Subgroups — #38767107

Order 25472.

520 conjugacy classes of subgroups.

The subgroup lattice has 256 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
2	C₂×C₁₉₉⋊#95	C₂	1	2	yes	maximal
3	C₂×C₁₉₉⋊#95	C₂	1	2	yes	maximal
4	C₂×C₁₉₉⋊#95	C₂	1	1	yes	maximal, characteristic
5	C₂×#95×C₁₉₉	C₂	1	1	yes	maximal, characteristic
6	C₂³×Dic₁₉₉	C₂²	1	2	yes	—
7	C₂³×Dic₁₉₉	C₂²	1	1	yes	characteristic
8	C₂³×Dic₁₉₉	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊Dic₁₉₉	—	2	4	no	—
10	C₂×C₄⋊Dic₁₉₉	—	2	2	no	—
11	C₁₉₉⋊#95	—	2	4	no	—
12	C₁₉₉⋊#95	—	2	4	no	—
13	C₁₉₉⋊#95	—	2	2	no	—
14	C₂³×C₄×C₁₉₉	C₂²	1	2	yes	—
15	C₂³×C₄×C₁₉₉	C₂²	1	1	yes	characteristic
16	#95×C₁₉₉	C₄	1	4	yes	complemented
17	C₂²×Dic₁₉₉	—	2	2	no	—
18	C₂²×Dic₁₉₉	—	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	1	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	1	no	—
30	C₂²×Dic₁₉₉	—	2	2	no	—
31	C₂²×Dic₁₉₉	D₄	1	2	yes	—
32	C₂²×Dic₁₉₉	D₄	1	4	yes	—
33	C₂²×Dic₁₉₉	Q₈	1	2	yes	—
34	C₄⋊Dic₁₉₉	—	2	8	no	—
35	C₂⁴×C₁₉₉	C₂³	1	1	yes	characteristic
36	C₂²×C₄×C₁₉₉	—	2	2	no	—
37	C₂²×C₄×C₁₉₉	—	2	2	no	—
38	C₂²×C₄×C₁₉₉	—	2	2	no	—
39	C₂²×C₄×C₁₉₉	—	2	2	no	—
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	C₂×C₄	1	4	yes	—
45	C₂²×C₄×C₁₉₉	C₂×C₄	1	2	yes	—
46	C₂²×C₄×C₁₉₉	D₄	1	2	yes	—
47	C₂²×C₄×C₁₉₉	Q₈	1	2	yes	—
48	C₂×Dic₁₉₉	—	2	4	no	—
49	C₂×Dic₁₉₉	—	2	4	no	—
50	C₂×Dic₁₉₉	—	2	2	no	—
51	C₂×Dic₁₉₉	—	2	4	no	—
52	C₂×Dic₁₉₉	—	2	2	no	—
53	C₂×Dic₁₉₉	—	2	4	no	—
54	C₂×Dic₁₉₉	—	2	4	no	—
55	C₂×Dic₁₉₉	—	4	1	no	—
56	C₂×Dic₁₉₉	—	4	2	no	—
57	C₂×Dic₁₉₉	—	4	2	no	—
58	C₂×Dic₁₉₉	—	4	1	no	—
59	C₂×Dic₁₉₉	—	4	2	no	—
60	C₂×Dic₁₉₉	—	4	2	no	—
61	C₂×Dic₁₉₉	—	4	1	no	—
62	C₂×Dic₁₉₉	—	4	2	no	—
63	C₂×Dic₁₉₉	—	4	1	no	—
64	C₂×Dic₁₉₉	—	4	2	no	—
65	C₂³×C₁₉₉	C₂²×C₄	1	1	yes	characteristic
66	C₂³×C₁₉₉	C₂×D₄	1	2	yes	—
67	C₂³×C₁₉₉	C₂×D₄	1	1	yes	characteristic
68	C₂³×C₁₉₉	C₂×D₄	1	1	yes	characteristic
69	C₂³×C₁₉₉	C₂×Q₈	1	1	yes	characteristic
70	C₂³×C₁₉₉	C₂×Q₈	1	1	yes	characteristic
71	C₂³×C₁₉₉	#41	1	2	yes	—
