Subgroups — #27180680

Order 12928.

472 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15692285	C₂	1	1	yes	maximal, characteristic, complemented
2	#20265748	C₂	1	1	yes	maximal, characteristic, complemented
3	#20265660	C₂	1	1	yes	maximal, characteristic, complemented
4	#20265397	C₂	1	2	yes	maximal, complemented
5	#20265390	C₂	1	2	yes	maximal, complemented
6	#20265385	C₂	1	2	yes	maximal, complemented
7	#20265380	C₂	1	2	yes	maximal, complemented
8	#20265203	C₂	1	1	yes	maximal, characteristic, complemented
9	#20265178	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#15692180	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×#135×C₁₀₁	C₂	1	1	yes	maximal, characteristic
12	C₂²×Dic₂₀₂	C₂²	1	1	yes	characteristic
13	#15692285	—	2	4	no	—
14	C₂×#13108064	C₂²	1	2	yes	—
15	C₂×#13108062	C₂²	1	2	yes	—
16	C₂×#13108058	C₂²	1	1	yes	characteristic
17	#15692253	C₂²	1	2	yes	—
18	#15692242	C₂²	1	2	yes	—
19	#15692236	—	2	2	no	—
20	#15692234	—	2	1	no	—
21	#15692234	—	2	1	no	—
22	Q₈×Dic₁₀₁	C₂²	1	2	yes	—
23	D₄×Dic₁₀₁	—	2	2	no	—
24	D₄×Dic₁₀₁	C₂²	1	2	yes	—
25	#15692214	—	2	1	no	—
26	#15692214	C₂²	1	1	yes	characteristic
27	C₂×C₄⋊Dic₁₀₁	—	2	1	no	—
28	C₂×C₄⋊Dic₁₀₁	C₂²	1	1	yes	characteristic
29	#15692184	C₂²	1	2	yes	—
30	#15692184	C₂²	1	2	yes	—
31	#15692180	C₂²	1	2	yes	—
32	#15692180	C₂²	1	2	yes	—
33	#15692179	C₂²	1	2	yes	—
34	#15692176	C₂²	1	2	yes	—
35	C₂×#41×C₁₀₁	C₂²	1	1	yes	characteristic
36	C₂×#41×C₁₀₁	C₂²	1	1	yes	characteristic
37	#135×C₁₀₁	—	2	4	no	—
38	C₂×Q₁₆×C₁₀₁	C₂²	1	1	yes	characteristic
39	C₂×SD₁₆×C₁₀₁	C₂²	1	1	yes	characteristic
40	C₂×SD₁₆×C₁₀₁	C₂²	1	1	yes	characteristic
41	C₂×D₈×C₁₀₁	C₂²	1	1	yes	characteristic
42	C₂²×C₈×C₁₀₁	C₂²	1	1	yes	characteristic
43	C₂×Dic₂₀₂	—	2	2	no	—
44	C₂×Dic₂₀₂	—	2	2	no	—
45	C₂×Dic₂₀₂	—	2	1	no	—
46	C₂×Dic₂₀₂	C₂³	1	2	yes	—
47	C₂²×Dic₁₀₁	—	2	1	no	—
48	C₂²×Dic₁₀₁	—	2	1	no	—
49	C₂²×Dic₁₀₁	—	2	2	no	—
50	#13108064	—	2	4	no	—
51	#13108062	—	2	4	no	—
52	#13108058	—	2	2	no	—
53	C₂×#11717579	C₂³	1	2	yes	—
54	C₄⋊Dic₁₀₁	—	2	1	no	—
55	C₄⋊Dic₁₀₁	—	2	2	no	—
56	C₄⋊Dic₁₀₁	D₄	1	1	yes	characteristic
57	C₄⋊Dic₁₀₁	D₄	1	1	yes	characteristic
58	C₄⋊Dic₁₀₁	C₂³	1	2	yes	—
59	#13108053	—	2	2	no	—
60	#13108053	—	2	2	no	—
61	C₂²⋊Dic₁₀₁	—	2	2	no	—
62	C₂²⋊Dic₁₀₁	—	2	2	no	—
63	C₂²⋊Dic₁₀₁	—	2	2	no	—
64	C₂²⋊Dic₁₀₁	D₄	1	2	yes	—
65	C₄×Dic₁₀₁	—	2	2	no	—
66	C₄×Dic₁₀₁	C₂³	1	2	yes	—
