Subgroups — #27187874

Order 12928.

492 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15692342	C₂	1	2	yes	maximal, complemented
2	#20266212	C₂	1	1	yes	maximal, characteristic, complemented
3	#20266211	C₂	1	1	yes	maximal, characteristic, complemented
4	#20266209	C₂	1	2	yes	maximal, complemented
5	#20266206	C₂	1	2	yes	maximal, complemented
6	#20266205	C₂	1	1	yes	maximal, characteristic, complemented
7	D₄×#11717581	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄⋊#11717581	C₂	1	1	yes	maximal, characteristic, complemented
9	#20266150	C₂	1	2	yes	maximal, complemented
10	C₄×#13108083	C₂	1	1	yes	maximal, characteristic, complemented
11	#20265722	C₂	1	1	yes	maximal, characteristic
12	#15692342	—	2	8	no	—
13	C₂×#13108084	C₂²	1	2	yes	—
14	C₂×#13108083	C₂²	1	2	yes	—
15	C₂×#13108083	C₂²	1	1	yes	characteristic
16	C₂×#13108083	C₂²	1	1	yes	characteristic
17	C₂×D₁₀₁⋊C₈	C₂²	1	2	yes	—
18	C₂²×#11717581	C₂²	1	2	yes	—
19	C₂²×#11717581	C₂²	1	2	yes	—
20	C₄⋊#11717581	C₂²	1	1	yes	characteristic
21	C₄⋊#11717581	C₂²	1	2	yes	—
22	C₄⋊#11717581	C₂²	1	1	yes	characteristic
23	#15692312	C₂²	1	2	yes	—
24	#15692312	C₂²	1	2	yes	—
25	C₂²⋊#11717581	C₂²	1	2	yes	—
26	C₂²⋊#11717581	C₂²	1	2	yes	—
27	#15692310	C₂²	1	1	yes	characteristic
28	#15692310	C₂²	1	1	yes	characteristic
29	C₄×#11717581	C₂²	1	1	yes	characteristic
30	C₄×#11717581	C₂²	1	1	yes	characteristic
31	C₂×#13108073	C₂²	1	2	yes	—
32	#15692254	C₄	1	1	yes	characteristic
33	#15692251	C₄	1	1	yes	characteristic
34	#15692241	C₄	1	2	yes	—
35	#15692229	C₄	1	1	yes	characteristic
36	#15692227	C₄	1	2	yes	—
37	D₄×Dic₁₀₁	C₂²	1	1	yes	characteristic
38	#15692217	C₂²	1	1	yes	characteristic
39	#15692215	C₂²	1	2	yes	—
40	C₂×C₄×Dic₁₀₁	C₂²	1	1	yes	characteristic
41	#121×C₁₀₁	C₄	1	1	yes	characteristic
42	#13108084	—	2	4	no	—
43	#13108083	—	2	2	no	—
44	#13108083	—	2	4	no	—
45	#13108083	D₄	1	4	yes	complemented
46	D₁₀₁⋊C₈	—	2	4	no	—
47	C₂×#11717581	—	2	4	no	—
48	C₂×#11717581	—	2	4	no	—
49	C₂×#11717581	C₂³	1	1	yes	characteristic
50	C₂×#11717581	C₂³	1	1	yes	characteristic
51	C₂×#11717581	C₂³	1	2	yes	—
52	C₂×#11717581	C₂³	1	2	yes	—
53	C₂×#11717581	C₂³	1	1	yes	characteristic
54	C₂×#11717581	C₂³	1	1	yes	characteristic
55	#13108073	—	2	4	no	—
56	#13108073	—	2	4	no	—
57	C₂×Dic₂₀₂	C₂×C₄	1	2	yes	—
