Subgroups — C₄×#15690835 #27150542

Order 12416.

548 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15690835	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×D₃₈₈	C₂	1	1	yes	maximal, characteristic, complemented
3	#20260845	C₂	1	2	yes	maximal, complemented
4	#20260833	C₂	1	2	yes	maximal, complemented
5	C₄×#13095498	C₂	1	2	yes	maximal, complemented
6	C₄×#13095496	C₂	1	2	yes	maximal, complemented
7	#20260778	C₂	1	1	yes	maximal, characteristic, complemented
8	#20260739	C₂	1	1	yes	maximal, characteristic, complemented
9	#20260736	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×#13095495	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×#41×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×D₃₈₈	C₂²	1	1	yes	characteristic
13	C₂²×C₄×D₉₇	—	2	1	no	—
14	#15690835	C₄	1	8	yes	direct product factor
15	C₂×#13095498	C₂²	1	2	yes	—
16	C₂×#13095496	C₂²	1	2	yes	—
17	C₂×#13095495	C₂²	1	1	yes	characteristic
18	C₂×#13095495	C₂²	1	1	yes	characteristic
19	C₄×D₃₈₈	—	2	1	no	—
20	C₄×D₃₈₈	—	2	1	no	—
21	C₄×D₃₈₈	C₂²	1	2	yes	—
22	C₄×D₃₈₈	C₂²	1	2	yes	—
23	C₂×C₄⋊Dic₉₇	C₂²	1	1	yes	characteristic
24	C₂×#13095488	—	2	1	no	—
25	#15690743	C₂²	1	2	yes	complemented
26	#15690741	C₂²	1	2	yes	complemented
27	#15690734	C₂²	1	2	yes	complemented
28	#15690731	C₂²	1	2	yes	complemented
29	#15690730	C₂²	1	2	yes	—
30	#15690730	C₂²	1	2	yes	—
31	#15690721	C₂²	1	2	yes	complemented
32	C₄×#11715895	C₂²	1	2	yes	complemented
33	C₂×#41×C₉₇	C₂²	1	1	yes	characteristic
34	C₄×Q₈×C₉₇	C₂²	1	1	yes	characteristic, complemented
35	C₄×Q₈×C₉₇	C₂²	1	1	yes	characteristic, complemented
36	C₄×D₄×C₉₇	—	2	1	no	—
37	C₄×D₄×C₉₇	—	2	1	no	—
38	C₄×D₄×C₉₇	C₂²	1	1	yes	characteristic, complemented
39	C₄×D₄×C₉₇	C₂²	1	1	yes	characteristic, complemented
40	#117×C₉₇	—	2	1	no	—
41	#117×C₉₇	C₂²	1	1	yes	characteristic
42	C₂×C₄²×C₉₇	—	2	1	no	—
43	C₂×C₄²×C₉₇	C₂²	1	1	yes	characteristic
44	C₂³×D₉₇	—	2	1	no	—
45	C₂×D₃₈₈	—	2	1	no	—
46	C₂×D₃₈₈	—	2	2	no	—
47	C₂×D₃₈₈	C₂×C₄	1	4	yes	complemented
48	C₂×D₃₈₈	C₂³	1	2	yes	—
49	C₂²×Dic₉₇	—	2	1	no	—
50	C₂×C₄×D₉₇	—	2	2	no	—
51	C₂×C₄×D₉₇	—	2	2	no	—
52	C₂×C₄×D₉₇	—	4	4	no	—
53	#13095498	C₂×C₄	1	8	yes	complemented
54	#13095496	C₂×C₄	1	8	yes	complemented
55	#13095495	—	2	2	no	—
56	#13095495	C₂×C₄	1	4	yes	complemented
