Subgroups — #38767072

Order 25472.

1088 conjugacy classes of subgroups.

The subgroup lattice has 266 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×Dic₁₉₉	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₂²⋊Dic₁₉₉	C₂	1	1	yes	maximal, characteristic
3	C₂×C₁₉₉⋊#95	C₂	1	4	yes	maximal, complemented
4	C₂²×C₂²⋊C₄×C₁₉₉	C₂	1	1	yes	maximal, characteristic
5	C₂³×Dic₁₉₉	—	2	8	no	—
6	C₂³×Dic₁₉₉	—	2	2	no	—
7	C₂³×Dic₁₉₉	—	2	2	no	—
8	C₂³×Dic₁₉₉	C₄	1	4	yes	complemented
9	C₂³×Dic₁₉₉	C₂²	1	2	yes	—
10	C₂³×Dic₁₉₉	C₂²	1	2	yes	—
11	C₂×C₂²⋊Dic₁₉₉	—	2	2	no	—
12	C₂×C₂²⋊Dic₁₉₉	—	2	2	no	—
13	C₂×C₂²⋊Dic₁₉₉	C₄	1	4	yes	complemented
14	C₁₉₉⋊#95	—	2	8	no	—
15	C₂⁵×C₁₉₉	C₂²	1	1	yes	characteristic
16	C₂³×C₄×C₁₉₉	C₂²	1	2	yes	—
17	C₂×C₂²⋊C₄×C₁₉₉	—	2	2	no	—
18	C₂×C₂²⋊C₄×C₁₉₉	—	2	2	no	—
19	C₂×C₂²⋊C₄×C₁₉₉	C₄	1	4	yes	complemented
20	C₂²×Dic₁₉₉	—	2	16	no	—
21	C₂²×Dic₁₉₉	—	2	16	no	—
22	C₂²×Dic₁₉₉	—	2	16	no	—
23	C₂²×Dic₁₉₉	—	2	8	no	—
24	C₂²×Dic₁₉₉	—	2	4	no	—
25	C₂²×Dic₁₉₉	—	2	2	no	—
26	C₂²×Dic₁₉₉	—	2	2	no	—
27	C₂²×Dic₁₉₉	—	2	4	no	—
28	C₂²×Dic₁₉₉	—	2	2	no	—
29	C₂²×Dic₁₉₉	—	2	2	no	—
30	C₂²×Dic₁₉₉	—	2	4	no	—
31	C₂²×Dic₁₉₉	C₂×C₄	1	4	yes	—
32	C₂²×Dic₁₉₉	C₂×C₄	1	4	yes	complemented
33	C₂²×Dic₁₉₉	D₄	1	8	yes	—
34	C₂²⋊Dic₁₉₉	—	2	8	no	—
35	C₂⁴×C₁₉₉	—	2	4	no	—
36	C₂⁴×C₁₉₉	—	2	4	no	—
37	C₂⁴×C₁₉₉	C₂×C₄	1	2	yes	—
38	C₂⁴×C₁₉₉	C₂×C₄	1	2	yes	—
39	C₂⁴×C₁₉₉	C₂×C₄	1	2	yes	—
40	C₂⁴×C₁₉₉	D₄	1	2	yes	—
41	C₂⁴×C₁₉₉	D₄	1	2	yes	—
42	C₂⁴×C₁₉₉	D₄	1	2	yes	—
43	C₂⁴×C₁₉₉	Q₈	1	2	yes	—
44	C₂⁴×C₁₉₉	C₂³	1	1	yes	characteristic
45	C₂²×C₄×C₁₉₉	—	2	2	no	—
46	C₂²×C₄×C₁₉₉	—	2	2	no	—
47	C₂²×C₄×C₁₉₉	—	2	4	no	—
48	C₂²×C₄×C₁₉₉	—	2	4	no	—
49	C₂²×C₄×C₁₉₉	C₂×C₄	1	4	yes	complemented
50	C₂²⋊C₄×C₁₉₉	—	2	8	no	—
51	C₂×Dic₁₉₉	—	2	32	no	—
52	C₂×Dic₁₉₉	—	2	4	no	—
