Subgroups — #27187870

Order 12928.

544 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₁₀₁⋊M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×#13108083	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	#20266180	C₂	1	1	yes	maximal, characteristic, complemented
8	#20266149	C₂	1	1	yes	maximal, characteristic, complemented
9	#20266147	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×#13108083	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×#15692215	C₂	1	1	yes	maximal, characteristic
12	C₂×#13108084	C₂²	1	2	yes	complemented
13	D₁₀₁⋊M₁₆	—	2	4	no	—
14	C₂×#13108083	—	2	4	no	—
15	C₂×#13108083	C₂²	1	1	yes	characteristic
16	C₂×#13108083	C₂²	1	1	yes	characteristic
17	C₂×#13108083	C₂²	1	1	yes	characteristic
18	C₂×#13108083	C₂²	1	1	yes	characteristic
19	C₂×#13108083	C₂²	1	2	yes	complemented
20	C₂×D₁₀₁⋊C₈	C₂²	1	2	yes	complemented
21	C₂²×#11717581	C₂²	1	2	yes	complemented
22	#15692322	C₂²	1	2	yes	complemented
23	C₄⋊#11717581	C₂²	1	2	yes	complemented
24	#15692312	C₂²	1	2	yes	complemented
25	#15692312	C₂²	1	2	yes	complemented
26	C₂²⋊#11717581	C₂²	1	2	yes	complemented
27	C₂²⋊#11717581	C₂²	1	2	yes	complemented
28	#15692310	C₂²	1	2	yes	complemented
29	C₄×#11717581	C₂²	1	2	yes	complemented
30	C₂²×C₂²⋊D₁₀₁	C₄	1	1	yes	characteristic
31	C₂³×Dic₁₀₁	C₂²	1	1	yes	characteristic
32	C₂²×C₄×D₁₀₁	C₂²	1	1	yes	characteristic
33	#15692215	C₄	1	1	yes	characteristic
34	#15692215	C₄	1	1	yes	characteristic
35	#15692215	C₄	1	1	yes	characteristic
36	#15692215	C₄	1	1	yes	characteristic
37	#15692215	C₂²	1	1	yes	characteristic
38	#15692215	C₂²	1	1	yes	characteristic
39	#15692215	C₂²	1	1	yes	characteristic
40	#15692215	C₂²	1	1	yes	characteristic
41	C₂×#13108053	C₄	1	1	yes	characteristic
42	C₂×#13108051	C₄	1	1	yes	characteristic
43	C₂×C₄×Dic₁₀₁	C₂²	1	1	yes	characteristic
44	C₂×C₂²⋊C₄×C₁₀₁	C₄	1	1	yes	characteristic
45	#13108084	—	2	4	no	—
46	#13108083	—	2	2	no	—
47	#13108083	—	2	2	no	—
48	#13108083	—	2	2	no	—
49	#13108083	—	2	4	no	—
50	#13108083	D₄	1	4	yes	complemented
51	D₁₀₁⋊C₈	—	2	4	no	—
52	C₂×#11717581	—	2	4	no	—
53	C₂×#11717581	C₂³	1	2	yes	—
