Subgroups — C₄×#15690843 #27150559

Order 12416.

484 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15690843	C₂	1	1	yes	maximal, characteristic
2	C₄×#13095511	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×Dic₁₉₄	C₂	1	1	yes	maximal, characteristic, complemented
4	#20260843	C₂	1	2	yes	maximal, complemented
5	#20260837	C₂	1	2	yes	maximal, complemented
6	C₄×Dic₃₈₈	C₂	1	2	yes	maximal, complemented
7	C₄×#13095500	C₂	1	2	yes	maximal, complemented
8	#20260777	C₂	1	1	yes	maximal, characteristic, complemented
9	#20260745	C₂	1	1	yes	maximal, characteristic, complemented
10	#20260737	C₂	1	1	yes	maximal, characteristic
11	C₄×M₁₆×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13095511	C₂²	1	1	yes	characteristic
13	C₂²×Dic₁₉₄	C₂²	1	1	yes	characteristic
14	#15690843	C₄	1	8	yes	direct product factor
15	C₂×Dic₃₈₈	C₂²	1	2	yes	—
16	C₂×#13095500	C₂²	1	2	yes	—
17	C₄×Dic₁₉₄	—	2	1	no	—
18	C₄×Dic₁₉₄	—	2	1	no	—
19	C₄×Dic₁₉₄	C₂²	1	2	yes	—
20	C₄×Dic₁₉₄	C₂²	1	2	yes	—
21	C₄×Dic₁₉₄	C₂²	1	1	yes	characteristic
22	C₄×Dic₁₉₄	C₂²	1	1	yes	characteristic
23	C₄×D₃₈₈	C₂²	1	1	yes	characteristic
24	C₄×D₃₈₈	C₂²	1	1	yes	characteristic
25	C₄×C₂²⋊D₉₇	—	2	1	no	—
26	C₄×C₂²⋊D₉₇	—	2	1	no	—
27	#15690769	C₂²	1	1	yes	characteristic
28	#15690767	—	2	1	no	—
29	C₂×C₄⋊Dic₉₇	C₂²	1	1	yes	characteristic
30	C₂×#13095490	—	2	1	no	—
31	C₂×C₄×Dic₉₇	—	2	1	no	—
32	C₄²×D₉₇	—	2	1	no	—
33	#15690744	C₂²	1	2	yes	—
34	#15690742	C₂²	1	2	yes	—
35	#15690735	C₂²	1	2	yes	—
36	#15690735	C₂²	1	2	yes	—
37	#15690735	C₂²	1	2	yes	—
38	#15690732	C₂²	1	2	yes	—
39	C₂×M₁₆×C₉₇	C₂²	1	1	yes	characteristic
40	C₂×M₁₆×C₉₇	C₂²	1	1	yes	characteristic
41	C₂×C₄²×C₉₇	C₂²	1	1	yes	characteristic
42	#97×C₉₇	C₂²	1	2	yes	—
43	C₄×C₈×C₉₇	C₂²	1	2	yes	—
44	#13095511	—	2	2	no	—
45	#13095511	C₂×C₄	1	4	yes	—
46	C₂×Dic₁₉₄	—	2	1	no	—
47	C₂×Dic₁₉₄	—	2	2	no	—
48	C₂×Dic₁₉₄	C₂×C₄	1	4	yes	complemented
49	C₂×Dic₁₉₄	C₂³	1	2	yes	—
50	C₂×Dic₁₉₄	C₂³	1	1	yes	characteristic
51	C₂×D₃₈₈	C₂³	1	1	yes	characteristic
52	C₂×C₂²⋊D₉₇	—	2	1	no	—
53	C₂²×Dic₉₇	—	2	1	no	—
54	C₂²×Dic₉₇	—	2	1	no	—
55	C₂×C₄×D₉₇	—	2	1	no	—
56	C₂×C₄×D₉₇	—	2	1	no	—
57	C₂×C₄×D₉₇	—	2	1	no	—
58	Dic₃₈₈	C₂×C₄	1	8	yes	complemented
59	#13095500	C₂×C₄	1	8	yes	complemented
60	C₄⋊Dic₉₇	—	2	1	no	—
61	C₄⋊Dic₉₇	C₂³	1	2	yes	—
62	C₄⋊Dic₉₇	C₂³	1	1	yes	characteristic
63	C₄⋊Dic₉₇	C₂³	1	1	yes	characteristic
64	#13095490	—	2	2	no	—
65	#13095490	—	2	2	no	—
66	#13095490	—	2	1	no	—
67	#13095490	—	2	1	no	—
68	C₂²⋊Dic₉₇	—	2	1	no	—
69	#13095488	—	2	1	no	—
70	#13095488	—	2	1	no	—
71	C₄×Dic₉₇	—	2	1	no	—
72	C₄×Dic₉₇	—	2	1	no	—
73	C₄×Dic₉₇	—	2	1	no	—
74	C₄×Dic₉₇	—	2	2	no	—
75	C₄×Dic₉₇	—	2	2	no	—
76	C₂²×C₄×C₉₇	D₄	1	1	yes	characteristic
77	C₂²×C₄×C₉₇	D₄	1	1	yes	characteristic
78	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
79	M₁₆×C₉₇	—	2	2	no	—
80	M₁₆×C₉₇	C₂×C₄	1	4	yes	complemented
81	C₂×C₈×C₉₇	C₂³	1	2	yes	—
82	C₂×C₈×C₉₇	C₂³	1	2	yes	—
83	C₄²×C₉₇	D₄	1	1	yes	characteristic
84	C₄²×C₉₇	D₄	1	1	yes	characteristic
