Subgroups — #27179647

Order 12928.

772 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#41×D₁₀₁	C₂	1	1	yes	maximal, characteristic
2	C₂²×#13108057	C₂	1	1	yes	maximal, characteristic, complemented
3	#117×D₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
4	#20265184	C₂	1	2	yes	maximal, complemented
5	#20265180	C₂	1	2	yes	maximal, complemented
6	#20265177	C₂	1	2	yes	maximal, complemented
7	#20265175	C₂	1	2	yes	maximal, complemented
8	#20265174	C₂	1	1	yes	maximal, characteristic, complemented
9	#20265173	C₂	1	1	yes	maximal, characteristic, complemented
10	#20265171	C₂	1	1	yes	maximal, characteristic, complemented
11	#416×C₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13108075	C₂²	1	1	yes	characteristic
13	C₂×#13108074	—	2	1	no	—
14	C₂×#13108074	C₂²	1	1	yes	characteristic
15	C₂×#13108073	—	2	1	no	—
16	C₂×#13108073	C₂²	1	1	yes	characteristic
17	#41×D₁₀₁	—	2	4	no	—
18	#41×D₁₀₁	C₄	1	8	yes	complemented
19	C₂×Q₈×D₁₀₁	C₂²	1	1	yes	characteristic
20	C₂×D₄×D₁₀₁	—	2	1	no	—
21	C₂×D₄×D₁₀₁	C₂²	1	1	yes	characteristic
22	C₂²×C₄×D₁₀₁	—	2	1	no	—
23	C₂²×C₄×D₁₀₁	C₂²	1	1	yes	characteristic
24	C₂×#13108057	—	2	4	no	—
25	C₂×#13108057	C₂²	1	2	yes	—
26	C₂×#13108057	C₂²	1	2	yes	complemented
27	C₂²×#11717579	C₂²	1	1	yes	characteristic, complemented
28	#15692214	C₂²	1	1	yes	characteristic, complemented
29	#15692213	C₂²	1	2	yes	—
30	#15692212	—	2	1	no	—
31	#15692211	—	2	1	no	—
32	C₄⋊C₄×D₁₀₁	C₂²	1	2	yes	complemented
33	C₂²⋊C₄×D₁₀₁	—	2	1	no	—
34	C₄²×D₁₀₁	—	2	1	no	—
35	#15692180	C₂²	1	2	yes	complemented
36	#15692179	C₂²	1	2	yes	complemented
37	#15692178	C₂²	1	2	yes	complemented
38	#15692177	C₂²	1	2	yes	complemented
39	#15692176	C₂²	1	2	yes	complemented
40	#15692175	C₂²	1	2	yes	complemented
41	C₂×#41×C₁₀₁	C₂²	1	1	yes	characteristic
42	C₂²×C₈×C₁₀₁	C₂²	1	1	yes	characteristic, complemented
43	#117×C₁₀₁	C₂²	1	1	yes	characteristic, complemented
44	#103×C₁₀₁	C₂²	1	2	yes	complemented
45	#102×C₁₀₁	C₂²	1	2	yes	complemented
46	C₂³×D₁₀₁	—	2	1	no	—
47	C₂³×D₁₀₁	D₄	1	1	yes	characteristic