58	C₂×D₄₀₄	C₂×C₄	1	1	yes	characteristic
59	C₂×C₂²⋊D₁₀₁	C₂×C₄	1	2	yes	—
60	C₂×C₂²⋊D₁₀₁	C₂×C₄	1	2	yes	—
61	C₂×C₂²⋊D₁₀₁	C₂×C₄	1	2	yes	—
62	C₂²×Dic₁₀₁	C₂×C₄	1	1	yes	characteristic
63	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
64	C₂²×Dic₁₀₁	C₂³	1	1	yes	characteristic
65	C₂×C₄×D₁₀₁	C₂³	1	2	yes	—
66	C₄⋊Dic₁₀₁	C₂×C₄	1	1	yes	characteristic
67	#13108053	C₂×C₄	1	2	yes	—
68	C₂²⋊Dic₁₀₁	C₂×C₄	1	2	yes	—
69	#13108051	C₂×C₄	1	2	yes	—
70	#13108051	C₂×C₄	1	2	yes	—
71	C₄×Dic₁₀₁	C₂×C₄	1	1	yes	characteristic
72	C₄×Dic₁₀₁	C₂×C₄	1	1	yes	characteristic
73	C₄×Dic₁₀₁	C₂³	1	1	yes	characteristic
74	C₄×Dic₁₀₁	C₂³	1	1	yes	characteristic
75	C₂×D₄×C₁₀₁	C₂×C₄	1	2	yes	—
76	C₂×D₄×C₁₀₁	C₂×C₄	1	1	yes	characteristic
77	C₂²×C₄×C₁₀₁	C₂×C₄	1	1	yes	characteristic
78	C₄⋊C₄×C₁₀₁	C₂×C₄	1	1	yes	characteristic
79	C₂²⋊C₄×C₁₀₁	C₂×C₄	1	2	yes	—
80	C₂³×C₁₀₁	C₂²×C₄	1	2	yes	—
81	C₂³×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
82	C₂²×D₁₀₁	C₂²×C₄	1	2	yes	—
83	D₄×C₁₀₁	—	2	4	no	—
84	D₄×C₁₀₁	—	2	2	no	—
85	C₂×C₄×C₁₀₁	—	2	2	no	—
86	C₂×C₄×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₄×C₁₀₁	C₂²×C₄	1	2	yes	—
88	C₂×C₄×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
89	C₂²⋊D₁₀₁	—	2	4	no	—
90	C₂²⋊D₁₀₁	—	2	4	no	—
91	C₂²⋊D₁₀₁	—	4	2	no	—
92	C₂×Dic₁₀₁	—	2	2	no	—
93	C₂×Dic₁₀₁	—	2	2	no	—
94	C₂×Dic₁₀₁	—	4	1	no	—
95	C₂×Dic₁₀₁	C₂²×C₄	1	1	yes	characteristic
96	C₂×Dic₁₀₁	C₂²×C₄	1	1	yes	characteristic
97	C₂×Dic₁₀₁	C₂²×C₄	1	2	yes	—
98	C₂×Dic₁₀₁	C₂²×C₄	1	1	yes	characteristic
99	C₂×Dic₁₀₁	C₂×D₄	1	2	yes	—
100	C₂×Dic₁₀₁	#41	1	2	yes	—
101	C₂×Dic₁₀₁	C₂⁴	1	1	yes	characteristic
102	D₄₀₄	—	2	2	no	—
103	C₄×D₁₀₁	—	2	2	no	—
104	C₄×D₁₀₁	—	2	2	no	—
105	Dic₂₀₂	—	2	4	no	—
106	#11717581	—	2	2	no	—
107	#11717581	—	2	2	no	—
108	#11717581	—	2	1	no	—
109	#11717581	—	2	1	no	—
110	#11717581	C₂×D₄	1	2	yes	—
111	#11717581	C₂×D₄	1	2	yes	—
112	C₂²×C₁₀₁	—	2	1	no	—
113	C₂²×C₁₀₁	—	2	1	no	—
114	C₂²×C₁₀₁	—	2	2	no	—
115	C₂²×C₁₀₁	—	2	2	no	—
116	C₂²×C₁₀₁	—	2	2	no	—
117	C₂²×C₁₀₁	C₄×D₄	1	2	yes	—
118	C₂²×C₁₀₁	C₂³×C₄	1	1	yes	characteristic
119	C₂×D₁₀₁	—	2	2	no	—
120	C₂×D₁₀₁	—	2	2	no	—
121	C₂×D₁₀₁	—	2	2	no	—
122	C₄×C₁₀₁	—	2	2	no	—
123	C₄×C₁₀₁	—	2	1	no	—
124	C₄×C₁₀₁	#131	1	2	yes	complemented
125	Dic₁₀₁	—	2	2	no	—
126	Dic₁₀₁	—	2	1	no	—
127	Dic₁₀₁	—	2	1	no	—
128	Dic₁₀₁	C₄×D₄	1	2	yes	—
129	Dic₁₀₁	C₂²×D₄	1	1	yes	characteristic