54	C₂×#11717581	C₂³	1	2	yes	—
55	C₂×#11717581	C₂³	1	2	yes	—
56	C₂×#11717581	C₂³	1	2	yes	—
57	C₂³×D₁₀₁	C₂×C₄	1	1	yes	characteristic
58	C₂×C₂²⋊D₁₀₁	—	2	4	no	—
59	C₂×C₂²⋊D₁₀₁	C₂×C₄	1	1	yes	characteristic
60	C₂×C₂²⋊D₁₀₁	C₂×C₄	1	1	yes	characteristic
61	C₂×C₂²⋊D₁₀₁	C₂×C₄	1	1	yes	characteristic
62	C₂×C₂²⋊D₁₀₁	C₂×C₄	1	1	yes	characteristic
63	C₂²×Dic₁₀₁	—	2	2	no	—
64	C₂²×Dic₁₀₁	—	2	2	no	—
65	C₂²×Dic₁₀₁	C₂×C₄	1	1	yes	characteristic
66	C₂²×Dic₁₀₁	C₂×C₄	1	1	yes	characteristic
67	C₂²×Dic₁₀₁	C₂×C₄	1	1	yes	characteristic
68	C₂²×Dic₁₀₁	C₂×C₄	1	1	yes	characteristic
69	C₂²×Dic₁₀₁	C₂³	1	1	yes	characteristic
70	C₂²×Dic₁₀₁	C₂³	1	1	yes	characteristic
71	C₂²×Dic₁₀₁	C₂³	1	1	yes	characteristic
72	C₂×C₄×D₁₀₁	—	2	2	no	—
73	C₂×C₄×D₁₀₁	—	2	2	no	—
74	C₂×C₄×D₁₀₁	C₂×C₄	1	1	yes	characteristic
75	C₂×C₄×D₁₀₁	C₂×C₄	1	1	yes	characteristic
76	C₂×C₄×D₁₀₁	C₂³	1	1	yes	characteristic
77	C₂×C₄×D₁₀₁	C₂³	1	1	yes	characteristic
78	#13108053	C₂×C₄	1	1	yes	characteristic
79	#13108053	C₂×C₄	1	1	yes	characteristic
80	#13108053	C₂×C₄	1	1	yes	characteristic
81	#13108053	C₂×C₄	1	1	yes	characteristic
82	#13108051	C₂×C₄	1	1	yes	characteristic
83	#13108051	C₂×C₄	1	1	yes	characteristic
84	#13108051	C₂×C₄	1	1	yes	characteristic
85	#13108051	C₂×C₄	1	1	yes	characteristic
86	C₄×Dic₁₀₁	C₂×C₄	1	1	yes	characteristic
87	C₄×Dic₁₀₁	C₂×C₄	1	1	yes	characteristic
88	C₄×Dic₁₀₁	C₂³	1	1	yes	characteristic
89	C₄×Dic₁₀₁	C₂³	1	1	yes	characteristic
90	C₂⁴×C₁₀₁	C₂×C₄	1	1	yes	characteristic
91	C₂²×C₄×C₁₀₁	C₂×C₄	1	1	yes	characteristic
92	C₂²×C₄×C₁₀₁	C₂×C₄	1	1	yes	characteristic
93	C₂²⋊C₄×C₁₀₁	C₂×C₄	1	1	yes	characteristic
94	C₂²⋊C₄×C₁₀₁	C₂×C₄	1	1	yes	characteristic
95	C₂²⋊C₄×C₁₀₁	C₂×C₄	1	1	yes	characteristic
96	C₂²⋊C₄×C₁₀₁	C₂×C₄	1	1	yes	characteristic
97	C₂³×C₁₀₁	—	2	2	no	—
98	C₂³×C₁₀₁	—	2	2	no	—
99	C₂³×C₁₀₁	—	4	1	no	—
100	C₂³×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
101	C₂³×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
102	C₂³×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
103	C₂²×D₁₀₁	—	2	2	no	—
104	C₂²×D₁₀₁	—	2	2	no	—
105	C₂²×D₁₀₁	—	4	1	no	—