85	C₄²×C₉₇	C₂³	1	1	yes	characteristic
86	C₄²×C₉₇	C₂³	1	1	yes	characteristic
87	C₂³×C₉₇	C₂×D₄	1	1	yes	characteristic
88	C₂²×D₉₇	—	2	1	no	—
89	C₂×C₄×C₉₇	—	2	2	no	—
90	C₂×C₄×C₉₇	—	4	1	no	—
91	C₂×C₄×C₉₇	C₂²×C₄	1	2	yes	—
92	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
93	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
94	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
95	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
96	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
97	C₂×C₄×C₉₇	#41	1	2	yes	—
98	C₂×C₄×C₉₇	C₂⁴	1	1	yes	characteristic
99	C₂²⋊D₉₇	—	2	4	no	—
100	C₂×Dic₉₇	—	2	4	no	—
101	C₂×Dic₉₇	—	2	1	no	—
102	C₂×Dic₉₇	—	2	2	no	—
103	C₂×Dic₉₇	—	2	1	no	—
104	C₂×Dic₉₇	—	2	1	no	—
105	C₂×Dic₉₇	—	2	2	no	—
106	C₂×Dic₉₇	—	4	2	no	—
107	D₃₈₈	—	2	1	no	—
108	D₃₈₈	C₂²×C₄	1	2	yes	—
109	C₄×D₉₇	—	2	4	no	—
110	C₄×D₉₇	—	4	2	no	—
111	C₄×D₉₇	—	4	2	no	—
112	Dic₁₉₄	—	2	4	no	—
113	Dic₁₉₄	—	2	2	no	—
114	Dic₁₉₄	—	2	1	no	—
115	Dic₁₉₄	C₂²×C₄	1	4	yes	—
116	Dic₁₉₄	C₂²×C₄	1	2	yes	—
117	C₈×C₉₇	—	2	2	no	—
118	C₈×C₉₇	C₂²×C₄	1	4	yes	—
119	C₂²×C₉₇	—	2	1	no	—
120	C₂²×C₉₇	—	2	1	no	—
121	C₂²×C₉₇	C₄×D₄	1	2	yes	—
122	C₂²×C₉₇	C₂²×D₄	1	1	yes	characteristic
123	C₂×D₉₇	—	2	2	no	—
124	C₂×D₉₇	—	4	1	no	—
125	C₂×D₉₇	—	4	1	no	—
126	C₄×C₉₇	—	2	1	no	—
127	C₄×C₉₇	—	2	1	no	—
128	C₄×C₉₇	—	2	1	no	—
129	C₄×C₉₇	C₄×D₄	1	2	yes	—
130	C₄×C₉₇	#137	1	2	yes	complemented
131	C₄×C₉₇	C₂³×C₄	1	1	yes	characteristic
132	C₄×C₉₇	C₂×#41	1	1	yes	characteristic
133	Dic₉₇	—	2	4	no	—
134	Dic₉₇	—	2	2	no	—
135	Dic₉₇	—	4	1	no	—
136	Dic₉₇	—	4	1	no	—
137	Dic₉₇	—	4	2	no	—
138	C₂×C₉₇	—	2	2	no	—
139	C₂×C₉₇	C₂×C₄×D₄	1	1	yes	characteristic
140	C₂×C₉₇	C₂×#137	1	1	yes	characteristic
141	C₂×C₉₇	#575	1	1	yes	characteristic
142	D₉₇	—	4	2	no	—
143	C₄×#137	—	97	1	no	maximal, Sylow
144	C₉₇	C₄×#137	1	1	yes	characteristic, complemented, Sylow
145	C₂×#137	—	97	1	no	—
146	C₄×#41	—	97	1	no	—
147	C₂×C₄×Q₈	—	97	1	no	—
148	#441	—	97	2	no	—
149	#440	—	97	2	no	—
150	C₄×Q₁₆	—	97	2	no	—
151	C₄×SD₁₆	—	97	2	no	—
152	#428	—	97	1	no	—
153	#419	—	97	1	no	—
154	#417	—	97	1	no	—
155	C₄×M₁₆	D₉₇	1	1	yes	characteristic, complemented
156	C₂×#41	—	97	1	no	—
157	C₂²×Q₈	—	97	1	no	—
158	#137	—	97	8	no	—
159	C₂×Q₁₆	—	97	2	no	—
160	C₂×SD₁₆	—	97	2	no	—
161	C₂×M₁₆	C₂×D₉₇	1	1	yes	characteristic
162	C₂×M₁₆	C₂×D₉₇	1	1	yes	characteristic
163	C₄×Q₈	—	97	2	no	—
164	C₄×Q₈	—	97	2	no	—
165	C₄×Q₈	—	97	1	no	—
166	C₄×Q₈	—	97	1	no	—
167	C₄×Q₈	—	194	1	no	—
168	C₄×Q₈	—	194	1	no	—
169	C₄×D₄	—	97	1	no	—
170	C₄×D₄	—	97	1	no	—
171	C₄×D₄	—	194	1	no	—
172	C₄×D₄	—	194	1	no	—
173	#117	—	97	1	no	—
174	#117	—	194	1	no	—
175	C₂×C₄⋊C₄	—	97	1	no	—
176	C₂×C₄⋊C₄	—	194	1	no	—
177	C₂×C₄²	—	194	1	no	—
178	C₂×C₄²	—	194	1	no	—
179	C₂×C₄²	C₂×D₉₇	1	1	yes	characteristic
180	#107	—	97	2	no	—
181	#106	—	97	2	no	—
182	#103	—	97	2	no	—
183	#103	—	97	2	no	—
184	#103	—	97	2	no	—
185	#102	—	97	2	no	—
186	#97	C₂×D₉₇	1	2	yes	—
187	C₄×C₈	C₂×D₉₇	1	2	yes	—
188	#41	—	97	4	no	—
