Subgroups — #27187877

Order 12928.

492 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15692343	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#15692342	C₂	1	1	yes	maximal, characteristic, complemented
3	#20266210	C₂	1	2	yes	maximal, complemented
4	#20266209	C₂	1	2	yes	maximal, complemented
5	#20266206	C₂	1	2	yes	maximal, complemented
6	#20266204	C₂	1	2	yes	maximal, complemented
7	D₁₀₁⋊(C₄⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
8	#20266152	C₂	1	1	yes	maximal, characteristic, complemented
9	#20266151	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×#13108084	C₂	1	1	yes	maximal, characteristic, complemented
11	#20265726	C₂	1	1	yes	maximal, characteristic
12	#15692343	—	2	4	no	—
13	#15692342	—	2	4	no	—
14	C₂×#13108084	C₂²	1	2	yes	—
15	C₂×#13108084	C₂²	1	1	yes	characteristic
16	C₂×#13108084	C₂²	1	1	yes	characteristic
17	C₂×#13108083	C₂²	1	2	yes	—
18	C₂×D₁₀₁⋊C₈	C₂²	1	1	yes	characteristic
19	C₂×D₁₀₁⋊C₈	C₂²	1	1	yes	characteristic
20	C₂×D₁₀₁⋊C₈	C₂²	1	2	yes	—
21	C₂²×#11717581	C₂²	1	2	yes	—
22	#15692322	C₂²	1	1	yes	characteristic
23	#15692322	C₂²	1	1	yes	characteristic
24	C₄⋊#11717581	C₂²	1	1	yes	characteristic
25	C₄⋊#11717581	C₂²	1	1	yes	characteristic
26	#15692312	C₂²	1	2	yes	—
27	#15692312	C₂²	1	2	yes	—
28	C₂²⋊#11717581	C₂²	1	2	yes	—
29	C₂²⋊#11717581	C₂²	1	2	yes	—
30	#15692310	C₂²	1	2	yes	—
31	C₄×#11717581	C₂²	1	2	yes	—
32	C₂×#13108075	C₂²	1	1	yes	characteristic
33	C₂×#13108073	C₂²	1	1	yes	characteristic
34	#15692253	C₄	1	1	yes	characteristic
35	#15692251	C₄	1	1	yes	characteristic
36	#15692243	C₄	1	1	yes	characteristic
37	#15692228	C₄	1	2	yes	—
38	#15692228	C₄	1	2	yes	—
39	#15692215	C₂²	1	2	yes	—
40	#15692215	C₂²	1	2	yes	—
41	C₄²×D₁₀₁	C₂²	1	1	yes	characteristic
42	#124×C₁₀₁	C₄	1	1	yes	characteristic
43	#13108084	—	2	2	no	—
44	#13108084	—	2	4	no	—
45	#13108084	D₄	1	4	yes	complemented
46	#13108083	—	2	4	no	—
47	D₁₀₁⋊C₈	—	2	2	no	—
48	D₁₀₁⋊C₈	—	2	2	no	—
49	D₁₀₁⋊C₈	—	2	4	no	—
50	C₂×#11717581	—	2	4	no	—
51	C₂×#11717581	C₂³	1	2	yes	—
52	C₂×#11717581	C₂³	1	1	yes	characteristic
53	C₂×#11717581	C₂³	1	1	yes	characteristic
54	C₂×#11717581	C₂³	1	1	yes	characteristic
55	C₂×#11717581	C₂³	1	1	yes	characteristic
56	C₂×#11717581	C₂³	1	2	yes	—
57	#13108075	—	2	2	no	—
58	#13108075	—	2	2	no	—
59	#13108073	—	2	2	no	—
60	#13108073	—	2	2	no	—