72	C₂³×C₁₉₉	#41	1	1	yes	characteristic
73	C₂³×C₁₉₉	#41	1	2	yes	—
74	C₂³×C₁₉₉	#41	1	2	yes	—
75	C₂³×C₁₉₉	#41	1	1	yes	characteristic
76	C₂×C₄×C₁₉₉	—	2	4	no	—
77	C₂×C₄×C₁₉₉	—	2	2	no	—
78	C₂×C₄×C₁₉₉	—	2	4	no	—
79	C₂×C₄×C₁₉₉	—	2	2	no	—
80	C₂×C₄×C₁₉₉	—	2	4	no	—
81	C₂×C₄×C₁₉₉	—	2	4	no	—
82	C₂×C₄×C₁₉₉	—	2	4	no	—
83	C₂×C₄×C₁₉₉	—	4	2	no	—
84	C₂×C₄×C₁₉₉	—	4	2	no	—
85	C₂×C₄×C₁₉₉	—	4	2	no	—
86	C₂×C₄×C₁₉₉	—	4	2	no	—
87	C₂×C₄×C₁₉₉	C₄⋊C₄	1	4	yes	—
88	C₂²×C₁₉₉	C₂×C₄⋊C₄	1	1	yes	characteristic
89	C₂²×C₁₉₉	#117	1	2	yes	—
90	C₂²×C₁₉₉	C₄×D₄	1	2	yes	—
91	C₂²×C₁₉₉	C₄×D₄	1	1	yes	characteristic
92	C₂²×C₁₉₉	C₄×Q₈	1	1	yes	characteristic
93	C₂²×C₁₉₉	#120	1	1	yes	characteristic
94	C₂²×C₁₉₉	#121	1	2	yes	—
95	C₂²×C₁₉₉	#121	1	1	yes	characteristic
96	C₂²×C₁₉₉	C₂²⋊Q₈	1	2	yes	—
97	C₂²×C₁₉₉	C₂²⋊Q₈	1	2	yes	—
98	C₂²×C₁₉₉	C₂²⋊Q₈	1	1	yes	characteristic
99	C₂²×C₁₉₉	#123	1	2	yes	—
100	C₂²×C₁₉₉	#123	1	1	yes	characteristic
101	C₂²×C₁₉₉	#123	1	2	yes	—
102	C₂²×C₁₉₉	#124	1	2	yes	—
103	C₂²×C₁₉₉	#124	1	1	yes	characteristic
104	C₂²×C₁₉₉	#124	1	1	yes	characteristic
105	C₂²×C₁₉₉	#125	1	2	yes	—
106	C₂²×C₁₉₉	#125	1	1	yes	characteristic
107	C₂²×C₁₉₉	#125	1	2	yes	—
108	C₂²×C₁₉₉	#126	1	2	yes	—
109	C₂²×C₁₉₉	#126	1	2	yes	—
110	C₂²×C₁₉₉	C₄⋊Q₈	1	1	yes	characteristic
111	C₄×C₁₉₉	—	2	4	no	—
112	C₄×C₁₉₉	—	4	4	no	—
113	Dic₁₉₉	—	4	2	no	—
114	Dic₁₉₉	—	4	4	no	—
115	Dic₁₉₉	—	4	2	no	—
116	C₂×C₁₉₉	#382	1	1	yes	characteristic
117	C₂×C₁₉₉	#385	1	1	yes	characteristic
118	C₂×C₁₉₉	#387	1	2	yes	—
119	C₂×C₁₉₉	#388	1	2	yes	—
120	C₂×C₁₉₉	#389	1	1	yes	characteristic
121	C₂×C₁₉₉	#393	1	1	yes	characteristic
122	C₂×C₁₉₉	#394	1	1	yes	characteristic
123	C₂×C₁₉₉	#397	1	1	yes	characteristic
124	C₂×C₁₉₉	#398	1	2	yes	—
125	C₂×C₁₉₉	#400	1	1	yes	characteristic
126	C₂×C₁₉₉	#400	1	2	yes	—
127	C₁₉₉	#1442	1	1	yes	characteristic, complemented, Sylow
128	#1442	—	199	1	no	maximal, Sylow
129	C₂²×C₄⋊C₄	—	199	1	no	—
130	C₂×#95	—	199	2	no	—
131	C₂×#95	—	199	2	no	—
132	C₂×#95	—	199	1	no	—
133	C₂×#95	D₁₉₉	1	1	yes	characteristic
