Subgroups — C₄×#20263305 #38745247

Order 25216.

656 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#20263305	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Dic₃₉₄	C₂	1	1	yes	maximal, characteristic, complemented
3	#27160985	C₂	1	1	yes	maximal, characteristic, complemented
4	#27160977	C₂	1	1	yes	maximal, characteristic, complemented
5	#27160963	C₂	1	2	yes	maximal, complemented
6	#27160952	C₂	1	2	yes	maximal
7	#27160939	C₂	1	1	yes	maximal, characteristic
8	C₄×#15691337	C₂	1	2	yes	maximal, complemented
9	C₄×C₄⋊Dic₁₉₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×C₂²⋊Dic₁₉₇	C₂	1	2	yes	maximal
11	C₂²×C₄²×C₁₉₇	C₂	1	1	yes	maximal, characteristic
12	C₂²×Dic₃₉₄	C₂²	1	1	yes	characteristic
13	#20263305	C₄	1	8	yes	direct product factor
14	C₄×Dic₃₉₄	—	2	4	no	—
15	C₂×C₄⋊Dic₁₉₇	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊Dic₁₉₇	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊Dic₁₉₇	C₂²	1	1	yes	characteristic
18	C₂×#15691337	C₂²	1	2	yes	—
19	C₂×#15691337	C₂²	1	2	yes	—
20	C₂×#15691337	C₂²	1	2	yes	—
21	C₂×C₂²⋊Dic₁₉₇	C₂²	1	2	yes	—
22	C₂×C₂²⋊Dic₁₉₇	C₂²	1	2	yes	—
23	C₂×C₄×Dic₁₉₇	C₂²	1	2	yes	—
24	C₂×C₄×Dic₁₉₇	C₂²	1	2	yes	—
25	#20263233	C₂²	1	2	yes	—
26	#20263233	C₂²	1	2	yes	—
27	#20263233	C₂²	1	2	yes	—
28	#20263233	C₂²	1	2	yes	—
29	#20263233	C₂²	1	2	yes	—
30	C₂³×C₄×C₁₉₇	C₂²	1	1	yes	characteristic
31	C₂³×C₄×C₁₉₇	C₂²	1	1	yes	characteristic
32	C₂³×C₄×C₁₉₇	C₂²	1	1	yes	characteristic
33	C₂×C₄²×C₁₉₇	—	2	4	no	—
34	C₂×C₄²×C₁₉₇	C₂²	1	1	yes	characteristic
35	C₂×C₄²×C₁₉₇	C₂²	1	1	yes	characteristic
36	C₂×C₄²×C₁₉₇	C₂²	1	1	yes	characteristic
37	C₂×C₄²×C₁₉₇	C₂²	1	1	yes	characteristic
38	C₂×Dic₃₉₄	—	2	2	no	—
39	C₂×Dic₃₉₄	—	2	2	no	—
40	C₂×Dic₃₉₄	C₂×C₄	1	4	yes	complemented
41	C₂²×Dic₁₉₇	C₂³	1	2	yes	—
42	C₂²×Dic₁₉₇	C₂³	1	2	yes	—
43	C₂²×Dic₁₉₇	C₂³	1	2	yes	—
44	C₂²×Dic₁₉₇	C₂³	1	2	yes	—
45	C₄⋊Dic₁₉₇	—	2	2	no	—
46	C₄⋊Dic₁₉₇	—	2	2	no	—
47	C₄⋊Dic₁₉₇	C₂×C₄	1	4	yes	complemented
48	#15691337	—	2	4	no	—
49	#15691337	—	2	4	no	—
50	#15691337	C₂×C₄	1	8	yes	complemented
51	C₂²⋊Dic₁₉₇	—	2	4	no	—
52	C₂²⋊Dic₁₉₇	C₂×C₄	1	8	yes	—