61	C₂×Dic₂₀₂	C₂×C₄	1	1	yes	characteristic
62	C₂×D₄₀₄	C₂×C₄	1	2	yes	—
63	C₂×D₄₀₄	C₂×C₄	1	1	yes	characteristic
64	C₂×C₂²⋊D₁₀₁	C₂×C₄	1	2	yes	—
65	C₂×C₂²⋊D₁₀₁	C₂×C₄	1	2	yes	—
66	C₂×C₂²⋊D₁₀₁	C₂×C₄	1	2	yes	—
67	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
68	C₂×C₄×D₁₀₁	C₂×C₄	1	1	yes	characteristic
69	C₂×C₄×D₁₀₁	C₂×C₄	1	1	yes	characteristic
70	C₂×C₄×D₁₀₁	C₂³	1	2	yes	—
71	C₂×C₄×D₁₀₁	C₂³	1	1	yes	characteristic
72	#13108053	C₂×C₄	1	2	yes	—
73	#13108053	C₂×C₄	1	2	yes	—
74	#13108051	C₂×C₄	1	2	yes	—
75	#13108051	C₂×C₄	1	2	yes	—
76	C₄×Dic₁₀₁	C₂×C₄	1	1	yes	characteristic
77	C₄×Dic₁₀₁	C₂³	1	1	yes	characteristic
78	C₄×Dic₁₀₁	C₂³	1	1	yes	characteristic
79	C₂×Q₈×C₁₀₁	C₂×C₄	1	1	yes	characteristic
80	C₂×D₄×C₁₀₁	C₂×C₄	1	1	yes	characteristic
81	C₂²⋊C₄×C₁₀₁	C₂×C₄	1	2	yes	—
82	C₂²⋊C₄×C₁₀₁	C₂×C₄	1	2	yes	—
83	C₄²×C₁₀₁	C₂×C₄	1	1	yes	characteristic
84	C₂³×C₁₀₁	C₂²×C₄	1	2	yes	—
85	C₂²×D₁₀₁	C₂²×C₄	1	2	yes	—
86	C₂²×D₁₀₁	C₂²×C₄	1	1	yes	characteristic
87	Q₈×C₁₀₁	—	2	2	no	—
88	D₄×C₁₀₁	—	2	2	no	—
89	C₂×C₄×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
90	C₂×C₄×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₄×C₁₀₁	C₂²×C₄	1	2	yes	—
92	C₂×C₄×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
93	C₂²⋊D₁₀₁	—	2	4	no	—
94	C₂²⋊D₁₀₁	—	2	4	no	—
95	C₂²⋊D₁₀₁	—	4	2	no	—
96	C₂×Dic₁₀₁	—	2	2	no	—
97	C₂×Dic₁₀₁	—	2	2	no	—
98	C₂×Dic₁₀₁	C₂²×C₄	1	1	yes	characteristic
99	C₂×Dic₁₀₁	C₂²×C₄	1	2	yes	—
100	C₂×Dic₁₀₁	C₂²×C₄	1	1	yes	characteristic
101	C₂×Dic₁₀₁	C₂⁴	1	1	yes	characteristic
102	D₄₀₄	—	2	4	no	—
103	D₄₀₄	—	2	2	no	—
104	C₄×D₁₀₁	—	2	2	no	—
105	C₄×D₁₀₁	—	2	2	no	—
106	C₄×D₁₀₁	—	2	2	no	—
107	C₄×D₁₀₁	—	4	1	no	—
108	C₄×D₁₀₁	C₂×D₄	1	2	yes	—
109	C₄×D₁₀₁	#41	1	2	yes	—
110	Dic₂₀₂	—	2	2	no	—
111	#11717581	—	2	1	no	—
112	#11717581	—	2	1	no	—
113	#11717581	—	2	1	no	—
114	#11717581	—	2	1	no	—
115	#11717581	—	2	2	no	—
116	#11717581	C₂×D₄	1	4	yes	—
117	C₂²×C₁₀₁	—	2	2	no	—
118	C₂²×C₁₀₁	—	2	2	no	—
119	C₂²×C₁₀₁	—	2	2	no	—
120	C₂²×C₁₀₁	C₂³×C₄	1	1	yes	characteristic
121	C₂×D₁₀₁	—	2	1	no	—
122	C₂×D₁₀₁	—	2	1	no	—
123	C₂×D₁₀₁	—	2	2	no	—
124	C₂×D₁₀₁	—	2	2	no	—
125	C₂×D₁₀₁	—	2	2	no	—
126	C₂×D₁₀₁	C₄×D₄	1	2	yes	—
127	C₄×C₁₀₁	—	2	2	no	—
128	C₄×C₁₀₁	—	2	1	no	—
129	C₄×C₁₀₁	—	2	1	no	—
130	C₄×C₁₀₁	C₄×D₄	1	2	yes	—
131	Dic₁₀₁	—	2	2	no	—
132	Dic₁₀₁	—	2	1	no	—
133	Dic₁₀₁	#131	1	2	yes	complemented
134	Dic₁₀₁	C₂²×D₄	1	1	yes	characteristic
