Subgroups — C₄×#11758519 #16854530

Order 3968.

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×#11758519	C₂	1	1	yes	maximal, characteristic
2	C₄×#92495	C₂	1	1	yes	maximal, characteristic
3	#14521154	C₂	1	2	yes	maximal, complemented
4	#14521147	C₂	1	2	yes	maximal, complemented
5	C₄×#92482	C₂	1	2	yes	maximal, complemented
6	C₄×#92481	C₂	1	2	yes	maximal, complemented
7	#14521089	C₂	1	1	yes	maximal, characteristic, complemented
8	#14521058	C₂	1	1	yes	maximal, characteristic, complemented
9	#14521048	C₂	1	1	yes	maximal, characteristic
10	C₄×#92474	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₄×Q₈×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×Q₈×C₃₁	C₂²	1	1	yes	characteristic
13	C₂×#92495	C₂²	1	1	yes	characteristic
14	C₄×Q₈×C₃₁	—	2	1	no	—
15	C₄×Q₈×C₃₁	—	2	1	no	—
16	C₄×Q₈×C₃₁	C₂²	1	2	yes	—
17	C₄×Q₈×C₃₁	C₂²	1	2	yes	—
18	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
19	C₂×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
20	C₂×C₄²×C₃₁	—	2	1	no	—
21	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
22	C₂×#92482	C₂²	1	2	yes	—
23	#11758519	C₄	1	8	yes	direct product factor
24	C₂×#92481	C₂²	1	2	yes	—
25	C₄×C₂²⋊D₃₁	—	2	1	no	—
26	C₄×C₂²⋊D₃₁	—	2	1	no	—
27	#11758503	C₂²	1	1	yes	characteristic
28	C₂×#92474	C₂²	1	1	yes	characteristic
29	C₂×#92474	C₂²	1	1	yes	characteristic
30	C₄×D₁₂₄	C₂²	1	1	yes	characteristic
31	C₄×D₁₂₄	C₂²	1	1	yes	characteristic
32	#11758449	—	2	1	no	—
33	C₄²×D₃₁	—	2	1	no	—
34	C₄×Dic₆₂	C₂²	1	1	yes	characteristic
35	C₄×Dic₆₂	C₂²	1	1	yes	characteristic
36	#11758423	C₂²	1	2	yes	—
37	#11758423	C₂²	1	2	yes	—
38	#11758398	C₂²	1	2	yes	—
39	#11758397	C₂²	1	2	yes	—
40	#11758396	C₂²	1	2	yes	—
41	#11758395	C₂²	1	2	yes	—
42	#11758391	C₂²	1	2	yes	—
43	C₄×C₃₁⋊C₈	C₂²	1	2	yes	—
44	C₂×Q₈×C₃₁	—	2	1	no	—
45	C₂×Q₈×C₃₁	—	2	2	no	—
46	C₂×Q₈×C₃₁	C₂×C₄	1	4	yes	complemented
47	C₂×Q₈×C₃₁	C₂³	1	2	yes	—
48	C₂²×C₄×C₃₁	—	2	1	no	—
49	C₂²×C₄×C₃₁	—	2	1	no	—
50	C₂²×C₄×C₃₁	D₄	1	1	yes	characteristic
51	C₂²×C₄×C₃₁	D₄	1	1	yes	characteristic
52	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
53	C₂×C₂²⋊D₃₁	—	2	1	no	—
54	#92495	—	2	2	no	—
55	#92495	C₂×C₄	1	4	yes	—
56	C₂×D₁₂₄	C₂³	1	1	yes	characteristic
57	C₂×C₄×D₃₁	—	2	1	no	—
58	C₂×C₄×D₃₁	—	2	1	no	—
59	C₂×C₄×D₃₁	—	2	1	no	—
60	C₂×Dic₆₂	C₂³	1	1	yes	characteristic
61	C₄⋊C₄×C₃₁	—	2	2	no	—
62	C₄⋊C₄×C₃₁	—	2	2	no	—
63	C₄⋊C₄×C₃₁	—	2	1	no	—
64	C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
65	C₄²×C₃₁	—	2	2	no	—
66	C₄²×C₃₁	—	2	2	no	—
67	C₄²×C₃₁	D₄	1	1	yes	characteristic
68	C₄²×C₃₁	D₄	1	1	yes	characteristic
69	C₄²×C₃₁	C₂³	1	1	yes	characteristic
70	C₄²×C₃₁	C₂³	1	1	yes	characteristic