53	C₄×Dic₁₉₇	—	2	4	no	—
54	C₄×Dic₁₉₇	—	2	4	no	—
55	C₂⁴×C₁₉₇	C₂³	1	1	yes	characteristic
56	C₂²×C₄×C₁₉₇	—	2	2	no	—
57	C₂²×C₄×C₁₉₇	—	2	2	no	—
58	C₂²×C₄×C₁₉₇	—	2	2	no	—
59	C₂²×C₄×C₁₉₇	—	2	2	no	—
60	C₂²×C₄×C₁₉₇	—	2	2	no	—
61	C₂²×C₄×C₁₉₇	—	2	2	no	—
62	C₂²×C₄×C₁₉₇	—	2	2	no	—
63	C₂²×C₄×C₁₉₇	C₂×C₄	1	4	yes	—
64	C₂²×C₄×C₁₉₇	Q₈	1	4	yes	complemented
65	C₂²×C₄×C₁₉₇	C₂³	1	1	yes	characteristic
66	C₂²×C₄×C₁₉₇	C₂³	1	1	yes	characteristic
67	C₂²×C₄×C₁₉₇	C₂³	1	1	yes	characteristic
68	C₂²×C₄×C₁₉₇	C₂³	1	1	yes	characteristic
69	C₂²×C₄×C₁₉₇	C₂³	1	1	yes	characteristic
70	C₂²×C₄×C₁₉₇	C₂³	1	1	yes	characteristic
71	C₄²×C₁₉₇	—	2	2	no	—
72	C₄²×C₁₉₇	—	2	2	no	—
73	C₄²×C₁₉₇	—	2	2	no	—
74	C₄²×C₁₉₇	D₄	1	4	yes	—
75	Dic₃₉₄	—	2	8	no	—
76	C₂×Dic₁₉₇	—	2	2	no	—
77	C₂×Dic₁₉₇	—	2	2	no	—
78	C₂×Dic₁₉₇	—	2	2	no	—
79	C₂×Dic₁₉₇	—	2	2	no	—
80	C₂×Dic₁₉₇	—	2	2	no	—
81	C₂×Dic₁₉₇	—	2	2	no	—
82	C₂×Dic₁₉₇	—	2	2	no	—
83	C₂×Dic₁₉₇	—	2	2	no	—
84	C₂×Dic₁₉₇	—	2	2	no	—
85	C₂×Dic₁₉₇	—	2	2	no	—
86	C₂×Dic₁₉₇	C₂²×C₄	1	4	yes	—
87	C₂×Dic₁₉₇	C₂²×C₄	1	4	yes	—
88	C₂³×C₁₉₇	—	2	1	no	—
89	C₂³×C₁₉₇	—	2	1	no	—
90	C₂³×C₁₉₇	—	2	1	no	—
91	C₂³×C₁₉₇	—	2	1	no	—
92	C₂³×C₁₉₇	C₂²×C₄	1	2	yes	—
93	C₂³×C₁₉₇	C₂×Q₈	1	2	yes	—
94	C₂³×C₁₉₇	#41	1	2	yes	—
95	C₂³×C₁₉₇	C₂⁴	1	1	yes	characteristic
96	C₂×C₄×C₁₉₇	—	2	8	no	—
97	C₂×C₄×C₁₉₇	—	2	1	no	—
98	C₂×C₄×C₁₉₇	—	2	1	no	—
99	C₂×C₄×C₁₉₇	—	2	1	no	—
100	C₂×C₄×C₁₉₇	—	2	1	no	—
101	C₂×C₄×C₁₉₇	—	2	1	no	—
102	C₂×C₄×C₁₉₇	—	2	1	no	—
103	C₂×C₄×C₁₉₇	—	2	1	no	—
104	C₂×C₄×C₁₉₇	—	2	1	no	—
105	C₂×C₄×C₁₉₇	—	2	8	no	—
106	C₂×C₄×C₁₉₇	—	2	8	no	—
107	C₂×C₄×C₁₉₇	C₂²×C₄	1	2	yes	—
108	C₂×C₄×C₁₉₇	C₂²×C₄	1	2	yes	—
109	C₂×C₄×C₁₉₇	C₂×D₄	1	2	yes	—
110	C₂×C₄×C₁₉₇	C₂×D₄	1	2	yes	—
111	C₂×C₄×C₁₉₇	C₂×D₄	1	2	yes	—
112	C₂×C₄×C₁₉₇	C₂×Q₈	1	2	yes	—
113	C₂×C₄×C₁₉₇	C₂×Q₈	1	2	yes	—
114	C₂×C₄×C₁₉₇	#41	1	2	yes	—
115	C₂×C₄×C₁₉₇	#41	1	2	yes	—
116	C₂×C₄×C₁₉₇	#41	1	2	yes	—
117	C₂²×C₁₉₇	—	2	2	no	—
118	C₂²×C₁₉₇	—	2	2	no	—
119	C₂²×C₁₉₇	—	2	2	no	—
120	C₂²×C₁₉₇	—	2	2	no	—
121	C₂²×C₁₉₇	—	2	2	no	—
122	C₂²×C₁₉₇	—	2	2	no	—
123	C₂²×C₁₉₇	C₄×Q₈	1	4	yes	complemented
124	C₂²×C₁₉₇	C₂³×C₄	1	1	yes	characteristic
