Subgroups — #38767095

Order 25472.

780 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄⋊Dic₁₉₉	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₂²⋊Dic₁₉₉	C₂	1	2	yes	maximal
3	C₂×C₁₉₉⋊#95	C₂	1	2	yes	maximal, complemented
4	C₂×C₁₉₉⋊#95	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×C₂²⋊C₄×C₁₉₉	C₂	1	1	yes	maximal, characteristic
6	C₂³×Dic₁₉₉	C₂²	1	2	yes	—
7	C₂³×Dic₁₉₉	C₂²	1	2	yes	—
8	C₂×C₄⋊Dic₁₉₉	—	2	4	no	—
9	C₂×C₄⋊Dic₁₉₉	—	2	2	no	—
10	C₂×C₂²⋊Dic₁₉₉	—	2	4	no	—
11	C₂×C₂²⋊Dic₁₉₉	—	2	4	no	—
12	C₂×C₂²⋊Dic₁₉₉	—	2	4	no	—
13	C₁₉₉⋊#95	—	2	4	no	—
14	C₁₉₉⋊#95	—	2	2	no	—
15	C₂⁵×C₁₉₉	C₂²	1	1	yes	characteristic
16	C₂³×C₄×C₁₉₉	C₂²	1	1	yes	characteristic
17	C₂³×C₄×C₁₉₉	C₂²	1	1	yes	characteristic
18	C₂×C₂²⋊C₄×C₁₉₉	—	2	2	no	—
19	C₂×C₂²⋊C₄×C₁₉₉	—	2	2	no	—
20	C₂×C₂²⋊C₄×C₁₉₉	C₄	1	4	yes	complemented
21	C₂²×Dic₁₉₉	—	2	2	no	—
22	C₂²×Dic₁₉₉	—	2	2	no	—
23	C₂²×Dic₁₉₉	—	2	2	no	—
24	C₂²×Dic₁₉₉	—	2	2	no	—
25	C₂²×Dic₁₉₉	—	2	2	no	—
26	C₂²×Dic₁₉₉	—	2	2	no	—
27	C₂²×Dic₁₉₉	—	2	2	no	—
28	C₂²×Dic₁₉₉	—	2	2	no	—
29	C₂²×Dic₁₉₉	—	2	2	no	—
30	C₂²×Dic₁₉₉	—	2	2	no	—
31	C₂²×Dic₁₉₉	—	2	2	no	—
32	C₂²×Dic₁₉₉	—	2	2	no	—
33	C₂²×Dic₁₉₉	D₄	1	4	yes	—
34	C₂²×Dic₁₉₉	D₄	1	4	yes	—
35	C₄⋊Dic₁₉₉	—	2	8	no	—
36	C₂²⋊Dic₁₉₉	—	2	16	no	—
37	C₂⁴×C₁₉₉	—	2	1	no	—
38	C₂⁴×C₁₉₉	—	2	1	no	—
39	C₂⁴×C₁₉₉	—	2	2	no	—
40	C₂⁴×C₁₉₉	—	2	2	no	—
41	C₂⁴×C₁₉₉	—	2	2	no	—
42	C₂⁴×C₁₉₉	—	2	2	no	—
43	C₂⁴×C₁₉₉	—	2	1	no	—
44	C₂⁴×C₁₉₉	—	2	1	no	—
45	C₂⁴×C₁₉₉	C₂×C₄	1	2	yes	—
46	C₂⁴×C₁₉₉	D₄	1	2	yes	—
47	C₂⁴×C₁₉₉	Q₈	1	2	yes	—
48	C₂⁴×C₁₉₉	C₂³	1	1	yes	characteristic
49	C₂²×C₄×C₁₉₉	—	2	2	no	—
50	C₂²×C₄×C₁₉₉	—	2	2	no	—
51	C₂²×C₄×C₁₉₉	—	2	1	no	—
52	C₂²×C₄×C₁₉₉	—	2	1	no	—
53	C₂²×C₄×C₁₉₉	—	2	2	no	—
54	C₂²×C₄×C₁₉₉	—	2	2	no	—
55	C₂²×C₄×C₁₉₉	C₂×C₄	1	2	yes	—
56	C₂²×C₄×C₁₉₉	C₂×C₄	1	2	yes	—
57	C₂²×C₄×C₁₉₉	D₄	1	2	yes	—
58	C₂²×C₄×C₁₉₉	D₄	1	2	yes	—
59	C₂²⋊C₄×C₁₉₉	—	2	8	no	—
60	C₂×Dic₁₉₉	—	2	4	no	—
61	C₂×Dic₁₉₉	—	2	4	no	—
62	C₂×Dic₁₉₉	—	2	4	no	—
63	C₂×Dic₁₉₉	—	2	4	no	—
64	C₂×Dic₁₉₉	—	2	4	no	—
65	C₂×Dic₁₉₉	—	2	4	no	—
66	C₂×Dic₁₉₉	—	4	2	no	—
67	C₂×Dic₁₉₉	—	4	2	no	—
68	C₂×Dic₁₉₉	—	4	2	no	—
69	C₂×Dic₁₉₉	—	4	2	no	—