71	C₂²⋊Dic₃₁	—	2	1	no	—
72	#92482	C₂×C₄	1	8	yes	complemented
73	#92481	C₂×C₄	1	8	yes	complemented
74	(C₂×D₃₁)⋊C₄	—	2	1	no	—
75	(C₂×D₃₁)⋊C₄	—	2	1	no	—
76	C₄⋊Dic₃₁	C₂³	1	1	yes	characteristic
77	C₄⋊Dic₃₁	C₂³	1	1	yes	characteristic
78	Dic₃₁⋊C₄	—	2	1	no	—
79	Dic₃₁⋊C₄	—	2	1	no	—
80	C₄×Dic₃₁	—	2	1	no	—
81	C₄×Dic₃₁	—	2	1	no	—
82	C₄×Dic₃₁	—	2	1	no	—
83	#92474	—	2	2	no	—
84	#92474	C₂×C₄	1	4	yes	complemented
85	C₂×C₃₁⋊C₈	C₂³	1	2	yes	—
86	C₂×C₃₁⋊C₈	C₂³	1	2	yes	—
87	C₂³×C₃₁	C₂×D₄	1	1	yes	characteristic
88	C₂²×D₃₁	—	2	1	no	—
89	Q₈×C₃₁	—	2	4	no	—
90	Q₈×C₃₁	—	2	2	no	—
91	Q₈×C₃₁	C₂²×C₄	1	4	yes	—
92	C₂×C₄×C₃₁	—	2	4	no	—
93	C₂×C₄×C₃₁	—	2	2	no	—
94	C₂×C₄×C₃₁	—	2	2	no	—
95	C₂×C₄×C₃₁	—	2	2	no	—
96	C₂×C₄×C₃₁	—	4	1	no	—
97	C₂×C₄×C₃₁	—	4	2	no	—
98	C₂×C₄×C₃₁	C₂²×C₄	1	2	yes	—
99	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
100	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
101	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
102	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
103	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
104	C₂×C₄×C₃₁	#41	1	2	yes	—
105	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
106	C₂²⋊D₃₁	—	2	4	no	—
107	C₂×Dic₃₁	—	2	1	no	—
108	C₂×Dic₃₁	—	2	1	no	—
109	C₂×Dic₃₁	—	2	1	no	—
110	D₁₂₄	—	2	1	no	—
111	D₁₂₄	C₂²×C₄	1	2	yes	—
112	C₄×D₃₁	—	2	4	no	—
113	C₄×D₃₁	—	4	2	no	—
114	C₄×D₃₁	—	4	2	no	—
115	Dic₆₂	—	2	1	no	—
116	Dic₆₂	C₂²×C₄	1	2	yes	—
117	C₃₁⋊C₈	—	2	2	no	—
118	C₃₁⋊C₈	C₂²×C₄	1	4	yes	—
119	C₄×#137	—	31	1	no	maximal, Sylow
120	C₂²×C₃₁	—	2	1	no	—
121	C₂²×C₃₁	—	2	1	no	—
122	C₂²×C₃₁	C₄×D₄	1	2	yes	—
123	C₂²×C₃₁	C₂²×D₄	1	1	yes	characteristic
124	C₂×D₃₁	—	2	2	no	—
125	C₂×D₃₁	—	4	1	no	—
126	C₂×D₃₁	—	4	1	no	—
127	C₄×C₃₁	—	2	4	no	—
128	C₄×C₃₁	—	2	1	no	—
129	C₄×C₃₁	—	2	1	no	—
130	C₄×C₃₁	—	2	1	no	—
131	C₄×C₃₁	—	4	2	no	—
132	C₄×C₃₁	C₄×D₄	1	2	yes	—
133	C₄×C₃₁	#137	1	2	yes	complemented
134	C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
135	C₄×C₃₁	C₂×#41	1	1	yes	characteristic
136	Dic₃₁	—	2	2	no	—
137	Dic₃₁	—	4	1	no	—
138	Dic₃₁	—	4	1	no	—
139	C₂×#137	—	31	1	no	—
140	C₄×#41	—	31	1	no	—
141	C₂×C₄×Q₈	D₃₁	1	1	yes	characteristic, complemented
142	#441	—	31	2	no	—
143	#440	—	31	2	no	—
144	C₄×Q₁₆	—	31	2	no	—
145	C₄×SD₁₆	—	31	2	no	—
146	#428	—	31	1	no	—
147	#419	—	31	1	no	—
148	#417	—	31	1	no	—
149	C₄×M₁₆	—	31	1	no	—
150	C₂×C₃₁	—	2	2	no	—
151	C₂×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
152	C₂×C₃₁	C₂×#137	1	1	yes	characteristic
153	C₂×C₃₁	#575	1	1	yes	characteristic
154	D₃₁	—	4	2	no	—
155	C₂×#41	—	31	1	no	—
156	C₂²×Q₈	C₂×D₃₁	1	1	yes	characteristic
157	#137	—	31	8	no	—
158	C₂×Q₁₆	—	31	2	no	—