125	C₂²×C₁₉₇	C₂²×D₄	1	1	yes	characteristic
126	C₂²×C₁₉₇	C₂²×Q₈	1	1	yes	characteristic
127	C₂²×C₁₉₇	C₂×#41	1	1	yes	characteristic
128	C₂²×C₁₉₇	C₂×#41	1	1	yes	characteristic
129	C₂²×C₁₉₇	C₂×#41	1	1	yes	characteristic
130	C₂²×C₁₉₇	C₂×#41	1	1	yes	characteristic
131	C₄×C₁₉₇	—	2	2	no	—
132	C₄×C₁₉₇	—	2	2	no	—
133	C₄×C₁₉₇	—	2	2	no	—
134	C₄×C₁₉₇	—	2	2	no	—
135	C₄×C₁₉₇	C₄×D₄	1	4	yes	—
136	C₄×C₁₉₇	C₂²⋊Q₈	1	4	yes	complemented
137	Dic₁₉₇	—	2	4	no	—
138	Dic₁₉₇	—	2	4	no	—
139	Dic₁₉₇	—	4	2	no	—
140	Dic₁₉₇	—	4	2	no	—
141	C₂×C₁₉₇	—	2	4	no	—
142	C₂×C₁₉₇	C₂×C₄×D₄	1	1	yes	characteristic
143	C₂×C₁₉₇	C₂×C₄×Q₈	1	1	yes	characteristic
144	C₂×C₁₉₇	C₄×#41	1	1	yes	characteristic
145	C₂×C₁₉₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
146	C₂×C₁₉₇	#525	1	1	yes	characteristic
147	C₂×C₁₉₇	#529	1	1	yes	characteristic
148	C₂×C₁₉₇	#533	1	1	yes	characteristic
149	C₁₉₇	C₄×C₂²⋊Q₈	1	1	yes	characteristic, complemented, Sylow
150	C₄×C₂²⋊Q₈	—	197	1	no	maximal, Sylow
151	C₂×C₂²⋊Q₈	—	197	1	no	—
152	C₂×C₄×Q₈	—	197	1	no	—
153	C₂²×C₄²	D₁₉₇	1	1	yes	characteristic
154	#391	—	197	1	no	—
155	#389	—	197	1	no	—
156	#387	—	197	2	no	—
157	#385	—	197	2	no	—
158	#380	—	197	1	no	—
159	C₄×C₄⋊C₄	—	197	1	no	—
160	C₄×C₄⋊C₄	—	197	2	no	—
161	C₄×C₂²⋊C₄	—	197	2	no	—
162	C₂²×Q₈	—	197	1	no	—
163	C₂³×C₄	C₂×D₁₉₇	1	1	yes	characteristic
164	C₂³×C₄	C₂×D₁₉₇	1	1	yes	characteristic
165	C₂³×C₄	C₂×D₁₉₇	1	1	yes	characteristic
166	C₂²⋊Q₈	—	197	8	no	—
167	C₄×Q₈	—	394	4	no	—
168	C₂×C₄⋊C₄	—	197	2	no	—
169	C₂×C₄⋊C₄	—	197	1	no	—
170	C₂×C₄⋊C₄	—	197	2	no	—
171	C₂×C₄⋊C₄	—	197	1	no	—
172	C₂×C₄⋊C₄	—	197	2	no	—
173	C₂×C₄⋊C₄	—	197	1	no	—
174	C₂×C₂²⋊C₄	—	197	2	no	—
175	C₂×C₂²⋊C₄	—	197	2	no	—
176	C₂×C₄²	—	2	4	no	—
177	C₂×C₄²	—	197	2	no	—
178	C₂×C₄²	—	197	2	no	—
179	C₂×C₄²	C₂×D₁₉₇	1	1	yes	characteristic
180	C₂×C₄²	C₂×D₁₉₇	1	1	yes	characteristic
181	C₂×C₄²	C₂×D₁₉₇	1	1	yes	characteristic
182	C₂×C₄²	C₂×D₁₉₇	1	1	yes	characteristic
183	#95	—	197	2	no	—
184	#95	—	197	2	no	—
185	#95	—	197	2	no	—
186	#95	—	197	2	no	—
187	#95	—	197	2	no	—
188	C₂⁴	C₂²×D₁₉₇	1	1	yes	characteristic
189	C₂×Q₈	—	394	2	no	—
190	C₂×Q₈	—	394	2	no	—
