Subgroups — C₂×#27161199 #38751606

Order 25216.

464 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	C₂×#20263346	C₂	1	1	yes	maximal, characteristic
2	C₂×#20263332	C₂	1	1	yes	maximal, characteristic, complemented
3	#27161199	C₂	1	8	yes	maximal, direct product factor
4	C₂×#20263261	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#20263253	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#20263252	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#20263236	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#131×C₁₉₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#15691358	C₂²	1	1	yes	characteristic
10	#20263346	C₄	1	4	yes	complemented
11	#20263346	C₂²	1	4	yes	—
12	C₂×#15691348	C₄	1	1	yes	characteristic, complemented
13	C₂×#15691346	C₄	1	1	yes	characteristic, complemented
14	C₂×#15691345	C₄	1	1	yes	characteristic, complemented
15	C₂×#15691343	C₄	1	1	yes	characteristic, complemented
16	#20263332	—	2	2	no	—
17	#20263332	C₂²	1	4	yes	complemented
18	C₂×#15691342	C₂²	1	1	yes	characteristic
19	C₂×#15691341	—	2	1	no	—
20	C₂²×#13096802	C₂²	1	1	yes	characteristic
21	C₂×C₈×D₁₉₇	—	2	1	no	—
22	C₂×C₄×Dic₁₉₇	C₂²	1	1	yes	characteristic
23	#20263261	C₂²	1	4	yes	complemented
24	#20263253	C₂²	1	4	yes	complemented
25	#20263252	C₂²	1	4	yes	complemented
26	#20263236	C₂²	1	4	yes	complemented
27	C₂×#41×C₁₉₇	C₂²	1	1	yes	characteristic
28	#131×C₁₉₇	—	2	2	no	—
29	#131×C₁₉₇	C₂²	1	4	yes	complemented
30	C₂×M₁₆×C₁₉₇	—	2	1	no	—
31	C₂×M₁₆×C₁₉₇	C₂²	1	1	yes	characteristic
32	C₂²×C₈×C₁₉₇	—	2	1	no	—
33	C₂²×C₈×C₁₉₇	C₂²	1	1	yes	characteristic
34	#15691358	—	2	1	no	—
35	#15691358	—	2	1	no	—
36	#15691358	C₂×C₄	1	2	yes	—
37	#15691358	C₂³	1	2	yes	—
38	C₂×Dic₃₉₄	C₂×C₄	1	1	yes	characteristic
39	C₂×D₇₈₈	C₂×C₄	1	1	yes	characteristic
40	C₂×C₂²⋊D₁₉₇	—	2	1	no	—
41	C₂²×Dic₁₉₇	—	2	1	no	—
42	C₂×C₄×D₁₉₇	—	2	1	no	—
43	#15691348	C₂×C₄	1	4	yes	complemented
44	#15691346	C₂×C₄	1	4	yes	complemented
45	#15691345	C₂×C₄	1	4	yes	complemented
46	#15691343	C₂×C₄	1	4	yes	complemented
47	#15691342	—	2	1	no	—
48	#15691342	C₂³	1	2	yes	—
49	#15691341	—	2	4	no	—
50	C₂×#13096802	—	2	1	no	—
51	C₂×#13096802	C₂×C₄	1	1	yes	characteristic
52	C₂×#13096802	C₂×C₄	1	1	yes	characteristic
53	C₂×#13096802	C₂×C₄	1	2	yes	—
54	C₂×#13096802	C₂³	1	2	yes	—
55	C₈×D₁₉₇	—	2	4	no	—
56	C₄×Dic₁₉₇	—	2	1	no	—
57	C₄×Dic₁₉₇	C₂³	1	2	yes	—
58	#41×C₁₉₇	—	2	1	no	—
59	#41×C₁₉₇	—	2	1	no	—
60	#41×C₁₉₇	C₂×C₄	1	2	yes	—
61	#41×C₁₉₇	C₂³	1	2	yes	—
62	C₂×Q₈×C₁₉₇	C₂×C₄	1	1	yes	characteristic
63	C₂×D₄×C₁₉₇	—	2	1	no	—
64	C₂×D₄×C₁₉₇	C₂×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₁₉₇	—	2	1	no	—
66	C₂²×C₄×C₁₉₇	C₂³	1	1	yes	characteristic
67	M₁₆×C₁₉₇	—	2	1	no	—
68	M₁₆×C₁₉₇	—	2	4	no	—
69	M₁₆×C₁₉₇	C₂³	1	2	yes	—
70	C₂×C₈×C₁₉₇	—	2	1	no	—
71	C₂×C₈×C₁₉₇	—	2	4	no	—
72	C₂×C₈×C₁₉₇	D₄	1	1	yes	characteristic
73	C₂×C₈×C₁₉₇	D₄	1	1	yes	characteristic
