Subgroups — #27186840

Order 12928.

388 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₁₀₁⋊M₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×D₁₀₁⋊M₁₆	C₂	1	1	yes	maximal, characteristic
3	#20266058	C₂	1	4	yes	maximal, complemented
4	#117×D₁₀₁	C₂	1	1	yes	maximal, characteristic
5	C₂×#13108084	—	2	1	no	—
6	C₂×#13108084	C₄	1	2	yes	complemented
7	D₁₀₁⋊M₁₆	—	2	1	no	—
8	D₁₀₁⋊M₁₆	—	2	1	no	—
9	D₁₀₁⋊M₁₆	—	2	1	no	—
10	D₁₀₁⋊M₁₆	—	2	1	no	—
11	D₁₀₁⋊M₁₆	—	2	1	no	—
12	D₁₀₁⋊M₁₆	—	2	1	no	—
13	D₁₀₁⋊M₁₆	C₄	1	2	yes	complemented
14	D₁₀₁⋊M₁₆	C₄	1	2	yes	complemented
15	C₂×#13108083	C₂²	1	2	yes	—
16	C₂×#13108083	C₂²	1	2	yes	—
17	C₂×D₁₀₁⋊C₈	—	2	1	no	—
18	C₂×D₁₀₁⋊C₈	C₄	1	2	yes	complemented
19	C₂²×C₄×D₁₀₁	C₂²	1	1	yes	characteristic
20	#15692214	C₄	1	1	yes	characteristic
21	#15692213	—	2	1	no	—
22	#15692212	—	2	1	no	—
23	#15692211	C₂²	1	2	yes	—
24	C₄⋊C₄×D₁₀₁	—	2	1	no	—
25	C₂²⋊C₄×D₁₀₁	C₄	1	2	yes	—
26	C₄²×D₁₀₁	—	2	1	no	—
27	#117×C₁₀₁	C₄	1	1	yes	characteristic
28	#13108084	—	2	2	no	—
29	#13108084	—	2	2	no	—
30	#13108084	—	2	2	no	—
31	#13108084	—	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	D₁₀₁⋊C₈	—	2	2	no	—
37	D₁₀₁⋊C₈	—	2	2	no	—
38	D₁₀₁⋊C₈	—	2	2	no	—
39	D₁₀₁⋊C₈	—	2	2	no	—
40	C₂×#11717581	—	2	2	no	—
41	C₂×#11717581	C₂×C₄	1	4	yes	complemented
42	C₂³×D₁₀₁	C₂×C₄	1	1	yes	characteristic
43	C₂²×Dic₁₀₁	C₂³	1	1	yes	characteristic
44	C₂×C₄×D₁₀₁	—	2	2	no	—
45	C₂×C₄×D₁₀₁	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄×D₁₀₁	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄×D₁₀₁	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄×D₁₀₁	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄×D₁₀₁	D₄	1	1	yes	characteristic
50	C₂×C₄×D₁₀₁	D₄	1	1	yes	characteristic
51	C₂×C₄×D₁₀₁	D₄	1	1	yes	characteristic
52	C₂×C₄×D₁₀₁	D₄	1	1	yes	characteristic
53	C₂×C₄×D₁₀₁	D₄	1	1	yes	characteristic
54	C₂×C₄×D₁₀₁	D₄	1	1	yes	characteristic
55	C₂×C₄×D₁₀₁	Q₈	1	1	yes	characteristic
56	C₂×C₄×D₁₀₁	Q₈	1	1	yes	characteristic
57	C₄⋊Dic₁₀₁	—	2	1	no	—
58	#13108053	—	2	2	no	—
59	C₂²⋊Dic₁₀₁	C₂×C₄	1	2	yes	—
60	#13108051	—	2	2	no	—
61	C₄×Dic₁₀₁	—	2	1	no	—
62	C₄×Dic₁₀₁	—	2	2	no	—
63	C₂²×C₄×C₁₀₁	C₂×C₄	1	1	yes	characteristic