191	C₂×Q₈	—	197	4	no	—
192	C₂²×C₄	—	2	2	no	—
193	C₂²×C₄	—	2	2	no	—
194	C₂²×C₄	—	2	2	no	—
195	C₂²×C₄	—	2	2	no	—
196	C₂²×C₄	—	2	2	no	—
197	C₂²×C₄	—	2	2	no	—
198	C₂²×C₄	—	2	2	no	—
199	C₂²×C₄	—	197	2	no	—
200	C₂²×C₄	—	197	2	no	—
201	C₂²×C₄	—	197	2	no	—
202	C₂²×C₄	—	197	2	no	—
203	C₂²×C₄	C₄×D₁₉₇	1	4	yes	—
204	C₂²×C₄	Dic₃₉₄	1	4	yes	complemented
205	C₂²×C₄	C₂²×D₁₉₇	1	1	yes	characteristic
206	C₂²×C₄	C₂²×D₁₉₇	1	1	yes	characteristic
207	C₂²×C₄	C₂²×D₁₉₇	1	1	yes	characteristic
208	C₂²×C₄	C₂²×D₁₉₇	1	1	yes	characteristic
209	C₂²×C₄	C₂²×D₁₉₇	1	1	yes	characteristic
210	C₂²×C₄	C₂²×D₁₉₇	1	1	yes	characteristic
211	C₄⋊C₄	—	394	4	no	—
212	C₄⋊C₄	—	394	2	no	—
213	C₄⋊C₄	—	394	4	no	—
214	C₄⋊C₄	—	394	2	no	—
215	C₄⋊C₄	—	197	8	no	—
216	C₄⋊C₄	—	197	4	no	—
217	C₂²⋊C₄	—	394	4	no	—
218	C₂²⋊C₄	—	197	8	no	—
219	C₄²	—	2	2	no	—
220	C₄²	—	2	2	no	—
221	C₄²	—	2	2	no	—
222	C₄²	—	394	4	no	—
223	C₄²	—	394	4	no	—
224	C₄²	C₂²⋊D₁₉₇	1	4	yes	—
225	C₂³	—	2	1	no	—
226	C₂³	—	2	1	no	—
227	C₂³	—	2	1	no	—
228	C₂³	—	2	1	no	—
229	C₂³	C₂×C₄×D₁₉₇	1	2	yes	—
230	C₂³	C₂×Dic₃₉₄	1	2	yes	—
231	C₂³	#15691358	1	2	yes	—
232	C₂³	C₂³×D₁₉₇	1	1	yes	characteristic
233	Q₈	—	394	8	no	—
234	C₂×C₄	—	2	8	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	2	1	no	—
237	C₂×C₄	—	2	1	no	—
238	C₂×C₄	—	2	1	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	2	8	no	—
244	C₂×C₄	—	2	8	no	—
245	C₂×C₄	—	394	2	no	—
246	C₂×C₄	—	394	2	no	—
247	C₂×C₄	—	394	2	no	—
248	C₂×C₄	—	394	2	no	—
249	C₂×C₄	—	394	2	no	—
250	C₂×C₄	—	394	2	no	—
251	C₂×C₄	—	394	2	no	—
252	C₂×C₄	—	394	2	no	—
253	C₂×C₄	—	394	2	no	—
254	C₂×C₄	—	394	2	no	—
255	C₂×C₄	—	197	4	no	—
256	C₂×C₄	—	197	4	no	—
257	C₂×C₄	C₂×C₄×D₁₉₇	1	2	yes	—
258	C₂×C₄	C₂×C₄×D₁₉₇	1	2	yes	—
259	C₂×C₄	C₂×C₂²⋊D₁₉₇	1	2	yes	—
260	C₂×C₄	C₂×C₂²⋊D₁₉₇	1	2	yes	—
261	C₂×C₄	C₂×C₂²⋊D₁₉₇	1	2	yes	—
262	C₂×C₄	C₂×Dic₃₉₄	1	2	yes	—
263	C₂×C₄	C₂×Dic₃₉₄	1	2	yes	—
264	C₂×C₄	#15691358	1	2	yes	—
265	C₂×C₄	#15691358	1	2	yes	—
266	C₂×C₄	#15691358	1	2	yes	—
267	C₂²	—	2	2	no	—
268	C₂²	—	2	2	no	—
269	C₂²	—	2	2	no	—
270	C₂²	—	2	2	no	—
271	C₂²	—	2	2	no	—
272	C₂²	—	2	2	no	—
