Subgroups — C₄×#20263324 #38745280

Order 25216.

588 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#20263324	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Dic₃₉₄	C₂	1	1	yes	maximal, characteristic
3	#27160986	C₂	1	1	yes	maximal, characteristic
4	#27160984	C₂	1	1	yes	maximal, characteristic
5	#27160974	C₂	1	4	yes	maximal
6	#27160948	C₂	1	1	yes	maximal, characteristic
7	C₄×#15691337	C₂	1	4	yes	maximal
8	C₄²×Dic₁₉₇	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄×Q₈×C₁₉₇	C₂	1	1	yes	maximal, characteristic
10	C₂²×Dic₃₉₄	C₂²	1	1	yes	characteristic
11	#20263324	C₄	1	8	yes	direct product factor
12	C₄×Dic₃₉₄	—	2	4	no	—
13	C₂×C₄⋊Dic₁₉₇	C₂²	1	1	yes	characteristic
14	C₂×#15691337	C₂²	1	4	yes	—
15	C₂×#15691337	C₂²	1	2	yes	—
16	C₂×#15691337	C₂²	1	4	yes	—
17	C₂×C₄×Dic₁₉₇	C₂²	1	2	yes	—
18	C₂×C₄×Dic₁₉₇	C₂²	1	1	yes	characteristic
19	C₂×C₄×Dic₁₉₇	C₂²	1	2	yes	—
20	C₂×C₄×Dic₁₉₇	C₂²	1	1	yes	characteristic
21	C₂×C₄×Dic₁₉₇	C₂²	1	2	yes	—
22	#20263233	C₂²	1	4	yes	—
23	#20263233	C₂²	1	4	yes	—
24	C₂²×Q₈×C₁₉₇	C₂²	1	1	yes	characteristic
25	C₄×Q₈×C₁₉₇	—	2	4	no	—
26	C₂×C₄⋊C₄×C₁₉₇	C₂²	1	1	yes	characteristic
27	C₂×C₄⋊C₄×C₁₉₇	C₂²	1	2	yes	—
28	C₂×C₄²×C₁₉₇	C₂²	1	2	yes	—
29	C₂×C₄²×C₁₉₇	C₂²	1	1	yes	characteristic
30	C₂×Dic₃₉₄	—	2	2	no	—
31	C₂×Dic₃₉₄	—	2	2	no	—
32	C₂×Dic₃₉₄	C₂×C₄	1	4	yes	—
33	C₂²×Dic₁₉₇	C₂³	1	2	yes	—
34	C₂²×Dic₁₉₇	C₂³	1	2	yes	—
35	C₂²×Dic₁₉₇	C₂³	1	2	yes	—
36	C₂²×Dic₁₉₇	C₂³	1	2	yes	—
37	C₄⋊Dic₁₉₇	—	2	2	no	—
38	#15691337	—	2	8	no	—
39	#15691337	—	2	4	no	—
40	#15691337	C₂×C₄	1	16	yes	—
41	C₄×Dic₁₉₇	—	2	4	no	—
42	C₄×Dic₁₉₇	—	2	4	no	—
43	C₄×Dic₁₉₇	—	2	2	no	—
44	C₄×Dic₁₉₇	C₂×C₄	1	4	yes	—
45	C₄×Dic₁₉₇	Q₈	1	8	yes	complemented
46	C₂×Q₈×C₁₉₇	—	2	2	no	—
47	C₂×Q₈×C₁₉₇	—	2	2	no	—
48	C₂×Q₈×C₁₉₇	C₂×C₄	1	4	yes	—
49	C₂²×C₄×C₁₉₇	C₂³	1	2	yes	—
50	C₂²×C₄×C₁₉₇	C₂³	1	2	yes	—
51	C₂²×C₄×C₁₉₇	C₂³	1	1	yes	characteristic
52	C₂²×C₄×C₁₉₇	C₂³	1	1	yes	characteristic
53	C₂²×C₄×C₁₉₇	C₂³	1	1	yes	characteristic
54	C₄⋊C₄×C₁₉₇	—	2	4	no	—
55	C₄⋊C₄×C₁₉₇	—	2	2	no	—
56	C₄²×C₁₉₇	—	2	4	no	—
57	C₄²×C₁₉₇	D₄	1	4	yes	—
58	Dic₃₉₄	—	2	8	no	—
59	C₂×Dic₁₉₇	—	2	2	no	—
60	C₂×Dic₁₉₇	—	2	2	no	—
61	C₂×Dic₁₉₇	—	2	2	no	—
62	C₂×Dic₁₉₇	—	2	2	no	—
63	C₂×Dic₁₉₇	—	2	2	no	—
64	C₂×Dic₁₉₇	—	2	4	no	—
65	C₂×Dic₁₉₇	C₂²×C₄	1	4	yes	—
66	C₂×Dic₁₉₇	C₂²×C₄	1	4	yes	—
67	C₂×Dic₁₉₇	C₂×Q₈	1	4	yes	—
68	C₂×Dic₁₉₇	C₂×Q₈	1	4	yes	—
69	C₂×Dic₁₉₇	#41	1	4	yes	—
70	C₂³×C₁₉₇	C₂⁴	1	1	yes	characteristic
