Subgroups — C₂×#20261365 #27155136

Order 12416.

1080 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20261365	C₂	1	16	yes	maximal, direct product factor
2	C₂×#15690810	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#15690809	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#15690809	C₂	1	2	yes	maximal, complemented
5	C₂×#15690801	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#15690800	C₂	1	2	yes	maximal, complemented
7	C₂×#15690790	C₂	1	2	yes	maximal
8	C₂×#15690788	C₂	1	2	yes	maximal
9	C₂×#15690768	C₂	1	1	yes	maximal, characteristic
10	C₂×#15690767	C₂	1	1	yes	maximal, characteristic
11	C₂×C₄⋊C₄×D₉₇	C₂	1	1	yes	maximal, characteristic
12	C₂×#125×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂²×Dic₁₉₄	C₂²	1	2	yes	—
14	C₂²×Dic₁₉₄	C₂²	1	2	yes	—
15	C₂²×C₄×D₉₇	C₂²	1	2	yes	—
16	C₂²×C₄×D₉₇	C₂²	1	1	yes	characteristic
17	#15690810	C₂²	1	8	yes	complemented
18	#15690809	C₂²	1	8	yes	complemented
19	#15690809	C₂²	1	16	yes	complemented
20	#15690801	C₂²	1	8	yes	complemented
21	#15690800	C₂²	1	16	yes	complemented
22	#15690790	C₂²	1	16	yes	—
23	#15690788	C₂²	1	16	yes	—
24	#15690768	C₂²	1	8	yes	—
25	#15690767	C₂²	1	8	yes	—
26	C₂×C₄⋊Dic₉₇	C₂²	1	1	yes	characteristic
27	C₂×C₄⋊Dic₉₇	C₂²	1	1	yes	characteristic
28	C₂×C₄⋊Dic₉₇	C₂²	1	2	yes	—
29	C₂×C₄⋊Dic₉₇	C₂²	1	2	yes	—
30	C₂×#13095490	C₂²	1	2	yes	—
31	C₂×#13095490	C₂²	1	1	yes	characteristic
32	C₂×#13095490	C₂²	1	1	yes	characteristic
33	C₂×#13095490	C₂²	1	2	yes	—
34	C₂×#13095490	C₂²	1	2	yes	—
35	C₄⋊C₄×D₉₇	C₂²	1	8	yes	—
36	C₂×#13095488	C₂²	1	1	yes	characteristic
37	C₂×#13095488	C₂²	1	1	yes	characteristic
38	C₂×#13095488	C₂²	1	2	yes	—
39	C₂×C₄×Dic₉₇	C₂²	1	1	yes	characteristic
40	C₂×C₄×Dic₉₇	C₂²	1	2	yes	—
41	#125×C₉₇	C₂²	1	8	yes	complemented
42	C₂×C₄⋊C₄×C₉₇	C₂²	1	1	yes	characteristic
43	C₂×C₄⋊C₄×C₉₇	C₂²	1	1	yes	characteristic
44	C₂×C₄⋊C₄×C₉₇	C₂²	1	2	yes	—
45	C₂×C₄⋊C₄×C₉₇	C₂²	1	2	yes	—
46	C₂×C₄²×C₉₇	C₂²	1	1	yes	characteristic
47	C₂³×D₉₇	C₂³	1	1	yes	characteristic
48	C₂×Dic₁₉₄	—	4	4	no	—
49	C₂×Dic₁₉₄	—	4	4	no	—
50	C₂×Dic₁₉₄	C₂³	1	8	yes	—
51	C₂×Dic₁₉₄	C₂³	1	8	yes	—
52	C₂²×Dic₉₇	C₂³	1	1	yes	characteristic
53	C₂²×Dic₉₇	C₂³	1	1	yes	characteristic
54	C₂²×Dic₉₇	C₂³	1	2	yes	—
55	C₂²×Dic₉₇	C₂³	1	2	yes	—
56	C₂²×Dic₉₇	C₂³	1	1	yes	characteristic
57	C₂×C₄×D₉₇	—	4	4	no	—
58	C₂×C₄×D₉₇	Q₈	1	8	yes	complemented
59	C₂×C₄×D₉₇	C₂³	1	8	yes	—
60	C₂×C₄×D₉₇	C₂³	1	4	yes	—
61	C₄⋊Dic₉₇	C₂³	1	4	yes	—
62	C₄⋊Dic₉₇	C₂³	1	4	yes	—
63	C₄⋊Dic₉₇	C₂³	1	8	yes	—
64	C₄⋊Dic₉₇	C₂³	1	8	yes	—
65	#13095490	C₂³	1	8	yes	—
66	#13095490	C₂³	1	4	yes	—
67	#13095490	C₂³	1	4	yes	—
68	#13095490	C₂³	1	8	yes	—
69	#13095490	C₂³	1	8	yes	—
70	#13095488	C₂³	1	4	yes	—
71	#13095488	C₂³	1	4	yes	—
72	#13095488	C₂³	1	8	yes	—
73	C₄×Dic₉₇	C₂³	1	4	yes	—
74	C₄×Dic₉₇	C₂³	1	8	yes	—
75	C₂²×C₄×C₉₇	C₂³	1	2	yes	—
76	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
