Subgroups — C₄×D₈×D₉₇ #27150483

Order 12416.

616 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₈×D₉₇	C₂	1	1	yes	maximal, characteristic
2	C₄×D₄×D₉₇	C₂	1	2	yes	maximal, complemented
3	C₄×D₇₇₆	C₂	1	1	yes	maximal, characteristic, complemented
4	D₈×Dic₉₇	C₂	1	1	yes	maximal, characteristic, complemented
5	#20260817	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×#13095496	C₂	1	2	yes	maximal, complemented
7	#20260815	C₂	1	2	yes	maximal, complemented
8	#107×D₉₇	C₂	1	1	yes	maximal, characteristic, complemented
9	#102×D₉₇	C₂	1	2	yes	maximal, complemented
10	C₄×C₈×D₉₇	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×D₈×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×D₄×D₉₇	C₂²	1	2	yes	—
13	C₂²×C₄×D₉₇	—	2	2	no	—
14	C₂×D₇₇₆	C₂²	1	1	yes	characteristic
15	C₂×#13095496	C₂²	1	2	yes	—
16	D₈×D₉₇	C₄	1	8	yes	direct product factor
17	C₂×C₈×D₉₇	C₂²	1	1	yes	characteristic
18	C₂×C₈×D₉₇	C₂²	1	1	yes	characteristic
19	C₄×D₃₈₈	C₂²	1	2	yes	complemented
20	D₄×Dic₉₇	C₂²	1	2	yes	complemented
21	#15690772	C₂²	1	2	yes	complemented
22	C₄×C₂²⋊D₉₇	—	2	2	no	—
23	#15690770	—	2	2	no	—
24	C₄⋊C₄×D₉₇	C₂²	1	2	yes	—
25	C₂²⋊C₄×D₉₇	—	2	2	no	—
26	C₄²×D₉₇	C₂²	1	1	yes	characteristic
27	#15690744	C₂²	1	1	yes	characteristic, complemented
28	#15690743	C₂²	1	2	yes	complemented
29	#15690732	C₂²	1	2	yes	complemented
30	#15690731	C₂²	1	2	yes	complemented
31	#15690730	C₂²	1	2	yes	complemented
32	C₄×#11715895	C₂²	1	1	yes	characteristic, complemented
33	C₈×Dic₉₇	C₂²	1	1	yes	characteristic, complemented
34	C₈×Dic₉₇	C₂²	1	1	yes	characteristic, complemented
35	C₂×D₈×C₉₇	C₂²	1	1	yes	characteristic
36	C₄×D₄×C₉₇	C₂²	1	2	yes	complemented
37	#107×C₉₇	C₂²	1	1	yes	characteristic, complemented
38	#102×C₉₇	C₂²	1	2	yes	complemented
39	C₄×C₈×C₉₇	C₂²	1	1	yes	characteristic, complemented
40	C₂³×D₉₇	—	2	2	no	—
41	C₂×D₃₈₈	C₂³	1	2	yes	—
42	C₂×C₂²⋊D₉₇	—	2	2	no	—
43	D₄×D₉₇	—	2	4	no	—
44	D₄×D₉₇	C₂×C₄	1	8	yes	complemented
45	C₂²×Dic₉₇	—	2	2	no	—
46	C₂×C₄×D₉₇	—	2	2	no	—
47	C₂×C₄×D₉₇	—	2	2	no	—
48	C₂×C₄×D₉₇	—	2	2	no	—
49	C₂×C₄×D₉₇	—	4	8	no	—