71	Q₈×C₁₉₇	—	2	8	no	—
72	C₂×C₄×C₁₉₇	—	2	2	no	—
73	C₂×C₄×C₁₉₇	—	2	2	no	—
74	C₂×C₄×C₁₉₇	—	2	2	no	—
75	C₂×C₄×C₁₉₇	—	2	2	no	—
76	C₂×C₄×C₁₉₇	—	2	4	no	—
77	C₂×C₄×C₁₉₇	C₂²×C₄	1	4	yes	—
78	C₂×C₄×C₁₉₇	C₂²×C₄	1	2	yes	—
79	C₂×C₄×C₁₉₇	C₂×D₄	1	2	yes	—
80	C₂×C₄×C₁₉₇	C₂×D₄	1	2	yes	—
81	C₂×C₄×C₁₉₇	C₂×D₄	1	2	yes	—
82	C₂×C₄×C₁₉₇	C₂×Q₈	1	4	yes	—
83	C₂×C₄×C₁₉₇	#41	1	2	yes	—
84	C₂²×C₁₉₇	C₂³×C₄	1	1	yes	characteristic
85	C₂²×C₁₉₇	C₂²×D₄	1	1	yes	characteristic
86	C₂²×C₁₉₇	C₂²×Q₈	1	2	yes	—
87	C₂²×C₁₉₇	C₂×#41	1	2	yes	—
88	C₂²×C₁₉₇	C₂×#41	1	1	yes	characteristic
89	C₄×C₁₉₇	—	2	4	no	—
90	C₄×C₁₉₇	—	2	2	no	—
91	C₄×C₁₉₇	—	4	2	no	—
92	C₄×C₁₉₇	C₄×D₄	1	4	yes	—
93	C₄×C₁₉₇	C₄⋊Q₈	1	4	yes	complemented
94	Dic₁₉₇	—	2	4	no	—
95	Dic₁₉₇	—	2	4	no	—
96	Dic₁₉₇	—	4	2	no	—
97	Dic₁₉₇	C₄×Q₈	1	8	yes	complemented
98	C₂×C₁₉₇	C₂×C₄×D₄	1	1	yes	characteristic
99	C₂×C₁₉₇	C₂×C₄×Q₈	1	2	yes	—
100	C₂×C₁₉₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
101	C₂×C₁₉₇	#532	1	1	yes	characteristic
102	C₂×C₁₉₇	#533	1	2	yes	—
103	C₁₉₇	C₄×C₄⋊Q₈	1	1	yes	characteristic, complemented, Sylow
104	C₄×C₄⋊Q₈	—	197	1	no	maximal, Sylow
105	C₂×C₄⋊Q₈	—	197	1	no	—
106	C₂×C₄×Q₈	—	197	1	no	—
107	C₂×C₄×Q₈	D₁₉₇	1	1	yes	characteristic
108	#391	—	197	1	no	—
109	#391	—	197	1	no	—
110	#389	—	197	4	no	—
111	#384	—	197	1	no	—
112	C₄×C₄⋊C₄	—	197	4	no	—
113	C₄³	—	197	1	no	—
114	C₂²×Q₈	—	197	1	no	—
115	C₂²×Q₈	C₂×D₁₉₇	1	1	yes	characteristic
116	C₄⋊Q₈	—	197	8	no	—
117	C₄×Q₈	—	2	4	no	—
118	C₄×Q₈	—	394	4	no	—
119	C₂×C₄⋊C₄	—	197	2	no	—
120	C₂×C₄⋊C₄	—	197	1	no	—
121	C₂×C₄⋊C₄	—	197	4	no	—
122	C₂×C₄⋊C₄	—	197	4	no	—
123	C₂×C₄⋊C₄	C₂×D₁₉₇	1	1	yes	characteristic
124	C₂×C₄⋊C₄	C₂×D₁₉₇	1	2	yes	—
125	C₂×C₄²	—	197	2	no	—
126	C₂×C₄²	—	197	1	no	—
127	C₂×C₄²	—	197	2	no	—
128	C₂×C₄²	—	197	1	no	—
129	C₂×C₄²	—	197	2	no	—
130	C₂×C₄²	C₂×D₁₉₇	1	2	yes	—
131	C₂×C₄²	C₂×D₁₉₇	1	1	yes	characteristic
132	#95	—	197	4	no	—
133	#95	—	197	4	no	—
134	C₂×Q₈	—	2	2	no	—
135	C₂×Q₈	—	2	2	no	—
136	C₂×Q₈	—	394	2	no	—
137	C₂×Q₈	—	394	2	no	—
138	C₂×Q₈	—	197	4	no	—
139	C₂×Q₈	C₄×D₁₉₇	1	4	yes	—
140	C₂²×C₄	—	197	2	no	—
141	C₂²×C₄	—	197	2	no	—
142	C₂²×C₄	—	197	2	no	—
143	C₂²×C₄	—	197	2	no	—
144	C₂²×C₄	C₂²×D₁₉₇	1	2	yes	—
145	C₂²×C₄	C₂²×D₁₉₇	1	2	yes	—
146	C₂²×C₄	C₂²×D₁₉₇	1	1	yes	characteristic
147	C₂²×C₄	C₂²×D₁₉₇	1	1	yes	characteristic
148	C₂²×C₄	C₂²×D₁₉₇	1	1	yes	characteristic
149	C₄⋊C₄	—	2	4	no	—
150	C₄⋊C₄	—	2	2	no	—