53	C₂×Dic₁₉₉	—	2	4	no	—
54	C₂×Dic₁₉₉	—	2	8	no	—
55	C₂×Dic₁₉₉	—	2	8	no	—
56	C₂×Dic₁₉₉	—	2	4	no	—
57	C₂×Dic₁₉₉	—	2	8	no	—
58	C₂×Dic₁₉₉	—	4	2	no	—
59	C₂×Dic₁₉₉	—	4	4	no	—
60	C₂×Dic₁₉₉	—	4	2	no	—
61	C₂×Dic₁₉₉	C₂²⋊C₄	1	8	yes	complemented
62	C₂³×C₁₉₉	—	2	4	no	—
63	C₂³×C₁₉₉	—	2	4	no	—
64	C₂³×C₁₉₉	—	2	8	no	—
65	C₂³×C₁₉₉	—	2	8	no	—
66	C₂³×C₁₉₉	—	2	4	no	—
67	C₂³×C₁₉₉	—	2	4	no	—
68	C₂³×C₁₉₉	—	2	8	no	—
69	C₂³×C₁₉₉	—	2	8	no	—
70	C₂³×C₁₉₉	—	2	4	no	—
71	C₂³×C₁₉₉	—	2	4	no	—
72	C₂³×C₁₉₉	C₄²	1	4	yes	—
73	C₂³×C₁₉₉	C₂²⋊C₄	1	4	yes	—
74	C₂³×C₁₉₉	C₂²⋊C₄	1	4	yes	—
75	C₂³×C₁₉₉	C₂²⋊C₄	1	4	yes	—
76	C₂³×C₁₉₉	C₄⋊C₄	1	4	yes	—
77	C₂³×C₁₉₉	C₄⋊C₄	1	4	yes	—
78	C₂³×C₁₉₉	C₄⋊C₄	1	4	yes	—
79	C₂³×C₁₉₉	C₂²×C₄	1	1	yes	characteristic
80	C₂³×C₁₉₉	C₂²×C₄	1	1	yes	characteristic
81	C₂³×C₁₉₉	C₂²×C₄	1	1	yes	characteristic
82	C₂³×C₁₉₉	C₂×D₄	1	4	yes	—
83	C₂³×C₁₉₉	C₂×D₄	1	1	yes	characteristic
84	C₂³×C₁₉₉	C₂×D₄	1	1	yes	characteristic
85	C₂³×C₁₉₉	C₂×D₄	1	1	yes	characteristic
86	C₂³×C₁₉₉	C₂×Q₈	1	1	yes	characteristic
87	C₂³×C₁₉₉	#41	1	4	yes	—
88	C₂×C₄×C₁₉₉	—	2	4	no	—
89	C₂×C₄×C₁₉₉	—	2	8	no	—
90	C₂×C₄×C₁₉₉	—	4	2	no	—
91	C₂×C₄×C₁₉₉	—	4	4	no	—
92	C₂×C₄×C₁₉₉	—	4	2	no	—
93	C₂²×C₁₉₉	—	2	8	no	—
94	C₂²×C₁₉₉	—	2	4	no	—
95	C₂²×C₁₉₉	—	2	8	no	—
96	C₂²×C₁₉₉	—	2	4	no	—
97	C₂²×C₁₉₉	—	2	8	no	—
98	C₂²×C₁₉₉	—	2	8	no	—
99	C₂²×C₁₉₉	—	2	8	no	—
100	C₂²×C₁₉₉	—	2	8	no	—
101	C₂²×C₁₉₉	#95	1	8	yes	complemented
102	C₂²×C₁₉₉	C₂×C₄²	1	1	yes	characteristic
103	C₂²×C₁₉₉	C₂×C₂²⋊C₄	1	2	yes	—
104	C₂²×C₁₉₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
105	C₂²×C₁₉₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
106	C₂²×C₁₉₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
107	C₂²×C₁₉₉	C₂×C₄⋊C₄	1	1	yes	characteristic
108	C₂²×C₁₉₉	C₂×C₄⋊C₄	1	1	yes	characteristic