64	C₄⋊C₄×C₁₀₁	—	2	1	no	—
65	C₂²⋊C₄×C₁₀₁	C₂×C₄	1	2	yes	—
66	C₄²×C₁₀₁	—	2	1	no	—
67	C₂³×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
68	C₂²×D₁₀₁	—	4	2	no	—
69	C₂²×D₁₀₁	C₄²	1	1	yes	characteristic
70	C₂²×D₁₀₁	C₄²	1	1	yes	characteristic
71	C₂²×D₁₀₁	C₂²⋊C₄	1	1	yes	characteristic
72	C₂²×D₁₀₁	C₂²⋊C₄	1	1	yes	characteristic
73	C₂²×D₁₀₁	C₄⋊C₄	1	1	yes	characteristic
74	C₂²×D₁₀₁	C₄⋊C₄	1	1	yes	characteristic
75	C₂×C₄×C₁₀₁	—	2	2	no	—
76	C₂×C₄×C₁₀₁	C₄²	1	1	yes	characteristic
77	C₂×C₄×C₁₀₁	C₄²	1	1	yes	characteristic
78	C₂×C₄×C₁₀₁	C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₄×C₁₀₁	C₂²⋊C₄	1	1	yes	characteristic
80	C₂×C₄×C₁₀₁	C₄⋊C₄	1	1	yes	characteristic
81	C₂×C₄×C₁₀₁	C₄⋊C₄	1	1	yes	characteristic
82	C₂×Dic₁₀₁	—	2	2	no	—
83	C₂×Dic₁₀₁	C₂²×C₄	1	1	yes	characteristic
84	C₂×Dic₁₀₁	C₂²×C₄	1	1	yes	characteristic
85	C₂×Dic₁₀₁	C₂×D₄	1	1	yes	characteristic
86	C₂×Dic₁₀₁	C₂×D₄	1	1	yes	characteristic
87	C₂×Dic₁₀₁	C₂×D₄	1	1	yes	characteristic
88	C₂×Dic₁₀₁	C₂×Q₈	1	1	yes	characteristic
89	C₄×D₁₀₁	—	4	4	no	—
90	C₄×D₁₀₁	C₂²⋊C₄	1	1	yes	characteristic
91	C₄×D₁₀₁	C₂²⋊C₄	1	1	yes	characteristic
92	C₄×D₁₀₁	C₂²⋊C₄	1	1	yes	characteristic
93	C₄×D₁₀₁	C₂²⋊C₄	1	1	yes	characteristic
94	C₄×D₁₀₁	C₂²⋊C₄	1	1	yes	characteristic
95	C₄×D₁₀₁	C₂²⋊C₄	1	1	yes	characteristic
96	C₄×D₁₀₁	C₂²⋊C₄	1	1	yes	characteristic
97	C₄×D₁₀₁	C₂²⋊C₄	1	1	yes	characteristic
98	C₄×D₁₀₁	C₄⋊C₄	1	1	yes	characteristic
99	C₄×D₁₀₁	C₄⋊C₄	1	1	yes	characteristic
100	C₄×D₁₀₁	C₄⋊C₄	1	1	yes	characteristic
101	C₄×D₁₀₁	C₄⋊C₄	1	1	yes	characteristic
102	C₄×D₁₀₁	C₄⋊C₄	1	1	yes	characteristic
103	C₄×D₁₀₁	C₄⋊C₄	1	1	yes	characteristic
104	C₄×D₁₀₁	C₄⋊C₄	1	1	yes	characteristic
105	C₄×D₁₀₁	C₄⋊C₄	1	1	yes	characteristic
106	#11717581	—	2	4	no	—
107	#11717581	—	2	4	no	—
108	C₂²×C₁₀₁	—	4	1	no	—
109	C₂²×C₁₀₁	C₂×C₄²	1	1	yes	characteristic
110	C₂²×C₁₀₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
111	C₂²×C₁₀₁	C₂×C₄⋊C₄	1	1	yes	characteristic
112	C₂×D₁₀₁	—	2	2	no	—
113	C₂×D₁₀₁	—	2	2	no	—
114	C₂×D₁₀₁	—	2	2	no	—
115	C₂×D₁₀₁	—	4	1	no	—
116	C₂×D₁₀₁	—	4	1	no	—
117	C₂×D₁₀₁	—	4	1	no	—
118	C₂×D₁₀₁	#95	1	1	yes	characteristic