70	C₂×Dic₁₉₉	—	4	2	no	—
71	C₂×Dic₁₉₉	—	4	2	no	—
72	C₂×Dic₁₉₉	—	4	2	no	—
73	C₂×Dic₁₉₉	—	4	2	no	—
74	C₂³×C₁₉₉	—	2	2	no	—
75	C₂³×C₁₉₉	—	2	2	no	—
76	C₂³×C₁₉₉	—	2	4	no	—
77	C₂³×C₁₉₉	—	2	4	no	—
78	C₂³×C₁₉₉	—	2	4	no	—
79	C₂³×C₁₉₉	—	2	2	no	—
80	C₂³×C₁₉₉	—	2	2	no	—
81	C₂³×C₁₉₉	—	2	4	no	—
82	C₂³×C₁₉₉	—	2	4	no	—
83	C₂³×C₁₉₉	—	2	2	no	—
84	C₂³×C₁₉₉	—	2	2	no	—
85	C₂³×C₁₉₉	—	2	4	no	—
86	C₂³×C₁₉₉	—	4	1	no	—
87	C₂³×C₁₉₉	—	4	2	no	—
88	C₂³×C₁₉₉	—	4	2	no	—
89	C₂³×C₁₉₉	—	4	2	no	—
90	C₂³×C₁₉₉	—	4	1	no	—
91	C₂³×C₁₉₉	—	4	2	no	—
92	C₂³×C₁₉₉	—	4	2	no	—
93	C₂³×C₁₉₉	—	4	1	no	—
94	C₂³×C₁₉₉	—	4	2	no	—
95	C₂³×C₁₉₉	—	4	1	no	—
96	C₂³×C₁₉₉	C₄⋊C₄	1	4	yes	complemented
97	C₂³×C₁₉₉	C₂²×C₄	1	1	yes	characteristic
98	C₂³×C₁₉₉	C₂×D₄	1	2	yes	—
99	C₂³×C₁₉₉	C₂×D₄	1	2	yes	—
100	C₂³×C₁₉₉	C₂×D₄	1	1	yes	characteristic
101	C₂³×C₁₉₉	C₂×D₄	1	1	yes	characteristic
102	C₂³×C₁₉₉	C₂×D₄	1	1	yes	characteristic
103	C₂³×C₁₉₉	C₂×Q₈	1	1	yes	characteristic
104	C₂³×C₁₉₉	#41	1	2	yes	—
105	C₂³×C₁₉₉	#41	1	2	yes	—
106	C₂³×C₁₉₉	#41	1	1	yes	characteristic
107	C₂³×C₁₉₉	#41	1	1	yes	characteristic
108	C₂×C₄×C₁₉₉	—	2	4	no	—
109	C₂×C₄×C₁₉₉	—	2	2	no	—
110	C₂×C₄×C₁₉₉	—	2	4	no	—
111	C₂×C₄×C₁₉₉	—	2	2	no	—
112	C₂×C₄×C₁₉₉	—	2	2	no	—
113	C₂×C₄×C₁₉₉	—	2	4	no	—
114	C₂×C₄×C₁₉₉	—	4	2	no	—
115	C₂×C₄×C₁₉₉	—	4	1	no	—
116	C₂×C₄×C₁₉₉	—	4	1	no	—
117	C₂×C₄×C₁₉₉	C₂²⋊C₄	1	4	yes	complemented
118	C₂²×C₁₉₉	—	2	4	no	—
119	C₂²×C₁₉₉	—	2	8	no	—
120	C₂²×C₁₉₉	—	4	2	no	—
121	C₂²×C₁₉₉	—	4	4	no	—
122	C₂²×C₁₉₉	—	4	2	no	—
123	C₂²×C₁₉₉	—	4	2	no	—
124	C₂²×C₁₉₉	—	4	2	no	—
125	C₂²×C₁₉₉	—	4	4	no	—
126	C₂²×C₁₉₉	—	4	4	no	—
127	C₂²×C₁₉₉	—	4	2	no	—
128	C₂²×C₁₉₉	—	4	2	no	—
129	C₂²×C₁₉₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
130	C₂²×C₁₉₉	C₂×C₄⋊C₄	1	1	yes	characteristic
131	C₂²×C₁₉₉	#117	1	1	yes	characteristic
132	C₂²×C₁₉₉	C₄×D₄	1	2	yes	—
133	C₂²×C₁₉₉	C₄×D₄	1	2	yes	—
134	C₂²×C₁₉₉	#120	1	1	yes	characteristic
135	C₂²×C₁₉₉	#120	1	2	yes	—
136	C₂²×C₁₉₉	#121	1	2	yes	—
137	C₂²×C₁₉₉	#121	1	2	yes	—
138	C₂²×C₁₉₉	#121	1	2	yes	—
139	C₂²×C₁₉₉	#121	1	1	yes	characteristic
140	C₂²×C₁₉₉	#121	1	1	yes	characteristic
