Subgroups — #38767100

Order 25472.

564 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Dic₁₉₉⋊C₄	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₉₉⋊#95	C₂	1	2	yes	maximal
3	C₂×C₁₉₉⋊#95	C₂	1	1	yes	maximal, characteristic
4	C₂×C₁₉₉⋊#95	C₂	1	2	yes	maximal
5	C₂²×C₄⋊C₄×C₁₉₉	C₂	1	1	yes	maximal, characteristic
6	C₂³×Dic₁₉₉	C₂²	1	1	yes	characteristic
7	C₂³×Dic₁₉₉	C₂²	1	2	yes	—
8	C₂³×Dic₁₉₉	C₂²	1	1	yes	characteristic
9	C₂×Dic₁₉₉⋊C₄	—	2	2	no	—
10	C₂×Dic₁₉₉⋊C₄	—	2	2	no	—
11	C₂×Dic₁₉₉⋊C₄	C₄	1	4	yes	complemented
12	C₁₉₉⋊#95	—	2	4	no	—
13	C₁₉₉⋊#95	—	2	4	no	—
14	C₁₉₉⋊#95	—	2	2	no	—
15	C₂³×C₄×C₁₉₉	C₂²	1	2	yes	—
16	C₂³×C₄×C₁₉₉	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊C₄×C₁₉₉	—	2	4	no	—
18	C₂×C₄⋊C₄×C₁₉₉	—	2	2	no	—
19	C₂²×Dic₁₉₉	—	2	1	no	—
20	C₂²×Dic₁₉₉	—	2	1	no	—
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	1	no	—
26	C₂²×Dic₁₉₉	—	2	1	no	—
27	C₂²×Dic₁₉₉	—	2	1	no	—
28	C₂²×Dic₁₉₉	—	2	1	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	C₂×C₄	1	2	yes	—
34	C₂²×Dic₁₉₉	C₂×C₄	1	2	yes	—
35	C₂²×Dic₁₉₉	D₄	1	2	yes	—
36	C₂²×Dic₁₉₉	Q₈	1	4	yes	—
37	C₂²×Dic₁₉₉	Q₈	1	2	yes	—
38	Dic₁₉₉⋊C₄	—	2	8	no	—
39	C₂⁴×C₁₉₉	C₂³	1	1	yes	characteristic
40	C₂²×C₄×C₁₉₉	—	2	2	no	—
41	C₂²×C₄×C₁₉₉	—	2	2	no	—
42	C₂²×C₄×C₁₉₉	—	2	2	no	—
43	C₂²×C₄×C₁₉₉	—	2	2	no	—
44	C₂²×C₄×C₁₉₉	—	2	2	no	—
45	C₂²×C₄×C₁₉₉	—	2	2	no	—
46	C₂²×C₄×C₁₉₉	—	2	2	no	—
47	C₂²×C₄×C₁₉₉	—	2	2	no	—
48	C₂²×C₄×C₁₉₉	C₂×C₄	1	2	yes	—
49	C₂²×C₄×C₁₉₉	D₄	1	4	yes	—
50	C₂²×C₄×C₁₉₉	D₄	1	2	yes	—
51	C₂²×C₄×C₁₉₉	D₄	1	2	yes	—
52	C₄⋊C₄×C₁₉₉	—	2	8	no	—
53	C₂×Dic₁₉₉	—	2	2	no	—
54	C₂×Dic₁₉₉	—	2	2	no	—
55	C₂×Dic₁₉₉	—	2	2	no	—
56	C₂×Dic₁₉₉	—	2	2	no	—
57	C₂×Dic₁₉₉	—	2	2	no	—
58	C₂×Dic₁₉₉	—	2	2	no	—
59	C₂×Dic₁₉₉	—	2	2	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	—	4	1	no	—
65	C₂×Dic₁₉₉	—	4	2	no	—
66	C₂×Dic₁₉₉	—	4	2	no	—
67	C₂×Dic₁₉₉	—	4	2	no	—
68	C₂×Dic₁₉₉	—	4	2	no	—
69	C₂×Dic₁₉₉	—	4	1	no	—
70	C₂×Dic₁₉₉	—	4	2	no	—
71	C₂×Dic₁₉₉	C₄⋊C₄	1	4	yes	complemented
72	C₂³×C₁₉₉	C₂²×C₄	1	1	yes	characteristic
73	C₂³×C₁₉₉	C₂×D₄	1	1	yes	characteristic
74	C₂³×C₁₉₉	C₂×D₄	1	1	yes	characteristic
75	C₂³×C₁₉₉	C₂×D₄	1	2	yes	—
