Subgroups — #27187904

Order 12928.

456 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₁₀₁⋊#131	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#15692342	C₂	1	1	yes	maximal, characteristic, complemented
3	#20266224	C₂	1	2	yes	maximal, complemented
4	#20266222	C₂	1	2	yes	maximal, complemented
5	#20266220	C₂	1	2	yes	maximal, complemented
6	#20266219	C₂	1	2	yes	maximal, complemented
7	#20266192	C₂	1	1	yes	maximal, characteristic, complemented
8	#20266175	C₂	1	1	yes	maximal, characteristic, complemented
9	#20266174	C₂	1	1	yes	maximal, characteristic, complemented
10	#20266143	C₂	1	1	yes	maximal, characteristic, complemented
11	#20265973	C₂	1	1	yes	maximal, characteristic
12	#15692343	C₂²	1	1	yes	characteristic, complemented
13	#15692343	C₂²	1	1	yes	characteristic, complemented
14	#15692342	—	2	1	no	—
15	#15692342	—	2	1	no	—
16	#15692342	—	2	1	no	—
17	#15692342	—	2	1	no	—
18	#15692342	C₂²	1	2	yes	complemented
19	#15692342	C₂²	1	2	yes	complemented
20	#15692342	C₂²	1	1	yes	characteristic, complemented
21	#15692342	C₂²	1	1	yes	characteristic, complemented
22	C₂×#13108084	C₂²	1	1	yes	characteristic
23	D₁₀₁⋊M₁₆	—	2	1	no	—
24	D₁₀₁⋊M₁₆	C₂²	1	1	yes	characteristic
25	C₂×#13108083	—	2	1	no	—
26	C₂×D₁₀₁⋊C₈	—	2	1	no	—
27	C₂×D₁₀₁⋊C₈	C₂²	1	1	yes	characteristic
28	C₂²×#11717581	—	2	1	no	—
29	#15692334	C₂²	1	1	yes	characteristic
30	#15692326	C₂²	1	2	yes	complemented
31	#15692325	C₂²	1	2	yes	complemented
32	#15692320	C₂²	1	2	yes	complemented
33	#15692319	C₂²	1	2	yes	complemented
34	#15692319	C₂²	1	2	yes	complemented
35	#15692319	C₂²	1	2	yes	complemented
36	#15692309	C₂²	1	2	yes	complemented
37	C₈×#87938	C₂²	1	2	yes	complemented
38	C₂×#13108073	C₂²	1	1	yes	characteristic
39	#41×D₁₀₁	C₂²	1	1	yes	characteristic
40	#15692288	C₄	1	1	yes	characteristic, complemented
41	#15692285	C₄	1	1	yes	characteristic, complemented
42	#15692284	C₄	1	2	yes	complemented
43	#15692281	C₄	1	1	yes	characteristic, complemented
44	#15692279	C₄	1	2	yes	complemented
45	#15692276	C₂²	1	2	yes	—
46	#15692274	C₂²	1	2	yes	—
47	M₁₆×D₁₀₁	C₂²	1	1	yes	characteristic
48	AGL(1,ℤ/8)×C₁₀₁	C₄	1	1	yes	characteristic, complemented
49	#13108084	—	2	1	no	—
50	#13108084	—	2	1	no	—
51	#13108084	—	2	1	no	—
52	#13108084	C₂³	1	1	yes	characteristic
53	#13108084	C₂³	1	1	yes	characteristic
54	#13108084	C₂³	1	2	yes	—
55	#13108083	—	2	1	no	—
56	#13108083	—	2	2	no	—