67	#41×C₁₀₁	—	2	2	no	—
68	#41×C₁₀₁	—	2	2	no	—
69	#41×C₁₀₁	—	2	2	no	—
70	#41×C₁₀₁	D₄	1	4	yes	—
71	C₂×Q₈×C₁₀₁	C₂³	1	1	yes	characteristic
72	C₂×Q₈×C₁₀₁	C₂³	1	1	yes	characteristic
73	C₂×D₄×C₁₀₁	—	2	1	no	—
74	C₂×D₄×C₁₀₁	—	2	1	no	—
75	C₂×D₄×C₁₀₁	C₂³	1	1	yes	characteristic
76	C₂×D₄×C₁₀₁	C₂³	1	1	yes	characteristic
77	C₂²×C₄×C₁₀₁	—	2	1	no	—
78	C₂²×C₄×C₁₀₁	—	2	1	no	—
79	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
80	Q₁₆×C₁₀₁	—	2	2	no	—
81	SD₁₆×C₁₀₁	—	2	2	no	—
82	SD₁₆×C₁₀₁	—	2	2	no	—
83	D₈×C₁₀₁	—	2	2	no	—
84	C₂×C₈×C₁₀₁	—	2	2	no	—
85	C₂×C₈×C₁₀₁	C₂³	1	1	yes	characteristic
86	C₂×C₈×C₁₀₁	C₂³	1	1	yes	characteristic
87	C₂³×C₁₀₁	—	2	1	no	—
88	C₂³×C₁₀₁	—	2	1	no	—
89	C₂³×C₁₀₁	C₂×D₄	1	1	yes	characteristic
90	Q₈×C₁₀₁	—	2	1	no	—
91	Q₈×C₁₀₁	—	2	1	no	—
92	Q₈×C₁₀₁	—	2	1	no	—
93	Q₈×C₁₀₁	C₂×D₄	1	2	yes	—
94	D₄×C₁₀₁	—	2	4	no	—
95	D₄×C₁₀₁	—	2	1	no	—
96	D₄×C₁₀₁	—	2	1	no	—
97	D₄×C₁₀₁	—	2	1	no	—
98	D₄×C₁₀₁	—	4	2	no	—
99	D₄×C₁₀₁	C₂×D₄	1	2	yes	—
100	C₂×C₄×C₁₀₁	—	2	1	no	—
101	C₂×C₄×C₁₀₁	—	2	4	no	—
102	C₂×C₄×C₁₀₁	—	2	1	no	—
103	C₂×C₄×C₁₀₁	—	4	2	no	—
104	C₂×C₄×C₁₀₁	C₂×D₄	1	1	yes	characteristic
105	C₂×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
106	C₂×C₄×C₁₀₁	#41	1	2	yes	—
107	C₂×C₄×C₁₀₁	C₂⁴	1	1	yes	characteristic
108	C₂×Dic₁₀₁	—	2	2	no	—
109	C₂×Dic₁₀₁	—	2	2	no	—
110	C₂×Dic₁₀₁	—	4	2	no	—
111	C₂×Dic₁₀₁	—	8	2	no	—
112	C₂×Dic₁₀₁	—	8	1	no	—
113	C₂×Dic₁₀₁	C₂×D₄	1	2	yes	—
114	C₂×Dic₁₀₁	C₂×D₄	1	2	yes	—
115	Dic₂₀₂	—	2	2	no	—
116	Dic₂₀₂	—	2	2	no	—
117	Dic₂₀₂	—	4	2	no	—
118	C₈×C₁₀₁	—	2	1	no	—
119	C₈×C₁₀₁	—	2	1	no	—
120	#11717579	—	2	2	no	—
121	C₂²×C₁₀₁	—	2	1	no	—
122	C₂²×C₁₀₁	—	2	1	no	—
123	C₂²×C₁₀₁	—	2	2	no	—
124	C₂²×C₁₀₁	—	4	1	no	—
125	C₂²×C₁₀₁	—	4	1	no	—
126	C₂²×C₁₀₁	—	4	1	no	—
127	C₂²×C₁₀₁	—	4	1	no	—
128	C₂²×C₁₀₁	—	4	1	no	—
129	C₂²×C₁₀₁	#121	1	2	yes	—
130	C₂²×C₁₀₁	C₂²×D₄	1	1	yes	characteristic
131	C₄×C₁₀₁	—	2	2	no	—
132	C₄×C₁₀₁	—	4	1	no	—
133	C₄×C₁₀₁	#121	1	2	yes	—
134	C₄×C₁₀₁	C₂²×D₄	1	1	yes	characteristic
135	C₄×C₁₀₁	C₂×#41	1	1	yes	characteristic
136	Dic₁₀₁	—	2	2	no	—
137	Dic₁₀₁	—	2	2	no	—
138	Dic₁₀₁	—	4	2	no	—
139	Dic₁₀₁	—	4	2	no	—
140	C₂×C₁₀₁	—	2	2	no	—
141	C₂×C₁₀₁	—	4	2	no	—
142	C₂×C₁₀₁	—	8	1	no	—
143	C₂×C₁₀₁	C₂×#121	1	1	yes	characteristic
144	C₂×C₁₀₁	#577	1	1	yes	characteristic
145	C₂×C₁₀₁	#578	1	1	yes	characteristic
146	#3046	—	101	1	no	maximal, Sylow