106	C₂²×D₁₀₁	C₂²×C₄	1	1	yes	characteristic
107	C₂²×D₁₀₁	C₂²×C₄	1	1	yes	characteristic
108	C₂×C₄×C₁₀₁	—	2	2	no	—
109	C₂×C₄×C₁₀₁	—	4	1	no	—
110	C₂×C₄×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
111	C₂×C₄×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
112	C₂×C₄×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
113	C₂×C₄×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
114	C₂²⋊D₁₀₁	—	2	2	no	—
115	C₂²⋊D₁₀₁	—	2	2	no	—
116	C₂²⋊D₁₀₁	—	2	2	no	—
117	C₂²⋊D₁₀₁	M₁₆	1	4	yes	complemented
118	C₂×Dic₁₀₁	—	2	2	no	—
119	C₂×Dic₁₀₁	—	2	2	no	—
120	C₂×Dic₁₀₁	—	2	2	no	—
121	C₂×Dic₁₀₁	—	2	2	no	—
122	C₂×Dic₁₀₁	—	2	1	no	—
123	C₂×Dic₁₀₁	—	2	1	no	—
124	C₂×Dic₁₀₁	—	2	1	no	—
125	C₂×Dic₁₀₁	—	2	1	no	—
126	C₂×Dic₁₀₁	C₂²×C₄	1	1	yes	characteristic
127	C₂×Dic₁₀₁	C₂²×C₄	1	1	yes	characteristic
128	C₂×Dic₁₀₁	C₂²×C₄	1	1	yes	characteristic
129	C₂×Dic₁₀₁	C₂²×C₄	1	1	yes	characteristic
130	C₂×Dic₁₀₁	C₂²×C₄	1	1	yes	characteristic
131	C₂×Dic₁₀₁	C₂×D₄	1	2	yes	—
132	C₂×Dic₁₀₁	#41	1	2	yes	—
133	C₂×Dic₁₀₁	C₂⁴	1	1	yes	characteristic
134	C₄×D₁₀₁	—	2	2	no	—
135	C₄×D₁₀₁	—	2	2	no	—
136	C₄×D₁₀₁	—	2	1	no	—
137	C₄×D₁₀₁	—	2	1	no	—
138	C₄×D₁₀₁	—	2	1	no	—
139	C₄×D₁₀₁	—	2	1	no	—
140	#11717581	—	2	2	no	—
141	#11717581	—	2	2	no	—
142	#11717581	—	2	2	no	—
143	#11717581	C₂×D₄	1	4	yes	complemented
144	C₂²×C₁₀₁	—	2	1	no	—
145	C₂²×C₁₀₁	—	2	1	no	—
146	C₂²×C₁₀₁	—	2	1	no	—
147	C₂²×C₁₀₁	—	2	1	no	—
148	C₂²×C₁₀₁	—	2	1	no	—
149	C₂²×C₁₀₁	—	2	1	no	—
150	C₂²×C₁₀₁	—	2	1	no	—
151	C₂²×C₁₀₁	—	4	4	no	—
152	C₂²×C₁₀₁	C₄×D₄	1	2	yes	—
153	C₂²×C₁₀₁	C₂×M₁₆	1	2	yes	complemented
154	C₂²×C₁₀₁	C₂³×C₄	1	1	yes	characteristic
155	C₂×D₁₀₁	—	2	1	no	—
156	C₂×D₁₀₁	—	2	1	no	—
157	C₂×D₁₀₁	—	2	1	no	—
158	C₂×D₁₀₁	—	2	1	no	—
159	C₂×D₁₀₁	—	2	1	no	—
160	C₂×D₁₀₁	—	4	4	no	—
161	C₂×D₁₀₁	C₂×M₁₆	1	2	yes	complemented
162	C₄×C₁₀₁	—	2	1	no	—
163	C₄×C₁₀₁	—	2	1	no	—
164	C₄×C₁₀₁	—	2	1	no	—
165	C₄×C₁₀₁	—	2	1	no	—
166	Dic₁₀₁	—	2	1	no	—
167	Dic₁₀₁	—	2	1	no	—
168	Dic₁₀₁	—	2	1	no	—
169	Dic₁₀₁	C₄×D₄	1	2	yes	—
170	Dic₁₀₁	C₂×M₁₆	1	2	yes	complemented
171	Dic₁₀₁	C₂²×D₄	1	1	yes	characteristic
172	Dic₁₀₁	C₂×#41	1	1	yes	characteristic