57	#13108083	—	2	2	no	—
58	#13108083	—	2	1	no	—
59	#13108083	—	2	1	no	—
60	D₁₀₁⋊C₈	—	2	1	no	—
61	D₁₀₁⋊C₈	—	2	1	no	—
62	D₁₀₁⋊C₈	—	2	1	no	—
63	D₁₀₁⋊C₈	D₄	1	1	yes	characteristic
64	D₁₀₁⋊C₈	D₄	1	1	yes	characteristic
65	D₁₀₁⋊C₈	C₂³	1	1	yes	characteristic
66	D₁₀₁⋊C₈	C₂³	1	1	yes	characteristic
67	C₂×#11717581	—	2	1	no	—
68	C₂×#11717581	—	2	2	no	—
69	C₂×#11717581	—	2	2	no	—
70	C₂×#11717581	—	2	1	no	—
71	C₂×#11717581	—	2	1	no	—
72	C₂×#11717581	D₄	1	1	yes	characteristic
73	C₂×#11717581	D₄	1	1	yes	characteristic
74	C₄⋊#87938	—	2	1	no	—
75	C₂²⋊#87938	—	2	1	no	—
76	C₄×#87938	C₂³	1	2	yes	—
77	#13108075	C₂³	1	1	yes	characteristic
78	#13108074	—	2	1	no	—
79	#13108074	C₂×C₄	1	1	yes	characteristic
80	#13108073	—	2	1	no	—
81	#13108073	—	2	1	no	—
82	#13108073	—	2	1	no	—
83	#13108073	C₂×C₄	1	2	yes	—
84	#13108073	C₂³	1	1	yes	characteristic
85	#13108073	C₂³	1	2	yes	—
86	C₂×Dic₂₀₂	C₂×C₄	1	1	yes	characteristic
87	Q₈×D₁₀₁	C₂×C₄	1	1	yes	characteristic
88	C₂×C₂²⋊D₁₀₁	—	2	1	no	—
89	D₄×D₁₀₁	—	2	1	no	—
90	D₄×D₁₀₁	C₂×C₄	1	1	yes	characteristic
91	C₂²×Dic₁₀₁	—	2	1	no	—
92	C₂×C₄×D₁₀₁	—	2	1	no	—
93	C₂×C₄×D₁₀₁	C₂³	1	1	yes	characteristic
94	Dic₄₀₄	C₂×C₄	1	2	yes	—
95	#13108064	C₂×C₄	1	2	yes	—
96	#13108063	C₂×C₄	1	2	yes	—
97	#13108062	C₂×C₄	1	2	yes	—
98	#13108062	C₂×C₄	1	2	yes	—
99	#13108059	C₂×C₄	1	2	yes	—
100	#13108058	C₂×C₄	1	1	yes	characteristic
101	#13108057	C₂×C₄	1	2	yes	—
102	C₈×D₁₀₁	C₂³	1	2	yes	—
103	#41×C₁₀₁	C₂×C₄	1	1	yes	characteristic
104	C₂×D₄×C₁₀₁	C₂×C₄	1	1	yes	characteristic
105	SD₁₆×C₁₀₁	C₂×C₄	1	2	yes	—
106	D₈×C₁₀₁	C₂×C₄	1	2	yes	—
107	M₁₆×C₁₀₁	C₂×C₄	1	1	yes	characteristic
108	C₂³×C₁₀₁	—	2	1	no	—
109	C₂²×D₁₀₁	—	2	1	no	—
110	C₂²×D₁₀₁	#41	1	1	yes	characteristic
111	C₂×#87938	—	2	2	no	—
112	Q₈×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
113	D₄×C₁₀₁	—	2	1	no	—
114	D₄×C₁₀₁	—	2	1	no	—
115	D₄×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
116	D₄×C₁₀₁	C₂²×C₄	1	2	yes	—
117	C₂×C₄×C₁₀₁	—	2	1	no	—
118	C₂×C₄×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
119	C₂²⋊D₁₀₁	—	2	1	no	—
120	C₂²⋊D₁₀₁	—	2	2	no	—
121	C₂²⋊D₁₀₁	—	2	1	no	—
122	C₂²⋊D₁₀₁	—	2	2	no	—
123	C₂²⋊D₁₀₁	—	2	1	no	—
124	C₂×Dic₁₀₁	—	2	1	no	—
125	C₂×Dic₁₀₁	—	2	2	no	—
126	C₂×Dic₁₀₁	—	2	1	no	—
127	C₂×Dic₁₀₁	—	2	2	no	—
128	C₂×Dic₁₀₁	C₂×D₄	1	1	yes	characteristic
129	D₄₀₄	—	2	1	no	—
130	D₄₀₄	C₂²×C₄	1	1	yes	characteristic
131	C₄×D₁₀₁	—	2	1	no	—