119	C₂×D₁₀₁	#95	1	1	yes	characteristic
120	C₂×D₁₀₁	#95	1	1	yes	characteristic
121	C₂×D₁₀₁	#95	1	1	yes	characteristic
122	C₄×C₁₀₁	—	4	2	no	—
123	C₄×C₁₀₁	#95	1	1	yes	characteristic
124	C₄×C₁₀₁	#95	1	1	yes	characteristic
125	C₄×C₁₀₁	#95	1	1	yes	characteristic
126	C₄×C₁₀₁	#95	1	1	yes	characteristic
127	Dic₁₀₁	—	4	2	no	—
128	Dic₁₀₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
129	Dic₁₀₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
130	Dic₁₀₁	C₂×C₄⋊C₄	1	1	yes	characteristic
131	Dic₁₀₁	C₂×C₄⋊C₄	1	1	yes	characteristic
132	C₂×C₁₀₁	—	2	1	no	—
133	C₂×C₁₀₁	—	2	1	no	—
134	C₂×C₁₀₁	—	2	1	no	—
135	C₂×C₁₀₁	C₂×#95	1	1	yes	characteristic
136	D₁₀₁	—	2	1	no	—
137	D₁₀₁	—	2	1	no	—
138	D₁₀₁	—	2	1	no	—
139	D₁₀₁	#344	1	2	yes	complemented
140	C₂×#344	—	101	1	no	maximal, Sylow
141	C₁₀₁	C₂×#344	1	1	yes	characteristic, complemented, Sylow
142	C₂²×M₁₆	—	101	1	no	—
143	C₂²×M₁₆	—	101	1	no	—
144	C₂×#117	—	101	1	no	—
145	#344	—	101	4	no	—
146	C₂³×C₄	—	101	1	no	—
147	C₂×M₁₆	—	101	2	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	—	202	1	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	C₂²×C₈	—	101	2	no	—
160	C₂²×C₈	—	202	1	no	—
161	#117	—	101	1	no	—
162	#117	—	101	2	no	—
163	#117	—	202	1	no	—
164	#117	—	202	1	no	—
165	#117	#87938	1	1	yes	characteristic
166	C₂×C₄⋊C₄	—	202	1	no	—
167	C₂×C₂²⋊C₄	—	101	2	no	—
168	C₂×C₄²	—	202	1	no	—
169	C₂⁴	—	101	1	no	—
170	C₂²×C₄	—	101	1	no	—
171	C₂²×C₄	—	101	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	—	202	2	no	—
184	C₂²×C₄	C₂×#87938	1	1	yes	characteristic
185	M₁₆	—	101	4	no	—
186	M₁₆	—	202	2	no	—
187	M₁₆	—	202	2	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	C₂×C₈	—	101	4	no	—
194	C₂×C₈	—	202	2	no	—
195	C₂×C₈	—	202	2	no	—
196	C₂×C₈	—	202	2	no	—
197	C₂×C₈	—	202	2	no	—
198	C₂×C₈	—	202	2	no	—
199	C₄⋊C₄	—	2	1	no	—
200	C₄⋊C₄	—	202	2	no	—
201	C₄⋊C₄	—	202	1	no	—
202	C₂²⋊C₄	—	101	2	no	—
203	C₂²⋊C₄	—	202	2	no	—
204	C₂²⋊C₄	C₂×#87938	1	2	yes	—
205	C₄²	—	2	1	no	—
206	C₄²	—	202	2	no	—
207	C₄²	—	202	1	no	—
208	C₂³	—	101	1	no	—
209	C₂³	—	101	1	no	—
210	C₂³	—	101	1	no	—
211	C₂³	—	101	1	no	—
212	C₂³	—	101	1	no	—
213	C₂³	—	101	1	no	—
214	C₂³	—	404	2	no	—
