Subgroups — C₈×#13095510 #27150691

Order 12416.

456 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#13095510	C₂	1	1	yes	maximal, characteristic
2	C₈×Dic₁₉₄	C₂	1	1	yes	maximal, characteristic, complemented
3	#20260859	C₂	1	1	yes	maximal, characteristic, complemented
4	D₄×#11715895	C₂	1	1	yes	maximal, characteristic, complemented
5	C₈×C₂²⋊D₉₇	C₂	1	2	yes	maximal, complemented
6	#20260798	C₂	1	2	yes	maximal, complemented
7	#20260787	C₂	1	1	yes	maximal, characteristic, complemented
8	#20260785	C₂	1	2	yes	maximal, complemented
9	C₂×C₈×Dic₉₇	C₂	1	2	yes	maximal, complemented
10	C₄×C₈×D₉₇	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×D₄×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13095510	C₄	1	1	yes	characteristic
13	C₂²×#11715895	C₂²	1	2	yes	—
14	C₂×C₈×D₉₇	C₂²	1	1	yes	characteristic
15	C₂×C₈×D₉₇	C₂²	1	1	yes	characteristic
16	C₄×Dic₁₉₄	C₂²	1	1	yes	characteristic
17	#15690775	C₄	1	1	yes	characteristic
18	D₄×Dic₉₇	C₄	1	1	yes	characteristic
19	C₄×C₂²⋊D₉₇	C₂²	1	2	yes	—
20	#15690770	C₄	1	2	yes	—
21	#15690768	C₄	1	1	yes	characteristic
22	#15690766	C₄	1	2	yes	—
23	C₂×C₄×Dic₉₇	C₂²	1	2	yes	—
24	C₄²×D₉₇	C₂²	1	1	yes	characteristic
25	C₄⋊#11715895	C₂²	1	1	yes	characteristic
26	#15690739	C₂²	1	2	yes	—
27	#15690739	C₂²	1	2	yes	—
28	C₂²⋊#11715895	C₂²	1	2	yes	—
29	#15690722	C₂²	1	2	yes	—
30	C₄×#11715895	C₂²	1	1	yes	characteristic
31	C₈×Dic₉₇	C₂²	1	1	yes	characteristic
32	C₈×Dic₉₇	C₂²	1	1	yes	characteristic
33	C₈×Dic₉₇	C₂²	1	2	yes	—
34	C₈×Dic₉₇	C₂²	1	2	yes	—
35	C₈×Dic₉₇	C₂²	1	2	yes	—
36	C₂²×C₈×C₉₇	C₂²	1	2	yes	—
37	C₄×D₄×C₉₇	C₂²	1	1	yes	characteristic
38	C₄⋊C₈×C₉₇	C₂²	1	1	yes	characteristic
39	C₂²⋊C₈×C₉₇	C₂²	1	2	yes	—
40	C₄×C₈×C₉₇	C₂²	1	1	yes	characteristic
41	#13095510	C₈	1	8	yes	direct product factor
42	C₂×Dic₁₉₄	C₂×C₄	1	1	yes	characteristic
43	C₂×C₂²⋊D₉₇	C₂×C₄	1	2	yes	—
44	C₂²×Dic₉₇	C₂×C₄	1	2	yes	—
45	C₂²×Dic₉₇	C₂×C₄	1	2	yes	—
46	C₂×C₄×D₉₇	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄×D₉₇	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄×D₉₇	C₂³	1	1	yes	characteristic
49	C₂×#11715895	—	2	4	no	—
50	C₂×#11715895	C₂³	1	1	yes	characteristic
51	C₂×#11715895	C₂³	1	2	yes	—
52	C₂×#11715895	C₂³	1	1	yes	characteristic
53	C₈×D₉₇	—	2	2	no	—
54	C₈×D₉₇	—	2	2	no	—
55	C₄⋊Dic₉₇	C₂×C₄	1	1	yes	characteristic
56	#13095490	C₂×C₄	1	2	yes	—
57	#13095490	C₂×C₄	1	2	yes	—
58	C₂²⋊Dic₉₇	C₂×C₄	1	2	yes	—
59	#13095488	C₂×C₄	1	2	yes	—
60	C₄×Dic₉₇	C₂×C₄	1	1	yes	characteristic
61	C₄×Dic₉₇	C₂×C₄	1	1	yes	characteristic
62	C₄×Dic₉₇	C₂×C₄	1	2	yes	—
63	C₄×Dic₉₇	C₂×C₄	1	2	yes	—
64	C₄×Dic₉₇	C₂³	1	1	yes	characteristic
65	C₄×Dic₉₇	C₂³	1	2	yes	—
66	C₂×D₄×C₉₇	C₂×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₉₇	C₂³	1	2	yes	—