48	#13108075	—	2	2	no	—
49	#13108075	C₂×C₄	1	4	yes	—
50	#13108074	—	2	2	no	—
51	#13108074	—	2	8	no	—
52	#13108074	C₂×C₄	1	4	yes	complemented
53	#13108073	—	2	8	no	—
54	#13108073	—	2	2	no	—
55	#13108073	C₂×C₄	1	4	yes	—
56	C₂×Dic₂₀₂	—	2	1	no	—
57	C₂×Dic₂₀₂	C₂³	1	1	yes	characteristic
58	Q₈×D₁₀₁	—	2	2	no	—
59	Q₈×D₁₀₁	C₂×C₄	1	4	yes	complemented
60	C₂×D₄₀₄	—	2	1	no	—
61	C₂×D₄₀₄	C₂³	1	1	yes	characteristic
62	C₂×C₂²⋊D₁₀₁	—	2	1	no	—
63	C₂×C₂²⋊D₁₀₁	—	2	1	no	—
64	C₂×C₂²⋊D₁₀₁	—	2	1	no	—
65	D₄×D₁₀₁	—	2	8	no	—
66	D₄×D₁₀₁	—	2	2	no	—
67	D₄×D₁₀₁	C₂×C₄	1	4	yes	complemented
68	C₂²×Dic₁₀₁	—	2	1	no	—
69	C₂²×Dic₁₀₁	D₄	1	1	yes	characteristic
70	C₂×C₄×D₁₀₁	—	2	1	no	—
71	C₂×C₄×D₁₀₁	—	2	1	no	—
72	C₂×C₄×D₁₀₁	—	2	1	no	—
73	C₂×C₄×D₁₀₁	—	2	8	no	—
74	C₂×C₄×D₁₀₁	—	2	2	no	—
75	C₂×C₄×D₁₀₁	C₂×C₄	1	4	yes	—
76	C₂×C₄×D₁₀₁	D₄	1	1	yes	characteristic
77	C₂×C₄×D₁₀₁	D₄	1	1	yes	characteristic
78	C₂×C₄×D₁₀₁	D₄	1	4	yes	—
79	C₂×C₄×D₁₀₁	C₂³	1	1	yes	characteristic
80	C₂×C₄×D₁₀₁	C₂³	1	1	yes	characteristic
81	#13108057	—	2	4	no	—
82	#13108057	—	2	4	no	—
83	C₂×#11717579	—	2	2	no	—
84	C₂×#11717579	C₂³	1	2	yes	—
85	C₄⋊Dic₁₀₁	C₂³	1	2	yes	—
86	#13108053	—	2	2	no	—
87	C₂²⋊Dic₁₀₁	—	2	1	no	—
88	#13108051	—	2	2	no	—
89	C₄×Dic₁₀₁	—	2	1	no	—
90	C₄×Dic₁₀₁	—	2	2	no	—
91	#41×C₁₀₁	—	2	2	no	—
92	#41×C₁₀₁	C₂×C₄	1	4	yes	complemented
93	C₂×Q₈×C₁₀₁	C₂³	1	1	yes	characteristic
94	C₂×D₄×C₁₀₁	—	2	1	no	—
95	C₂×D₄×C₁₀₁	C₂³	1	1	yes	characteristic
96	C₂²×C₄×C₁₀₁	—	2	1	no	—
97	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
98	C₂×C₈×C₁₀₁	—	2	2	no	—
99	C₂×C₈×C₁₀₁	C₂³	1	2	yes	—
100	C₄⋊C₄×C₁₀₁	C₂³	1	2	yes	—
101	C₂²⋊C₄×C₁₀₁	—	2	1	no	—
102	C₄²×C₁₀₁	—	2	1	no	—
103	C₂³×C₁₀₁	—	2	1	no	—
104	C₂³×C₁₀₁	C₂×D₄	1	1	yes	characteristic
105	C₂²×D₁₀₁	—	2	1	no	—
106	C₂²×D₁₀₁	—	2	4	no	—
107	C₂²×D₁₀₁	—	4	2	no	—
108	C₂²×D₁₀₁	—	4	4	no	—
109	C₂²×D₁₀₁	C₂²⋊C₄	1	4	yes	—
110	C₂²×D₁₀₁	C₂×D₄	1	1	yes	characteristic
111	C₂²×D₁₀₁	C₂×D₄	1	1	yes	characteristic
112	Q₈×C₁₀₁	—	2	1	no	—
113	Q₈×C₁₀₁	C₂²×C₄	1	2	yes	—
114	D₄×C₁₀₁	—	2	4	no	—
115	D₄×C₁₀₁	—	2	1	no	—