76	C₂³×C₁₉₉	C₂×D₄	1	1	yes	characteristic
77	C₂³×C₁₉₉	C₂×Q₈	1	1	yes	characteristic
78	C₂³×C₁₉₉	C₂×Q₈	1	2	yes	—
79	C₂³×C₁₉₉	#41	1	1	yes	characteristic
80	C₂³×C₁₉₉	#41	1	2	yes	—
81	C₂³×C₁₉₉	#41	1	2	yes	—
82	C₂³×C₁₉₉	#41	1	1	yes	characteristic
83	C₂×C₄×C₁₉₉	—	2	4	no	—
84	C₂×C₄×C₁₉₉	—	2	2	no	—
85	C₂×C₄×C₁₉₉	—	2	4	no	—
86	C₂×C₄×C₁₉₉	—	2	2	no	—
87	C₂×C₄×C₁₉₉	—	2	4	no	—
88	C₂×C₄×C₁₉₉	—	2	4	no	—
89	C₂×C₄×C₁₉₉	—	2	4	no	—
90	C₂×C₄×C₁₉₉	—	4	2	no	—
91	C₂×C₄×C₁₉₉	—	4	2	no	—
92	C₂×C₄×C₁₉₉	—	4	2	no	—
93	C₂×C₄×C₁₉₉	—	4	2	no	—
94	C₂×C₄×C₁₉₉	C₂²⋊C₄	1	4	yes	—
95	C₂²×C₁₉₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
96	C₂²×C₁₉₉	C₂×C₄⋊C₄	1	1	yes	characteristic
97	C₂²×C₁₉₉	#117	1	1	yes	characteristic
98	C₂²×C₁₉₉	C₄×D₄	1	2	yes	—
99	C₂²×C₁₉₉	C₄×Q₈	1	2	yes	—
100	C₂²×C₁₉₉	#120	1	1	yes	characteristic
101	C₂²×C₁₉₉	#121	1	1	yes	characteristic
102	C₂²×C₁₉₉	#121	1	2	yes	—
103	C₂²×C₁₉₉	#121	1	2	yes	—
104	C₂²×C₁₉₉	#121	1	1	yes	characteristic
105	C₂²×C₁₉₉	C₂²⋊Q₈	1	1	yes	characteristic
106	C₂²×C₁₉₉	C₂²⋊Q₈	1	2	yes	—
107	C₂²×C₁₉₉	C₂²⋊Q₈	1	2	yes	—
108	C₂²×C₁₉₉	C₂²⋊Q₈	1	1	yes	characteristic
109	C₂²×C₁₉₉	C₂²⋊Q₈	1	2	yes	—
110	C₂²×C₁₉₉	#123	1	2	yes	—
111	C₂²×C₁₉₉	#123	1	1	yes	characteristic
112	C₂²×C₁₉₉	#124	1	1	yes	characteristic
113	C₂²×C₁₉₉	#125	1	2	yes	—
114	C₂²×C₁₉₉	#125	1	2	yes	—
115	C₂²×C₁₉₉	#126	1	2	yes	—
116	C₂²×C₁₉₉	C₄⋊Q₈	1	2	yes	—
117	C₂²×C₁₉₉	C₄⋊Q₈	1	1	yes	characteristic
118	C₄×C₁₉₉	—	2	4	no	—
119	C₄×C₁₉₉	—	4	4	no	—
120	Dic₁₉₉	—	2	4	no	—
121	Dic₁₉₉	—	4	2	no	—
122	Dic₁₉₉	—	4	4	no	—
123	C₂×C₁₉₉	#380	1	1	yes	characteristic
124	C₂×C₁₉₉	#386	1	1	yes	characteristic
125	C₂×C₁₉₉	#387	1	2	yes	—
126	C₂×C₁₉₉	#389	1	2	yes	—
127	C₂×C₁₉₉	#391	1	1	yes	characteristic
128	C₂×C₁₉₉	#394	1	1	yes	characteristic
129	C₂×C₁₉₉	#395	1	1	yes	characteristic
130	C₂×C₁₉₉	#396	1	2	yes	—
131	C₂×C₁₉₉	#398	1	2	yes	—
132	C₂×C₁₉₉	#398	1	1	yes	characteristic
133	C₂×C₁₉₉	#400	1	1	yes	characteristic
134	C₁₉₉	#1441	1	1	yes	characteristic, complemented, Sylow
135	#1441	—	199	1	no	maximal, Sylow
136	C₂²×C₄⋊C₄	—	199	1	no	—
137	C₂²×C₄⋊C₄	D₁₉₉	1	1	yes	characteristic
138	C₂×#95	—	199	2	no	—