68	C₂×C₈×C₉₇	—	2	4	no	—
69	C₂×C₈×C₉₇	C₂³	1	1	yes	characteristic
70	C₂×C₈×C₉₇	C₂³	1	2	yes	—
71	C₂×C₈×C₉₇	C₂³	1	1	yes	characteristic
72	C₄⋊C₄×C₉₇	C₂×C₄	1	1	yes	characteristic
73	C₂²⋊C₄×C₉₇	C₂×C₄	1	2	yes	—
74	C₄²×C₉₇	C₂³	1	1	yes	characteristic
75	C₂³×C₉₇	C₂²×C₄	1	2	yes	—
76	C₂²×D₉₇	C₂²×C₄	1	1	yes	characteristic
77	D₄×C₉₇	C₂×C₈	1	4	yes	complemented
78	C₂×C₄×C₉₇	—	2	2	no	—
79	C₂×C₄×C₉₇	—	2	2	no	—
80	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₄×C₉₇	C₂²×C₄	1	2	yes	—
82	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₄×C₉₇	C₂⁴	1	1	yes	characteristic
84	C₂²⋊D₉₇	C₂×C₈	1	8	yes	complemented
85	C₂×Dic₉₇	—	2	4	no	—
86	C₂×Dic₉₇	C₂×C₈	1	8	yes	complemented
87	C₂×Dic₉₇	C₂²×C₄	1	1	yes	characteristic
88	C₂×Dic₉₇	C₂²×C₄	1	2	yes	—
89	C₂×Dic₉₇	C₂²×C₄	1	1	yes	characteristic
90	C₂×Dic₉₇	C₂²×C₄	1	2	yes	—
91	C₂×Dic₉₇	C₂²×C₄	1	1	yes	characteristic
92	C₄×D₉₇	—	2	1	no	—
93	C₄×D₉₇	—	2	1	no	—
94	C₄×D₉₇	—	2	2	no	—
95	C₄×D₉₇	C₂×C₈	1	4	yes	complemented
96	Dic₁₉₄	C₂×C₈	1	4	yes	complemented
97	C₈×C₉₇	—	2	2	no	—
98	C₈×C₉₇	—	2	1	no	—
99	C₈×C₉₇	#41	1	2	yes	complemented
100	#11715895	—	2	1	no	—
101	#11715895	—	2	2	no	—
102	#11715895	#41	1	2	yes	complemented
103	C₂²×C₉₇	—	2	2	no	—
104	C₂²×C₉₇	—	2	2	no	—
105	C₂²×C₉₇	C₂²×C₈	1	4	yes	—
106	C₂²×C₉₇	C₂³×C₄	1	1	yes	characteristic
107	C₂×D₉₇	—	2	1	no	—
108	C₂×D₉₇	—	2	1	no	—
109	C₂×D₉₇	C₂²×C₈	1	2	yes	—
110	C₄×C₉₇	—	2	2	no	—
111	C₄×C₉₇	—	2	1	no	—
112	C₄×C₉₇	C₂²×C₈	1	2	yes	—
113	C₄×C₉₇	C₂×#41	1	1	yes	characteristic
114	C₄×C₉₇	C₂×#41	1	1	yes	characteristic
115	Dic₉₇	—	2	1	no	—
116	Dic₉₇	—	2	2	no	—
117	Dic₉₇	C₂²×C₈	1	4	yes	—
118	Dic₉₇	C₂²×C₈	1	2	yes	—
119	Dic₉₇	#131	1	2	yes	—
120	C₂×C₉₇	—	2	4	no	—
121	C₂×C₉₇	C₄×#41	1	1	yes	characteristic
122	C₂×C₉₇	C₂³×C₈	1	1	yes	characteristic
123	C₂×C₉₇	C₂×#131	1	1	yes	characteristic
124	D₉₇	—	2	2	no	—
125	C₈×#41	—	97	1	no	maximal, Sylow
126	C₉₇	C₈×#41	1	1	yes	characteristic, complemented, Sylow
127	C₄×#41	—	97	1	no	—
128	C₈×Q₈	—	97	1	no	—
129	C₈×Q₈	—	97	1	no	—
130	C₈×D₄	—	97	2	no	—
131	C₈×D₄	—	97	1	no	—
132	C₈×D₄	—	97	2	no	—
133	C₈×D₄	D₉₇	1	1	yes	characteristic, complemented
134	#431	—	97	1	no	—
135	#431	—	97	2	no	—
136	C₂×C₄×C₈	—	97	2	no	—
137	C₂×C₄×C₈	—	97	1	no	—
138	C₂×#41	—	97	1	no	—
139	C₂²×C₈	—	97	2	no	—
140	C₂²×C₈	—	97	1	no	—
141	C₂²×C₈	—	97	1	no	—
142	C₂²×C₈	C₂×D₉₇	1	2	yes	—
143	C₄×Q₈	—	97	1	no	—
144	C₄×Q₈	—	97	1	no	—
145	C₄×D₄	—	97	1	no	—
146	C₄×D₄	—	97	2	no	—
147	C₄×D₄	—	97	2	no	—
148	C₄×D₄	C₂×D₉₇	1	1	yes	characteristic
149	#117	—	97	1	no	—
150	#117	—	97	2	no	—
151	C₂×C₄²	—	97	2	no	—
152	C₂×C₄²	—	97	1	no	—
153	C₄⋊C₈	—	97	2	no	—
154	C₄⋊C₈	—	97	2	no	—
155	C₄⋊C₈	—	97	1	no	—
156	C₄⋊C₈	C₂×D₉₇	1	1	yes	characteristic
157	C₂²⋊C₈	—	97	2	no	—
