Subgroups — #27180751

Order 12928.

548 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15692283	C₂	1	1	yes	maximal, characteristic, complemented
2	#20265688	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#15692229	C₂	1	1	yes	maximal, characteristic, complemented
4	#20265430	C₂	1	2	yes	maximal, complemented
5	#20265423	C₂	1	2	yes	maximal, complemented
6	#20265413	C₂	1	2	yes	maximal, complemented
7	#20265406	C₂	1	2	yes	maximal, complemented
8	#20265219	C₂	1	1	yes	maximal, characteristic, complemented
9	#20265183	C₂	1	1	yes	maximal, characteristic, complemented
10	#20265179	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂²×M₁₆×C₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13108074	C₂²	1	1	yes	characteristic
13	C₂²×D₄₀₄	C₂²	1	1	yes	characteristic
14	C₂²×C₂²⋊D₁₀₁	—	2	1	no	—
15	#15692283	—	2	4	no	—
16	C₂×#13108063	C₂²	1	2	yes	complemented
17	C₂×D₈₀₈	C₂²	1	2	yes	complemented
18	#15692238	—	2	1	no	—
19	#15692234	C₂²	1	1	yes	characteristic, complemented
20	#15692234	C₂²	1	1	yes	characteristic, complemented
21	#15692229	—	2	1	no	—
22	#15692229	—	2	1	no	—
23	#15692229	C₂²	1	2	yes	—
24	#15692229	C₂²	1	2	yes	—
25	#15692229	C₂²	1	1	yes	characteristic, complemented
26	#15692229	C₂²	1	1	yes	characteristic, complemented
27	#15692224	—	2	1	no	—
28	C₄×C₂²⋊D₁₀₁	—	2	1	no	—
29	C₄×C₂²⋊D₁₀₁	—	2	1	no	—
30	#15692214	C₂²	1	1	yes	characteristic
31	C₂×C₄⋊Dic₁₀₁	C₂²	1	1	yes	characteristic
32	C₂×#13108051	—	2	1	no	—
33	#15692189	C₂²	1	2	yes	complemented
34	#15692187	C₂²	1	2	yes	complemented
35	#15692180	C₂²	1	2	yes	complemented
36	#15692177	C₂²	1	2	yes	complemented
37	#15692177	C₂²	1	2	yes	complemented
38	#15692177	C₂²	1	2	yes	complemented
39	C₂³×C₄×C₁₀₁	C₂²	1	1	yes	characteristic
40	C₂×M₁₆×C₁₀₁	—	2	4	no	—
41	C₂×M₁₆×C₁₀₁	C₂²	1	1	yes	characteristic
42	C₂×M₁₆×C₁₀₁	C₂²	1	1	yes	characteristic
43	C₂×M₁₆×C₁₀₁	C₂²	1	2	yes	complemented
44	C₂²×C₈×C₁₀₁	C₂²	1	2	yes	complemented
45	C₂³×D₁₀₁	—	2	1	no	—
46	#13108074	—	2	2	no	—
47	#13108074	—	2	2	no	—
48	C₂×Dic₂₀₂	C₂³	1	1	yes	characteristic
49	C₂×D₄₀₄	—	2	2	no	—
50	C₂×D₄₀₄	—	2	2	no	—
51	C₂×D₄₀₄	—	2	1	no	—
52	C₂×D₄₀₄	C₂³	1	2	yes	—
53	C₂×D₄₀₄	C₂³	1	1	yes	characteristic
54	C₂×C₂²⋊D₁₀₁	—	2	2	no	—
55	C₂×C₂²⋊D₁₀₁	—	2	2	no	—
56	C₂×C₂²⋊D₁₀₁	—	2	1	no	—
57	C₂×C₂²⋊D₁₀₁	—	2	1	no	—
58	C₂×C₂²⋊D₁₀₁	—	2	1	no	—
59	C₂×C₂²⋊D₁₀₁	—	4	4	no	—
