁₀₁	C₂²×D₄	1	1	yes	characteristic
113	C₂²×C₁₀₁	C₂×#41	1	2	yes	—
114	C₂²×C₁₀₁	C₂×#41	1	2	yes	—
115	C₂×D₁₀₁	—	2	4	no	—
116	C₂×D₁₀₁	—	2	4	no	—
117	C₂×D₁₀₁	—	2	2	no	—
118	C₂×D₁₀₁	—	2	2	no	—
119	C₂×D₁₀₁	C₂×C₂²⋊C₄	1	4	yes	—
120	C₄×C₁₀₁	—	2	4	no	—
121	C₄×C₁₀₁	—	4	2	no	—
122	C₄×C₁₀₁	C₂×C₂²⋊C₄	1	4	yes	—
123	Dic₁₀₁	—	2	4	no	—
124	Dic₁₀₁	—	2	4	no	—
125	Dic₁₀₁	—	4	2	no	—
126	Dic₁₀₁	C₂×C₂²⋊C₄	1	4	yes	—
127	C₂×C₁₀₁	—	4	4	no	—
128	C₂×C₁₀₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
129	C₂×C₁₀₁	C₄×#41	1	2	yes	—
130	C₂×C₁₀₁	#525	1	2	yes	—
131	C₂×C₁₀₁	#532	1	1	yes	characteristic
132	C₂×C₁₀₁	#532	1	1	yes	characteristic
133	D₁₀₁	—	2	4	no	—
134	#2297	—	101	1	no	maximal, Sylow
135	C₁₀₁	#2297	1	1	yes	characteristic, complemented, Sylow
136	C₂²×#41	—	101	1	no	—
137	C₂×#117	—	101	1	no	—
138	C₂²×C₄²	—	101	1	no	—
139	#391	—	101	1	no	—
140	#391	—	101	1	no	—
141	#390	—	101	1	no	—
142	#390	D₁₀₁	1	1	yes	characteristic, complemented
143	#386	—	101	4	no	—
144	C₄×C₂²⋊C₄	—	101	4	no	—
145	C₂×#41	—	101	8	no	—
146	C₂×#41	—	202	4	no	—
147	C₂×#41	—	202	4	no	—
148	C₂²×Q₈	—	101	1	no	—
149	C₂²×D₄	—	101	2	no	—
150	C₂²×D₄	C₂×D₁₀₁	1	1	yes	characteristic
151	C₂³×C₄	—	101	1	no	—
152	C₂³×C₄	—	101	2	no	—
153	C₂³×C₄	—	101	2	no	—
154	#117	—	202	4	no	—
155	C₂×C₄⋊C₄	—	101	1	no	—
156	C₂×C₄⋊C₄	C₂×D₁₀₁	1	1	yes	characteristic
157	C₂×C₂²⋊C₄	—	101	4	no	—
158	C₂×C₂²⋊C₄	—	101	2	no	—
159	C₂×C₂²⋊C₄	C₂×D₁₀₁	1	4	yes	—
160	C₂×C₄²	—	101	2	no	—
161	C₂×C₄²	—	101	2	no	—
162	C₂×C₄²	—	101	1	no	—
163	C₂×C₄²	—	202	4	no	—
164	C₂×C₄²	C₂×D₁₀₁	1	1	yes	characteristic
165	#95	—	101	4	no	—
166	#95	—	101	4	no	—
167	C₂⁴	—	101	1	no	—
168	C₂⁴	C₂²×D₁₀₁	1	2	yes	—
169	#41	—	202	32	no	—
170	C₂×Q₈	—	101	4	no	—
171	C₂×Q₈	—	202	2	no	—
172	C₂×Q₈	—	202	2	no	—
173	C₂×D₄	—	2	2	no	—
174	C₂×D₄	—	2	2	no	—
175	C₂×D₄	—	101	8	no	—
176	C₂×D₄	—	202	4	no	—
177	C₂×D₄	—	202	4	no	—
178	C₂×D₄	C₄×D₁₀₁	1	4	yes	complemented
179	C₂²×C₄	—	101	1	no	—
180	C₂²×C₄	—	101	4	no	—
181	C₂²×C₄	—	101	4	no	—
182	C₂²×C₄	—	101	4	no	—
183	C₂²×C₄	—	101	8	no	—
184	C₂²×C₄	—	101	2	no	—
185	C₂²×C₄	—	101	2	no	—
186	C₂²×C₄	—	101	2	no	—
187	C₂²×C₄	—	202	4	no	—
188	C₂²×C₄	—	202	4	no	—
189	C₂²×C₄	—	202	4	no	—
190	C₂²×C₄	—	202	4	no	—
191	C₂²×C₄	—	202	4	no	—
192	C₂²×C₄	C₂²×D₁₀₁	1	2	yes	—
193	C₂²×C₄	C₂²×D₁₀₁	1	2	yes	—
194	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
195	C₄⋊C₄	—	2	2	no	—
196	C₄⋊C₄	—	202	2	no	—
197	C₂²⋊C₄	—	2	8	no	—
198	C₂²⋊C₄	—	202	4	no	—
199	C₂²⋊C₄	—	202	8	no	—
200	C₄²	—	2	2	no	—
201	C₄²	—	202	4	no	—
202	C₄²	—	202	4	no	—
203	C₄²	—	202	2	no	—
204	C₂³	—	2	2	no	—
205	C₂³	—	2	2	no	—
206	C₂³	—	2	4	no	—
207	C₂³	—	2	4	no	—
208	C₂³	—	101	2	no	—
209	C₂³	—	101	2	no	—
210	C₂³	—	101	2	no	—
211	C₂³	—	202	2	no	—
212	C₂³	—	202	2	no	—
213	C₂³	C₂×C₄×D₁₀₁	1	4	yes	—
214	C₂³	C₂³×D₁₀₁	1	1	yes	characteristic
215	Q₈	—	202	8	no	—
216	D₄	—	2	8	no	—
217	D₄	—	202	16	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	4	no	—
222	C₂×C₄	—	101	2	no	—
223	C₂×C₄	—	101	2	no	—
224	C₂×C₄	—	101	2	no	—
225	C₂×C₄	—	101	16	no	—
226	C₂×C₄	—	101	4	no	—
227	C₂×C₄	—	101	4	no	—
228	C₂×C₄	—	202	16	no	—
229	C₂×C₄	—	202	16	no	—
230	C₂×C₄	—	202	2	no	—
231	C₂×C₄	—	202	4	no	—
232	C₂×C₄	—	202	2	no	—
233	C₂×C₄	—	202	2	no	—
234	C₂×C₄	—	202	2	no	—
235	C₂×C₄	—	202	2	no	—
236	C₂×C₄	C₂×C₄×D₁₀₁	1	2	yes	—
237	C₂×C₄	D₄×D₁₀₁	1	2	yes	—
238	C₂×C₄	D₄×D₁₀₁	1	2	yes	—
239	C₂×C₄	#13108073	1	4	yes	complemented
240	C₂²	—	2	4	no	—
241	C₂²	—	2	8	no	—
242	C₂²	—	4	4	no	—
243	C₂²	—	4	2	no	—
244	C₂²	—	4	2	no	—
245	C₂²	—	101	4	no	—
246	C₂²	—	202	4	no	—
247	C₂²	—	202	4	no	—
248	C₂²	—	202	2	no	—
249	C₂²	—	202	2	no	—
250	C₂²	C₂²×C₄×D₁₀₁	1	1	yes	characteristic
251	C₂²	C₂×D₄×D₁₀₁	1	1	yes	characteristic
252	C₂²	C₂×D₄×D₁₀₁	1	1	yes	characteristic
253	C₂²	#41×D₁₀₁	1	2	yes	—
254	C₂²	C₂×#13108073	1	2	yes	—
255	C₄	—	2	4	no	—
256	C₄	—	4	2	no	—
257	C₄	—	101	4	no	—
258	C₄	—	404	2	no	—
259	C₄	—	202	4	no	—
260	C₄	—	202	4	no	—
261	C₄	C₂²⋊C₄×D₁₀₁	1	4	yes	—
262	C₂	—	4	4	no	—
263	C₂	—	202	4	no	—
264	C₂	C₂×C₂²⋊C₄×D₁₀₁	1	1	yes	characteristic
265	C₂	C₄×#13108073	1	2	yes	—
266	C₂	#20265687	1	2	yes	—
267	C₂	#20265725	1	1	yes	characteristic
268	C₂	#20265726	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₀₁
commutator	C₂²×C₁₀₁	C₂³×C₄
Fitting	#390×C₁₀₁	C₂
socle, cosocle	C₂³×C₁₀₁	C₂⁴
prime radical	C₂²×#13108073	C₂
hypercenter	#390	D₁₀₁
hypocenter	C₁₀₁	#2297
LCS	G ▷ C₂²×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂³ ◁ #390

Complements

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

N	H ≅ G/N	type
2. C₂×#15692213	C₂	semidirect
3. C₂×C₄²×D₁₀₁	C₂	semidirect
4. #20265083	C₂	semidirect
5. #20265082	C₂	semidirect
6. #20265080	C₂	semidirect
7. #20265053	C₂	semidirect
8. C₂²⋊C₄×Dic₁₀₁	C₂	semidirect
9. #390×C₁₀₁	C₂	semidirect
12. C₂×#13108073	C₄	semidirect
36. C₂×Dic₂₀₂	C₂×C₄	semidirect
42. C₂²×Dic₁₀₁	C₂×C₄	semidirect
62. C₂×D₄×C₁₀₁	C₂×C₄	semidirect
88. C₂×C₄×C₁₀₁	#41	semidirect
101. C₂×Dic₁₀₁	#41	semidirect
103. C₄×D₁₀₁	C₂²⋊C₄	semidirect
135. C₁₀₁	#2297	semidirect
142. #390	D₁₀₁	semidirect
178. C₂×D₄	C₄×D₁₀₁	semidirect
239. C₂×C₄	#13108073	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#13108073	2
C₂×#15692213	2
C₂×C₄²×D₁₀₁	2
#20265083	2
#20265082	2
#20265080	2
#20265053	2
C₂²⋊C₄×Dic₁₀₁	2
#390×C₁₀₁	2
#2297	101