141	C₂²×C₁₉₉	C₂²⋊Q₈	1	2	yes	—
142	C₂²×C₁₉₉	C₂²⋊Q₈	1	2	yes	—
143	C₂²×C₁₉₉	C₂²⋊Q₈	1	1	yes	characteristic
144	C₂²×C₁₉₉	C₂²⋊Q₈	1	1	yes	characteristic
145	C₂²×C₁₉₉	#123	1	1	yes	characteristic
146	C₂²×C₁₉₉	#123	1	2	yes	—
147	C₂²×C₁₉₉	#123	1	2	yes	—
148	C₂²×C₁₉₉	#124	1	2	yes	—
149	C₂²×C₁₉₉	#124	1	1	yes	characteristic
150	C₂²×C₁₉₉	#126	1	2	yes	—
151	C₂²×C₁₉₉	#127	1	1	yes	characteristic
152	C₄×C₁₉₉	—	2	4	no	—
153	C₄×C₁₉₉	—	4	2	no	—
154	Dic₁₉₉	—	4	4	no	—
155	Dic₁₉₉	—	4	4	no	—
156	C₂×C₁₉₉	—	4	4	no	—
157	C₂×C₁₉₉	#380	1	1	yes	characteristic
158	C₂×C₁₉₉	#385	1	2	yes	—
159	C₂×C₁₉₉	#386	1	1	yes	characteristic
160	C₂×C₁₉₉	#388	1	2	yes	—
161	C₂×C₁₉₉	#390	1	1	yes	characteristic
162	C₂×C₁₉₉	#392	1	1	yes	characteristic
163	C₂×C₁₉₉	#393	1	1	yes	characteristic
164	C₂×C₁₉₉	#394	1	2	yes	—
165	C₂×C₁₉₉	#396	1	2	yes	—
166	C₂×C₁₉₉	#397	1	1	yes	characteristic
167	C₂×C₁₉₉	#399	1	1	yes	characteristic
168	C₁₉₉	#1439	1	1	yes	characteristic, complemented, Sylow
169	#1439	—	199	1	no	maximal, Sylow
170	C₂²×C₄⋊C₄	—	199	1	no	—
171	C₂²×C₂²⋊C₄	—	199	2	no	—
172	C₂²×C₂²⋊C₄	D₁₉₉	1	1	yes	characteristic
173	C₂×#95	—	199	2	no	—
174	C₂×#95	—	199	1	no	—
175	C₂⁵	C₂×D₁₉₉	1	1	yes	characteristic
176	C₂³×C₄	—	199	2	no	—
177	C₂³×C₄	—	199	2	no	—
178	C₂³×C₄	C₂×D₁₉₉	1	1	yes	characteristic
179	C₂³×C₄	C₂×D₁₉₉	1	1	yes	characteristic
180	C₂×C₄⋊C₄	—	398	4	no	—
181	C₂×C₄⋊C₄	—	398	2	no	—
182	C₂×C₂²⋊C₄	—	2	2	no	—
183	C₂×C₂²⋊C₄	—	2	2	no	—
184	C₂×C₂²⋊C₄	—	398	4	no	—
185	C₂×C₂²⋊C₄	—	398	4	no	—
186	C₂×C₂²⋊C₄	—	398	4	no	—
187	C₂×C₂²⋊C₄	Dic₁₉₉	1	4	yes	complemented
188	#95	—	398	4	no	—
189	#95	—	398	2	no	—
190	C₂⁴	—	2	2	no	—
191	C₂⁴	—	2	2	no	—
192	C₂⁴	—	2	1	no	—
193	C₂⁴	—	2	1	no	—
194	C₂⁴	—	2	2	no	—
195	C₂⁴	—	2	2	no	—
196	C₂⁴	—	2	1	no	—
197	C₂⁴	—	2	1	no	—
198	C₂⁴	C₂×Dic₁₉₉	1	2	yes	—
199	C₂⁴	D₇₉₆	1	2	yes	—
200	C₂⁴	Dic₃₉₈	1	2	yes	—
201	C₂⁴	C₂²×D₁₉₉	1	1	yes	characteristic
202	C₂²×C₄	—	2	2	no	—
203	C₂²×C₄	—	2	2	no	—
204	C₂²×C₄	—	2	2	no	—
205	C₂²×C₄	—	2	1	no	—
206	C₂²×C₄	—	2	1	no	—
207	C₂²×C₄	—	2	2	no	—
208	C₂²×C₄	—	398	2	no	—
209	C₂²×C₄	—	398	2	no	—
210	C₂²×C₄	—	398	2	no	—
211	C₂²×C₄	—	398	2	no	—