60	C₂²×Dic₁₀₁	—	2	1	no	—
61	C₂×C₄×D₁₀₁	—	2	1	no	—
62	#13108063	—	2	4	no	—
63	D₈₀₈	—	2	4	no	—
64	C₄⋊Dic₁₀₁	—	2	1	no	—
65	C₄⋊Dic₁₀₁	C₂³	1	1	yes	characteristic
66	C₄⋊Dic₁₀₁	C₂³	1	1	yes	characteristic
67	C₄⋊Dic₁₀₁	C₂³	1	2	yes	—
68	#13108053	—	2	1	no	—
69	#13108053	—	2	1	no	—
70	C₂²⋊Dic₁₀₁	—	2	1	no	—
71	C₂²⋊Dic₁₀₁	—	2	1	no	—
72	C₂²⋊Dic₁₀₁	—	2	1	no	—
73	#13108051	—	2	2	no	—
74	#13108051	—	2	2	no	—
75	#13108051	—	2	1	no	—
76	#13108051	—	2	1	no	—
77	C₄×Dic₁₀₁	—	2	1	no	—
78	C₂⁴×C₁₀₁	D₄	1	1	yes	characteristic
79	C₂²×C₄×C₁₀₁	—	2	2	no	—
80	C₂²×C₄×C₁₀₁	—	2	2	no	—
81	C₂²×C₄×C₁₀₁	D₄	1	1	yes	characteristic
82	C₂²×C₄×C₁₀₁	D₄	1	1	yes	characteristic
83	C₂²×C₄×C₁₀₁	D₄	1	1	yes	characteristic
84	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
85	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
86	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
87	M₁₆×C₁₀₁	—	2	2	no	—
88	M₁₆×C₁₀₁	—	2	4	no	—
89	M₁₆×C₁₀₁	D₄	1	4	yes	complemented
90	C₂×C₈×C₁₀₁	—	2	4	no	—
91	C₂×C₈×C₁₀₁	C₂³	1	2	yes	—
92	C₂×C₈×C₁₀₁	C₂³	1	2	yes	—
93	C₂³×C₁₀₁	—	2	2	no	—
94	C₂³×C₁₀₁	—	2	2	no	—
95	C₂³×C₁₀₁	—	4	1	no	—
96	C₂³×C₁₀₁	C₂×D₄	1	1	yes	characteristic
97	C₂³×C₁₀₁	C₂×D₄	1	1	yes	characteristic
98	C₂³×C₁₀₁	C₂×D₄	1	1	yes	characteristic
99	C₂²×D₁₀₁	—	2	1	no	—
100	C₂²×D₁₀₁	—	2	2	no	—
101	C₂²×D₁₀₁	—	4	2	no	—
102	C₂²×D₁₀₁	—	4	2	no	—
103	C₂²×D₁₀₁	—	4	2	no	—
104	C₂×C₄×C₁₀₁	—	2	2	no	—
105	C₂×C₄×C₁₀₁	—	2	2	no	—
106	C₂×C₄×C₁₀₁	—	2	2	no	—
107	C₂×C₄×C₁₀₁	—	2	1	no	—
108	C₂×C₄×C₁₀₁	—	2	1	no	—
109	C₂×C₄×C₁₀₁	—	2	1	no	—
110	C₂×C₄×C₁₀₁	—	2	1	no	—
111	C₂×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
112	C₂×C₄×C₁₀₁	C₂×D₄	1	1	yes	characteristic
113	C₂×C₄×C₁₀₁	C₂×D₄	1	1	yes	characteristic
114	C₂×C₄×C₁₀₁	C₂×D₄	1	1	yes	characteristic
115	C₂×C₄×C₁₀₁	#41	1	2	yes	—
116	C₂×C₄×C₁₀₁	C₂⁴	1	1	yes	characteristic
117	C₂²⋊D₁₀₁	—	4	2	no	—
118	C₂²⋊D₁₀₁	—	4	4	no	—
119	C₂²⋊D₁₀₁	—	4	2	no	—
120	C₂²⋊D₁₀₁	—	4	2	no	—
121	C₂²⋊D₁₀₁	—	4	4	no	—
122	C₂×Dic₁₀₁	—	2	1	no	—
123	C₂×Dic₁₀₁	—	2	1	no	—
124	C₂×Dic₁₀₁	—	2	2	no	—
125	C₂×Dic₁₀₁	—	2	1	no	—
126	C₂×Dic₁₀₁	—	4	2	no	—
127	D₄₀₄	—	2	1	no	—
128	D₄₀₄	—	2	2	no	—
129	D₄₀₄	—	2	2	no	—
130	D₄₀₄	—	2	1	no	—
131	D₄₀₄	—	4	2	no	—
132	C₄×D₁₀₁	—	4	2	no	—
133	Dic₂₀₂	—	2	1	no	—
134	Dic₂₀₂	—	2	1	no	—
135	C₈×C₁₀₁	—	2	2	no	—
136	C₈×C₁₀₁	C₂×D₄	1	4	yes	complemented
137	C₂²×C₁₀₁	—	2	1	no	—
138	C₂²×C₁₀₁	—	2	1	no	—