116	D₄×C₁₀₁	C₂²×C₄	1	2	yes	—
117	C₂×C₄×C₁₀₁	—	2	1	no	—
118	C₂×C₄×C₁₀₁	—	2	4	no	—
119	C₂×C₄×C₁₀₁	—	2	2	no	—
120	C₂×C₄×C₁₀₁	C₂²×C₄	1	2	yes	—
121	C₂×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
122	C₂×C₄×C₁₀₁	C₂×D₄	1	1	yes	characteristic
123	C₂×C₄×C₁₀₁	C₂⁴	1	1	yes	characteristic
124	C₂²⋊D₁₀₁	—	2	4	no	—
125	C₂²⋊D₁₀₁	—	2	4	no	—
126	C₂²⋊D₁₀₁	—	2	4	no	—
127	C₂×Dic₁₀₁	—	2	1	no	—
128	C₂×Dic₁₀₁	—	2	4	no	—
129	C₂×Dic₁₀₁	—	2	2	no	—
130	C₂×Dic₁₀₁	C₂²⋊C₄	1	4	yes	—
131	C₂×Dic₁₀₁	C₂×D₄	1	1	yes	characteristic
132	C₂×Dic₁₀₁	C₂×D₄	1	1	yes	characteristic
133	D₄₀₄	—	2	4	no	—
134	D₄₀₄	—	2	1	no	—
135	D₄₀₄	C₂²×C₄	1	2	yes	—
136	C₄×D₁₀₁	—	2	4	no	—
137	C₄×D₁₀₁	—	2	4	no	—
138	C₄×D₁₀₁	—	2	4	no	—
139	C₄×D₁₀₁	—	4	4	no	—
140	C₄×D₁₀₁	C₂²⋊C₄	1	4	yes	—
141	C₄×D₁₀₁	C₂²⋊C₄	1	4	yes	—
142	C₄×D₁₀₁	C₂²×C₄	1	2	yes	—
143	C₄×D₁₀₁	C₂²×C₄	1	2	yes	—
144	C₄×D₁₀₁	C₂×D₄	1	2	yes	—
145	C₄×D₁₀₁	C₂×D₄	1	2	yes	—
146	Dic₂₀₂	—	2	1	no	—
147	Dic₂₀₂	—	2	4	no	—
148	Dic₂₀₂	C₂²×C₄	1	2	yes	—
149	C₈×C₁₀₁	—	2	2	no	—
150	#11717579	—	2	2	no	—
151	C₂²×C₁₀₁	—	2	2	no	—
152	C₂²×C₁₀₁	—	2	2	no	—
153	C₂²×C₁₀₁	—	4	2	no	—
154	C₂²×C₁₀₁	C₂×C₂²⋊C₄	1	2	yes	—
155	C₂²×C₁₀₁	C₂²×D₄	1	1	yes	characteristic
156	C₂×D₁₀₁	—	2	2	no	—
157	C₂×D₁₀₁	—	2	2	no	—
158	C₂×D₁₀₁	—	2	2	no	—
159	C₂×D₁₀₁	—	4	4	no	—
160	C₂×D₁₀₁	—	4	8	no	—
161	C₂×D₁₀₁	—	4	2	no	—
162	C₂×D₁₀₁	C₂×C₂²⋊C₄	1	2	yes	—
163	C₂×D₁₀₁	C₂×C₂²⋊C₄	1	2	yes	—
164	C₄×C₁₀₁	—	2	2	no	—
165	C₄×C₁₀₁	—	4	2	no	—
166	C₄×C₁₀₁	C₂×C₂²⋊C₄	1	2	yes	—
167	C₄×C₁₀₁	C₂³×C₄	1	1	yes	characteristic
168	C₄×C₁₀₁	C₂²×D₄	1	1	yes	characteristic
169	Dic₁₀₁	—	2	2	no	—
170	Dic₁₀₁	—	4	2	no	—
171	Dic₁₀₁	C₂×C₂²⋊C₄	1	2	yes	—
172	Dic₁₀₁	C₂×C₂²⋊C₄	1	2	yes	—
173	C₂×C₁₀₁	—	2	2	no	—
174	C₂×C₁₀₁	—	4	2	no	—
175	C₂×C₁₀₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
176	C₂×C₁₀₁	#575	1	2	yes	—
177	D₁₀₁	—	2	2	no	—
178	D₁₀₁	—	2	2	no	—
179	D₁₀₁	—	4	2	no	—
180	#2896	—	101	1	no	maximal, Sylow
181	C₁₀₁	#2896	1	1	yes	characteristic, complemented, Sylow
182	C₂²×#41	—	101	1	no	—
183	C₂²×M₁₆	—	101	1	no	—