173	C₂×C₁₀₁	—	2	2	no	—
174	C₂×C₁₀₁	—	2	2	no	—
175	C₂×C₁₀₁	—	4	1	no	—
176	C₂×C₁₀₁	C₂×C₄×D₄	1	1	yes	characteristic
177	C₂×C₁₀₁	C₂²×M₁₆	1	1	yes	characteristic
178	C₂×C₁₀₁	#568	1	1	yes	characteristic
179	D₁₀₁	—	2	2	no	—
180	D₁₀₁	—	4	1	no	—
181	D₄×M₁₆	—	101	1	no	maximal, Sylow
182	C₁₀₁	D₄×M₁₆	1	1	yes	characteristic, complemented, Sylow
183	C₂²×M₁₆	—	101	1	no	—
184	C₂²×M₁₆	—	101	1	no	—
185	C₂×C₄×D₄	—	101	1	no	—
186	#436	—	101	2	no	—
187	#435	—	101	2	no	—
188	#435	—	101	2	no	—
189	C₈×D₄	—	101	2	no	—
190	#423	—	101	1	no	—
191	#407	—	101	1	no	—
192	#407	—	101	1	no	—
193	C₄×M₁₆	—	101	1	no	—
194	C₂²×D₄	—	101	1	no	—
195	C₂³×C₄	—	101	1	no	—
196	C₂³×C₄	—	101	1	no	—
197	C₂×M₁₆	—	101	1	no	—
198	C₂×M₁₆	—	101	1	no	—
199	C₂×M₁₆	—	101	1	no	—
200	C₂×M₁₆	—	101	1	no	—
201	C₂×M₁₆	—	101	2	no	—
202	C₂×M₁₆	—	101	2	no	—
203	C₂×M₁₆	—	202	4	no	—
204	C₂×M₁₆	—	202	4	no	—
205	C₂²×C₈	—	101	2	no	—
206	C₂²×C₈	—	101	2	no	—
207	C₄×D₄	—	101	1	no	—
208	C₄×D₄	—	101	1	no	—
209	C₄×D₄	—	101	1	no	—
210	C₄×D₄	—	101	1	no	—
211	C₄×D₄	—	101	1	no	—
212	C₄×D₄	—	101	1	no	—
213	C₄×D₄	—	101	1	no	—
214	C₄×D₄	—	101	1	no	—
215	C₂×C₄⋊C₄	—	101	1	no	—
216	C₂×C₂²⋊C₄	—	101	1	no	—
217	C₂×C₂²⋊C₄	#87938	1	1	yes	characteristic
218	C₂×C₄²	—	101	1	no	—
219	C₄⋊C₈	—	101	2	no	—
220	C₄⋊C₈	—	101	2	no	—
221	C₂²⋊C₈	—	101	2	no	—
222	C₂²⋊C₈	—	101	2	no	—
223	C₂²⋊C₈	—	101	2	no	—
224	C₂²⋊C₈	—	101	2	no	—
225	#97	—	101	2	no	—
226	C₄×C₈	—	101	2	no	—
227	C₂⁴	—	101	1	no	—
228	C₂⁴	C₂×#87938	1	1	yes	characteristic
229	C₂×D₄	—	101	1	no	—
230	C₂×D₄	—	101	1	no	—
231	C₂×D₄	—	101	1	no	—
232	C₂×D₄	—	101	1	no	—
233	C₂×D₄	—	202	4	no	—
234	C₂²×C₄	—	101	1	no	—
235	C₂²×C₄	—	101	1	no	—
236	C₂²×C₄	—	101	1	no	—
237	C₂²×C₄	—	101	1	no	—
238	C₂²×C₄	—	101	1	no	—
239	C₂²×C₄	—	101	1	no	—
240	C₂²×C₄	—	101	1	no	—
241	C₂²×C₄	—	101	1	no	—
242	C₂²×C₄	—	101	1	no	—
243	C₂²×C₄	—	101	1	no	—
244	C₂²×C₄	—	101	1	no	—
245	C₂²×C₄	—	202	2	no	—
246	C₂²×C₄	—	202	2	no	—
247	C₂²×C₄	—	202	2	no	—
248	C₂²×C₄	—	202	2	no	—
249	C₂²×C₄	C₂×#87938	1	1	yes	characteristic
250	C₂²×C₄	C₂×#87938	1	1	yes	characteristic
251	M₁₆	—	101	4	no	—
252	M₁₆	—	202	4	no	—