135	Dic₁₀₁	C₂×#41	1	1	yes	characteristic
136	C₂×C₁₀₁	—	4	2	no	—
137	C₂×C₁₀₁	C₂×C₄×D₄	1	1	yes	characteristic
138	C₂×C₁₀₁	C₂×#131	1	1	yes	characteristic
139	C₂×C₁₀₁	#568	1	1	yes	characteristic
140	D₁₀₁	—	2	2	no	—
141	D₁₀₁	—	4	2	no	—
142	#2935	—	101	1	no	maximal, Sylow
143	C₁₀₁	#2935	1	1	yes	characteristic, complemented, Sylow
144	C₂×#131	—	101	1	no	—
145	C₂×#131	—	101	1	no	—
146	#532	—	101	1	no	—
147	#436	—	101	2	no	—
148	#435	—	101	2	no	—
149	#435	—	101	2	no	—
150	C₈×D₄	—	101	2	no	—
151	C₂×C₄⋊C₈	—	101	1	no	—
152	#408	—	101	1	no	—
153	#408	—	101	1	no	—
154	C₄×M₁₆	—	101	1	no	—
155	C₂×#41	—	101	1	no	—
156	C₂×#41	—	101	1	no	—
157	#131	—	202	4	no	—
158	#131	—	202	4	no	—
159	C₂×M₁₆	—	101	1	no	—
160	C₂×M₁₆	—	101	2	no	—
161	C₂×M₁₆	—	101	2	no	—
162	C₂×M₁₆	—	101	1	no	—
163	C₂²×C₈	—	101	2	no	—
164	C₂²×C₈	—	101	1	no	—
165	C₂²×C₈	—	101	1	no	—
166	C₂²×C₈	—	101	2	no	—
167	C₄⋊Q₈	—	101	1	no	—
168	#127	—	101	1	no	—
169	#124	—	101	1	no	—
170	#124	#87938	1	1	yes	characteristic
171	#121	—	101	2	no	—
172	#121	—	101	2	no	—
173	C₄×D₄	—	101	2	no	—
174	C₄×D₄	—	101	2	no	—
175	C₂×C₄²	—	101	1	no	—
176	C₄⋊C₈	—	101	1	no	—
177	C₄⋊C₈	—	101	1	no	—
178	C₄⋊C₈	—	101	1	no	—
179	C₄⋊C₈	—	101	1	no	—
180	C₂²⋊C₈	—	101	2	no	—
181	C₂²⋊C₈	—	101	2	no	—
182	C₂²⋊C₈	—	101	2	no	—
183	C₂²⋊C₈	—	101	2	no	—
184	#97	—	101	2	no	—
185	C₄×C₈	—	101	2	no	—
186	#41	—	202	2	no	—
187	#41	—	202	2	no	—
188	#41	—	202	2	no	—
189	#41	—	202	2	no	—
190	C₂×Q₈	—	101	1	no	—
191	C₂×Q₈	C₂×#87938	1	1	yes	characteristic
192	C₂×D₄	—	101	2	no	—
193	C₂×D₄	—	101	2	no	—
194	C₂×D₄	—	101	2	no	—
195	C₂×D₄	—	101	2	no	—
196	C₂×D₄	—	101	1	no	—
197	C₂×D₄	C₂×#87938	1	1	yes	characteristic
198	C₂²×C₄	—	101	1	no	—
199	C₂²×C₄	—	101	2	no	—
200	C₂²×C₄	—	101	1	no	—
201	C₂²×C₄	—	101	2	no	—
202	C₂²×C₄	—	101	1	no	—
203	M₁₆	—	101	4	no	—
204	M₁₆	—	202	4	no	—
205	M₁₆	—	202	4	no	—
206	M₁₆	—	202	2	no	—
207	C₂×C₈	—	101	2	no	—
208	C₂×C₈	—	101	2	no	—
209	C₂×C₈	—	101	1	no	—
210	C₂×C₈	—	101	1	no	—
211	C₂×C₈	—	101	1	no	—
212	C₂×C₈	—	101	1	no	—
213	C₂×C₈	—	202	4	no	—
214	C₂×C₈	—	202	4	no	—
215	C₂×C₈	—	202	2	no	—
216	C₂×C₈	—	202	2	no	—
217	C₄⋊C₄	—	101	2	no	—
218	C₄⋊C₄	—	101	2	no	—
219	C₂²⋊C₄	—	101	2	no	—
220	C₂²⋊C₄	—	101	2	no	—
221	C₂²⋊C₄	C₂×#87938	1	2	yes	—
222	C₂²⋊C₄	C₂×#87938	1	2	yes	—
223	C₄²	—	101	1	no	—
224	C₄²	—	101	1	no	—
225	C₄²	—	101	1	no	—