184	C₂×#117	—	101	1	no	—
185	#419	—	101	2	no	—
186	#418	—	101	2	no	—
187	#417	—	101	2	no	—
188	#417	—	101	2	no	—
189	#416	—	101	1	no	—
190	#416	—	101	1	no	—
191	#416	—	101	1	no	—
192	#416	D₁₀₁	1	1	yes	characteristic, complemented
193	C₂×#41	—	101	8	no	—
194	C₂×#41	—	101	1	no	—
195	C₂×#41	—	101	1	no	—
196	C₂×#41	—	101	1	no	—
197	C₂×#41	—	202	1	no	—
198	C₂×#41	—	202	1	no	—
199	C₂×#41	—	202	4	no	—
200	C₂×#41	C₂×D₁₀₁	1	1	yes	characteristic
201	C₂²×Q₈	—	101	1	no	—
202	C₂²×D₄	—	101	1	no	—
203	C₂²×D₄	—	202	1	no	—
204	C₂³×C₄	—	101	1	no	—
205	C₂³×C₄	—	202	1	no	—
206	C₂×M₁₆	—	101	2	no	—
207	C₂×M₁₆	—	101	2	no	—
208	C₂×M₁₆	—	202	4	no	—
209	C₂²×C₈	—	101	1	no	—
210	C₂²×C₈	C₂×D₁₀₁	1	1	yes	characteristic, complemented
211	#117	—	101	1	no	—
212	#117	—	101	2	no	—
213	#117	—	202	1	no	—
214	#117	—	202	1	no	—
215	#117	C₂×D₁₀₁	1	1	yes	characteristic, complemented
216	C₂×C₄⋊C₄	—	101	2	no	—
217	C₂×C₂²⋊C₄	—	202	1	no	—
218	C₂×C₄²	—	202	1	no	—
219	#103	—	101	2	no	—
220	#103	—	101	2	no	—
221	#103	—	101	2	no	—
222	#103	C₂×D₁₀₁	1	2	yes	complemented
223	#102	—	101	2	no	—
224	#102	—	101	2	no	—
225	#102	—	101	2	no	—
226	#102	C₂×D₁₀₁	1	2	yes	complemented
227	C₂⁴	—	101	1	no	—
228	C₂⁴	—	202	1	no	—
229	#41	—	2	2	no	—
230	#41	—	101	4	no	—
231	#41	—	101	4	no	—
232	#41	—	101	4	no	—
233	#41	—	202	8	no	—
234	#41	—	202	8	no	—
235	#41	—	202	2	no	—
236	#41	—	202	2	no	—
237	#41	—	202	2	no	—
238	#41	C₄×D₁₀₁	1	4	yes	complemented
239	C₂×Q₈	—	101	4	no	—
240	C₂×Q₈	—	101	1	no	—
241	C₂×Q₈	—	202	1	no	—
242	C₂×Q₈	—	202	2	no	—
243	C₂×Q₈	C₂²×D₁₀₁	1	1	yes	characteristic
244	C₂×D₄	—	2	1	no	—
245	C₂×D₄	—	101	4	no	—
246	C₂×D₄	—	101	1	no	—
247	C₂×D₄	—	202	1	no	—
248	C₂×D₄	—	202	2	no	—
249	C₂×D₄	—	202	1	no	—
250	C₂×D₄	—	202	8	no	—
251	C₂×D₄	—	202	1	no	—
252	C₂×D₄	—	202	1	no	—
253	C₂×D₄	C₂²×D₁₀₁	1	1	yes	characteristic
254	C₂²×C₄	—	2	1	no	—
255	C₂²×C₄	—	101	1	no	—
256	C₂²×C₄	—	101	4	no	—
257	C₂²×C₄	—	101	4	no	—
258	C₂²×C₄	—	101	1	no	—
259	C₂²×C₄	—	101	1	no	—
260	C₂²×C₄	—	101	1	no	—
261	C₂²×C₄	—	101	1	no	—
262	C₂²×C₄	—	202	1	no	—