139	C₂×#95	—	199	2	no	—
140	C₂×#95	—	199	1	no	—
141	C₂³×C₄	—	199	1	no	—
142	C₂³×C₄	—	199	2	no	—
143	C₂³×C₄	—	199	1	no	—
144	C₂³×C₄	C₂×D₁₉₉	1	2	yes	—
145	C₂³×C₄	C₂×D₁₉₉	1	1	yes	characteristic
146	C₂×C₄⋊C₄	—	2	4	no	—
147	C₂×C₄⋊C₄	—	2	2	no	—
148	C₂×C₄⋊C₄	—	398	2	no	—
149	C₂×C₄⋊C₄	—	398	2	no	—
150	C₂×C₄⋊C₄	—	199	4	no	—
151	#95	—	398	4	no	—
152	#95	—	398	4	no	—
153	#95	—	398	2	no	—
154	C₂⁴	C₂²×D₁₉₉	1	1	yes	characteristic
155	C₂²×C₄	—	2	2	no	—
156	C₂²×C₄	—	2	2	no	—
157	C₂²×C₄	—	2	2	no	—
158	C₂²×C₄	—	2	2	no	—
159	C₂²×C₄	—	2	2	no	—
160	C₂²×C₄	—	2	2	no	—
161	C₂²×C₄	—	2	2	no	—
162	C₂²×C₄	—	2	2	no	—
163	C₂²×C₄	—	398	2	no	—
164	C₂²×C₄	—	398	2	no	—
165	C₂²×C₄	—	398	2	no	—
166	C₂²×C₄	—	398	1	no	—
167	C₂²×C₄	—	398	2	no	—
168	C₂²×C₄	—	398	1	no	—
169	C₂²×C₄	—	398	2	no	—
170	C₂²×C₄	—	398	2	no	—
171	C₂²×C₄	—	398	2	no	—
172	C₂²×C₄	—	398	2	no	—
173	C₂²×C₄	—	398	1	no	—
174	C₂²×C₄	—	398	1	no	—
175	C₂²×C₄	—	398	1	no	—
176	C₂²×C₄	—	398	1	no	—
177	C₂²×C₄	—	199	2	no	—
178	C₂²×C₄	—	199	4	no	—
179	C₂²×C₄	—	199	2	no	—
180	C₂²×C₄	—	199	2	no	—
181	C₂²×C₄	—	199	2	no	—
182	C₂²×C₄	C₄×D₁₉₉	1	2	yes	—
183	C₂²×C₄	C₂²⋊D₁₉₉	1	4	yes	—
184	C₂²×C₄	C₂²⋊D₁₉₉	1	2	yes	—
185	C₂²×C₄	D₇₉₆	1	2	yes	—
186	C₄⋊C₄	—	2	8	no	—
187	C₄⋊C₄	—	398	8	no	—
188	C₂³	C₂×C₄×D₁₉₉	1	1	yes	characteristic
189	C₂³	D₄×D₁₉₉	1	1	yes	characteristic
190	C₂³	C₂×C₂²⋊D₁₉₉	1	2	yes	—
191	C₂³	C₂×C₂²⋊D₁₉₉	1	1	yes	characteristic
192	C₂³	C₂×D₇₉₆	1	1	yes	characteristic
193	C₂³	Q₈×D₁₉₉	1	1	yes	characteristic
194	C₂³	Q₈×D₁₉₉	1	2	yes	—
195	C₂³	#15691885	1	1	yes	characteristic
196	C₂³	#15691885	1	2	yes	—
197	C₂³	#15691886	1	2	yes	—
198	C₂³	#15691887	1	1	yes	characteristic
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	2	4	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	2	4	no	—
203	C₂×C₄	—	2	4	no	—
204	C₂×C₄	—	2	4	no	—
205	C₂×C₄	—	2	4	no	—
206	C₂×C₄	—	4	2	no	—
207	C₂×C₄	—	4	2	no	—
208	C₂×C₄	—	4	2	no	—
209	C₂×C₄	—	4	2	no	—
210	C₂×C₄	—	796	1	no	—
211	C₂×C₄	—	796	2	no	—
212	C₂×C₄	—	796	1	no	—
213	C₂×C₄	—	796	2	no	—
214	C₂×C₄	—	796	2	no	—
215	C₂×C₄	—	796	2	no	—
216	C₂×C₄	—	796	2	no	—
217	C₂×C₄	—	398	2	no	—