226	C₄²	C₂×#87938	1	1	yes	characteristic
227	C₂³	—	101	1	no	—
228	C₂³	—	101	2	no	—
229	C₂³	C₂²×#87938	1	2	yes	—
230	Q₈	—	2	2	no	—
231	Q₈	—	202	2	no	—
232	D₄	—	2	2	no	—
233	D₄	—	404	2	no	—
234	D₄	—	202	4	no	—
235	D₄	—	202	4	no	—
236	D₄	—	202	4	no	—
237	D₄	—	202	2	no	—
238	C₂×C₄	—	101	1	no	—
239	C₂×C₄	—	101	1	no	—
240	C₂×C₄	—	101	2	no	—
241	C₂×C₄	—	101	2	no	—
242	C₂×C₄	—	101	1	no	—
243	C₂×C₄	—	101	2	no	—
244	C₂×C₄	—	404	1	no	—
245	C₂×C₄	—	202	2	no	—
246	C₂×C₄	—	202	2	no	—
247	C₂×C₄	—	202	2	no	—
248	C₂×C₄	—	202	2	no	—
249	C₂×C₄	—	202	2	no	—
250	C₂×C₄	C₂²×#87938	1	1	yes	characteristic
251	C₂×C₄	C₂²×#87938	1	1	yes	characteristic
252	C₂×C₄	C₂²×#87938	1	2	yes	—
253	C₂×C₄	C₂²×#87938	1	1	yes	characteristic
254	C₈	—	101	4	no	—
255	C₈	—	202	1	no	—
256	C₈	—	202	1	no	—
257	C₈	—	202	1	no	—
258	C₈	—	202	1	no	—
259	C₈	—	202	2	no	—
260	C₂²	—	2	2	no	—
261	C₂²	—	2	2	no	—
262	C₂²	—	2	2	no	—
263	C₂²	—	101	2	no	—
264	C₂²	—	202	2	no	—
265	C₂²	—	202	2	no	—
266	C₂²	—	202	2	no	—
267	C₂²	—	202	1	no	—
268	C₂²	—	202	1	no	—
269	C₂²	C₂³×#87938	1	1	yes	characteristic
270	C₄	—	2	2	no	—
271	C₄	—	2	1	no	—
272	C₄	—	2	1	no	—
273	C₄	—	101	1	no	—
274	C₄	—	101	1	no	—
275	C₄	—	101	2	no	—
276	C₄	—	202	2	no	—
277	C₄	—	202	1	no	—
278	C₄	D₄×#87938	1	2	yes	—
279	C₂	—	4	2	no	—
280	C₂	—	404	2	no	—
281	C₂	—	202	2	no	—
282	C₂	C₂×D₄×#87938	1	1	yes	characteristic
283	C₂	D₁₀₁⋊#131	1	1	yes	characteristic
284	C₂	#20266256	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×#87938
commutator, socle	C₂²×C₁₀₁	C₂³×C₄
Fitting	#124×C₁₀₁	C₄
prime radical	#20265726	C₂
cosocle	C₂×Dic₁₀₁	C₂⁴
hypercenter	#124	#87938
hypocenter	C₁₀₁	#2935
LCS	G ▷ C₂²×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂² ◁ #124

Complements

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

N	H ≅ G/N	type
1. C₂×#15692343	C₂	semidirect
2. C₂×#15692342	C₂	semidirect
3. #20266210	C₂	semidirect
4. #20266209	C₂	semidirect
5. #20266206	C₂	semidirect
6. #20266204	C₂	semidirect
7. D₁₀₁⋊(C₄⋊C₈)	C₂	semidirect
8. #20266152	C₂	semidirect
9. #20266151	C₂	semidirect
10. C₄×#13108084	C₂	semidirect
45. #13108084	D₄	semidirect
133. Dic₁₀₁	#131	semidirect
143. C₁₀₁	#2935	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15692343	2
C₂×#15692342	2
#20266210	2
#20266209	2
#20266206	2
#20266204	2
D₁₀₁⋊(C₄⋊C₈)	2
#20266152	2
#20266151	2
C₄×#13108084	2
#20265726	2
#2935	101