130	Dic₁₀₁	C₂×#41	1	1	yes	characteristic
131	C₂×C₁₀₁	—	2	2	no	—
132	C₂×C₁₀₁	—	4	2	no	—
133	C₂×C₁₀₁	C₂×C₄×D₄	1	1	yes	characteristic
134	C₂×C₁₀₁	C₂×#131	1	1	yes	characteristic
135	C₂×C₁₀₁	#568	1	1	yes	characteristic
136	D₁₀₁	—	4	2	no	—
137	#2935	—	101	1	no	maximal, Sylow
138	C₁₀₁	#2935	1	1	yes	characteristic, complemented, Sylow
139	C₂×#131	—	101	2	no	—
140	#532	—	101	1	no	—
141	#436	—	101	1	no	—
142	#436	—	101	1	no	—
143	#435	—	101	2	no	—
144	#435	—	101	2	no	—
145	C₈×D₄	—	101	1	no	—
146	C₈×D₄	—	101	1	no	—
147	C₂×C₄⋊C₈	—	101	1	no	—
148	#408	—	101	2	no	—
149	C₄×M₁₆	—	101	1	no	—
150	C₂×#41	—	101	2	no	—
151	#131	—	202	8	no	—
152	C₂×M₁₆	—	101	1	no	—
153	C₂×M₁₆	—	101	2	no	—
154	C₂×M₁₆	—	101	1	no	—
155	C₂×M₁₆	—	101	2	no	—
156	C₂²×C₈	—	101	2	no	—
157	C₂²×C₈	—	101	2	no	—
158	C₂²×C₈	—	101	2	no	—
159	C₄⋊Q₈	—	101	1	no	—
160	#127	—	101	1	no	—
161	#124	—	101	2	no	—
162	#121	—	101	2	no	—
163	#121	—	101	1	no	—
164	#121	#87938	1	1	yes	characteristic
165	C₄×D₄	—	101	1	no	—
166	C₄×D₄	—	101	1	no	—
167	C₄×D₄	—	101	2	no	—
168	C₂×C₄²	—	101	1	no	—
169	C₄⋊C₈	—	101	1	no	—
170	C₄⋊C₈	—	101	2	no	—
171	C₄⋊C₈	—	101	1	no	—
172	C₂²⋊C₈	—	101	2	no	—
173	C₂²⋊C₈	—	101	2	no	—
174	C₂²⋊C₈	—	101	2	no	—
175	C₂²⋊C₈	—	101	2	no	—
176	#97	—	101	1	no	—
177	#97	—	101	1	no	—
178	C₄×C₈	—	101	1	no	—
179	C₄×C₈	—	101	1	no	—
180	#41	—	202	4	no	—
181	#41	—	202	4	no	—
182	C₂×Q₈	—	101	2	no	—
183	C₂×D₄	—	101	2	no	—
184	C₂×D₄	—	101	2	no	—
185	C₂×D₄	—	101	2	no	—
186	C₂×D₄	—	101	1	no	—
187	C₂×D₄	C₂×#87938	1	1	yes	characteristic
188	C₂×D₄	C₂×#87938	1	2	yes	—
189	C₂²×C₄	—	101	1	no	—
190	C₂²×C₄	—	101	1	no	—
191	C₂²×C₄	—	101	2	no	—
192	C₂²×C₄	—	101	2	no	—
193	C₂²×C₄	C₂×#87938	1	1	yes	characteristic
194	M₁₆	—	101	4	no	—
195	M₁₆	—	202	4	no	—
196	M₁₆	—	202	4	no	—
197	M₁₆	—	202	2	no	—
198	C₂×C₈	—	101	1	no	—
199	C₂×C₈	—	101	1	no	—
200	C₂×C₈	—	101	1	no	—
201	C₂×C₈	—	101	1	no	—
202	C₂×C₈	—	101	2	no	—
203	C₂×C₈	—	101	2	no	—
204	C₂×C₈	—	202	4	no	—
205	C₂×C₈	—	202	4	no	—
206	C₂×C₈	—	202	4	no	—
207	C₄⋊C₄	—	101	2	no	—
208	C₄⋊C₄	—	101	1	no	—
209	C₄⋊C₄	C₂×#87938	1	1	yes	characteristic
210	C₂²⋊C₄	—	101	2	no	—
211	C₂²⋊C₄	—	101	2	no	—
212	C₂²⋊C₄	—	101	2	no	—
213	C₂²⋊C₄	C₂×#87938	1	2	yes	—
214	C₄²	—	101	1	no	—
215	C₄²	—	101	1	no	—
216	C₄²	—	101	1	no	—
217	C₄²	—	101	1	no	—
218	C₂³	—	101	2	no	—