253	M₁₆	—	202	4	no	—
254	M₁₆	—	202	2	no	—
255	M₁₆	—	202	2	no	—
256	M₁₆	—	202	2	no	—
257	C₂×C₈	—	101	2	no	—
258	C₂×C₈	—	101	2	no	—
259	C₂×C₈	—	101	2	no	—
260	C₂×C₈	—	101	2	no	—
261	C₂×C₈	—	202	4	no	—
262	C₂×C₈	—	202	4	no	—
263	C₄⋊C₄	—	101	1	no	—
264	C₄⋊C₄	—	101	1	no	—
265	C₄⋊C₄	—	101	1	no	—
266	C₄⋊C₄	—	101	1	no	—
267	C₂²⋊C₄	—	101	1	no	—
268	C₂²⋊C₄	—	101	1	no	—
269	C₂²⋊C₄	—	101	1	no	—
270	C₂²⋊C₄	—	101	1	no	—
271	C₂²⋊C₄	C₂×#87938	1	1	yes	characteristic
272	C₂²⋊C₄	C₂×#87938	1	1	yes	characteristic
273	C₂²⋊C₄	C₂×#87938	1	1	yes	characteristic
274	C₂²⋊C₄	C₂×#87938	1	1	yes	characteristic
275	C₄²	—	101	1	no	—
276	C₄²	—	101	1	no	—
277	C₄²	—	101	1	no	—
278	C₄²	—	101	1	no	—
279	C₂³	—	2	2	no	—
280	C₂³	—	2	2	no	—
281	C₂³	—	4	1	no	—
282	C₂³	—	101	1	no	—
283	C₂³	—	101	1	no	—
284	C₂³	—	404	1	no	—
285	C₂³	—	202	2	no	—
286	C₂³	—	202	2	no	—
287	C₂³	C₂²×#87938	1	1	yes	characteristic
288	C₂³	C₂²×#87938	1	1	yes	characteristic
289	C₂³	C₂²×#87938	1	1	yes	characteristic
290	D₄	—	101	4	no	—
291	D₄	—	202	2	no	—
292	D₄	—	202	2	no	—
293	D₄	—	202	2	no	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	4	1	no	—
296	C₂×C₄	—	101	2	no	—
297	C₂×C₄	—	101	2	no	—
298	C₂×C₄	—	101	1	no	—
299	C₂×C₄	—	101	1	no	—
300	C₂×C₄	—	101	1	no	—
301	C₂×C₄	—	101	1	no	—
302	C₂×C₄	—	101	1	no	—
303	C₂×C₄	—	101	1	no	—
304	C₂×C₄	—	202	1	no	—
305	C₂×C₄	—	202	1	no	—
306	C₂×C₄	—	202	1	no	—
307	C₂×C₄	—	202	1	no	—
308	C₂×C₄	—	202	2	no	—
309	C₂×C₄	—	202	2	no	—
310	C₂×C₄	—	202	1	no	—
311	C₂×C₄	—	202	1	no	—
312	C₂×C₄	—	202	1	no	—
313	C₂×C₄	—	202	1	no	—
314	C₂×C₄	—	202	2	no	—
315	C₂×C₄	—	202	2	no	—
316	C₂×C₄	—	202	2	no	—
317	C₂×C₄	—	202	2	no	—
318	C₂×C₄	C₂²×#87938	1	1	yes	characteristic
319	C₂×C₄	C₂²×#87938	1	1	yes	characteristic
320	C₂×C₄	C₂²×#87938	1	1	yes	characteristic
321	C₂×C₄	C₂²×#87938	1	1	yes	characteristic
322	C₈	—	101	4	no	—
323	C₈	—	202	2	no	—
324	C₈	—	202	2	no	—
325	C₈	—	202	2	no	—
326	C₂²	—	2	1	no	—
327	C₂²	—	2	1	no	—
328	C₂²	—	2	1	no	—
329	C₂²	—	2	1	no	—
330	C₂²	—	2	1	no	—
331	C₂²	—	2	1	no	—
332	C₂²	—	2	1	no	—
333	C₂²	—	4	4	no	—
334	C₂²	—	101	2	no	—
335	C₂²	—	404	4	no	—