50	C₂×C₄×D₉₇	D₄	1	1	yes	characteristic
51	C₂×C₄×D₉₇	D₄	1	1	yes	characteristic
52	C₂×C₄×D₉₇	C₂³	1	1	yes	characteristic
53	D₇₇₆	C₂×C₄	1	4	yes	complemented
54	#13095496	C₂×C₄	1	8	yes	complemented
55	C₂×#11715895	C₂³	1	1	yes	characteristic
56	C₂×#11715895	C₂³	1	1	yes	characteristic
57	C₈×D₉₇	—	2	2	no	—
58	C₈×D₉₇	C₂×C₄	1	4	yes	complemented
59	C₄⋊Dic₉₇	C₂³	1	2	yes	—
60	#13095490	—	2	2	no	—
61	C₂²⋊Dic₉₇	—	2	2	no	—
62	#13095488	—	2	2	no	—
63	#13095488	—	2	2	no	—
64	C₄×Dic₉₇	D₄	1	1	yes	characteristic
65	C₄×Dic₉₇	D₄	1	1	yes	characteristic
66	C₄×Dic₉₇	C₂³	1	1	yes	characteristic
67	C₂×D₄×C₉₇	C₂³	1	2	yes	—
68	C₂²×C₄×C₉₇	—	2	2	no	—
69	D₈×C₉₇	C₂×C₄	1	4	yes	complemented
70	C₂×C₈×C₉₇	C₂³	1	1	yes	characteristic
71	C₂×C₈×C₉₇	C₂³	1	1	yes	characteristic
72	C₄⋊C₄×C₉₇	C₂³	1	2	yes	—
73	C₂²⋊C₄×C₉₇	—	2	2	no	—
74	C₄²×C₉₇	C₂³	1	1	yes	characteristic
75	C₂³×C₉₇	—	2	2	no	—
76	C₂²×D₉₇	—	2	2	no	—
77	C₂²×D₉₇	—	2	8	no	—
78	C₂²×D₉₇	—	4	4	no	—
79	C₂²×D₉₇	—	4	4	no	—
80	C₂²×D₉₇	C₂×D₄	1	1	yes	characteristic
81	D₄×C₉₇	—	2	2	no	—
82	D₄×C₉₇	C₂²×C₄	1	4	yes	complemented
83	C₂×C₄×C₉₇	—	2	2	no	—
84	C₂×C₄×C₉₇	—	4	4	no	—
85	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
86	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
87	C₂×C₄×C₉₇	C₂⁴	1	1	yes	characteristic
88	C₂²⋊D₉₇	—	2	8	no	—
89	C₂×Dic₉₇	—	2	2	no	—
90	C₂×Dic₉₇	—	4	4	no	—
91	C₂×Dic₉₇	C₂×D₄	1	1	yes	characteristic
92	C₂×Dic₉₇	C₂×D₄	1	1	yes	characteristic
93	C₂×Dic₉₇	C₂×D₄	1	1	yes	characteristic
94	D₃₈₈	—	2	2	no	—
95	D₃₈₈	C₂²×C₄	1	4	yes	complemented
96	C₄×D₉₇	—	2	2	no	—
97	C₄×D₉₇	—	4	4	no	—
98	C₄×D₉₇	—	4	4	no	—
99	C₄×D₉₇	—	4	4	no	—
100	C₄×D₉₇	D₈	1	4	yes	direct product factor
101	C₄×D₉₇	C₂²×C₄	1	2	yes	—
102	C₄×D₉₇	#41	1	2	yes	—
103	C₈×C₉₇	—	2	1	no	—
104	C₈×C₉₇	C₂²×C₄	1	2	yes	complemented
105	#11715895	—	2	1	no	—
106	#11715895	C₂²×C₄	1	2	yes	complemented
107	C₂²×C₉₇	—	2	4	no	—
108	C₂²×C₉₇	—	4	2	no	—
109	C₂²×C₉₇	—	4	2	no	—
110	C₂²×C₉₇	C₂²×D₄	1	1	yes	characteristic
111	C₂×D₉₇	—	2	4	no	—
112	C₂×D₉₇	—	4	2	no	—
113	C₂×D₉₇	—	4	2	no	—
114	C₂×D₉₇	—	4	16	no	—
115	C₂×D₉₇	C₄×D₄	1	2	yes	—