215	C₂³	C₂²×#87938	1	1	yes	characteristic
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	101	1	no	—
218	C₂×C₄	—	101	1	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	—	404	4	no	—
240	C₂×C₄	—	202	2	no	—
241	C₂×C₄	C₄×#87938	1	1	yes	characteristic
242	C₂×C₄	C₄×#87938	1	1	yes	characteristic
243	C₂×C₄	C₂²⋊#87938	1	1	yes	characteristic
244	C₂×C₄	C₂²⋊#87938	1	1	yes	characteristic
245	C₂×C₄	C₄⋊#87938	1	1	yes	characteristic
246	C₂×C₄	C₄⋊#87938	1	1	yes	characteristic
247	C₈	—	202	4	no	—
248	C₈	—	202	4	no	—
249	C₂²	—	4	1	no	—
250	C₂²	—	101	1	no	—
251	C₂²	—	101	1	no	—
252	C₂²	—	101	1	no	—
253	C₂²	—	101	1	no	—
254	C₂²	—	404	1	no	—
255	C₂²	—	404	1	no	—
256	C₂²	—	404	1	no	—
257	C₂²	—	202	2	no	—
258	C₂²	—	202	2	no	—
259	C₂²	—	202	2	no	—
260	C₂²	C₂×C₄×#87938	1	1	yes	characteristic
261	C₂²	C₂×C₂²⋊#87938	1	1	yes	characteristic
262	C₂²	C₂×C₄⋊#87938	1	1	yes	characteristic
263	C₄	—	4	2	no	—
264	C₄	—	101	1	no	—
265	C₄	—	101	1	no	—
266	C₄	—	101	1	no	—
267	C₄	—	101	1	no	—
268	C₄	—	404	2	no	—
269	C₄	(C₂×#87938)⋊C₄	1	1	yes	characteristic
270	C₄	(C₂×#87938)⋊C₄	1	1	yes	characteristic
271	C₄	(C₂×#87938)⋊C₄	1	1	yes	characteristic
272	C₄	(C₂×#87938)⋊C₄	1	1	yes	characteristic
273	C₂	—	2	1	no	—
274	C₂	—	2	1	no	—
275	C₂	—	2	1	no	—
276	C₂	—	101	2	no	—
277	C₂	—	202	1	no	—
278	C₂	—	202	1	no	—
279	C₂	—	202	1	no	—
280	C₂	C₂×(C₂×#87938)⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₄	(C₂×#87938)⋊C₄
commutator	C₂²×C₁₀₁	C₂×C₄²
Frattini	C₂³	C₂²×#87938
Fitting	#117×C₁₀₁	C₄
socle	C₂×C₁₀₁	C₂×#95
prime radical	C₂³×D₁₀₁	C₂×C₄
cosocle	C₂²×Dic₁₀₁	C₂³
hypercenter	#117	#87938
hypocenter	C₁₀₁	C₂×#344
LCS	G ▷ C₂²×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #117

Complements

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

N	H ≅ G/N	type
3. #20266058	C₂	semidirect
6. C₂×#13108084	C₄	semidirect
13. D₁₀₁⋊M₁₆	C₄	semidirect
14. D₁₀₁⋊M₁₆	C₄	semidirect
18. C₂×D₁₀₁⋊C₈	C₄	semidirect
35. #13108083	C₂×C₄	semidirect
41. C₂×#11717581	C₂×C₄	semidirect
139. D₁₀₁	#344	semidirect
141. C₁₀₁	C₂×#344	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₁₀₁⋊M₁₆	2
#20266058	2
#117×D₁₀₁	2
C₂×#344	101