336	C₂²	—	202	1	no	—
337	C₂²	—	202	1	no	—
338	C₂²	—	202	1	no	—
339	C₂²	—	202	1	no	—
340	C₂²	—	202	1	no	—
341	C₂²	D₄×#87938	1	2	yes	—
342	C₂²	D₁₀₁⋊M₁₆	1	2	yes	complemented
343	C₂²	C₂³×#87938	1	1	yes	characteristic
344	C₄	—	2	1	no	—
345	C₄	—	2	1	no	—
346	C₄	—	2	1	no	—
347	C₄	—	2	1	no	—
348	C₄	—	101	1	no	—
349	C₄	—	101	1	no	—
350	C₄	—	101	2	no	—
351	C₄	—	101	2	no	—
352	C₄	—	202	1	no	—
353	C₄	—	202	1	no	—
354	C₄	—	202	1	no	—
355	C₂	—	2	2	no	—
356	C₂	—	2	2	no	—
357	C₂	—	4	1	no	—
358	C₂	—	404	1	no	—
359	C₂	—	202	2	no	—
360	C₂	C₂×D₄×#87938	1	1	yes	characteristic
361	C₂	C₂×D₁₀₁⋊M₁₆	1	1	yes	characteristic
362	C₂	#20266255	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×#87938
commutator, socle	C₂²×C₁₀₁	C₂³×C₄
Fitting	C₂×C₂²⋊C₄×C₁₀₁	C₄
prime radical	C₂²×C₂²⋊D₁₀₁	C₄
cosocle	C₂×Dic₁₀₁	C₂⁴
hypercenter	C₂×C₂²⋊C₄	#87938
hypocenter	C₁₀₁	D₄×M₁₆
LCS	G ▷ C₂²×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂² ◁ C₂×C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
1. C₂×D₁₀₁⋊M₁₆	C₂	semidirect
2. C₂²×#13108083	C₂	semidirect
3. #20266210	C₂	semidirect
4. #20266209	C₂	semidirect
5. #20266206	C₂	semidirect
6. #20266204	C₂	semidirect
7. #20266180	C₂	semidirect
8. #20266149	C₂	semidirect
9. #20266147	C₂	semidirect
10. C₄×#13108083	C₂	semidirect
12. C₂×#13108084	C₂²	semidirect
19. C₂×#13108083	C₂²	semidirect
20. C₂×D₁₀₁⋊C₈	C₂²	semidirect
21. C₂²×#11717581	C₂²	semidirect
22. #15692322	C₂²	semidirect
23. C₄⋊#11717581	C₂²	semidirect
24. #15692312	C₂²	semidirect
25. #15692312	C₂²	semidirect
26. C₂²⋊#11717581	C₂²	semidirect
27. C₂²⋊#11717581	C₂²	semidirect
28. #15692310	C₂²	semidirect
29. C₄×#11717581	C₂²	semidirect
50. #13108083	D₄	semidirect
117. C₂²⋊D₁₀₁	M₁₆	semidirect
143. #11717581	C₂×D₄	semidirect
153. C₂²×C₁₀₁	C₂×M₁₆	semidirect
161. C₂×D₁₀₁	C₂×M₁₆	semidirect
170. Dic₁₀₁	C₂×M₁₆	semidirect
182. C₁₀₁	D₄×M₁₆	semidirect
342. C₂²	D₁₀₁⋊M₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₁₀₁⋊M₁₆	2
C₂²×#13108083	2
#20266210	2
#20266209	2
#20266206	2
#20266204	2
#20266180	2
#20266149	2
#20266147	2
C₄×#13108083	2
C₂×#15692215	2
D₄×M₁₆	101