158	C₂²⋊C₈	—	97	2	no	—
159	C₂²⋊C₈	C₂×D₉₇	1	2	yes	—
160	C₄×C₈	—	97	2	no	—
161	C₄×C₈	—	97	1	no	—
162	C₄×C₈	—	97	2	no	—
163	C₄×C₈	—	97	1	no	—
164	C₄×C₈	—	97	2	no	—
165	C₄×C₈	—	97	1	no	—
166	C₄×C₈	C₂×D₉₇	1	1	yes	characteristic
167	#41	—	97	8	no	—
168	C₂×Q₈	—	97	1	no	—
169	C₂×D₄	—	97	2	no	—
170	C₂×D₄	C₄×D₉₇	1	1	yes	characteristic
171	C₂²×C₄	—	97	2	no	—
172	C₂²×C₄	—	97	1	no	—
173	C₂²×C₄	—	97	2	no	—
174	C₂²×C₄	—	97	1	no	—
175	C₂²×C₄	—	97	1	no	—
176	C₂²×C₄	C₂²×D₉₇	1	2	yes	—
177	C₂×C₈	—	2	4	no	—
178	C₂×C₈	—	97	1	no	—
179	C₂×C₈	—	97	2	no	—
180	C₂×C₈	—	97	1	no	—
181	C₂×C₈	—	194	2	no	—
182	C₂×C₈	—	194	4	no	—
183	C₂×C₈	—	194	2	no	—
184	C₂×C₈	C₂²×D₉₇	1	2	yes	—
185	C₂×C₈	C₂²×D₉₇	1	1	yes	characteristic
186	C₂×C₈	C₂²×D₉₇	1	1	yes	characteristic
187	C₄⋊C₄	—	97	2	no	—
188	C₄⋊C₄	—	97	2	no	—
189	C₄⋊C₄	—	97	1	no	—
190	C₄⋊C₄	C₄×D₉₇	1	1	yes	characteristic
191	C₂²⋊C₄	—	97	2	no	—
192	C₂²⋊C₄	—	97	2	no	—
193	C₂²⋊C₄	C₄×D₉₇	1	2	yes	—
194	C₄²	—	97	2	no	—
195	C₄²	—	97	1	no	—
196	C₄²	—	97	2	no	—
197	C₄²	—	97	1	no	—
198	C₄²	—	97	1	no	—
199	C₄²	—	97	2	no	—
200	C₄²	C₂²×D₉₇	1	1	yes	characteristic
201	C₂³	—	97	1	no	—
202	C₂³	C₂×C₄×D₉₇	1	2	yes	—
203	Q₈	—	97	4	no	—
204	D₄	—	97	8	no	—
205	D₄	C₈×D₉₇	1	4	yes	complemented
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	97	8	no	—
209	C₂×C₄	—	97	1	no	—
210	C₂×C₄	—	97	1	no	—
211	C₂×C₄	—	97	2	no	—
212	C₂×C₄	—	97	4	no	—
213	C₂×C₄	—	97	1	no	—
214	C₂×C₄	—	97	2	no	—
215	C₂×C₄	—	194	2	no	—
216	C₂×C₄	—	194	4	no	—
217	C₂×C₄	—	194	1	no	—
218	C₂×C₄	—	194	1	no	—
219	C₂×C₄	C₂×C₄×D₉₇	1	2	yes	—
220	C₂×C₄	C₂×C₄×D₉₇	1	1	yes	characteristic
221	C₂×C₄	C₂×C₄×D₉₇	1	1	yes	characteristic
222	C₂×C₄	C₂³×D₉₇	1	1	yes	characteristic
223	C₈	—	2	1	no	—
224	C₈	—	2	2	no	—
225	C₈	—	97	2	no	—
226	C₈	—	194	2	no	—
227	C₈	—	194	1	no	—
228	C₈	#13095510	1	2	yes	direct product factor
229	C₂²	—	2	2	no	—
230	C₂²	—	2	2	no	—
231	C₂²	—	97	2	no	—
232	C₂²	—	194	1	no	—
233	C₂²	—	194	1	no	—
234	C₂²	C₂×C₈×D₉₇	1	4	yes	—
235	C₂²	C₂²×C₄×D₉₇	1	1	yes	characteristic
236	C₄	—	2	2	no	—
237	C₄	—	2	1	no	—
238	C₄	—	97	2	no	—
239	C₄	—	97	4	no	—
240	C₄	—	97	2	no	—
241	C₄	—	194	1	no	—
242	C₄	—	194	2	no	—
243	C₄	C₂×C₈×D₉₇	1	2	yes	—
244	C₄	#41×D₉₇	1	1	yes	characteristic
245	C₄	C₂×#13095510	1	1	yes	characteristic
246	C₂	—	2	4	no	—
247	C₂	—	194	2	no	—
248	C₂	C₄×#13095510	1	1	yes	characteristic
249	C₂	C₂²×C₈×D₉₇	1	1	yes	characteristic
250	C₂	#131×D₉₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂²×D₉₇
commutator	C₂×C₉₇	C₂³×C₈
Frattini	C₂×C₄	C₂³×D₉₇
Fitting	C₈×D₄×C₉₇	C₂
socle	C₂²×C₉₇	C₂³×C₄
prime radical	C₂×#13095510	C₄
cosocle	C₂×C₄×C₉₇	C₂⁴
hypercenter	C₈×D₄	D₉₇
hypocenter	C₉₇	C₈×#41
LCS	G ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂×C₈ ◁ C₈×D₄