132	C₄×D₁₀₁	—	2	1	no	—
133	C₄×D₁₀₁	—	2	1	no	—
134	C₄×D₁₀₁	C₂²×C₄	1	1	yes	characteristic
135	C₄×D₁₀₁	C₂×D₄	1	1	yes	characteristic
136	C₄×D₁₀₁	#41	1	1	yes	characteristic
137	C₄×D₁₀₁	C₂⁴	1	1	yes	characteristic
138	Dic₂₀₂	—	2	1	no	—
139	Dic₂₀₂	—	2	1	no	—
140	Dic₂₀₂	C₂²×C₄	1	1	yes	characteristic
141	Dic₂₀₂	C₂²×C₄	1	2	yes	—
142	#11717581	—	2	2	no	—
143	#11717581	—	2	1	no	—
144	#11717581	—	2	1	no	—
145	#11717581	C₂×D₄	1	1	yes	characteristic
146	#11717581	C₂×D₄	1	1	yes	characteristic
147	#11717581	C₂×D₄	1	1	yes	characteristic
148	#11717581	C₂×D₄	1	1	yes	characteristic
149	C₈×C₁₀₁	C₂²×C₄	1	2	yes	—
150	#11717579	C₂²×C₄	1	2	yes	—
151	C₂²×C₁₀₁	—	2	2	no	—
152	C₂²×C₁₀₁	—	2	1	no	—
153	C₂²×C₁₀₁	—	4	2	no	—
154	C₂²×C₁₀₁	C₄×D₄	1	1	yes	characteristic
155	C₂×D₁₀₁	—	2	1	no	—
156	C₂×D₁₀₁	—	4	1	no	—
157	C₂×D₁₀₁	—	4	2	no	—
158	C₂×D₁₀₁	C₄×D₄	1	1	yes	characteristic
159	C₂×D₁₀₁	C₂×#41	1	1	yes	characteristic
160	#87938	—	4	2	no	—
161	C₄×C₁₀₁	—	2	1	no	—
162	C₄×C₁₀₁	C₄×D₄	1	1	yes	characteristic
163	C₄×C₁₀₁	C₂³×C₄	1	1	yes	characteristic
164	Dic₁₀₁	—	2	1	no	—
165	Dic₁₀₁	—	2	2	no	—
166	Dic₁₀₁	C₄×D₄	1	1	yes	characteristic
167	Dic₁₀₁	C₂²×D₄	1	1	yes	characteristic
168	C₂×C₁₀₁	—	2	1	no	—
169	C₂×C₁₀₁	—	4	1	no	—
170	C₂×C₁₀₁	—	4	2	no	—
171	C₂×C₁₀₁	C₂×C₄×D₄	1	1	yes	characteristic
172	D₁₀₁	—	2	1	no	—
173	D₁₀₁	—	2	1	no	—
174	D₁₀₁	—	4	1	no	—
175	#2955	—	101	1	no	maximal, Sylow
176	C₁₀₁	#2955	1	1	yes	characteristic, complemented, Sylow
177	#575	—	101	1	no	—
178	C₂×#131	—	101	1	no	—
179	C₂×#131	—	101	1	no	—
180	#444	—	101	2	no	—
181	#444	—	101	2	no	—
182	#443	—	101	2	no	—
183	#443	—	101	2	no	—
184	#430	—	101	1	no	—
185	#421	—	101	1	no	—
186	#421	—	101	1	no	—
187	#405	—	101	1	no	—
188	C₂×#41	—	101	1	no	—
189	C₂×#41	—	101	1	no	—
190	#137	—	101	1	no	—
191	#137	—	101	2	no	—
192	#137	—	101	1	no	—
193	AGL(1,ℤ/8)	—	101	1	no	—
194	AGL(1,ℤ/8)	—	101	2	no	—
195	AGL(1,ℤ/8)	#87938	1	1	yes	characteristic, complemented
196	#135	—	101	2	no	—
197	#135	—	101	2	no	—
198	#131	—	101	2	no	—
199	#131	—	101	1	no	—
200	#131	—	101	2	no	—
201	#131	—	101	1	no	—
202	#131	—	101	1	no	—
203	#131	—	101	1	no	—
204	#131	—	202	1	no	—
205	#131	—	202	1	no	—
206	#131	—	202	1	no	—
207	#131	—	202	1	no	—
208	C₂×M₁₆	—	101	1	no	—
209	C₂×M₁₆	—	101	1	no	—
210	C₂×M₁₆	—	101	1	no	—
211	C₂×M₁₆	—	202	1	no	—
212	C₂×M₁₆	—	202	1	no	—
213	C₂²×C₈	—	101	1	no	—
214	C₂²×C₈	—	202	1	no	—
215	C₂²×C₈	—	202	1	no	—