151	C₄⋊C₄	—	394	8	no	—
152	C₄⋊C₄	—	394	4	no	—
153	C₄⋊C₄	—	394	2	no	—
154	C₄⋊C₄	—	197	16	no	—
155	C₄²	—	2	4	no	—
156	C₄²	—	394	4	no	—
157	C₄²	—	394	4	no	—
158	C₄²	—	394	2	no	—
159	C₄²	—	197	4	no	—
160	C₄²	—	197	8	no	—
161	C₄²	C₂²⋊D₁₉₇	1	4	yes	—
162	C₂³	C₂³×D₁₉₇	1	1	yes	characteristic
163	Q₈	—	2	8	no	—
164	Q₈	—	394	8	no	—
165	C₂×C₄	—	2	2	no	—
166	C₂×C₄	—	2	2	no	—
167	C₂×C₄	—	2	2	no	—
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	2	4	no	—
170	C₂×C₄	—	394	2	no	—
171	C₂×C₄	—	394	4	no	—
172	C₂×C₄	—	394	2	no	—
173	C₂×C₄	—	394	2	no	—
174	C₂×C₄	—	394	2	no	—
175	C₂×C₄	—	394	2	no	—
176	C₂×C₄	—	197	4	no	—
177	C₂×C₄	—	197	4	no	—
178	C₂×C₄	—	197	4	no	—
179	C₂×C₄	—	197	4	no	—
180	C₂×C₄	—	197	4	no	—
181	C₂×C₄	C₂×C₄×D₁₉₇	1	4	yes	—
182	C₂×C₄	C₂×C₄×D₁₉₇	1	2	yes	—
183	C₂×C₄	C₂×C₂²⋊D₁₉₇	1	2	yes	—
184	C₂×C₄	C₂×C₂²⋊D₁₉₇	1	2	yes	—
185	C₂×C₄	C₂×C₂²⋊D₁₉₇	1	2	yes	—
186	C₂×C₄	Q₈×D₁₉₇	1	4	yes	—
187	C₂×C₄	#15691358	1	2	yes	—
188	C₂²	C₂²×C₄×D₁₉₇	1	1	yes	characteristic
189	C₂²	C₂²×C₂²⋊D₁₉₇	1	1	yes	characteristic
190	C₂²	C₂×Q₈×D₁₉₇	1	2	yes	—
191	C₂²	#41×D₁₉₇	1	2	yes	—
192	C₂²	C₂×#15691358	1	1	yes	characteristic
193	C₄	—	2	2	no	—
194	C₄	—	2	4	no	—
195	C₄	—	4	2	no	—
196	C₄	—	788	2	no	—
197	C₄	—	394	4	no	—
198	C₄	—	394	4	no	—
199	C₄	—	197	8	no	—
200	C₄	C₄×C₂²⋊D₁₉₇	1	4	yes	—
201	C₄	#20263324	1	4	yes	direct product factor
202	C₂	C₂×C₄×C₂²⋊D₁₉₇	1	1	yes	characteristic
203	C₂	C₄×Q₈×D₁₉₇	1	2	yes	—
204	C₂	C₂×#20263324	1	1	yes	characteristic
205	C₂	#27161625	1	1	yes	characteristic
206	C₂	#27161633	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂²×D₁₉₇
commutator	C₂²×C₁₉₇	C₂³×C₄
Frattini	C₂³	C₂³×D₁₉₇
Fitting	C₂×C₄×Q₈×C₁₉₇	C₂
socle, prime radical, cosocle	C₂³×C₁₉₇	C₂⁴
hypercenter	C₂×C₄×Q₈	D₁₉₇
hypocenter	C₁₉₇	C₄×C₄⋊Q₈
LCS	G ▷ C₂²×C₁₉₇ ▷ C₁₉₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×Q₈

Complements

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

N	H ≅ G/N	type
11. #20263324	C₄	direct (N × H)
45. C₄×Dic₁₉₇	Q₈	semidirect
93. C₄×C₁₉₇	C₄⋊Q₈	semidirect
97. Dic₁₉₇	C₄×Q₈	semidirect
103. C₁₉₇	C₄×C₄⋊Q₈	semidirect
201. C₄	#20263324	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#20263324	2
C₂×C₄×Dic₃₉₄	2
#27160986	2
#27160984	2
#27160974	2
#27160948	2
C₄×#15691337	2
C₄²×Dic₁₉₇	2
C₂×C₄×Q₈×C₁₉₇	2
C₄×C₄⋊Q₈	197

Subgroups — C4×#20263324 #38745280

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#20263324 #38745280