212	C₂²×C₄	—	398	2	no	—
213	C₂²×C₄	—	398	2	no	—
214	C₂²×C₄	—	398	2	no	—
215	C₂²×C₄	—	398	2	no	—
216	C₂²×C₄	—	398	2	no	—
217	C₂²×C₄	—	398	2	no	—
218	C₂²×C₄	—	398	2	no	—
219	C₂²×C₄	—	398	2	no	—
220	C₂²×C₄	—	199	4	no	—
221	C₂²×C₄	—	199	4	no	—
222	C₂²×C₄	C₂×Dic₁₉₉	1	2	yes	—
223	C₂²×C₄	C₂×Dic₁₉₉	1	2	yes	—
224	C₂²×C₄	C₂²⋊D₁₉₉	1	2	yes	—
225	C₂²×C₄	C₂²⋊D₁₉₉	1	2	yes	—
226	C₄⋊C₄	—	398	8	no	—
227	C₂²⋊C₄	—	2	8	no	—
228	C₂²⋊C₄	—	398	16	no	—
229	C₂³	—	2	4	no	—
230	C₂³	—	2	2	no	—
231	C₂³	—	2	2	no	—
232	C₂³	—	2	4	no	—
233	C₂³	—	2	4	no	—
234	C₂³	—	2	4	no	—
235	C₂³	—	2	2	no	—
236	C₂³	—	2	2	no	—
237	C₂³	—	2	4	no	—
238	C₂³	—	2	4	no	—
239	C₂³	—	2	2	no	—
240	C₂³	—	2	2	no	—
241	C₂³	—	4	1	no	—
242	C₂³	—	4	2	no	—
243	C₂³	—	4	2	no	—
244	C₂³	—	4	1	no	—
245	C₂³	—	4	2	no	—
246	C₂³	—	4	1	no	—
247	C₂³	—	4	1	no	—
248	C₂³	—	4	2	no	—
249	C₂³	—	4	2	no	—
250	C₂³	—	4	2	no	—
251	C₂³	C₄⋊Dic₁₉₉	1	4	yes	complemented
252	C₂³	C₂²×Dic₁₉₉	1	1	yes	characteristic
253	C₂³	D₄×D₁₉₉	1	2	yes	—
254	C₂³	D₄×D₁₉₉	1	2	yes	—
255	C₂³	C₂×C₂²⋊D₁₉₉	1	1	yes	characteristic
256	C₂³	C₂×C₂²⋊D₁₉₉	1	1	yes	characteristic
257	C₂³	C₂×D₇₉₆	1	1	yes	characteristic
258	C₂³	C₂×Dic₃₉₈	1	1	yes	characteristic
259	C₂³	#15691885	1	2	yes	—
260	C₂³	#15691885	1	2	yes	—
261	C₂³	#15691886	1	1	yes	characteristic
262	C₂³	#15691886	1	1	yes	characteristic
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	4	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	4	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	4	no	—
269	C₂×C₄	—	4	1	no	—
270	C₂×C₄	—	4	1	no	—
271	C₂×C₄	—	4	2	no	—
272	C₂×C₄	—	796	2	no	—
273	C₂×C₄	—	796	2	no	—
274	C₂×C₄	—	796	2	no	—
275	C₂×C₄	—	796	2	no	—
276	C₂×C₄	—	796	2	no	—
277	C₂×C₄	—	796	2	no	—
278	C₂×C₄	—	796	2	no	—
279	C₂×C₄	—	796	2	no	—
280	C₂×C₄	—	398	4	no	—
281	C₂×C₄	—	398	4	no	—
282	C₂×C₄	—	398	4	no	—
283	C₂×C₄	—	398	4	no	—
284	C₂×C₄	—	398	4	no	—
285	C₂×C₄	—	398	4	no	—
286	C₂×C₄	C₂²⋊Dic₁₉₉	1	4	yes	complemented
287	C₂²	—	2	4	no	—
288	C₂²	—	2	8	no	—
289	C₂²	—	4	4	no	—
290	C₂²	—	4	2	no	—
291	C₂²	—	4	2	no	—
292	C₂²	—	4	2	no	—
293	C₂²	—	4	4	no	—
294	C₂²	—	4	4	no	—