216	#117	—	101	1	no	—
217	#108	—	101	2	no	—
218	#108	—	101	2	no	—
219	#104	—	101	2	no	—
220	#104	—	101	2	no	—
221	#104	—	101	2	no	—
222	#104	—	101	2	no	—
223	#97	—	101	2	no	—
224	C₄×C₈	—	101	2	no	—
225	#41	—	101	1	no	—
226	#41	—	101	2	no	—
227	#41	—	101	1	no	—
228	#41	—	101	2	no	—
229	#41	—	101	1	no	—
230	#41	—	202	1	no	—
231	#41	—	202	1	no	—
232	#41	—	202	1	no	—
233	#41	—	202	1	no	—
234	#41	C₂×#87938	1	1	yes	characteristic
235	C₂×Q₈	—	101	1	no	—
236	C₂×Q₈	—	101	1	no	—
237	C₂×D₄	—	101	1	no	—
238	C₂×D₄	—	202	1	no	—
239	C₂×D₄	—	202	1	no	—
240	C₂×D₄	C₂×#87938	1	1	yes	characteristic
241	C₂²×C₄	—	101	1	no	—
242	C₂²×C₄	—	202	1	no	—
243	C₂²×C₄	—	202	1	no	—
244	Q₁₆	—	101	2	no	—
245	Q₁₆	—	101	2	no	—
246	SD₁₆	—	101	2	no	—
247	SD₁₆	—	101	2	no	—
248	SD₁₆	—	101	2	no	—
249	SD₁₆	C₂×#87938	1	2	yes	—
250	D₈	—	101	2	no	—
251	D₈	C₂×#87938	1	2	yes	—
252	M₁₆	—	101	1	no	—
253	M₁₆	—	101	2	no	—
254	M₁₆	—	101	1	no	—
255	M₁₆	—	101	1	no	—
256	M₁₆	—	101	2	no	—
257	M₁₆	—	202	2	no	—
258	M₁₆	—	202	1	no	—
259	M₁₆	—	202	2	no	—
260	M₁₆	—	202	1	no	—
261	M₁₆	—	202	1	no	—
262	M₁₆	—	202	1	no	—
263	M₁₆	—	202	1	no	—
264	M₁₆	—	202	1	no	—
265	M₁₆	C₂×#87938	1	1	yes	characteristic
266	C₂×C₈	—	101	1	no	—
267	C₂×C₈	—	101	1	no	—
268	C₂×C₈	—	101	1	no	—
269	C₂×C₈	—	101	1	no	—
270	C₂×C₈	—	101	1	no	—
271	C₂×C₈	—	101	1	no	—
272	C₂×C₈	—	101	2	no	—
273	C₂×C₈	—	202	2	no	—
274	C₂×C₈	—	202	1	no	—
275	C₂×C₈	—	202	2	no	—
276	C₂×C₈	—	202	1	no	—
277	C₂×C₈	—	202	1	no	—
278	C₂×C₈	—	202	1	no	—
279	C₂×C₈	—	202	1	no	—
280	C₂×C₈	—	202	1	no	—
281	C₄⋊C₄	—	202	1	no	—
282	C₂²⋊C₄	—	202	1	no	—
283	C₄²	—	101	2	no	—
284	C₂³	—	2	1	no	—
285	C₂³	—	101	1	no	—
286	C₂³	—	202	1	no	—
287	Q₈	—	101	1	no	—
288	Q₈	—	101	2	no	—
289	Q₈	—	202	1	no	—
290	Q₈	—	202	1	no	—
291	Q₈	C₂²×#87938	1	1	yes	characteristic
292	D₄	—	2	1	no	—
293	D₄	—	2	1	no	—
294	D₄	—	101	1	no	—
295	D₄	—	202	1	no	—
296	D₄	—	202	1	no	—
297	D₄	—	202	1	no	—
298	D₄	—	202	2	no	—
299	D₄	—	202	2	no	—
300	D₄	—	202	1	no	—
301	D₄	C₂²×#87938	1	2	yes	—
302	D₄	C₂²×#87938	1	1	yes	characteristic
303	C₂×C₄	—	2	1	no	—
304	C₂×C₄	—	101	1	no	—
305	C₂×C₄	—	101	1	no	—
306	C₂×C₄	—	101	1	no	—
307	C₂×C₄	—	101	1	no	—
308	C₂×C₄	—	101	1	no	—
309	C₂×C₄	—	202	1	no	—
310	C₂×C₄	—	202	1	no	—
311	C₂×C₄	—	202	1	no	—
312	C₂×C₄	—	202	2	no	—
313	C₂×C₄	—	202	2	no	—
314	C₂×C₄	—	202	2	no	—
315	C₂×C₄	—	202	1	no	—
316	C₂×C₄	—	202	1	no	—