134	C₂³×C₄	—	199	2	no	—
135	C₂³×C₄	—	199	1	no	—
136	C₂³×C₄	—	199	1	no	—
137	C₂³×C₄	C₂×D₁₉₉	1	2	yes	—
138	C₂³×C₄	C₂×D₁₉₉	1	1	yes	characteristic
139	C₂×C₄⋊C₄	—	398	4	no	—
140	C₂×C₄⋊C₄	—	398	2	no	—
141	#95	—	398	4	no	—
142	#95	—	398	4	no	—
143	#95	—	398	2	no	—
144	#95	Dic₁₉₉	1	4	yes	complemented
145	C₂⁴	C₂²×D₁₉₉	1	1	yes	characteristic
146	C₂²×C₄	—	2	2	no	—
147	C₂²×C₄	—	2	2	no	—
148	C₂²×C₄	—	2	2	no	—
149	C₂²×C₄	—	2	2	no	—
150	C₂²×C₄	—	2	2	no	—
151	C₂²×C₄	—	2	2	no	—
152	C₂²×C₄	—	2	2	no	—
153	C₂²×C₄	—	2	2	no	—
154	C₂²×C₄	—	398	2	no	—
155	C₂²×C₄	—	398	2	no	—
156	C₂²×C₄	—	398	2	no	—
157	C₂²×C₄	—	398	1	no	—
158	C₂²×C₄	—	398	1	no	—
159	C₂²×C₄	—	398	2	no	—
160	C₂²×C₄	—	398	2	no	—
161	C₂²×C₄	—	398	1	no	—
162	C₂²×C₄	—	398	2	no	—
163	C₂²×C₄	—	398	2	no	—
164	C₂²×C₄	—	398	2	no	—
165	C₂²×C₄	—	398	2	no	—
166	C₂²×C₄	—	398	2	no	—
167	C₂²×C₄	—	398	1	no	—
168	C₂²×C₄	—	199	2	no	—
169	C₂²×C₄	—	199	4	no	—
170	C₂²×C₄	—	199	2	no	—
171	C₂²×C₄	C₂×Dic₁₉₉	1	4	yes	—
172	C₂²×C₄	C₂×Dic₁₉₉	1	2	yes	—
173	C₂²×C₄	D₇₉₆	1	2	yes	—
174	C₂²×C₄	Dic₃₉₈	1	2	yes	—
175	C₄⋊C₄	—	398	8	no	—
176	C₂³	C₂²×Dic₁₉₉	1	1	yes	characteristic
177	C₂³	D₄×D₁₉₉	1	1	yes	characteristic
178	C₂³	D₄×D₁₉₉	1	2	yes	—
179	C₂³	C₂×D₇₉₆	1	1	yes	characteristic
180	C₂³	Q₈×D₁₉₉	1	1	yes	characteristic
181	C₂³	C₂×Dic₃₉₈	1	1	yes	characteristic
182	C₂³	#15691885	1	2	yes	—
183	C₂³	#15691885	1	1	yes	characteristic
184	C₂³	#15691886	1	2	yes	—
185	C₂³	#15691886	1	2	yes	—
186	C₂³	#15691887	1	1	yes	characteristic
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	2	4	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	4	no	—
191	C₂×C₄	—	2	4	no	—
192	C₂×C₄	—	2	4	no	—
193	C₂×C₄	—	2	4	no	—
194	C₂×C₄	—	4	2	no	—
195	C₂×C₄	—	4	2	no	—
196	C₂×C₄	—	4	2	no	—
197	C₂×C₄	—	4	2	no	—
198	C₂×C₄	—	796	1	no	—
199	C₂×C₄	—	796	2	no	—
200	C₂×C₄	—	796	2	no	—
201	C₂×C₄	—	796	1	no	—
202	C₂×C₄	—	796	2	no	—
203	C₂×C₄	—	796	1	no	—
204	C₂×C₄	—	796	2	no	—
205	C₂×C₄	—	796	1	no	—
206	C₂×C₄	—	796	2	no	—
207	C₂×C₄	—	796	2	no	—