74	C₂×C₈×C₁₉₇	D₄	1	2	yes	—
75	C₂×C₈×C₁₉₇	C₂³	1	2	yes	—
76	C₂²×D₁₉₇	—	2	1	no	—
77	Dic₃₉₄	—	2	1	no	—
78	Dic₃₉₄	C₂²×C₄	1	2	yes	—
79	D₇₈₈	—	2	1	no	—
80	D₇₈₈	C₂²×C₄	1	2	yes	—
81	C₂²⋊D₁₉₇	—	2	2	no	—
82	C₂²⋊D₁₉₇	—	2	2	no	—
83	C₂×Dic₁₉₇	—	2	1	no	—
84	C₂×Dic₁₉₇	—	2	2	no	—
85	C₂×Dic₁₉₇	—	4	2	no	—
86	C₄×D₁₉₇	—	2	2	no	—
87	C₄×D₁₉₇	—	2	2	no	—
88	#13096802	—	2	2	no	—
89	#13096802	C₂²×C₄	1	2	yes	—
90	#13096802	C₂²×C₄	1	2	yes	—
91	C₂³×C₁₉₇	—	2	1	no	—
92	C₂³×C₁₉₇	#41	1	1	yes	characteristic
93	Q₈×C₁₉₇	—	2	1	no	—
94	Q₈×C₁₉₇	C₂²×C₄	1	2	yes	—
95	D₄×C₁₉₇	—	2	2	no	—
96	D₄×C₁₉₇	—	2	2	no	—
97	D₄×C₁₉₇	—	2	1	no	—
98	D₄×C₁₉₇	C₂²×C₄	1	2	yes	—
99	C₂×C₄×C₁₉₇	—	2	1	no	—
100	C₂×C₄×C₁₉₇	—	2	2	no	—
101	C₂×C₄×C₁₉₇	—	2	2	no	—
102	C₂×C₄×C₁₉₇	C₂²×C₄	1	1	yes	characteristic
103	C₂×C₄×C₁₉₇	C₂²×C₄	1	1	yes	characteristic
104	C₂×C₄×C₁₉₇	C₂×D₄	1	1	yes	characteristic
105	C₂×C₄×C₁₉₇	C₂×D₄	1	1	yes	characteristic
106	C₂×C₄×C₁₉₇	#41	1	1	yes	characteristic
107	C₂×C₄×C₁₉₇	C₂⁴	1	1	yes	characteristic
108	C₈×C₁₉₇	—	2	2	no	—
109	C₈×C₁₉₇	C₂×D₄	1	2	yes	—
110	C₈×C₁₉₇	C₂×D₄	1	2	yes	—
111	C₂²×C₁₉₇	—	2	2	no	—
112	C₂²×C₁₉₇	—	2	2	no	—
113	C₂²×C₁₉₇	—	4	2	no	—
114	C₂²×C₁₉₇	C₄×D₄	1	1	yes	characteristic
115	C₂²×C₁₉₇	C₄×D₄	1	1	yes	characteristic
116	C₂²×C₁₉₇	C₂×#41	1	1	yes	characteristic
117	C₂×D₁₉₇	—	2	2	no	—
118	C₂×D₁₉₇	—	4	2	no	—
119	C₄×C₁₉₇	—	2	2	no	—
120	C₄×C₁₉₇	C₄×D₄	1	1	yes	characteristic
121	C₄×C₁₉₇	C₄×D₄	1	1	yes	characteristic
122	C₄×C₁₉₇	C₂³×C₄	1	1	yes	characteristic
123	C₄×C₁₉₇	C₂²×D₄	1	1	yes	characteristic
124	Dic₁₉₇	—	2	2	no	—
125	Dic₁₉₇	—	4	2	no	—
126	C₂×C₁₉₇	—	2	1	no	—
127	C₂×C₁₉₇	—	2	1	no	—
128	C₂×C₁₉₇	—	4	2	no	—
129	C₂×C₁₉₇	#444	1	2	yes	complemented
130	C₂×C₁₉₇	C₂×C₄×D₄	1	1	yes	characteristic
131	D₁₉₇	—	4	2	no	—
132	C₁₉₇	C₂×#444	1	1	yes	characteristic, complemented, Sylow
133	C₂×#444	—	197	1	no	maximal, Sylow
134	C₂×#135	—	197	1	no	—
135	C₂×#131	—	197	1	no	—
136	C₂×#131	D₁₉₇	1	1	yes	characteristic, complemented
137	#444	—	197	8	no	—
138	C₂×#108	—	197	1	no	—
139	C₂×#104	—	197	1	no	—
140	C₂×#104	—	197	1	no	—
141	C₂×#97	—	197	1	no	—
142	C₂×#41	—	197	1	no	—
143	C₂×#41	C₂×D₁₉₇	1	1	yes	characteristic
144	#135	—	197	4	no	—
145	#135	—	197	4	no	—
146	C₂×Q₁₆	—	197	1	no	—
147	C₂×SD₁₆	—	197	1	no	—
148	C₂×SD₁₆	—	197	1	no	—
149	C₂×D₈	—	197	1	no	—
150	#131	—	2	2	no	—
151	#131	—	394	2	no	—
152	#131	—	197	4	no	—
153	#131	C₂×D₁₉₇	1	4	yes	complemented
154	C₂×M₁₆	—	2	1	no	—
155	C₂×M₁₆	—	394	1	no	—
156	C₂×M₁₆	—	197	1	no	—
157	C₂×M₁₆	C₂×D₁₉₇	1	1	yes	characteristic
158	C₂²×C₈	—	2	1	no	—
159	C₂²×C₈	—	394	1	no	—
160	C₂²×C₈	—	197	1	no	—
161	C₂²×C₈	C₂×D₁₉₇	1	1	yes	characteristic
162	C₂×C₄²	—	197	1	no	—
163	#108	—	197	4	no	—
164	#104	—	197	4	no	—
165	#104	—	197	4	no	—
166	#97	—	197	4	no	—
167	#41	—	2	1	no	—
168	#41	—	2	1	no	—
169	#41	—	394	1	no	—
170	#41	—	394	1	no	—
171	#41	—	197	2	no	—