139	C₂²×C₁₀₁	—	2	1	no	—
140	C₂²×C₁₀₁	—	2	1	no	—
141	C₂²×C₁₀₁	—	2	1	no	—
142	C₂²×C₁₀₁	—	2	1	no	—
143	C₂²×C₁₀₁	—	2	1	no	—
144	C₂²×C₁₀₁	—	4	4	no	—
145	C₂²×C₁₀₁	#121	1	2	yes	—
146	C₂²×C₁₀₁	AGL(1,ℤ/8)	1	2	yes	complemented
147	C₂²×C₁₀₁	C₂²×D₄	1	1	yes	characteristic
148	C₂×D₁₀₁	—	4	1	no	—
149	C₂×D₁₀₁	—	4	2	no	—
150	C₂×D₁₀₁	—	4	2	no	—
151	C₂×D₁₀₁	—	4	1	no	—
152	C₂×D₁₀₁	—	4	1	no	—
153	C₂×D₁₀₁	—	4	2	no	—
154	C₂×D₁₀₁	—	8	4	no	—
155	C₄×C₁₀₁	—	2	1	no	—
156	C₄×C₁₀₁	—	2	1	no	—
157	C₄×C₁₀₁	#121	1	2	yes	—
158	C₄×C₁₀₁	C₂²×D₄	1	1	yes	characteristic
159	C₄×C₁₀₁	C₂×#41	1	1	yes	characteristic
160	Dic₁₀₁	—	4	1	no	—
161	Dic₁₀₁	—	4	2	no	—
162	Dic₁₀₁	—	4	1	no	—
163	Dic₁₀₁	—	4	1	no	—
164	C₂×C₁₀₁	—	2	2	no	—
165	C₂×C₁₀₁	—	2	2	no	—
166	C₂×C₁₀₁	—	4	1	no	—
167	C₂×C₁₀₁	C₂×#121	1	1	yes	characteristic
168	C₂×C₁₀₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
169	C₂×C₁₀₁	#575	1	1	yes	characteristic
170	D₁₀₁	—	8	1	no	—
171	D₁₀₁	—	8	2	no	—
172	#3055	—	101	1	no	maximal, Sylow
173	C₁₀₁	#3055	1	1	yes	characteristic, complemented, Sylow
174	C₂×AGL(1,ℤ/8)	—	101	1	no	—
175	C₂²×M₁₆	D₁₀₁	1	1	yes	characteristic, complemented
176	#525	—	101	1	no	—
177	C₂×#121	—	101	1	no	—
178	#469	—	101	2	no	—
179	#468	—	101	2	no	—
180	#466	—	101	2	no	—
181	#465	—	101	2	no	—
182	#428	—	101	1	no	—
183	#418	—	101	1	no	—
184	#417	—	101	1	no	—
185	C₂×#41	—	101	1	no	—
186	C₂²×D₄	—	101	1	no	—
187	C₂²×D₄	—	202	1	no	—
188	C₂³×C₄	C₂×D₁₀₁	1	1	yes	characteristic
189	AGL(1,ℤ/8)	—	202	4	no	—
190	C₂×SD₁₆	—	101	2	no	—
191	C₂×D₈	—	101	2	no	—
192	C₂×M₁₆	—	2	4	no	—
193	C₂×M₁₆	C₂×D₁₀₁	1	1	yes	characteristic
194	C₂×M₁₆	C₂×D₁₀₁	1	1	yes	characteristic
195	C₂×M₁₆	C₂×D₁₀₁	1	2	yes	complemented
196	C₂²×C₈	C₂×D₁₀₁	1	2	yes	complemented
197	#123	—	202	1	no	—
198	C₂²⋊Q₈	—	101	1	no	—
199	C₂²⋊Q₈	—	101	1	no	—
200	#121	—	101	2	no	—
201	#121	—	101	2	no	—
202	#121	—	101	1	no	—
203	#121	—	101	1	no	—
204	#121	—	202	1	no	—
205	#121	—	202	1	no	—
206	#120	—	202	1	no	—
207	C₄×D₄	—	202	1	no	—
208	C₄×D₄	—	202	1	no	—
209	#117	—	101	1	no	—
210	C₂×C₄⋊C₄	—	101	1	no	—
211	C₂×C₂²⋊C₄	—	202	1	no	—
212	#107	—	101	2	no	—
213	#106	—	101	2	no	—
214	#103	—	101	2	no	—
215	#102	—	101	2	no	—
216	#102	—	101	2	no	—
217	#102	—	101	2	no	—
218	C₂⁴	—	202	1	no	—
219	C₂⁴	D₄₀₄	1	1	yes	characteristic
220	#41	—	202	2	no	—
221	#41	—	202	2	no	—
222	C₂×Q₈	—	101	1	no	—
223	C₂×D₄	—	101	1	no	—