57	C₂×#11715895	C₂³	1	2	yes	—
58	C₂×#11715895	C₂³	1	2	yes	—
59	C₄⋊Dic₉₇	—	2	1	no	—
60	C₄⋊Dic₉₇	C₂³	1	2	yes	—
61	#13095488	—	2	2	no	—
62	#13095488	—	2	2	no	—
63	#41×C₉₇	—	2	2	no	—
64	#41×C₉₇	C₂×C₄	1	4	yes	complemented
65	C₂×Q₈×C₉₇	C₂³	1	1	yes	characteristic
66	C₂×D₄×C₉₇	—	2	1	no	—
67	C₂×D₄×C₉₇	C₂³	1	1	yes	characteristic
68	C₂²×C₄×C₉₇	—	2	1	no	—
69	C₂²×C₄×C₉₇	—	2	1	no	—
70	C₂²×C₄×C₉₇	—	2	1	no	—
71	C₂²×C₄×C₉₇	D₄	1	1	yes	characteristic
72	C₂²×C₄×C₉₇	D₄	1	1	yes	characteristic
73	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
74	C₄⋊C₄×C₉₇	—	2	1	no	—
75	C₄⋊C₄×C₉₇	—	2	1	no	—
76	C₄⋊C₄×C₉₇	C₂³	1	1	yes	characteristic
77	C₄⋊C₄×C₉₇	C₂³	1	1	yes	characteristic
78	C₂²⋊C₄×C₉₇	—	2	1	no	—
79	C₂²⋊C₄×C₉₇	—	2	1	no	—
80	C₂²⋊C₄×C₉₇	—	2	1	no	—
81	C₄²×C₉₇	—	2	1	no	—
82	C₄²×C₉₇	—	2	1	no	—
83	C₄²×C₉₇	—	2	1	no	—
84	C₄²×C₉₇	D₄	1	1	yes	characteristic
85	C₄²×C₉₇	D₄	1	1	yes	characteristic
86	C₄²×C₉₇	C₂³	1	1	yes	characteristic
87	C₄²×C₉₇	C₂³	1	1	yes	characteristic
88	C₂³×C₉₇	—	2	1	no	—
89	C₂³×C₉₇	C₂×D₄	1	1	yes	characteristic
90	C₂²×D₉₇	—	2	4	no	—
91	C₂²×D₉₇	—	2	2	no	—
92	C₂²×D₉₇	—	4	2	no	—
93	C₂²×D₉₇	—	4	2	no	—
94	Q₈×C₉₇	—	2	1	no	—
95	Q₈×C₉₇	C₂²×C₄	1	2	yes	complemented
96	D₄×C₉₇	—	2	4	no	—
97	D₄×C₉₇	—	2	1	no	—
98	D₄×C₉₇	C₂²×C₄	1	2	yes	complemented
99	C₂×C₄×C₉₇	—	2	1	no	—
100	C₂×C₄×C₉₇	—	2	4	no	—
101	C₂×C₄×C₉₇	—	2	2	no	—
102	C₂×C₄×C₉₇	—	2	1	no	—
103	C₂×C₄×C₉₇	—	2	1	no	—
104	C₂×C₄×C₉₇	—	4	1	no	—
105	C₂×C₄×C₉₇	—	4	2	no	—
106	C₂×C₄×C₉₇	—	4	2	no	—
107	C₂×C₄×C₉₇	C₂²×C₄	1	2	yes	—
108	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
109	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
110	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
111	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
112	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
113	C₂×C₄×C₉₇	#41	1	2	yes	—
114	C₂×C₄×C₉₇	C₂⁴	1	1	yes	characteristic
115	C₂×Dic₉₇	—	2	2	no	—
116	C₂×Dic₉₇	—	4	2	no	—
117	D₃₈₈	—	2	4	no	—
118	D₃₈₈	—	2	2	no	—
119	D₃₈₈	C₂²×C₄	1	4	yes	—
120	C₄×D₉₇	—	4	4	no	—
121	C₄×D₉₇	—	4	4	no	—
122	#11715895	—	2	2	no	—
123	#11715895	C₂²×C₄	1	4	yes	complemented
124	C₂²×C₉₇	—	2	1	no	—
125	C₂²×C₉₇	—	2	1	no	—
126	C₂²×C₉₇	—	2	2	no	—
127	C₂²×C₉₇	—	4	1	no	—
128	C₂²×C₉₇	—	4	1	no	—
129	C₂²×C₉₇	C₄×D₄	1	2	yes	—