109	C₂²×C₁₉₉	C₂×C₄⋊C₄	1	1	yes	characteristic
110	C₂²×C₁₉₉	#117	1	2	yes	—
111	C₂²×C₁₉₉	C₄×D₄	1	4	yes	—
112	C₂²×C₁₉₉	C₄×D₄	1	4	yes	—
113	C₂²×C₁₉₉	#120	1	2	yes	—
114	C₂²×C₁₉₉	#120	1	2	yes	—
115	C₂²×C₁₉₉	#121	1	4	yes	—
116	C₂²×C₁₉₉	C₂²⋊Q₈	1	4	yes	—
117	C₂²×C₁₉₉	#123	1	2	yes	—
118	C₂²×C₁₉₉	#123	1	2	yes	—
119	C₄×C₁₉₉	—	4	4	no	—
120	Dic₁₉₉	—	2	8	no	—
121	Dic₁₉₉	—	4	4	no	—
122	C₂×C₁₉₉	—	2	8	no	—
123	C₂×C₁₉₉	C₂×#95	1	1	yes	characteristic
124	C₂×C₁₉₉	C₄×C₂²⋊C₄	1	2	yes	—
125	C₂×C₁₉₉	#379	1	1	yes	characteristic
126	C₂×C₁₉₉	#380	1	2	yes	—
127	C₂×C₁₉₉	#381	1	1	yes	characteristic
128	C₂×C₁₉₉	#385	1	2	yes	—
129	C₂×C₁₉₉	#385	1	2	yes	—
130	C₂×C₁₉₉	#386	1	2	yes	—
131	C₂×C₁₉₉	#386	1	2	yes	—
132	C₁₉₉	C₂²⋊#95	1	1	yes	characteristic, complemented, Sylow
133	C₂²⋊#95	—	199	1	no	maximal, Sylow
134	C₂⁴×C₄	—	199	1	no	—
135	C₂²×C₂²⋊C₄	—	199	1	no	—
136	C₂²×C₂²⋊C₄	D₁₉₉	1	1	yes	characteristic
137	C₂×#95	—	199	4	no	—
138	C₂⁵	C₂×D₁₉₉	1	1	yes	characteristic
139	C₂³×C₄	—	398	2	no	—
140	C₂³×C₄	—	398	2	no	—
141	C₂³×C₄	—	398	8	no	—
142	C₂³×C₄	—	199	4	no	—
143	C₂³×C₄	—	199	2	no	—
144	C₂³×C₄	—	199	2	no	—
145	C₂³×C₄	C₂×D₁₉₉	1	2	yes	—
146	C₂×C₂²⋊C₄	—	2	2	no	—
147	C₂×C₂²⋊C₄	—	2	2	no	—
148	C₂×C₂²⋊C₄	—	398	2	no	—
149	C₂×C₂²⋊C₄	—	398	2	no	—
150	C₂×C₂²⋊C₄	—	199	4	no	—
151	C₂×C₂²⋊C₄	Dic₁₉₉	1	4	yes	complemented
152	#95	—	398	8	no	—
153	C₂⁴	—	2	4	no	—
154	C₂⁴	—	2	4	no	—
155	C₂⁴	C₄×D₁₉₉	1	2	yes	—
156	C₂⁴	C₄×D₁₉₉	1	2	yes	—
157	C₂⁴	C₂×Dic₁₉₉	1	2	yes	—
158	C₂⁴	C₂²⋊D₁₉₉	1	2	yes	—
159	C₂⁴	C₂²⋊D₁₉₉	1	2	yes	—
160	C₂⁴	D₇₉₆	1	2	yes	—
161	C₂⁴	Dic₃₉₈	1	2	yes	—
162	C₂⁴	C₂²×D₁₉₉	1	1	yes	characteristic
163	C₂²×C₄	—	2	4	no	—
164	C₂²×C₄	—	2	2	no	—
165	C₂²×C₄	—	2	2	no	—
166	C₂²×C₄	—	2	4	no	—
167	C₂²×C₄	—	398	4	no	—
168	C₂²×C₄	—	398	2	no	—
169	C₂²×C₄	—	398	2	no	—