172	#41	—	197	2	no	—
173	#41	C₄×D₁₉₇	1	2	yes	—
174	#41	C₂²×D₁₉₇	1	2	yes	—
175	C₂×Q₈	—	197	1	no	—
176	C₂×Q₈	C₄×D₁₉₇	1	1	yes	characteristic
177	C₂×D₄	—	2	1	no	—
178	C₂×D₄	—	394	1	no	—
179	C₂×D₄	—	197	1	no	—
180	C₂×D₄	C₄×D₁₉₇	1	1	yes	characteristic
181	C₂²×C₄	—	2	1	no	—
182	C₂²×C₄	—	394	1	no	—
183	C₂²×C₄	—	394	1	no	—
184	C₂²×C₄	C₂²×D₁₉₇	1	1	yes	characteristic
185	Q₁₆	—	197	4	no	—
186	SD₁₆	—	197	4	no	—
187	SD₁₆	—	197	4	no	—
188	D₈	—	197	4	no	—
189	M₁₆	—	2	1	no	—
190	M₁₆	—	2	4	no	—
191	M₁₆	—	394	1	no	—
192	M₁₆	—	394	4	no	—
193	M₁₆	—	197	2	no	—
194	M₁₆	C₂²×D₁₉₇	1	2	yes	—
195	C₂×C₈	—	2	1	no	—
196	C₂×C₈	—	2	4	no	—
197	C₂×C₈	—	394	1	no	—
198	C₂×C₈	—	394	4	no	—
199	C₂×C₈	—	197	1	no	—
200	C₂×C₈	—	197	1	no	—
201	C₂×C₈	—	197	2	no	—
202	C₂×C₈	—	197	2	no	—
203	C₂×C₈	C₂²⋊D₁₉₇	1	2	yes	—
204	C₂×C₈	C₂²⋊D₁₉₇	1	1	yes	characteristic
205	C₂×C₈	C₂²⋊D₁₉₇	1	1	yes	characteristic
206	C₂×C₈	C₂²×D₁₉₇	1	2	yes	—
207	C₄²	—	394	1	no	—
208	C₄²	—	197	2	no	—
209	C₂³	—	2	1	no	—
210	C₂³	—	394	1	no	—
211	C₂³	#15691358	1	1	yes	characteristic
212	Q₈	—	2	1	no	—
213	Q₈	—	394	1	no	—
214	Q₈	—	197	2	no	—
215	Q₈	C₂×C₄×D₁₉₇	1	2	yes	—
216	D₄	—	2	2	no	—
217	D₄	—	2	2	no	—
218	D₄	—	2	1	no	—
219	D₄	—	394	2	no	—
220	D₄	—	394	2	no	—
221	D₄	—	394	1	no	—
222	D₄	—	197	2	no	—
223	D₄	C₂×C₄×D₁₉₇	1	2	yes	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	788	2	no	—
228	C₂×C₄	—	394	2	no	—
229	C₂×C₄	—	394	1	no	—
230	C₂×C₄	—	394	2	no	—
231	C₂×C₄	—	394	2	no	—
232	C₂×C₄	C₂×C₄×D₁₉₇	1	1	yes	characteristic
233	C₂×C₄	C₂×C₄×D₁₉₇	1	1	yes	characteristic
234	C₂×C₄	C₂×C₂²⋊D₁₉₇	1	1	yes	characteristic
235	C₂×C₄	C₂×C₂²⋊D₁₉₇	1	1	yes	characteristic
236	C₂×C₄	#15691358	1	1	yes	characteristic
237	C₂×C₄	C₂³×D₁₉₇	1	1	yes	characteristic
238	C₈	—	2	2	no	—
239	C₈	—	394	2	no	—
240	C₈	—	197	2	no	—
241	C₈	—	197	2	no	—
242	C₈	C₂×C₂²⋊D₁₉₇	1	2	yes	—
243	C₈	C₂×C₂²⋊D₁₉₇	1	2	yes	—
244	C₂²	—	2	2	no	—
245	C₂²	—	2	2	no	—
246	C₂²	—	4	2	no	—
247	C₂²	—	788	2	no	—
248	C₂²	—	394	2	no	—
249	C₂²	C₄×C₂²⋊D₁₉₇	1	1	yes	characteristic
250	C₂²	C₄×C₂²⋊D₁₉₇	1	1	yes	characteristic
251	C₂²	C₂×#15691358	1	1	yes	characteristic
252	C₄	—	2	2	no	—
253	C₄	—	788	2	no	—
254	C₄	—	394	2	no	—
255	C₄	C₄×C₂²⋊D₁₉₇	1	1	yes	characteristic
256	C₄	C₄×C₂²⋊D₁₉₇	1	1	yes	characteristic
257	C₄	C₂²×C₄×D₁₉₇	1	1	yes	characteristic
258	C₄	C₂²×C₂²⋊D₁₉₇	1	1	yes	characteristic
259	C₂	—	2	1	no	—
260	C₂	—	2	1	no	—
261	C₂	—	4	2	no	—
262	C₂	—	788	2	no	—
263	C₂	#27161199	1	2	yes	direct product factor
264	C₂	C₂×C₄×C₂²⋊D₁₉₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₁₉₇
commutator	C₄×C₁₉₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₉₇
Fitting	C₂×#131×C₁₉₇	C₂
socle	C₂²×C₁₉₇	C₄×D₄
prime radical	C₂×#20263346	C₂
cosocle	C₂×C₄×C₁₉₇	C₂⁴
hypercenter	C₂×#131	D₁₉₇
hypocenter	C₁₉₇	C₂×#444
LCS	G ▷ C₄×C₁₉₇ ▷ C₂×C₁₉₇ ▷ C₁₉₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂×#131