77	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
78	C₂²×C₄×C₉₇	C₂³	1	2	yes	—
79	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
80	C₄⋊C₄×C₉₇	C₂³	1	4	yes	—
81	C₄⋊C₄×C₉₇	C₂³	1	4	yes	—
82	C₄⋊C₄×C₉₇	C₂³	1	8	yes	—
83	C₄⋊C₄×C₉₇	C₂³	1	8	yes	—
84	C₄²×C₉₇	C₂³	1	4	yes	—
85	C₂³×C₉₇	C₂⁴	1	1	yes	characteristic
86	C₂²×D₉₇	—	2	2	no	—
87	C₂²×D₉₇	—	2	2	no	—
88	C₂²×D₉₇	C₂×Q₈	1	4	yes	—
89	C₂²×D₉₇	C₂⁴	1	2	yes	—
90	C₂×C₄×C₉₇	—	2	4	no	—
91	C₂×C₄×C₉₇	—	2	2	no	—
92	C₂×C₄×C₉₇	—	2	4	no	—
93	C₂×C₄×C₉₇	—	2	2	no	—
94	C₂×C₄×C₉₇	C₂×Q₈	1	4	yes	—
95	C₂×C₄×C₉₇	C₂⁴	1	2	yes	—
96	C₂×C₄×C₉₇	C₂⁴	1	2	yes	—
97	C₂×C₄×C₉₇	C₂⁴	1	4	yes	—
98	C₂×C₄×C₉₇	C₂⁴	1	4	yes	—
99	C₂×C₄×C₉₇	C₂⁴	1	2	yes	—
100	C₂×Dic₉₇	—	2	4	no	—
101	C₂×Dic₉₇	—	2	2	no	—
102	C₂×Dic₉₇	—	2	4	no	—
103	C₂×Dic₉₇	—	2	2	no	—
104	C₂×Dic₉₇	C₂×Q₈	1	4	yes	—
105	C₂×Dic₉₇	C₂⁴	1	2	yes	—
106	C₂×Dic₉₇	C₂⁴	1	2	yes	—
107	C₂×Dic₉₇	C₂⁴	1	4	yes	—
108	C₂×Dic₉₇	C₂⁴	1	4	yes	—
109	C₂×Dic₉₇	C₂⁴	1	2	yes	—
110	C₄×D₉₇	—	4	8	no	—
111	C₄×D₉₇	C₂×Q₈	1	16	yes	complemented
112	Dic₁₉₄	—	4	8	no	—
113	Dic₁₉₄	—	4	8	no	—
114	C₂²×C₉₇	C₂²×Q₈	1	2	yes	—
115	C₂²×C₉₇	H(5,𝔽₂)	1	2	yes	—
116	C₂²×C₉₇	2₋¹⁺⁴	1	2	yes	—
117	C₂²×C₉₇	C₂⁵	1	1	yes	characteristic
118	C₂×D₉₇	—	2	8	no	—
119	C₂×D₉₇	—	2	2	no	—
120	C₂×D₉₇	—	2	2	no	—
121	C₂×D₉₇	C₂²×Q₈	1	4	yes	—
122	C₄×C₉₇	—	2	2	no	—
123	C₄×C₉₇	—	2	4	no	—
124	C₄×C₉₇	—	2	2	no	—
125	C₄×C₉₇	—	2	4	no	—
126	C₄×C₉₇	C₂²×Q₈	1	4	yes	—
127	Dic₉₇	—	2	2	no	—
128	Dic₉₇	—	2	4	no	—
129	Dic₉₇	—	2	4	no	—
130	Dic₉₇	—	2	2	no	—
131	Dic₉₇	C₂²×Q₈	1	4	yes	—
132	C₂×C₉₇	#544	1	4	yes	complemented
133	C₂×C₉₇	C₂³×Q₈	1	1	yes	characteristic
134	C₂×C₉₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
135	C₂×C₉₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
136	D₉₇	—	2	4	no	—
137	C₂×#544	—	97	1	no	maximal, Sylow
138	C₉₇	C₂×#544	1	1	yes	characteristic, complemented, Sylow
139	#544	—	97	16	no	—
140	C₂×C₄⋊Q₈	—	97	2	no	—
141	C₂×C₄⋊Q₈	—	97	1	no	—
142	C₂×C₄⋊Q₈	—	97	1	no	—
143	C₂×#125	—	97	2	no	—
144	C₂×#125	—	97	1	no	—
145	C₂×#125	D₉₇	1	1	yes	characteristic, complemented
146	C₂×C₂²⋊Q₈	—	97	2	no	—
147	C₂×C₂²⋊Q₈	—	97	2	no	—
148	C₂×#117	—	97	1	no	—
149	C₂×#117	—	97	1	no	—
150	C₂²×C₄⋊C₄	—	97	1	no	—
151	C₂²×Q₈	—	97	2	no	—
152	C₂²×Q₈	—	97	2	no	—
153	C₂³×C₄	—	97	1	no	—
154	C₂³×C₄	—	97	2	no	—
155	C₄⋊Q₈	—	97	16	no	—
156	C₄⋊Q₈	—	97	8	no	—
157	C₄⋊Q₈	—	97	8	no	—
158	#125	—	97	16	no	—
159	#125	—	97	8	no	—
160	#125	C₂×D₉₇	1	8	yes	complemented
161	C₂²⋊Q₈	—	97	16	no	—
162	C₂²⋊Q₈	—	97	16	no	—
163	#117	—	97	8	no	—
164	#117	—	97	8	no	—
165	C₂×C₄⋊C₄	—	97	2	no	—
166	C₂×C₄⋊C₄	—	97	1	no	—
167	C₂×C₄⋊C₄	—	97	1	no	—
168	C₂×C₄⋊C₄	—	97	1	no	—
169	C₂×C₄⋊C₄	—	97	1	no	—
170	C₂×C₄⋊C₄	—	97	2	no	—
171	C₂×C₄⋊C₄	—	97	2	no	—
172	C₂×C₄⋊C₄	—	97	2	no	—
173	C₂×C₄⋊C₄	—	97	2	no	—
174	C₂×C₄⋊C₄	—	97	8	no	—
175	C₂×C₄⋊C₄	C₂×D₉₇	1	1	yes	characteristic
176	C₂×C₄⋊C₄	C₂×D₉₇	1	1	yes	characteristic