219	C₂³	C₂²×#87938	1	2	yes	—
220	C₂³	C₂²×#87938	1	1	yes	characteristic
221	Q₈	—	202	4	no	—
222	D₄	—	2	4	no	—
223	D₄	—	2	2	no	—
224	D₄	—	404	2	no	—
225	D₄	—	202	4	no	—
226	D₄	—	202	4	no	—
227	D₄	—	202	2	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	101	2	no	—
230	C₂×C₄	—	101	2	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	2	no	—
236	C₂×C₄	—	404	1	no	—
237	C₂×C₄	—	202	2	no	—
238	C₂×C₄	—	202	2	no	—
239	C₂×C₄	—	202	2	no	—
240	C₂×C₄	—	202	2	no	—
241	C₂×C₄	C₂²×#87938	1	1	yes	characteristic
242	C₂×C₄	C₂²×#87938	1	2	yes	—
243	C₂×C₄	C₂²×#87938	1	1	yes	characteristic
244	C₈	—	101	2	no	—
245	C₈	—	101	2	no	—
246	C₈	—	202	2	no	—
247	C₈	—	202	2	no	—
248	C₈	—	202	1	no	—
249	C₈	—	202	1	no	—
250	C₂²	—	2	1	no	—
251	C₂²	—	2	1	no	—
252	C₂²	—	2	2	no	—
253	C₂²	—	2	2	no	—
254	C₂²	—	2	2	no	—
255	C₂²	—	202	2	no	—
256	C₂²	—	202	2	no	—
257	C₂²	—	202	2	no	—
258	C₂²	D₄×#87938	1	2	yes	—
259	C₂²	C₂³×#87938	1	1	yes	characteristic
260	C₄	—	2	2	no	—
261	C₄	—	2	1	no	—
262	C₄	—	101	1	no	—
263	C₄	—	101	1	no	—
264	C₄	—	101	2	no	—
265	C₄	—	202	2	no	—
266	C₄	—	202	1	no	—
267	C₄	—	202	1	no	—
268	C₄	#15692342	1	2	yes	complemented
269	C₂	—	2	2	no	—
270	C₂	—	4	2	no	—
271	C₂	—	404	2	no	—
272	C₂	C₂×D₄×#87938	1	1	yes	characteristic
273	C₂	C₂×#15692342	1	1	yes	characteristic
274	C₂	#20266256	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×#87938
commutator, socle	C₂²×C₁₀₁	C₂³×C₄
Fitting	#121×C₁₀₁	C₄
prime radical	#20265722	C₂
cosocle	C₂×Dic₁₀₁	C₂⁴
hypercenter	#121	#87938
hypocenter	C₁₀₁	#2935
LCS	G ▷ C₂²×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂² ◁ #121

Complements

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

N	H ≅ G/N	type
1. C₂×#15692342	C₂	semidirect
2. #20266212	C₂	semidirect
3. #20266211	C₂	semidirect
4. #20266209	C₂	semidirect
5. #20266206	C₂	semidirect
6. #20266205	C₂	semidirect
7. D₄×#11717581	C₂	semidirect
8. C₂×C₄⋊#11717581	C₂	semidirect
9. #20266150	C₂	semidirect
10. C₄×#13108083	C₂	semidirect
45. #13108083	D₄	semidirect
124. C₄×C₁₀₁	#131	semidirect
138. C₁₀₁	#2935	semidirect
268. C₄	#15692342	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15692342	2
#20266212	2
#20266211	2
#20266209	2
#20266206	2
#20266205	2
D₄×#11717581	2
C₂×C₄⋊#11717581	2
#20266150	2
C₄×#13108083	2
#20265722	2
#2935	101