170	C₂²×C₄	—	398	4	no	—
171	C₂²×C₄	—	398	16	no	—
172	C₂²×C₄	—	398	16	no	—
173	C₂²×C₄	—	398	8	no	—
174	C₂²×C₄	—	398	4	no	—
175	C₂²×C₄	—	398	2	no	—
176	C₂²×C₄	—	398	2	no	—
177	C₂²×C₄	—	398	16	no	—
178	C₂²×C₄	—	199	4	no	—
179	C₂²×C₄	—	199	4	no	—
180	C₂²×C₄	—	199	8	no	—
181	C₂²×C₄	C₂×Dic₁₉₉	1	4	yes	complemented
182	C₂²⋊C₄	—	2	8	no	—
183	C₂²⋊C₄	—	398	8	no	—
184	C₂³	—	2	4	no	—
185	C₂³	—	2	4	no	—
186	C₂³	—	2	4	no	—
187	C₂³	—	2	4	no	—
188	C₂³	—	2	8	no	—
189	C₂³	—	2	8	no	—
190	C₂³	—	2	4	no	—
191	C₂³	—	2	4	no	—
192	C₂³	—	2	8	no	—
193	C₂³	—	2	8	no	—
194	C₂³	C₄×Dic₁₉₉	1	4	yes	—
195	C₂³	(C₂×D₁₉₉)⋊C₄	1	4	yes	—
196	C₂³	(C₂×D₁₉₉)⋊C₄	1	4	yes	—
197	C₂³	C₂²⋊Dic₁₉₉	1	4	yes	—
198	C₂³	Dic₁₉₉⋊C₄	1	4	yes	—
199	C₂³	Dic₁₉₉⋊C₄	1	4	yes	—
200	C₂³	C₄⋊Dic₁₉₉	1	4	yes	—
201	C₂³	C₂×C₄×D₁₉₉	1	1	yes	characteristic
202	C₂³	C₂×C₄×D₁₉₉	1	1	yes	characteristic
203	C₂³	C₂²×Dic₁₉₉	1	1	yes	characteristic
204	C₂³	D₄×D₁₉₉	1	4	yes	—
205	C₂³	C₂×C₂²⋊D₁₉₉	1	1	yes	characteristic
206	C₂³	C₂×C₂²⋊D₁₉₉	1	1	yes	characteristic
207	C₂³	C₂×D₇₉₆	1	1	yes	characteristic
208	C₂³	C₂×Dic₃₉₈	1	1	yes	characteristic
209	C₂³	#15691885	1	4	yes	—
210	C₂×C₄	—	2	4	no	—
211	C₂×C₄	—	2	8	no	—
212	C₂×C₄	—	4	2	no	—
213	C₂×C₄	—	4	2	no	—
214	C₂×C₄	—	4	4	no	—
215	C₂×C₄	—	796	2	no	—
216	C₂×C₄	—	796	2	no	—
217	C₂×C₄	—	796	4	no	—
218	C₂×C₄	—	398	4	no	—
219	C₂×C₄	—	398	8	no	—
220	C₂×C₄	—	398	32	no	—
221	C₂×C₄	—	398	4	no	—
222	C₂×C₄	—	398	4	no	—
223	C₂×C₄	—	398	8	no	—
224	C₂×C₄	—	398	8	no	—
225	C₂×C₄	—	199	8	no	—
226	C₂²	—	2	8	no	—
227	C₂²	—	2	4	no	—
228	C₂²	—	2	8	no	—
229	C₂²	—	2	4	no	—
230	C₂²	—	2	8	no	—
231	C₂²	—	2	8	no	—
232	C₂²	—	2	8	no	—
233	C₂²	—	2	8	no	—
234	C₂²	C₁₉₉⋊#95	1	8	yes	complemented
235	C₂²	C₂×C₄×Dic₁₉₉	1	1	yes	characteristic
236	C₂²	C₂²⋊C₄×D₁₉₉	1	2	yes	—