Complements

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

N	H ≅ G/N	type
2. C₂×#20263332	C₂	semidirect
3. #27161199	C₂	direct (N × H)
4. C₂×#20263261	C₂	semidirect
5. C₂×#20263253	C₂	semidirect
6. C₂×#20263252	C₂	semidirect
7. C₂×#20263236	C₂	semidirect
8. C₂×#131×C₁₉₇	C₂	semidirect
10. #20263346	C₄	semidirect
12. C₂×#15691348	C₄	semidirect
13. C₂×#15691346	C₄	semidirect
14. C₂×#15691345	C₄	semidirect
15. C₂×#15691343	C₄	semidirect
17. #20263332	C₂²	semidirect
23. #20263261	C₂²	semidirect
24. #20263253	C₂²	semidirect
25. #20263252	C₂²	semidirect
26. #20263236	C₂²	semidirect
29. #131×C₁₉₇	C₂²	semidirect
43. #15691348	C₂×C₄	semidirect
44. #15691346	C₂×C₄	semidirect
45. #15691345	C₂×C₄	semidirect
46. #15691343	C₂×C₄	semidirect
129. C₂×C₁₉₇	#444	semidirect
132. C₁₉₇	C₂×#444	semidirect
136. C₂×#131	D₁₉₇	semidirect
153. #131	C₂×D₁₉₇	semidirect
263. C₂	#27161199	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×#20263346	2
C₂×#20263332	2
#27161199	2
C₂×#20263261	2
C₂×#20263253	2
C₂×#20263252	2
C₂×#20263236	2
C₂×#131×C₁₉₇	2
C₂×#444	197

Subgroups — C2×#27161199 #38751606

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#27161199 #38751606