224	C₂×D₄	—	101	2	no	—
225	C₂×D₄	—	404	4	no	—
226	C₂×D₄	—	202	1	no	—
227	C₂×D₄	—	202	1	no	—
228	C₂×D₄	—	202	1	no	—
229	C₂×D₄	—	202	2	no	—
230	C₂×D₄	—	202	2	no	—
231	C₂×D₄	—	202	2	no	—
232	C₂×D₄	—	202	1	no	—
233	C₂×D₄	—	202	2	no	—
234	C₂²×C₄	—	2	2	no	—
235	C₂²×C₄	—	2	2	no	—
236	C₂²×C₄	—	202	1	no	—
237	C₂²×C₄	—	202	1	no	—
238	C₂²×C₄	D₄₀₄	1	1	yes	characteristic
239	C₂²×C₄	D₄₀₄	1	1	yes	characteristic
240	C₂²×C₄	D₄₀₄	1	1	yes	characteristic
241	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
242	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
243	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
244	SD₁₆	—	202	4	no	—
245	D₈	—	202	4	no	—
246	M₁₆	—	2	2	no	—
247	M₁₆	—	2	4	no	—
248	M₁₆	C₂²⋊D₁₀₁	1	4	yes	complemented
249	C₂×C₈	—	2	4	no	—
250	C₂×C₈	C₂²×D₁₀₁	1	2	yes	—
251	C₂×C₈	C₂²×D₁₀₁	1	2	yes	—
252	C₄⋊C₄	—	101	1	no	—
253	C₄⋊C₄	—	101	1	no	—
254	C₄⋊C₄	—	101	2	no	—
255	C₄⋊C₄	—	202	1	no	—
256	C₄⋊C₄	—	202	1	no	—
257	C₄⋊C₄	—	202	1	no	—
258	C₂²⋊C₄	—	202	1	no	—
259	C₂²⋊C₄	—	202	1	no	—
260	C₂²⋊C₄	—	202	1	no	—
261	C₂²⋊C₄	—	202	2	no	—
262	C₂²⋊C₄	—	202	2	no	—
263	C₂²⋊C₄	—	202	1	no	—
264	C₂²⋊C₄	—	202	1	no	—
265	C₄²	—	202	1	no	—
266	C₂³	—	2	2	no	—
267	C₂³	—	2	2	no	—
268	C₂³	—	4	1	no	—
269	C₂³	—	404	2	no	—
270	C₂³	—	404	2	no	—
271	C₂³	—	404	2	no	—
272	C₂³	—	202	1	no	—
273	C₂³	—	202	2	no	—
274	C₂³	C₂×D₄₀₄	1	1	yes	characteristic
275	C₂³	C₂×D₄₀₄	1	1	yes	characteristic
276	C₂³	C₂×D₄₀₄	1	1	yes	characteristic
277	Q₈	—	202	1	no	—
278	Q₈	—	202	1	no	—
279	D₄	—	404	4	no	—
280	D₄	—	404	2	no	—
281	D₄	—	404	2	no	—
282	D₄	—	404	4	no	—
283	D₄	—	404	2	no	—
284	D₄	—	404	2	no	—
285	D₄	—	202	1	no	—
286	D₄	—	202	2	no	—
287	D₄	—	202	2	no	—
288	D₄	—	202	1	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	404	2	no	—
297	C₂×C₄	—	404	2	no	—
298	C₂×C₄	—	202	1	no	—
299	C₂×C₄	—	202	1	no	—
300	C₂×C₄	—	202	2	no	—
301	C₂×C₄	—	202	1	no	—
302	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	2	yes	—
303	C₂×C₄	C₂×D₄₀₄	1	1	yes	characteristic
304	C₂×C₄	C₂×D₄₀₄	1	1	yes	characteristic
305	C₂×C₄	C₂×D₄₀₄	1	1	yes	characteristic
306	C₂×C₄	#13108074	1	2	yes	—
307	C₂×C₄	C₂³×D₁₀₁	1	1	yes	characteristic
308	C₈	—	2	2	no	—
309	C₈	C₂×C₂²⋊D₁₀₁	1	4	yes	complemented
310	C₂²	—	2	1	no	—
311	C₂²	—	2	1	no	—
312	C₂²	—	2	1	no	—
313	C₂²	—	2	1	no	—
314	C₂²	—	2	1	no	—
315	C₂²	—	2	1	no	—
316	C₂²	—	2	1	no	—