116	C₂×D₉₇	C₂×D₈	1	2	yes	—
117	C₂×D₉₇	#135	1	2	yes	—
118	C₄×C₉₇	—	2	1	no	—
119	C₄×C₉₇	—	4	2	no	—
120	C₄×C₉₇	C₂×D₈	1	2	yes	complemented
121	C₄×C₉₇	C₂³×C₄	1	1	yes	characteristic
122	C₄×C₉₇	C₂×#41	1	1	yes	characteristic
123	Dic₉₇	—	2	1	no	—
124	Dic₉₇	—	4	2	no	—
125	Dic₉₇	C₄×D₄	1	2	yes	—
126	Dic₉₇	C₂×D₈	1	2	yes	complemented
127	C₂×C₉₇	—	4	4	no	—
128	C₂×C₉₇	C₂×C₄×D₄	1	1	yes	characteristic
129	C₂×C₉₇	C₂²×D₈	1	1	yes	characteristic
130	C₂×C₉₇	C₂×#135	1	1	yes	characteristic
131	D₉₇	—	4	4	no	—
132	D₉₇	C₄×D₈	1	4	yes	direct product factor
133	C₂×C₄×D₈	—	97	1	no	maximal, Sylow
134	C₉₇	C₂×C₄×D₈	1	1	yes	characteristic, complemented, Sylow
135	C₂²×D₈	—	97	1	no	—
136	C₂×C₄×D₄	—	97	2	no	—
137	C₄×D₈	—	97	1	no	—
138	C₄×D₈	—	97	2	no	—
139	C₄×D₈	—	97	1	no	—
140	C₄×D₈	—	97	2	no	—
141	C₄×D₈	—	97	1	no	—
142	C₄×D₈	D₉₇	1	1	yes	characteristic, direct product factor
143	C₂×#107	—	97	1	no	—
144	C₂×#102	—	97	2	no	—
145	C₂×C₄×C₈	—	97	1	no	—
146	C₂²×D₄	—	97	2	no	—
147	C₂³×C₄	—	194	2	no	—
148	C₂×D₈	—	97	2	no	—
149	C₂×D₈	—	97	8	no	—
150	C₂×D₈	—	97	1	no	—
151	C₂×D₈	C₂×D₉₇	1	1	yes	characteristic
152	C₂²×C₈	—	97	1	no	—
153	C₂²×C₈	—	97	1	no	—
154	C₄×D₄	—	97	2	no	—
155	C₄×D₄	—	97	2	no	—
156	C₄×D₄	—	97	2	no	—
157	C₄×D₄	—	194	2	no	—
158	C₄×D₄	—	194	2	no	—
159	C₄×D₄	C₂×D₉₇	1	2	yes	complemented
160	C₂×C₄⋊C₄	—	97	2	no	—
161	C₂×C₂²⋊C₄	—	194	2	no	—
162	C₂×C₄²	—	97	1	no	—
163	#107	—	97	1	no	—
164	#107	—	97	2	no	—
165	#107	C₂×D₉₇	1	1	yes	characteristic, complemented
166	#102	—	97	2	no	—
167	#102	—	97	2	no	—
168	#102	—	97	2	no	—
169	#102	C₂×D₉₇	1	2	yes	complemented
170	C₄×C₈	—	97	1	no	—
171	C₄×C₈	—	97	1	no	—
172	C₄×C₈	—	97	1	no	—
173	C₄×C₈	C₂×D₉₇	1	1	yes	characteristic, complemented
174	C₂⁴	—	194	2	no	—
175	C₂×D₄	—	97	8	no	—
176	C₂×D₄	—	97	2	no	—
177	C₂×D₄	—	194	4	no	—
178	C₂×D₄	—	194	2	no	—
179	C₂×D₄	C₂²×D₉₇	1	2	yes	—
180	C₂²×C₄	—	2	2	no	—
181	C₂²×C₄	—	97	1	no	—
182	C₂²×C₄	—	97	1	no	—
183	C₂²×C₄	—	97	1	no	—
184	C₂²×C₄	—	388	8	no	—
185	C₂²×C₄	—	194	2	no	—
186	C₂²×C₄	—	194	2	no	—
187	C₂²×C₄	—	194	2	no	—
188	C₂²×C₄	—	194	2	no	—