189	#41	—	194	2	no	—
190	C₂×Q₈	—	97	4	no	—
191	C₂×Q₈	—	97	2	no	—
192	C₂×Q₈	—	97	1	no	—
193	C₂×Q₈	—	194	1	no	—
194	C₂×Q₈	—	194	2	no	—
195	C₂×D₄	—	97	1	no	—
196	C₂×D₄	—	194	1	no	—
197	C₂²×C₄	—	194	1	no	—
198	C₂²×C₄	—	194	1	no	—
199	C₂²×C₄	—	194	1	no	—
200	C₂²×C₄	—	194	1	no	—
201	C₂²×C₄	—	194	1	no	—
202	C₂²×C₄	D₃₈₈	1	1	yes	characteristic
203	C₂²×C₄	D₃₈₈	1	1	yes	characteristic
204	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
205	Q₁₆	—	97	8	no	—
206	SD₁₆	—	97	8	no	—
207	M₁₆	—	2	2	no	—
208	M₁₆	C₄×D₉₇	1	4	yes	complemented
209	C₂×C₈	C₂²×D₉₇	1	2	yes	—
210	C₂×C₈	C₂²×D₉₇	1	2	yes	—
211	C₄⋊C₄	—	97	2	no	—
212	C₄⋊C₄	—	97	1	no	—
213	C₄⋊C₄	—	97	1	no	—
214	C₄⋊C₄	—	194	2	no	—
215	C₄⋊C₄	—	194	2	no	—
216	C₄⋊C₄	—	194	1	no	—
217	C₄⋊C₄	—	194	1	no	—
218	C₄⋊C₄	—	194	1	no	—
219	C₂²⋊C₄	—	194	1	no	—
220	C₂²⋊C₄	—	194	1	no	—
221	C₂²⋊C₄	—	194	1	no	—
222	C₄²	—	194	1	no	—
223	C₄²	—	194	1	no	—
224	C₄²	—	194	2	no	—
225	C₄²	—	194	1	no	—
226	C₄²	—	194	2	no	—
227	C₄²	D₃₈₈	1	1	yes	characteristic
228	C₄²	D₃₈₈	1	1	yes	characteristic
229	C₄²	C₂²×D₉₇	1	1	yes	characteristic
230	C₄²	C₂²×D₉₇	1	1	yes	characteristic
231	C₂³	—	194	1	no	—
232	C₂³	C₂×D₃₈₈	1	1	yes	characteristic
233	Q₈	—	97	4	no	—
234	Q₈	—	97	2	no	—
235	Q₈	—	194	4	no	—
236	Q₈	—	194	2	no	—
237	Q₈	—	194	1	no	—
238	D₄	—	97	2	no	—
239	D₄	—	194	4	no	—
240	D₄	—	194	1	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	4	1	no	—
243	C₂×C₄	—	388	2	no	—
244	C₂×C₄	—	388	2	no	—
245	C₂×C₄	—	388	2	no	—
246	C₂×C₄	—	194	1	no	—
247	C₂×C₄	—	194	1	no	—
248	C₂×C₄	—	194	2	no	—
249	C₂×C₄	—	194	4	no	—
250	C₂×C₄	—	194	1	no	—
251	C₂×C₄	—	194	2	no	—
252	C₂×C₄	—	194	4	no	—
253	C₂×C₄	C₂×C₄×D₉₇	1	2	yes	—
254	C₂×C₄	C₂×D₃₈₈	1	1	yes	characteristic
255	C₂×C₄	C₂×D₃₈₈	1	1	yes	characteristic
256	C₂×C₄	C₂×D₃₈₈	1	1	yes	characteristic
257	C₂×C₄	C₂×D₃₈₈	1	1	yes	characteristic
258	C₂×C₄	C₂×D₃₈₈	1	1	yes	characteristic
259	C₂×C₄	#13095511	1	2	yes	—
260	C₂×C₄	C₂³×D₉₇	1	1	yes	characteristic
261	C₈	—	2	2	no	—
262	C₈	C₂×C₄×D₉₇	1	4	yes	—
263	C₂²	—	2	1	no	—
264	C₂²	—	2	1	no	—
265	C₂²	—	388	1	no	—
266	C₂²	—	388	1	no	—
267	C₂²	—	194	2	no	—
268	C₂²	C₄×D₃₈₈	1	2	yes	—
269	C₂²	C₂²×D₃₈₈	1	1	yes	characteristic
270	C₄	—	2	1	no	—
271	C₄	—	2	1	no	—
272	C₄	—	2	1	no	—
273	C₄	—	388	1	no	—
274	C₄	—	388	1	no	—
275	C₄	—	388	2	no	—
276	C₄	—	194	2	no	—
277	C₄	—	194	4	no	—
278	C₄	C₄×D₃₈₈	1	2	yes	—
279	C₄	#15690843	1	2	yes	direct product factor
280	C₄	C₂²×C₄×D₉₇	1	1	yes	characteristic
281	C₄	C₂×#13095511	1	1	yes	characteristic
282	C₂	—	2	2	no	—
283	C₂	—	388	2	no	—
284	C₂	C₂×C₄×D₃₈₈	1	1	yes	characteristic
285	C₂	C₂×#15690843	1	1	yes	characteristic
286	C₂	#20261535	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₃₈₈
commutator	C₄×C₉₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₉₇
Fitting	C₄×M₁₆×C₉₇	C₂
socle	C₂²×C₉₇	C₂²×D₄
prime radical	C₂×#13095511	C₂²
cosocle	C₂×C₄×C₉₇	C₂⁴
hypercenter	C₄×M₁₆	D₉₇
hypocenter	C₉₇	C₄×#137
LCS	G ▷ C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×M₁₆