317	C₂²	—	4	4	no	—
318	C₂²	—	808	4	no	—
319	C₂²	—	404	1	no	—
320	C₂²	—	404	2	no	—
321	C₂²	—	404	2	no	—
322	C₂²	—	404	1	no	—
323	C₂²	—	404	1	no	—
324	C₂²	—	404	2	no	—
325	C₂²	#15692229	1	2	yes	—
326	C₂²	#15692283	1	2	yes	complemented
327	C₂²	C₂²×D₄₀₄	1	1	yes	characteristic
328	C₄	—	2	1	no	—
329	C₄	—	2	1	no	—
330	C₄	—	404	1	no	—
331	C₄	—	404	2	no	—
332	C₄	—	404	1	no	—
333	C₄	—	404	1	no	—
334	C₄	#15692229	1	2	yes	—
335	C₄	C₂²×C₂²⋊D₁₀₁	1	1	yes	characteristic
336	C₄	C₂×#13108074	1	1	yes	characteristic
337	C₂	—	2	2	no	—
338	C₂	—	2	2	no	—
339	C₂	—	4	1	no	—
340	C₂	—	808	2	no	—
341	C₂	—	808	1	no	—
342	C₂	C₂×#15692229	1	1	yes	characteristic
343	C₂	C₂×#15692283	1	1	yes	characteristic
344	C₂	#20265977	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₄₀₄
commutator, cosocle	C₂×C₄×C₁₀₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₀₁
Fitting	C₂²×M₁₆×C₁₀₁	C₂
socle	C₂²×C₁₀₁	C₂²×D₄
hypercenter	C₂²×M₁₆	D₁₀₁
hypocenter	C₁₀₁	#3055
LCS	G ▷ C₂×C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×M₁₆

Complements

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

N	H ≅ G/N	type
1. C₂×#15692283	C₂	semidirect
2. #20265688	C₂	semidirect
3. C₂×#15692229	C₂	semidirect
4. #20265430	C₂	semidirect
5. #20265423	C₂	semidirect
6. #20265413	C₂	semidirect
7. #20265406	C₂	semidirect
8. #20265219	C₂	semidirect
9. #20265183	C₂	semidirect
10. #20265179	C₂	semidirect
11. C₂²×M₁₆×C₁₀₁	C₂	semidirect
16. C₂×#13108063	C₂²	semidirect
17. C₂×D₈₀₈	C₂²	semidirect
19. #15692234	C₂²	semidirect
20. #15692234	C₂²	semidirect
25. #15692229	C₂²	semidirect
26. #15692229	C₂²	semidirect
33. #15692189	C₂²	semidirect
34. #15692187	C₂²	semidirect
35. #15692180	C₂²	semidirect
36. #15692177	C₂²	semidirect
37. #15692177	C₂²	semidirect
38. #15692177	C₂²	semidirect
43. C₂×M₁₆×C₁₀₁	C₂²	semidirect
44. C₂²×C₈×C₁₀₁	C₂²	semidirect
89. M₁₆×C₁₀₁	D₄	semidirect
136. C₈×C₁₀₁	C₂×D₄	semidirect
146. C₂²×C₁₀₁	AGL(1,ℤ/8)	semidirect
173. C₁₀₁	#3055	semidirect
175. C₂²×M₁₆	D₁₀₁	semidirect
195. C₂×M₁₆	C₂×D₁₀₁	semidirect
196. C₂²×C₈	C₂×D₁₀₁	semidirect
248. M₁₆	C₂²⋊D₁₀₁	semidirect
309. C₈	C₂×C₂²⋊D₁₀₁	semidirect
326. C₂²	#15692283	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15692283	2
#20265688	2
C₂×#15692229	2
#20265430	2
#20265423	2
#20265413	2
#20265406	2
#20265219	2
#20265183	2
#20265179	2
C₂²×M₁₆×C₁₀₁	2
#3055	101