189	D₈	—	97	8	no	—
190	D₈	—	97	4	no	—
191	D₈	C₄×D₉₇	1	4	yes	direct product factor
192	C₂×C₈	—	97	1	no	—
193	C₂×C₈	—	97	4	no	—
194	C₂×C₈	—	97	1	no	—
195	C₂×C₈	—	194	2	no	—
196	C₂×C₈	C₂²×D₉₇	1	1	yes	characteristic
197	C₂×C₈	C₂²×D₉₇	1	1	yes	characteristic
198	C₄⋊C₄	—	97	2	no	—
199	C₄⋊C₄	—	194	2	no	—
200	C₄⋊C₄	C₂²×D₉₇	1	2	yes	—
201	C₂²⋊C₄	—	2	2	no	—
202	C₂²⋊C₄	—	194	2	no	—
203	C₂²⋊C₄	—	194	2	no	—
204	C₂²⋊C₄	—	194	2	no	—
205	C₄²	—	97	1	no	—
206	C₄²	—	97	1	no	—
207	C₄²	—	97	1	no	—
208	C₄²	C₂²×D₉₇	1	1	yes	characteristic
209	C₂³	—	2	2	no	—
210	C₂³	—	97	1	no	—
211	C₂³	—	388	4	no	—
212	C₂³	—	388	4	no	—
213	C₂³	—	194	8	no	—
214	C₂³	—	194	2	no	—
215	D₄	—	2	2	no	—
216	D₄	—	97	4	no	—
217	D₄	—	194	8	no	—
218	D₄	—	194	2	no	—
219	D₄	C₂×C₄×D₉₇	1	4	yes	complemented
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	4	4	no	—
222	C₂×C₄	—	97	1	no	—
223	C₂×C₄	—	97	1	no	—
224	C₂×C₄	—	97	4	no	—
225	C₂×C₄	—	97	2	no	—
226	C₂×C₄	—	97	2	no	—
227	C₂×C₄	—	97	1	no	—
228	C₂×C₄	—	388	4	no	—
229	C₂×C₄	—	388	4	no	—
230	C₂×C₄	—	388	4	no	—
231	C₂×C₄	—	388	4	no	—
232	C₂×C₄	—	194	2	no	—
233	C₂×C₄	—	194	2	no	—
234	C₂×C₄	D₄×D₉₇	1	1	yes	characteristic
235	C₂×C₄	D₄×D₉₇	1	1	yes	characteristic
236	C₂×C₄	C₂³×D₉₇	1	1	yes	characteristic
237	C₈	—	2	1	no	—
238	C₈	—	97	2	no	—
239	C₈	—	194	1	no	—
240	C₈	C₂×C₄×D₉₇	1	2	yes	complemented
241	C₂²	—	2	4	no	—
242	C₂²	—	4	2	no	—
243	C₂²	—	4	2	no	—
244	C₂²	—	97	2	no	—
245	C₂²	—	97	2	no	—
246	C₂²	—	97	2	no	—
247	C₂²	—	388	16	no	—
248	C₂²	—	388	2	no	—
249	C₂²	—	388	2	no	—
250	C₂²	—	194	4	no	—
251	C₂²	C₂×D₄×D₉₇	1	1	yes	characteristic
252	C₄	—	2	1	no	—
253	C₄	—	4	2	no	—
254	C₄	—	97	2	no	—
255	C₄	—	97	2	no	—
256	C₄	—	388	2	no	—
257	C₄	—	194	1	no	—
258	C₄	D₈×D₉₇	1	2	yes	direct product factor
259	C₄	C₂²×C₄×D₉₇	1	1	yes	characteristic
260	C₄	#41×D₉₇	1	1	yes	characteristic
261	C₂	—	4	4	no	—
262	C₂	—	97	4	no	—
263	C₂	—	388	4	no	—
264	C₂	C₄×D₄×D₉₇	1	1	yes	characteristic
265	C₂	C₂×D₈×D₉₇	1	1	yes	characteristic
266	C₂	#135×D₉₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₄×D₉₇