263	C₂²×C₄	—	202	8	no	—
264	C₂²×C₄	—	202	2	no	—
265	C₂²×C₄	—	202	1	no	—
266	C₂²×C₄	—	202	1	no	—
267	C₂²×C₄	—	202	1	no	—
268	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
269	M₁₆	—	202	4	no	—
270	M₁₆	—	202	4	no	—
271	C₂×C₈	—	2	2	no	—
272	C₂×C₈	—	101	2	no	—
273	C₂×C₈	—	202	2	no	—
274	C₂×C₈	C₂²×D₁₀₁	1	2	yes	—
275	C₄⋊C₄	—	101	2	no	—
276	C₄⋊C₄	—	202	2	no	—
277	C₄⋊C₄	C₂²×D₁₀₁	1	2	yes	—
278	C₂²⋊C₄	—	2	1	no	—
279	C₂²⋊C₄	—	202	2	no	—
280	C₂²⋊C₄	—	202	1	no	—
281	C₄²	—	2	1	no	—
282	C₄²	—	202	2	no	—
283	C₄²	—	202	1	no	—
284	C₂³	—	2	1	no	—
285	C₂³	—	101	4	no	—
286	C₂³	—	101	1	no	—
287	C₂³	—	101	1	no	—
288	C₂³	—	404	2	no	—
289	C₂³	—	404	4	no	—
290	C₂³	—	202	4	no	—
291	C₂³	—	202	1	no	—
292	C₂³	D₄×D₁₀₁	1	1	yes	characteristic
293	Q₈	—	2	1	no	—
294	Q₈	—	101	2	no	—
295	Q₈	—	202	4	no	—
296	Q₈	—	202	1	no	—
297	Q₈	C₂×C₄×D₁₀₁	1	2	yes	—
298	D₄	—	2	4	no	—
299	D₄	—	2	1	no	—
300	D₄	—	101	2	no	—
301	D₄	—	202	4	no	—
302	D₄	—	202	4	no	—
303	D₄	—	202	4	no	—
304	D₄	—	202	4	no	—
305	D₄	—	202	1	no	—
306	D₄	C₂×C₄×D₁₀₁	1	2	yes	—
307	C₂×C₄	—	2	1	no	—
308	C₂×C₄	—	2	4	no	—
309	C₂×C₄	—	2	2	no	—
310	C₂×C₄	—	101	1	no	—
311	C₂×C₄	—	101	1	no	—
312	C₂×C₄	—	101	4	no	—
313	C₂×C₄	—	101	2	no	—
314	C₂×C₄	—	101	2	no	—
315	C₂×C₄	—	101	2	no	—
316	C₂×C₄	—	101	2	no	—
317	C₂×C₄	—	101	4	no	—
318	C₂×C₄	—	101	4	no	—
319	C₂×C₄	—	404	4	no	—
320	C₂×C₄	—	202	4	no	—
321	C₂×C₄	—	202	4	no	—
322	C₂×C₄	—	202	4	no	—
323	C₂×C₄	—	202	2	no	—
324	C₂×C₄	—	202	1	no	—
325	C₂×C₄	—	202	4	no	—
326	C₂×C₄	C₂×C₄×D₁₀₁	1	2	yes	—
327	C₂×C₄	D₄×D₁₀₁	1	2	yes	—
328	C₂×C₄	D₄×D₁₀₁	1	1	yes	characteristic
329	C₂×C₄	C₂³×D₁₀₁	1	1	yes	characteristic
330	C₈	—	2	2	no	—
331	C₈	—	202	2	no	—
332	C₂²	—	2	2	no	—
333	C₂²	—	2	2	no	—
334	C₂²	—	4	2	no	—
335	C₂²	—	101	2	no	—
336	C₂²	—	101	2	no	—
337	C₂²	—	404	4	no	—
338	C₂²	—	404	8	no	—
339	C₂²	—	404	2	no	—
340	C₂²	—	202	2	no	—
341	C₂²	—	202	2	no	—
342	C₂²	—	202	2	no	—
343	C₂²	C₂²⋊C₄×D₁₀₁	1	2	yes	—
344	C₂²	C₂×D₄×D₁₀₁	1	1	yes	characteristic
345	C₄	—	2	2	no	—
346	C₄	—	4	2	no	—
347	C₄	—	101	2	no	—