237	C₂²	C₂×(C₂×D₁₉₉)⋊C₄	1	1	yes	characteristic
238	C₂²	C₂×(C₂×D₁₉₉)⋊C₄	1	1	yes	characteristic
239	C₂²	C₂×C₂²⋊Dic₁₉₉	1	1	yes	characteristic
240	C₂²	C₂×Dic₁₉₉⋊C₄	1	1	yes	characteristic
241	C₂²	C₂×Dic₁₉₉⋊C₄	1	1	yes	characteristic
242	C₂²	C₂×C₄⋊Dic₁₉₉	1	1	yes	characteristic
243	C₂²	#20263911	1	2	yes	—
244	C₂²	(C₂²⋊D₁₉₉)⋊C₄	1	4	yes	—
245	C₂²	D₄×Dic₁₉₉	1	4	yes	—
246	C₂²	#20263923	1	2	yes	—
247	C₂²	#20263925	1	2	yes	—
248	C₂²	#20263927	1	4	yes	—
249	C₂²	#20263932	1	4	yes	—
250	C₂²	#20263936	1	2	yes	—
251	C₂²	#20263939	1	2	yes	—
252	C₄	—	4	4	no	—
253	C₄	—	796	4	no	—
254	C₄	—	398	8	no	—
255	C₂	—	2	8	no	—
256	C₂	C₂×C₁₉₉⋊#95	1	1	yes	characteristic
257	C₂	C₂²⋊C₄×Dic₁₉₉	1	2	yes	—
258	C₂	#27163873	1	1	yes	characteristic
259	C₂	#27163876	1	2	yes	—
260	C₂	#27163879	1	1	yes	characteristic
261	C₂	#27163890	1	2	yes	—
262	C₂	#27163893	1	2	yes	—
263	C₂	#27163895	1	2	yes	—
264	C₂	#27163898	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂⁴	C₂²×D₁₉₉
commutator	C₂²×C₁₉₉	C₂×C₄²
Fitting	C₂²×C₂²⋊C₄×C₁₉₉	C₂
socle, cosocle	C₂⁴×C₁₉₉	C₂³
prime radical	C₂⁵×C₁₉₉	C₂²
hypercenter	C₂²×C₂²⋊C₄	D₁₉₉
hypocenter	C₁₉₉	C₂²⋊#95
LCS	G ▷ C₂²×C₁₉₉ ▷ C₁₉₉
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
3. C₂×C₁₉₉⋊#95	C₂	semidirect
8. C₂³×Dic₁₉₉	C₄	semidirect
13. C₂×C₂²⋊Dic₁₉₉	C₄	semidirect
19. C₂×C₂²⋊C₄×C₁₉₉	C₄	semidirect
32. C₂²×Dic₁₉₉	C₂×C₄	semidirect
49. C₂²×C₄×C₁₉₉	C₂×C₄	semidirect
61. C₂×Dic₁₉₉	C₂²⋊C₄	semidirect
101. C₂²×C₁₉₉	#95	semidirect
132. C₁₉₉	C₂²⋊#95	semidirect
151. C₂×C₂²⋊C₄	Dic₁₉₉	semidirect
181. C₂²×C₄	C₂×Dic₁₉₉	semidirect
234. C₂²	C₁₉₉⋊#95	semidirect

Maximal subgroups

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

subgroup	index
C₂⁴×Dic₁₉₉	2
C₂²×C₂²⋊Dic₁₉₉	2
C₂×C₁₉₉⋊#95	2
C₂²×C₂²⋊C₄×C₁₉₉	2
C₂²⋊#95	199