177	C₂×C₄⋊C₄	C₂×D₉₇	1	2	yes	—
178	C₂×C₄⋊C₄	C₂×D₉₇	1	2	yes	—
179	C₂×C₂²⋊C₄	—	97	1	no	—
180	C₂×C₂²⋊C₄	—	97	2	no	—
181	C₂×C₂²⋊C₄	—	97	1	no	—
182	C₂×C₄²	—	97	2	no	—
183	C₂×C₄²	—	97	1	no	—
184	C₂×C₄²	C₂×D₉₇	1	1	yes	characteristic
185	C₂⁴	—	97	1	no	—
186	C₂×Q₈	—	97	8	no	—
187	C₂×Q₈	—	97	8	no	—
188	C₂×Q₈	—	388	4	no	—
189	C₂×Q₈	—	388	4	no	—
190	C₂²×C₄	—	97	1	no	—
191	C₂²×C₄	—	97	8	no	—
192	C₂²×C₄	—	97	4	no	—
193	C₂²×C₄	—	97	1	no	—
194	C₂²×C₄	—	97	1	no	—
195	C₂²×C₄	—	97	2	no	—
196	C₂²×C₄	—	97	2	no	—
197	C₂²×C₄	—	97	8	no	—
198	C₂²×C₄	—	388	4	no	—
199	C₂²×C₄	C₂²×D₉₇	1	2	yes	—
200	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
201	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
202	C₂²×C₄	C₂²×D₉₇	1	2	yes	—
203	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
204	C₄⋊C₄	—	97	8	no	—
205	C₄⋊C₄	—	97	4	no	—
206	C₄⋊C₄	—	97	4	no	—
207	C₄⋊C₄	—	97	8	no	—
208	C₄⋊C₄	—	97	8	no	—
209	C₄⋊C₄	—	97	4	no	—
210	C₄⋊C₄	—	97	4	no	—
211	C₄⋊C₄	—	97	8	no	—
212	C₄⋊C₄	—	97	8	no	—
213	C₄⋊C₄	C₂²×D₉₇	1	4	yes	—
214	C₄⋊C₄	C₂²×D₉₇	1	4	yes	—
215	C₄⋊C₄	C₂²×D₉₇	1	8	yes	—
216	C₄⋊C₄	C₂²×D₉₇	1	8	yes	—
217	C₂²⋊C₄	—	97	4	no	—
218	C₂²⋊C₄	—	97	8	no	—
219	C₂²⋊C₄	—	97	4	no	—
220	C₄²	—	97	8	no	—
221	C₄²	—	97	4	no	—
222	C₄²	C₂²×D₉₇	1	4	yes	—
223	C₂³	—	97	4	no	—
224	C₂³	—	97	2	no	—
225	C₂³	—	194	2	no	—
226	C₂³	—	194	2	no	—
227	C₂³	C₂³×D₉₇	1	1	yes	characteristic
228	Q₈	—	388	8	no	—
229	Q₈	—	388	8	no	—
230	C₂×C₄	—	2	4	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	4	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	97	4	no	—
235	C₂×C₄	—	97	2	no	—
236	C₂×C₄	—	97	16	no	—
237	C₂×C₄	—	97	2	no	—
238	C₂×C₄	—	97	2	no	—
239	C₂×C₄	—	97	4	no	—
240	C₂×C₄	—	97	4	no	—
241	C₂×C₄	—	388	8	no	—
242	C₂×C₄	—	194	4	no	—
243	C₂×C₄	—	194	2	no	—
244	C₂×C₄	—	194	4	no	—
245	C₂×C₄	—	194	2	no	—
246	C₂×C₄	Q₈×D₉₇	1	4	yes	—
247	C₂×C₄	C₂³×D₉₇	1	4	yes	—
248	C₂×C₄	C₂³×D₉₇	1	2	yes	—
249	C₂×C₄	C₂³×D₉₇	1	2	yes	—
250	C₂×C₄	C₂³×D₉₇	1	4	yes	—
251	C₂×C₄	C₂³×D₉₇	1	2	yes	—
252	C₂²	—	97	4	no	—
253	C₂²	—	194	2	no	—
254	C₂²	—	194	2	no	—
255	C₂²	—	194	8	no	—
256	C₂²	C₂×Q₈×D₉₇	1	2	yes	—
257	C₂²	#15690856	1	2	yes	—
258	C₂²	#15690857	1	2	yes	—
259	C₂²	C₂⁴×D₉₇	1	1	yes	characteristic
260	C₄	—	2	4	no	—
261	C₄	—	2	2	no	—
262	C₄	—	2	4	no	—
263	C₄	—	2	2	no	—
264	C₄	—	97	4	no	—
265	C₄	—	194	4	no	—
266	C₄	—	194	2	no	—
267	C₄	—	194	4	no	—
268	C₄	—	194	2	no	—
269	C₄	C₂×Q₈×D₉₇	1	4	yes	—
270	C₂	—	194	4	no	—
271	C₂	#20261365	1	4	yes	direct product factor
272	C₂	C₂²×Q₈×D₉₇	1	1	yes	characteristic
273	C₂	C₂×#15690856	1	1	yes	characteristic
274	C₂	C₂×#15690857	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂³×D₉₇
commutator, cosocle	C₂²×C₉₇	C₂⁵
Frattini	C₂²	C₂⁴×D₉₇
Fitting	C₂×#125×C₉₇	C₂
socle	C₂³×C₉₇	C₂⁴
prime radical	C₂³×D₉₇	C₂³
hypercenter	C₂×#125	D₉₇
hypocenter	C₉₇	C₂×#544
LCS	G ▷ C₂²×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂³ ◁ C₂×#125