348	C₄	—	101	2	no	—
349	C₄	—	404	2	no	—
350	C₄	—	202	2	no	—
351	C₄	C₂²⋊C₄×D₁₀₁	1	2	yes	—
352	C₄	C₂²×C₄×D₁₀₁	1	1	yes	characteristic
353	C₄	C₂×D₄×D₁₀₁	1	1	yes	characteristic
354	C₂	—	2	2	no	—
355	C₂	—	4	2	no	—
356	C₂	—	404	2	no	—
357	C₂	—	202	2	no	—
358	C₂	—	202	2	no	—
359	C₂	C₂×C₂²⋊C₄×D₁₀₁	1	1	yes	characteristic
360	C₂	#20265974	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	D₄×D₁₀₁
commutator	C₄×C₁₀₁	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₀₁
Fitting	#416×C₁₀₁	C₂
socle	C₂²×C₁₀₁	C₂²×D₄
prime radical	C₂×#41×D₁₀₁	C₂
cosocle	C₂×C₄×C₁₀₁	C₂⁴
hypercenter	#416	D₁₀₁
hypocenter	C₁₀₁	#2896
LCS	G ▷ C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #416

Complements

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

N	H ≅ G/N	type
2. C₂²×#13108057	C₂	semidirect
3. #117×D₁₀₁	C₂	semidirect
4. #20265184	C₂	semidirect
5. #20265180	C₂	semidirect
6. #20265177	C₂	semidirect
7. #20265175	C₂	semidirect
8. #20265174	C₂	semidirect
9. #20265173	C₂	semidirect
10. #20265171	C₂	semidirect
11. #416×C₁₀₁	C₂	semidirect
18. #41×D₁₀₁	C₄	semidirect
26. C₂×#13108057	C₂²	semidirect
27. C₂²×#11717579	C₂²	semidirect
28. #15692214	C₂²	semidirect
32. C₄⋊C₄×D₁₀₁	C₂²	semidirect
35. #15692180	C₂²	semidirect
36. #15692179	C₂²	semidirect
37. #15692178	C₂²	semidirect
38. #15692177	C₂²	semidirect
39. #15692176	C₂²	semidirect
40. #15692175	C₂²	semidirect
42. C₂²×C₈×C₁₀₁	C₂²	semidirect
43. #117×C₁₀₁	C₂²	semidirect
44. #103×C₁₀₁	C₂²	semidirect
45. #102×C₁₀₁	C₂²	semidirect
52. #13108074	C₂×C₄	semidirect
59. Q₈×D₁₀₁	C₂×C₄	semidirect
67. D₄×D₁₀₁	C₂×C₄	semidirect
92. #41×C₁₀₁	C₂×C₄	semidirect
181. C₁₀₁	#2896	semidirect
192. #416	D₁₀₁	semidirect
210. C₂²×C₈	C₂×D₁₀₁	semidirect
215. #117	C₂×D₁₀₁	semidirect
222. #103	C₂×D₁₀₁	semidirect
226. #102	C₂×D₁₀₁	semidirect
238. #41	C₄×D₁₀₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×#41×D₁₀₁	2
C₂²×#13108057	2
#117×D₁₀₁	2
#20265184	2
#20265180	2
#20265177	2
#20265175	2
#20265174	2
#20265173	2
#20265171	2
#416×C₁₀₁	2
#2896	101