208	C₂×C₄	—	398	4	no	—
209	C₂×C₄	—	398	4	no	—
210	C₂×C₄	—	398	4	no	—
211	C₂×C₄	—	398	4	no	—
212	C₂×C₄	—	398	2	no	—
213	C₂×C₄	—	398	4	no	—
214	C₂×C₄	—	398	2	no	—
215	C₂×C₄	C₄⋊Dic₁₉₉	1	4	yes	—
216	C₂²	C₂×C₄⋊Dic₁₉₉	1	1	yes	characteristic
217	C₂²	#20263914	1	2	yes	—
218	C₂²	D₄×Dic₁₉₉	1	2	yes	—
219	C₂²	D₄×Dic₁₉₉	1	1	yes	characteristic
220	C₂²	Q₈×Dic₁₉₉	1	1	yes	characteristic
221	C₂²	#20263925	1	1	yes	characteristic
222	C₂²	#20263928	1	2	yes	—
223	C₂²	#20263930	1	1	yes	characteristic
224	C₂²	#20263932	1	2	yes	—
225	C₂²	#20263933	1	2	yes	—
226	C₂²	#20263935	1	1	yes	characteristic
227	C₂²	#20263937	1	2	yes	—
228	C₂²	#20263939	1	1	yes	characteristic
229	C₂²	#20263940	1	2	yes	—
230	C₂²	#20263941	1	2	yes	—
231	C₂²	#20263943	1	1	yes	characteristic
232	C₂²	#20263943	1	1	yes	characteristic
233	C₂²	#20263945	1	2	yes	—
234	C₂²	#20263946	1	1	yes	characteristic
235	C₂²	#20263947	1	2	yes	—
236	C₂²	#20263948	1	2	yes	—
237	C₂²	#20263950	1	2	yes	—
238	C₂²	#20263954	1	1	yes	characteristic
239	C₄	—	2	4	no	—
240	C₄	—	4	4	no	—
241	C₄	—	796	2	no	—
242	C₄	—	796	4	no	—
243	C₄	—	796	2	no	—
244	C₂	#27163884	1	1	yes	characteristic
245	C₂	#27163893	1	1	yes	characteristic
246	C₂	#27163904	1	2	yes	—
247	C₂	#27163911	1	2	yes	—
248	C₂	#27163917	1	1	yes	characteristic
249	C₂	#27163931	1	1	yes	characteristic
250	C₂	#27163936	1	1	yes	characteristic
251	C₂	#27163945	1	1	yes	characteristic
252	C₂	#27163952	1	2	yes	—
253	C₂	#27163958	1	1	yes	characteristic
254	C₂	#27163960	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂⁴	C₂²×D₁₉₉
commutator	C₂³×C₁₉₉	C₂²×C₄
Fitting	C₂×#95×C₁₉₉	C₂
socle, prime radical, cosocle	C₂⁴×C₁₉₉	C₂³
hypercenter	C₂×#95	D₁₉₉
hypocenter	C₁₉₉	#1442
LCS	G ▷ C₂³×C₁₉₉ ▷ C₁₉₉
UCS	C₁ ◁ C₂⁴ ◁ C₂×#95

Complements

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

N	H ≅ G/N	type
16. #95×C₁₉₉	C₄	semidirect
127. C₁₉₉	#1442	semidirect
144. #95	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
C₂×C₁₉₉⋊#95	2
C₂×#95×C₁₉₉	2
#1442	199