273	C₂²	C₄×Dic₃₉₄	1	4	yes	complemented
274	C₂²	C₂²×C₄×D₁₉₇	1	1	yes	characteristic
275	C₂²	C₂²×C₂²⋊D₁₉₇	1	1	yes	characteristic
276	C₂²	C₂²×Dic₃₉₄	1	1	yes	characteristic
277	C₂²	C₂×#15691358	1	1	yes	characteristic
278	C₂²	C₂×#15691358	1	1	yes	characteristic
279	C₂²	C₂×#15691358	1	1	yes	characteristic
280	C₂²	C₂×#15691358	1	1	yes	characteristic
281	C₄	—	2	2	no	—
282	C₄	—	2	2	no	—
283	C₄	—	2	2	no	—
284	C₄	—	2	2	no	—
285	C₄	—	788	2	no	—
286	C₄	—	788	2	no	—
287	C₄	—	394	4	no	—
288	C₄	—	394	4	no	—
289	C₄	C₄×C₂²⋊D₁₉₇	1	4	yes	—
290	C₄	#20263305	1	4	yes	direct product factor
291	C₂	—	2	4	no	—
292	C₂	C₂×C₄×C₂²⋊D₁₉₇	1	1	yes	characteristic
293	C₂	C₂×C₄×Dic₃₉₄	1	1	yes	characteristic
294	C₂	C₄×#15691358	1	1	yes	characteristic
295	C₂	C₂×#20263305	1	1	yes	characteristic
296	C₂	#27161590	1	1	yes	characteristic
297	C₂	#27161613	1	1	yes	characteristic
298	C₂	#27161636	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²×D₁₉₇
commutator	C₂²×C₁₉₇	C₂³×C₄
Frattini	C₂³	C₂³×D₁₉₇
Fitting	C₂²×C₄²×C₁₉₇	C₂
socle, cosocle	C₂³×C₁₉₇	C₂⁴
prime radical	C₂⁴×C₁₉₇	C₂³
hypercenter	C₂²×C₄²	D₁₉₇
hypocenter	C₁₉₇	C₄×C₂²⋊Q₈
LCS	G ▷ C₂²×C₁₉₇ ▷ C₁₉₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄²

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×Dic₃₉₄	C₂	semidirect
3. #27160985	C₂	semidirect
4. #27160977	C₂	semidirect
5. #27160963	C₂	semidirect
8. C₄×#15691337	C₂	semidirect
9. C₄×C₄⋊Dic₁₉₇	C₂	semidirect
13. #20263305	C₄	direct (N × H)
40. C₂×Dic₃₉₄	C₂×C₄	semidirect
47. C₄⋊Dic₁₉₇	C₂×C₄	semidirect
50. #15691337	C₂×C₄	semidirect
64. C₂²×C₄×C₁₉₇	Q₈	semidirect
123. C₂²×C₁₉₇	C₄×Q₈	semidirect
136. C₄×C₁₉₇	C₂²⋊Q₈	semidirect
149. C₁₉₇	C₄×C₂²⋊Q₈	semidirect
204. C₂²×C₄	Dic₃₉₄	semidirect
273. C₂²	C₄×Dic₃₉₄	semidirect
290. C₄	#20263305	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×#20263305	2
C₂×C₄×Dic₃₉₄	2
#27160985	2
#27160977	2
#27160963	2
#27160952	2
#27160939	2
C₄×#15691337	2
C₄×C₄⋊Dic₁₉₇	2
C₄×C₂²⋊Dic₁₉₇	2
C₂²×C₄²×C₁₉₇	2
C₄×C₂²⋊Q₈	197

Subgroups — C4×#20263305 #38745247

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#20263305 #38745247