147	C₁₀₁	#3046	1	1	yes	characteristic, complemented, Sylow
148	C₂×#137	—	101	1	no	—
149	C₂×#135	D₁₀₁	1	1	yes	characteristic
150	#536	—	101	1	no	—
151	#520	—	101	1	no	—
152	#464	—	101	2	no	—
153	#463	—	101	2	no	—
154	#462	—	101	2	no	—
155	#461	—	101	2	no	—
156	#424	—	101	1	no	—
157	#417	—	101	1	no	—
158	C₂×#103	—	101	1	no	—
159	C₂×#41	C₂×D₁₀₁	1	1	yes	characteristic
160	C₂×#41	C₂×D₁₀₁	1	1	yes	characteristic
161	C₂²×Q₈	—	101	1	no	—
162	#137	—	202	4	no	—
163	#135	—	2	4	no	—
164	C₂×Q₁₆	—	101	2	no	—
165	C₂×Q₁₆	C₂×D₁₀₁	1	1	yes	characteristic
166	C₂×SD₁₆	—	101	2	no	—
167	C₂×SD₁₆	C₂×D₁₀₁	1	1	yes	characteristic
168	C₂×SD₁₆	C₂×D₁₀₁	1	1	yes	characteristic
169	C₂×D₈	C₂×D₁₀₁	1	1	yes	characteristic
170	C₂×M₁₆	—	101	1	no	—
171	C₂²×C₈	C₂×D₁₀₁	1	1	yes	characteristic
172	C₄⋊Q₈	—	101	2	no	—
173	#124	—	101	2	no	—
174	#123	—	202	2	no	—
175	C₂²⋊Q₈	—	202	1	no	—
176	C₂²⋊Q₈	—	202	1	no	—
177	C₄×Q₈	—	101	2	no	—
178	C₄×D₄	—	101	2	no	—
179	C₄×D₄	—	202	2	no	—
180	#117	—	101	1	no	—
181	#117	—	202	1	no	—
182	C₂×C₄⋊C₄	—	101	1	no	—
183	C₂×C₄⋊C₄	—	202	1	no	—
184	C₄⋊C₈	—	101	2	no	—
185	C₄⋊C₈	—	101	2	no	—
186	#103	—	101	2	no	—
187	#103	—	101	2	no	—
188	#103	—	101	2	no	—
189	#102	—	101	2	no	—
190	#41	—	2	2	no	—
191	#41	—	2	2	no	—
192	#41	—	2	2	no	—
193	#41	C₂²⋊D₁₀₁	1	4	yes	—
194	C₂×Q₈	—	101	2	no	—
195	C₂×Q₈	—	202	2	no	—
196	C₂×Q₈	—	202	2	no	—
197	C₂×Q₈	—	202	1	no	—
198	C₂×Q₈	C₂²×D₁₀₁	1	1	yes	characteristic
199	C₂×Q₈	C₂²×D₁₀₁	1	1	yes	characteristic
200	C₂×D₄	—	2	1	no	—
201	C₂×D₄	—	2	1	no	—
202	C₂×D₄	C₂²×D₁₀₁	1	1	yes	characteristic
203	C₂×D₄	C₂²×D₁₀₁	1	1	yes	characteristic
204	C₂²×C₄	—	2	1	no	—
205	C₂²×C₄	—	2	1	no	—
206	C₂²×C₄	—	202	2	no	—
207	C₂²×C₄	—	202	1	no	—
208	C₂²×C₄	—	202	1	no	—
209	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
210	Q₁₆	—	2	2	no	—
211	Q₁₆	—	202	4	no	—
212	SD₁₆	—	2	2	no	—
213	SD₁₆	—	2	2	no	—
214	SD₁₆	—	202	4	no	—
215	D₈	—	2	2	no	—
216	M₁₆	—	202	2	no	—
217	C₂×C₈	—	2	2	no	—
218	C₂×C₈	—	101	2	no	—
219	C₂×C₈	C₂²×D₁₀₁	1	1	yes	characteristic
220	C₂×C₈	C₂²×D₁₀₁	1	1	yes	characteristic
221	C₄⋊C₄	—	101	1	no	—
222	C₄⋊C₄	—	101	1	no	—
223	C₄⋊C₄	—	101	2	no	—
224	C₄⋊C₄	—	202	2	no	—
225	C₄⋊C₄	—	202	2	no	—
226	C₄⋊C₄	—	202	2	no	—
227	C₄⋊C₄	—	202	1	no	—
228	C₂²⋊C₄	—	101	2	no	—
229	C₂²⋊C₄	—	202	2	no	—
230	C₂²⋊C₄	—	202	2	no	—
231	C₂²⋊C₄	—	202	2	no	—
232	C₄²	—	101	2	no	—
233	C₄²	—	202	2	no	—
234	C₂³	—	2	1	no	—