Complements

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

N	H ≅ G/N	type
3. C₂×C₄×Dic₁₉₄	C₂	semidirect
4. #20260843	C₂	semidirect
5. #20260837	C₂	semidirect
6. C₄×Dic₃₈₈	C₂	semidirect
7. C₄×#13095500	C₂	semidirect
8. #20260777	C₂	semidirect
9. #20260745	C₂	semidirect
11. C₄×M₁₆×C₉₇	C₂	semidirect
14. #15690843	C₄	direct (N × H)
48. C₂×Dic₁₉₄	C₂×C₄	semidirect
58. Dic₃₈₈	C₂×C₄	semidirect
59. #13095500	C₂×C₄	semidirect
80. M₁₆×C₉₇	C₂×C₄	semidirect
130. C₄×C₉₇	#137	semidirect
144. C₉₇	C₄×#137	semidirect
155. C₄×M₁₆	D₉₇	semidirect
208. M₁₆	C₄×D₉₇	semidirect
279. C₄	#15690843	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×#15690843	2
C₄×#13095511	2
C₂×C₄×Dic₁₉₄	2
#20260843	2
#20260837	2
C₄×Dic₃₈₈	2
C₄×#13095500	2
#20260777	2
#20260745	2
#20260737	2
C₄×M₁₆×C₉₇	2
C₄×#137	97

Subgroups — C4×#15690843 #27150559

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#15690843 #27150559