Complements

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

N	H ≅ G/N	type
1. #20261365	C₂	direct (N × H)
2. C₂×#15690810	C₂	semidirect
3. C₂×#15690809	C₂	semidirect
4. C₂×#15690809	C₂	semidirect
5. C₂×#15690801	C₂	semidirect
6. C₂×#15690800	C₂	semidirect
12. C₂×#125×C₉₇	C₂	semidirect
17. #15690810	C₂²	semidirect
18. #15690809	C₂²	semidirect
19. #15690809	C₂²	semidirect
20. #15690801	C₂²	semidirect
21. #15690800	C₂²	semidirect
41. #125×C₉₇	C₂²	semidirect
58. C₂×C₄×D₉₇	Q₈	semidirect
111. C₄×D₉₇	C₂×Q₈	semidirect
132. C₂×C₉₇	#544	semidirect
138. C₉₇	C₂×#544	semidirect
145. C₂×#125	D₉₇	semidirect
160. #125	C₂×D₉₇	semidirect
271. C₂	#20261365	direct (N × H)

Maximal subgroups

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

subgroup	index
#20261365	2
C₂×#15690810	2
C₂×#15690809	2
C₂×#15690801	2
C₂×#15690800	2
C₂×#15690790	2
C₂×#15690788	2
C₂×#15690768	2
C₂×#15690767	2
C₂×C₄⋊C₄×D₉₇	2
C₂×#125×C₉₇	2
C₂×#544	97

Subgroups — C2×#20261365 #27155136

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20261365 #27155136