235	C₂³	—	2	1	no	—
236	C₂³	D₄×D₁₀₁	1	1	yes	characteristic
237	Q₈	—	2	1	no	—
238	Q₈	—	2	1	no	—
239	Q₈	—	2	1	no	—
240	Q₈	—	404	2	no	—
241	Q₈	—	202	2	no	—
242	Q₈	—	202	2	no	—
243	Q₈	C₂×C₂²⋊D₁₀₁	1	2	yes	—
244	D₄	—	2	4	no	—
245	D₄	—	2	1	no	—
246	D₄	—	2	1	no	—
247	D₄	—	2	1	no	—
248	D₄	—	4	2	no	—
249	D₄	C₂×C₂²⋊D₁₀₁	1	2	yes	—
250	C₂×C₄	—	2	1	no	—
251	C₂×C₄	—	2	4	no	—
252	C₂×C₄	—	2	1	no	—
253	C₂×C₄	—	4	2	no	—
254	C₂×C₄	—	808	1	no	—
255	C₂×C₄	—	808	2	no	—
256	C₂×C₄	—	101	2	no	—
257	C₂×C₄	—	101	2	no	—
258	C₂×C₄	—	404	2	no	—
259	C₂×C₄	—	202	2	no	—
260	C₂×C₄	—	202	2	no	—
261	C₂×C₄	D₄×D₁₀₁	1	1	yes	characteristic
262	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	2	yes	—
263	C₂×C₄	#13108073	1	2	yes	—
264	C₂×C₄	C₂³×D₁₀₁	1	1	yes	characteristic
265	C₈	—	2	1	no	—
266	C₈	—	2	1	no	—
267	C₈	—	202	2	no	—
268	C₂²	—	2	1	no	—
269	C₂²	—	2	1	no	—
270	C₂²	—	2	2	no	—
271	C₂²	—	4	1	no	—
272	C₂²	—	4	1	no	—
273	C₂²	—	4	1	no	—
274	C₂²	—	4	1	no	—
275	C₂²	—	4	1	no	—
276	C₂²	#15692227	1	2	yes	—
277	C₂²	C₂×D₄×D₁₀₁	1	1	yes	characteristic
278	C₄	—	2	2	no	—
279	C₄	—	4	1	no	—
280	C₄	—	404	2	no	—
281	C₄	—	404	2	no	—
282	C₄	—	202	2	no	—
283	C₄	—	202	2	no	—
284	C₄	#15692227	1	2	yes	—
285	C₄	C₂²×C₂²⋊D₁₀₁	1	1	yes	characteristic
286	C₄	C₂×#13108073	1	1	yes	characteristic
287	C₂	—	2	2	no	—
288	C₂	—	4	2	no	—
289	C₂	—	8	1	no	—
290	C₂	C₂×#15692227	1	1	yes	characteristic
291	C₂	#20265984	1	1	yes	characteristic
292	C₂	#20265991	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₀₁
commutator, cosocle	C₂×C₄×C₁₀₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₀₁
Fitting, prime radical	C₂×#135×C₁₀₁	C₂
socle	C₂²×C₁₀₁	C₂²×D₄
hypercenter	C₂×#135	D₁₀₁
hypocenter	C₁₀₁	#3046
LCS	G ▷ C₂×C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#135

Complements

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

N	H ≅ G/N	type
1. C₂×#15692285	C₂	semidirect
2. #20265748	C₂	semidirect
3. #20265660	C₂	semidirect
4. #20265397	C₂	semidirect
5. #20265390	C₂	semidirect
6. #20265385	C₂	semidirect
7. #20265380	C₂	semidirect
8. #20265203	C₂	semidirect
9. #20265178	C₂	semidirect
10. C₂×#15692180	C₂	semidirect
147. C₁₀₁	#3046	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15692285	2
#20265748	2
#20265660	2
#20265397	2
#20265390	2
#20265385	2
#20265380	2
#20265203	2
#20265178	2
C₂×#15692180	2
C₂×#135×C₁₀₁	2
#3046	101