130	C₂²×C₉₇	C₂²×D₄	1	1	yes	characteristic
131	C₂×D₉₇	—	2	4	no	—
132	C₂×D₉₇	—	4	8	no	—
133	C₂×D₉₇	—	4	2	no	—
134	C₂×D₉₇	—	4	2	no	—
135	C₄×C₉₇	—	2	2	no	—
136	C₄×C₉₇	—	2	1	no	—
137	C₄×C₉₇	—	2	1	no	—
138	C₄×C₉₇	—	2	1	no	—
139	C₄×C₉₇	—	4	1	no	—
140	C₄×C₉₇	—	4	1	no	—
141	C₄×C₉₇	C₄×D₄	1	2	yes	—
142	C₄×C₉₇	AGL(1,ℤ/8)	1	2	yes	complemented
143	C₄×C₉₇	C₂³×C₄	1	1	yes	characteristic
144	C₄×C₉₇	C₂×#41	1	1	yes	characteristic
145	Dic₉₇	—	4	2	no	—
146	C₂×C₉₇	—	2	2	no	—
147	C₂×C₉₇	—	4	2	no	—
148	C₂×C₉₇	C₂×C₄×D₄	1	1	yes	characteristic
149	C₂×C₉₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
150	C₂×C₉₇	#575	1	1	yes	characteristic
151	D₉₇	—	4	4	no	—
152	C₄×AGL(1,ℤ/8)	—	97	1	no	maximal, Sylow
153	C₉₇	C₄×AGL(1,ℤ/8)	1	1	yes	characteristic, complemented, Sylow
154	C₂×AGL(1,ℤ/8)	—	97	1	no	—
155	C₄×#41	D₉₇	1	1	yes	characteristic, complemented
156	C₂×C₄×D₄	—	97	1	no	—
157	#442	—	97	2	no	—
158	#440	—	97	2	no	—
159	C₄×SD₁₆	—	97	2	no	—
160	C₄×D₈	—	97	2	no	—
161	#428	—	97	1	no	—
162	#418	—	97	1	no	—
163	#417	—	97	1	no	—
164	C₄×M₁₆	—	97	1	no	—
165	C₂×#41	C₂×D₉₇	1	1	yes	characteristic
166	C₂²×D₄	—	97	1	no	—
167	C₂³×C₄	—	194	1	no	—
168	AGL(1,ℤ/8)	—	97	8	no	—
169	C₂×SD₁₆	—	97	2	no	—
170	C₂×D₈	—	97	2	no	—
171	C₂×M₁₆	—	97	1	no	—
172	C₂×M₁₆	—	97	1	no	—
173	C₄×Q₈	C₂×D₉₇	1	1	yes	characteristic, complemented
174	C₄×Q₈	C₂×D₉₇	1	1	yes	characteristic, complemented
175	C₄×D₄	—	2	1	no	—
176	C₄×D₄	—	2	1	no	—
177	C₄×D₄	—	97	2	no	—
178	C₄×D₄	—	97	2	no	—
179	C₄×D₄	—	194	1	no	—
180	C₄×D₄	—	194	1	no	—
181	C₄×D₄	C₂×D₉₇	1	1	yes	characteristic, complemented
182	C₄×D₄	C₂×D₉₇	1	1	yes	characteristic, complemented
183	#117	—	2	1	no	—
184	#117	C₂×D₉₇	1	1	yes	characteristic
185	C₂×C₄⋊C₄	—	97	1	no	—
186	C₂×C₂²⋊C₄	—	194	1	no	—
187	C₂×C₄²	—	2	1	no	—
188	C₂×C₄²	C₂×D₉₇	1	1	yes	characteristic
189	#107	—	97	2	no	—
190	#106	—	97	2	no	—
191	#103	—	97	2	no	—
192	#102	—	97	2	no	—
193	#102	—	97	2	no	—
194	#102	—	97	2	no	—
195	#97	—	97	2	no	—
196	C₄×C₈	—	97	2	no	—
197	C₂⁴	—	194	1	no	—
198	#41	—	2	2	no	—
199	#41	C₄×D₉₇	1	4	yes	complemented
200	C₂×Q₈	C₂²×D₉₇	1	1	yes	characteristic
201	C₂×D₄	—	2	1	no	—
202	C₂×D₄	—	97	4	no	—
203	C₂×D₄	—	97	2	no	—
204	C₂×D₄	—	194	1	no	—
205	C₂×D₄	—	194	2	no	—
206	C₂×D₄	C₂²×D₉₇	1	1	yes	characteristic
207	C₂²×C₄	—	2	1	no	—