218	C₂×C₄	—	398	4	no	—
219	C₂×C₄	—	398	4	no	—
220	C₂×C₄	—	398	4	no	—
221	C₂×C₄	—	398	4	no	—
222	C₂×C₄	—	398	2	no	—
223	C₂×C₄	—	398	2	no	—
224	C₂×C₄	—	398	2	no	—
225	C₂×C₄	—	398	2	no	—
226	C₂×C₄	—	398	2	no	—
227	C₂×C₄	—	398	2	no	—
228	C₂×C₄	—	199	4	no	—
229	C₂×C₄	(C₂×D₁₉₉)⋊C₄	1	4	yes	—
230	C₂²	C₂×(C₂×D₁₉₉)⋊C₄	1	1	yes	characteristic
231	C₂²	C₄⋊C₄×D₁₉₉	1	1	yes	characteristic
232	C₂²	#20263913	1	1	yes	characteristic
233	C₂²	C₄×C₂²⋊D₁₉₉	1	2	yes	—
234	C₂²	#20263920	1	2	yes	—
235	C₂²	#20263924	1	1	yes	characteristic
236	C₂²	#20263926	1	1	yes	characteristic
237	C₂²	#20263927	1	2	yes	—
238	C₂²	#20263929	1	2	yes	—
239	C₂²	#20263930	1	1	yes	characteristic
240	C₂²	#20263931	1	2	yes	—
241	C₂²	#20263931	1	1	yes	characteristic
242	C₂²	#20263933	1	2	yes	—
243	C₂²	#20263935	1	2	yes	—
244	C₂²	#20263935	1	1	yes	characteristic
245	C₂²	#20263936	1	2	yes	—
246	C₂²	#20263938	1	1	yes	characteristic
247	C₂²	#20263942	1	1	yes	characteristic
248	C₂²	#20263945	1	2	yes	—
249	C₂²	#20263945	1	2	yes	—
250	C₂²	#20263950	1	2	yes	—
251	C₂²	#20263953	1	2	yes	—
252	C₂²	#20263953	1	1	yes	characteristic
253	C₄	—	2	4	no	—
254	C₄	—	4	4	no	—
255	C₄	—	796	4	no	—
256	C₄	—	796	2	no	—
257	C₄	—	398	4	no	—
258	C₂	#27163875	1	1	yes	characteristic
259	C₂	#27163897	1	1	yes	characteristic
260	C₂	#27163899	1	2	yes	—
261	C₂	#27163913	1	2	yes	—
262	C₂	#27163923	1	1	yes	characteristic
263	C₂	#27163932	1	1	yes	characteristic
264	C₂	#27163937	1	1	yes	characteristic
265	C₂	#27163940	1	2	yes	—
266	C₂	#27163948	1	2	yes	—
267	C₂	#27163949	1	1	yes	characteristic
268	C₂	#27163956	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂⁴	C₂²×D₁₉₉
commutator	C₂³×C₁₉₉	C₂²×C₄
Fitting	C₂²×C₄⋊C₄×C₁₉₉	C₂
socle, prime radical, cosocle	C₂⁴×C₁₉₉	C₂³
hypercenter	C₂²×C₄⋊C₄	D₁₉₉
hypocenter	C₁₉₉	#1441
LCS	G ▷ C₂³×C₁₉₉ ▷ C₁₉₉
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
11. C₂×Dic₁₉₉⋊C₄	C₄	semidirect
71. C₂×Dic₁₉₉	C₄⋊C₄	semidirect
134. C₁₉₉	#1441	semidirect

Maximal subgroups

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

subgroup	index
C₂²×Dic₁₉₉⋊C₄	2
C₂×C₁₉₉⋊#95	2
C₂²×C₄⋊C₄×C₁₉₉	2
#1441	199