Complements

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

N	H ≅ G/N	type
2. C₈×Dic₁₉₄	C₂	semidirect
3. #20260859	C₂	semidirect
4. D₄×#11715895	C₂	semidirect
5. C₈×C₂²⋊D₉₇	C₂	semidirect
6. #20260798	C₂	semidirect
7. #20260787	C₂	semidirect
8. #20260785	C₂	semidirect
9. C₂×C₈×Dic₉₇	C₂	semidirect
10. C₄×C₈×D₉₇	C₂	semidirect
11. C₈×D₄×C₉₇	C₂	semidirect
41. #13095510	C₈	direct (N × H)
77. D₄×C₉₇	C₂×C₈	semidirect
84. C₂²⋊D₉₇	C₂×C₈	semidirect
86. C₂×Dic₉₇	C₂×C₈	semidirect
95. C₄×D₉₇	C₂×C₈	semidirect
96. Dic₁₉₄	C₂×C₈	semidirect
99. C₈×C₉₇	#41	semidirect
102. #11715895	#41	semidirect
126. C₉₇	C₈×#41	semidirect
133. C₈×D₄	D₉₇	semidirect
205. D₄	C₈×D₉₇	semidirect
228. C₈	#13095510	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×#13095510	2
C₈×Dic₁₉₄	2
#20260859	2
D₄×#11715895	2
C₈×C₂²⋊D₉₇	2
#20260798	2
#20260787	2
#20260785	2
C₂×C₈×Dic₉₇	2
C₄×C₈×D₉₇	2
C₈×D₄×C₉₇	2
C₈×#41	97

Subgroups — C8×#13095510 #27150691

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈×#13095510 #27150691