208	C₂²×C₄	—	2	1	no	—
209	C₂²×C₄	—	2	1	no	—
210	C₂²×C₄	—	388	4	no	—
211	C₂²×C₄	—	194	1	no	—
212	C₂²×C₄	—	194	2	no	—
213	C₂²×C₄	—	194	2	no	—
214	C₂²×C₄	C₂²⋊D₉₇	1	1	yes	characteristic
215	C₂²×C₄	C₂²⋊D₉₇	1	1	yes	characteristic
216	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
217	SD₁₆	—	97	8	no	—
218	D₈	—	97	8	no	—
219	M₁₆	—	97	4	no	—
220	M₁₆	—	194	2	no	—
221	C₂×C₈	—	97	2	no	—
222	C₂×C₈	—	97	2	no	—
223	C₄⋊C₄	—	2	1	no	—
224	C₄⋊C₄	—	2	1	no	—
225	C₄⋊C₄	—	97	2	no	—
226	C₄⋊C₄	—	194	1	no	—
227	C₄⋊C₄	C₂²×D₉₇	1	1	yes	characteristic
228	C₄⋊C₄	C₂²×D₉₇	1	1	yes	characteristic
229	C₂²⋊C₄	—	2	1	no	—
230	C₂²⋊C₄	—	2	1	no	—
231	C₂²⋊C₄	—	2	1	no	—
232	C₂²⋊C₄	—	194	2	no	—
233	C₂²⋊C₄	—	194	2	no	—
234	C₄²	—	2	1	no	—
235	C₄²	—	2	1	no	—
236	C₄²	—	2	1	no	—
237	C₄²	C₂²⋊D₉₇	1	1	yes	characteristic
238	C₄²	C₂²⋊D₉₇	1	1	yes	characteristic
239	C₄²	C₂²×D₉₇	1	1	yes	characteristic
240	C₄²	C₂²×D₉₇	1	1	yes	characteristic
241	C₂³	—	2	1	no	—
242	C₂³	—	388	2	no	—
243	C₂³	—	388	2	no	—
244	C₂³	—	194	4	no	—
245	C₂³	—	194	2	no	—
246	C₂³	C₂×C₂²⋊D₉₇	1	1	yes	characteristic
247	Q₈	—	2	1	no	—
248	Q₈	C₂×C₄×D₉₇	1	2	yes	complemented
249	D₄	—	2	4	no	—
250	D₄	—	2	1	no	—
251	D₄	—	97	4	no	—
252	D₄	—	194	4	no	—
253	D₄	—	194	2	no	—
254	D₄	C₂×C₄×D₉₇	1	2	yes	complemented
255	C₂×C₄	—	2	1	no	—
256	C₂×C₄	—	2	4	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	2	1	no	—
260	C₂×C₄	—	4	1	no	—
261	C₂×C₄	—	4	2	no	—
262	C₂×C₄	—	4	2	no	—
263	C₂×C₄	—	388	4	no	—
264	C₂×C₄	—	388	2	no	—
265	C₂×C₄	—	388	4	no	—
266	C₂×C₄	—	194	2	no	—
267	C₂×C₄	C₂×C₄×D₉₇	1	2	yes	—
268	C₂×C₄	C₂×C₂²⋊D₉₇	1	1	yes	characteristic
269	C₂×C₄	C₂×C₂²⋊D₉₇	1	1	yes	characteristic
270	C₂×C₄	C₂×C₂²⋊D₉₇	1	1	yes	characteristic
271	C₂×C₄	C₂×C₂²⋊D₉₇	1	1	yes	characteristic
272	C₂×C₄	C₂×C₂²⋊D₉₇	1	1	yes	characteristic
273	C₂×C₄	#13095511	1	2	yes	—
274	C₂×C₄	C₂³×D₉₇	1	1	yes	characteristic
275	C₈	—	97	4	no	—
276	C₈	—	194	2	no	—
277	C₂²	—	2	1	no	—
278	C₂²	—	2	1	no	—
279	C₂²	—	2	2	no	—
280	C₂²	—	4	1	no	—
281	C₂²	—	4	1	no	—
282	C₂²	—	388	8	no	—
283	C₂²	—	388	2	no	—
284	C₂²	—	388	2	no	—
285	C₂²	—	194	4	no	—
286	C₂²	C₄×C₂²⋊D₉₇	1	2	yes	—
287	C₂²	C₂²×C₂²⋊D₉₇	1	1	yes	characteristic
288	C₄	—	2	2	no	—
289	C₄	—	2	1	no	—
290	C₄	—	2	1	no	—
291	C₄	—	2	1	no	—
292	C₄	—	4	1	no	—
293	C₄	—	4	1	no	—