317	C₂×C₄	C₂²×#87938	1	1	yes	characteristic
318	C₈	—	101	1	no	—
319	C₈	—	101	1	no	—
320	C₈	—	101	1	no	—
321	C₈	—	101	1	no	—
322	C₈	—	101	2	no	—
323	C₈	—	202	2	no	—
324	C₈	—	202	1	no	—
325	C₈	—	202	1	no	—
326	C₈	C₂²×#87938	1	2	yes	—
327	C₂²	—	2	1	no	—
328	C₂²	—	2	2	no	—
329	C₂²	—	4	2	no	—
330	C₂²	—	101	1	no	—
331	C₂²	—	101	1	no	—
332	C₂²	—	404	1	no	—
333	C₂²	—	404	2	no	—
334	C₂²	—	202	1	no	—
335	C₂²	D₄×#87938	1	1	yes	characteristic
336	C₄	—	2	1	no	—
337	C₄	—	101	1	no	—
338	C₄	—	101	1	no	—
339	C₄	—	404	2	no	—
340	C₄	—	202	1	no	—
341	C₄	—	202	2	no	—
342	C₄	D₄×#87938	1	1	yes	characteristic
343	C₄	C₂³×#87938	1	1	yes	characteristic
344	C₂	—	2	1	no	—
345	C₂	—	4	2	no	—
346	C₂	—	4	1	no	—
347	C₂	—	404	1	no	—
348	C₂	—	202	1	no	—
349	C₂	—	202	1	no	—
350	C₂	C₂×D₄×#87938	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×D₄×#87938
commutator	C₄×C₁₀₁	C₂³×C₄
Frattini	C₄	C₂³×#87938
Fitting	AGL(1,ℤ/8)×C₁₀₁	C₄
socle	C₂×C₁₀₁	C₂×C₄×D₄
prime radical	#20265973	C₂
cosocle	C₄×D₁₀₁	C₂⁴
hypercenter	AGL(1,ℤ/8)	#87938
hypocenter	C₁₀₁	#2955
LCS	G ▷ C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ AGL(1,ℤ/8)

Complements

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

N	H ≅ G/N	type
1. D₁₀₁⋊#131	C₂	semidirect
2. C₂×#15692342	C₂	semidirect
3. #20266224	C₂	semidirect
4. #20266222	C₂	semidirect
5. #20266220	C₂	semidirect
6. #20266219	C₂	semidirect
7. #20266192	C₂	semidirect
8. #20266175	C₂	semidirect
9. #20266174	C₂	semidirect
10. #20266143	C₂	semidirect
12. #15692343	C₂²	semidirect
13. #15692343	C₂²	semidirect
18. #15692342	C₂²	semidirect
19. #15692342	C₂²	semidirect
20. #15692342	C₂²	semidirect
21. #15692342	C₂²	semidirect
30. #15692326	C₂²	semidirect
31. #15692325	C₂²	semidirect
32. #15692320	C₂²	semidirect
33. #15692319	C₂²	semidirect
34. #15692319	C₂²	semidirect
35. #15692319	C₂²	semidirect
36. #15692309	C₂²	semidirect
37. C₈×#87938	C₂²	semidirect
40. #15692288	C₄	semidirect
41. #15692285	C₄	semidirect
42. #15692284	C₄	semidirect
43. #15692281	C₄	semidirect
44. #15692279	C₄	semidirect
48. AGL(1,ℤ/8)×C₁₀₁	C₄	semidirect
176. C₁₀₁	#2955	semidirect
195. AGL(1,ℤ/8)	#87938	semidirect

Maximal subgroups

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

subgroup	index
D₁₀₁⋊#131	2
C₂×#15692342	2
#20266224	2
#20266222	2
#20266220	2
#20266219	2
#20266192	2
#20266175	2
#20266174	2
#20266143	2
#20265973	2
#2955	101