295	C₂²	—	4	2	no	—
296	C₂²	—	4	2	no	—
297	C₂²	—	4	2	no	—
298	C₂²	C₂×C₂²⋊Dic₁₉₉	1	1	yes	characteristic
299	C₂²	C₂×C₄⋊Dic₁₉₉	1	1	yes	characteristic
300	C₂²	#20263914	1	1	yes	characteristic
301	C₂²	D₄×Dic₁₉₉	1	2	yes	—
302	C₂²	D₄×Dic₁₉₉	1	2	yes	—
303	C₂²	#20263923	1	1	yes	characteristic
304	C₂²	#20263925	1	2	yes	—
305	C₂²	#20263926	1	2	yes	—
306	C₂²	#20263927	1	2	yes	—
307	C₂²	#20263928	1	2	yes	—
308	C₂²	#20263929	1	1	yes	characteristic
309	C₂²	#20263929	1	1	yes	characteristic
310	C₂²	#20263932	1	2	yes	—
311	C₂²	#20263932	1	2	yes	—
312	C₂²	#20263934	1	1	yes	characteristic
313	C₂²	#20263934	1	1	yes	characteristic
314	C₂²	#20263936	1	1	yes	characteristic
315	C₂²	#20263937	1	2	yes	—
316	C₂²	#20263939	1	2	yes	—
317	C₂²	#20263941	1	2	yes	—
318	C₂²	#20263942	1	1	yes	characteristic
319	C₂²	#20263948	1	2	yes	—
320	C₂²	#20263951	1	1	yes	characteristic
321	C₄	—	2	4	no	—
322	C₄	—	4	2	no	—
323	C₄	—	796	4	no	—
324	C₄	—	796	4	no	—
325	C₂	—	4	4	no	—
326	C₂	#27163878	1	1	yes	characteristic
327	C₂	#27163893	1	2	yes	—
328	C₂	#27163898	1	1	yes	characteristic
329	C₂	#27163911	1	2	yes	—
330	C₂	#27163921	1	1	yes	characteristic
331	C₂	#27163927	1	1	yes	characteristic
332	C₂	#27163930	1	1	yes	characteristic
333	C₂	#27163934	1	2	yes	—
334	C₂	#27163941	1	2	yes	—
335	C₂	#27163943	1	1	yes	characteristic
336	C₂	#27163954	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂⁴	C₂²×D₁₉₉
commutator	C₂³×C₁₉₉	C₂²×C₄
Fitting	C₂²×C₂²⋊C₄×C₁₉₉	C₂
socle, cosocle	C₂⁴×C₁₉₉	C₂³
prime radical	C₂⁵×C₁₉₉	C₂²
hypercenter	C₂²×C₂²⋊C₄	D₁₉₉
hypocenter	C₁₉₉	#1439
LCS	G ▷ C₂³×C₁₉₉ ▷ C₁₉₉
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
1. C₂²×C₄⋊Dic₁₉₉	C₂	semidirect
3. C₂×C₁₉₉⋊#95	C₂	semidirect
4. C₂×C₁₉₉⋊#95	C₂	semidirect
20. C₂×C₂²⋊C₄×C₁₉₉	C₄	semidirect
96. C₂³×C₁₉₉	C₄⋊C₄	semidirect
117. C₂×C₄×C₁₉₉	C₂²⋊C₄	semidirect
168. C₁₉₉	#1439	semidirect
187. C₂×C₂²⋊C₄	Dic₁₉₉	semidirect
251. C₂³	C₄⋊Dic₁₉₉	semidirect
286. C₂×C₄	C₂²⋊Dic₁₉₉	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₄⋊Dic₁₉₉	2
C₂²×C₂²⋊Dic₁₉₉	2
C₂×C₁₉₉⋊#95	2
C₂²×C₂²⋊C₄×C₁₉₉	2
#1439	199