294	C₄	—	388	2	no	—
295	C₄	C₄×C₂²⋊D₉₇	1	2	yes	—
296	C₄	#15690835	1	2	yes	direct product factor
297	C₄	C₂²×C₄×D₉₇	1	1	yes	characteristic
298	C₄	C₂×#13095511	1	1	yes	characteristic
299	C₂	—	2	2	no	—
300	C₂	—	4	2	no	—
301	C₂	—	388	4	no	—
302	C₂	C₂×C₄×C₂²⋊D₉₇	1	1	yes	characteristic
303	C₂	C₂×#15690835	1	1	yes	characteristic
304	C₂	#20261533	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₉₇
commutator	C₄×C₉₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₉₇
Fitting	C₄×#41×C₉₇	C₂
socle	C₂²×C₉₇	C₂²×D₄
prime radical	C₂×#15690835	C₂
cosocle	C₂×C₄×C₉₇	C₂⁴
hypercenter	C₄×#41	D₉₇
hypocenter	C₉₇	C₄×AGL(1,ℤ/8)
LCS	G ▷ C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×#41

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×D₃₈₈	C₂	semidirect
3. #20260845	C₂	semidirect
4. #20260833	C₂	semidirect
5. C₄×#13095498	C₂	semidirect
6. C₄×#13095496	C₂	semidirect
7. #20260778	C₂	semidirect
8. #20260739	C₂	semidirect
9. #20260736	C₂	semidirect
10. C₄×#13095495	C₂	semidirect
11. C₄×#41×C₉₇	C₂	semidirect
14. #15690835	C₄	direct (N × H)
25. #15690743	C₂²	semidirect
26. #15690741	C₂²	semidirect
27. #15690734	C₂²	semidirect
28. #15690731	C₂²	semidirect
31. #15690721	C₂²	semidirect
32. C₄×#11715895	C₂²	semidirect
34. C₄×Q₈×C₉₇	C₂²	semidirect
35. C₄×Q₈×C₉₇	C₂²	semidirect
38. C₄×D₄×C₉₇	C₂²	semidirect
39. C₄×D₄×C₉₇	C₂²	semidirect
47. C₂×D₃₈₈	C₂×C₄	semidirect
53. #13095498	C₂×C₄	semidirect
54. #13095496	C₂×C₄	semidirect
56. #13095495	C₂×C₄	semidirect
64. #41×C₉₇	C₂×C₄	semidirect
95. Q₈×C₉₇	C₂²×C₄	semidirect
98. D₄×C₉₇	C₂²×C₄	semidirect
123. #11715895	C₂²×C₄	semidirect
142. C₄×C₉₇	AGL(1,ℤ/8)	semidirect
153. C₉₇	C₄×AGL(1,ℤ/8)	semidirect
155. C₄×#41	D₉₇	semidirect
173. C₄×Q₈	C₂×D₉₇	semidirect
174. C₄×Q₈	C₂×D₉₇	semidirect
181. C₄×D₄	C₂×D₉₇	semidirect
182. C₄×D₄	C₂×D₉₇	semidirect
199. #41	C₄×D₉₇	semidirect
248. Q₈	C₂×C₄×D₉₇	semidirect
254. D₄	C₂×C₄×D₉₇	semidirect
296. C₄	#15690835	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×#15690835	2
C₂×C₄×D₃₈₈	2
#20260845	2
#20260833	2
C₄×#13095498	2
C₄×#13095496	2
#20260778	2
#20260739	2
#20260736	2
C₄×#13095495	2
C₄×#41×C₉₇	2
C₄×AGL(1,ℤ/8)	97

Subgroups — C4×#15690835 #27150542

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#15690835 #27150542