commutator	C₄×C₉₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₉₇
Fitting	C₄×D₈×C₉₇	C₂
socle	C₂²×C₉₇	C₂²×D₄
prime radical	C₂×D₈×D₉₇	C₂
cosocle	C₂×C₄×C₉₇	C₂⁴
hypercenter	C₄×D₈	D₉₇
hypocenter	C₉₇	C₂×C₄×D₈
LCS	G ▷ C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×D₈

Complements

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

N	H ≅ G/N	type
2. C₄×D₄×D₉₇	C₂	semidirect
3. C₄×D₇₇₆	C₂	semidirect
4. D₈×Dic₉₇	C₂	semidirect
5. #20260817	C₂	semidirect
6. C₄×#13095496	C₂	semidirect
7. #20260815	C₂	semidirect
8. #107×D₉₇	C₂	semidirect
9. #102×D₉₇	C₂	semidirect
10. C₄×C₈×D₉₇	C₂	semidirect
11. C₄×D₈×C₉₇	C₂	semidirect
16. D₈×D₉₇	C₄	direct (N × H)
19. C₄×D₃₈₈	C₂²	semidirect
20. D₄×Dic₉₇	C₂²	semidirect
21. #15690772	C₂²	semidirect
27. #15690744	C₂²	semidirect
28. #15690743	C₂²	semidirect
29. #15690732	C₂²	semidirect
30. #15690731	C₂²	semidirect
31. #15690730	C₂²	semidirect
32. C₄×#11715895	C₂²	semidirect
33. C₈×Dic₉₇	C₂²	semidirect
34. C₈×Dic₉₇	C₂²	semidirect
36. C₄×D₄×C₉₇	C₂²	semidirect
37. #107×C₉₇	C₂²	semidirect
38. #102×C₉₇	C₂²	semidirect
39. C₄×C₈×C₉₇	C₂²	semidirect
44. D₄×D₉₇	C₂×C₄	semidirect
53. D₇₇₆	C₂×C₄	semidirect
54. #13095496	C₂×C₄	semidirect
58. C₈×D₉₇	C₂×C₄	semidirect
69. D₈×C₉₇	C₂×C₄	semidirect
82. D₄×C₉₇	C₂²×C₄	semidirect
95. D₃₈₈	C₂²×C₄	semidirect
100. C₄×D₉₇	D₈	direct (N × H)
104. C₈×C₉₇	C₂²×C₄	semidirect
106. #11715895	C₂²×C₄	semidirect
120. C₄×C₉₇	C₂×D₈	semidirect
126. Dic₉₇	C₂×D₈	semidirect
132. D₉₇	C₄×D₈	direct (N × H)
134. C₉₇	C₂×C₄×D₈	semidirect
142. C₄×D₈	D₉₇	direct (N × H)
159. C₄×D₄	C₂×D₉₇	semidirect
165. #107	C₂×D₉₇	semidirect
169. #102	C₂×D₉₇	semidirect
173. C₄×C₈	C₂×D₉₇	semidirect
191. D₈	C₄×D₉₇	direct (N × H)
219. D₄	C₂×C₄×D₉₇	semidirect
240. C₈	C₂×C₄×D₉₇	semidirect
258. C₄	D₈×D₉₇	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×D₈×D₉₇	2
C₄×D₄×D₉₇	2
C₄×D₇₇₆	2
D₈×Dic₉₇	2
#20260817	2
C₄×#13095496	2
#20260815	2
#107×D₉₇	2
#102×D₉₇	2
C₄×C₈×D₉₇	2
C₄×D₈×C₉₇	2
C₂×C₄×D₈	97

Subgroups — C4×D8×D97 #27150483

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×D₈×D₉₇ #27150483