159	C₂×SD₁₆	—	31	2	no	—
160	C₂×M₁₆	—	31	1	no	—
161	C₂×M₁₆	—	31	1	no	—
162	C₄×Q₈	—	2	1	no	—
163	C₄×Q₈	—	2	1	no	—
164	C₄×Q₈	—	31	1	no	—
165	C₄×Q₈	—	31	1	no	—
166	C₄×Q₈	C₂×D₃₁	1	2	yes	—
167	C₄×Q₈	C₂×D₃₁	1	2	yes	—
168	C₄×D₄	—	31	1	no	—
169	C₄×D₄	—	31	1	no	—
170	C₄×D₄	—	62	1	no	—
171	C₄×D₄	—	62	1	no	—
172	#117	—	31	1	no	—
173	#117	—	62	1	no	—
174	C₂×C₄⋊C₄	—	2	1	no	—
175	C₂×C₄⋊C₄	C₂×D₃₁	1	1	yes	characteristic
176	C₂×C₄²	—	2	1	no	—
177	C₂×C₄²	—	62	1	no	—
178	C₂×C₄²	C₂×D₃₁	1	1	yes	characteristic
179	#107	—	31	2	no	—
180	#106	—	31	2	no	—
181	#103	—	31	2	no	—
182	#103	—	31	2	no	—
183	#103	—	31	2	no	—
184	#102	—	31	2	no	—
185	#97	—	31	2	no	—
186	C₄×C₈	—	31	2	no	—
187	C₃₁	C₄×#137	1	1	yes	characteristic, complemented, Sylow
188	#41	—	31	4	no	—
189	#41	—	62	2	no	—
190	C₂×Q₈	—	2	2	no	—
191	C₂×Q₈	—	2	1	no	—
192	C₂×Q₈	—	31	1	no	—
193	C₂×Q₈	C₄×D₃₁	1	4	yes	complemented
194	C₂×Q₈	C₂²×D₃₁	1	2	yes	—
195	C₂×D₄	—	31	1	no	—
196	C₂×D₄	—	62	1	no	—
197	C₂²×C₄	—	2	1	no	—
198	C₂²×C₄	—	2	1	no	—
199	C₂²×C₄	—	62	1	no	—
200	C₂²×C₄	—	62	1	no	—
201	C₂²×C₄	—	62	1	no	—
202	C₂²×C₄	C₂²⋊D₃₁	1	1	yes	characteristic
203	C₂²×C₄	C₂²⋊D₃₁	1	1	yes	characteristic
204	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
205	Q₁₆	—	31	8	no	—
206	SD₁₆	—	31	8	no	—
207	M₁₆	—	31	4	no	—
208	M₁₆	—	62	2	no	—
209	C₂×C₈	—	31	2	no	—
210	C₂×C₈	—	31	2	no	—
211	C₄⋊C₄	—	2	2	no	—
212	C₄⋊C₄	—	2	2	no	—
213	C₄⋊C₄	—	2	1	no	—
214	C₄⋊C₄	—	31	1	no	—
215	C₄⋊C₄	—	31	1	no	—
216	C₄⋊C₄	—	62	1	no	—
217	C₄⋊C₄	—	62	1	no	—
218	C₄⋊C₄	C₂²×D₃₁	1	2	yes	—
219	C₂²⋊C₄	—	62	1	no	—
220	C₂²⋊C₄	—	62	1	no	—
221	C₂²⋊C₄	—	62	1	no	—
222	C₄²	—	2	2	no	—
223	C₄²	—	2	2	no	—
224	C₄²	—	62	1	no	—
225	C₄²	—	62	1	no	—
226	C₄²	—	62	1	no	—
227	C₄²	C₂²⋊D₃₁	1	1	yes	characteristic
228	C₄²	C₂²⋊D₃₁	1	1	yes	characteristic
229	C₄²	C₂²×D₃₁	1	1	yes	characteristic
230	C₄²	C₂²×D₃₁	1	1	yes	characteristic
231	C₂³	—	62	1	no	—
232	C₂³	C₂×C₂²⋊D₃₁	1	1	yes	characteristic
233	Q₈	—	2	4	no	—
234	Q₈	—	2	2	no	—
235	Q₈	—	31	2	no	—
236	Q₈	—	62	1	no	—
237	Q₈	C₂×C₄×D₃₁	1	4	yes	—
238	D₄	—	31	2	no	—
239	D₄	—	62	4	no	—
240	D₄	—	62	1	no	—
241	C₂×C₄	—	2	4	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	4	1	no	—
246	C₂×C₄	—	4	2	no	—
247	C₂×C₄	—	62	1	no	—
248	C₂×C₄	—	62	4	no	—
249	C₂×C₄	—	62	1	no	—
250	C₂×C₄	—	62	1	no	—
251	C₂×C₄	—	124	2	no	—
252	C₂×C₄	—	124	2	no	—
253	C₂×C₄	C₂×C₄×D₃₁	1	2	yes	—
254	C₂×C₄	#92495	1	2	yes	—
255	C₂×C₄	C₂×C₂²⋊D₃₁	1	1	yes	characteristic
256	C₂×C₄	C₂×C₂²⋊D₃₁	1	1	yes	characteristic
257	C₂×C₄	C₂×C₂²⋊D₃₁	1	1	yes	characteristic
258	C₂×C₄	C₂×C₂²⋊D₃₁	1	1	yes	characteristic
259	C₂×C₄	C₂×C₂²⋊D₃₁	1	1	yes	characteristic
260	C₂×C₄	C₂³×D₃₁	1	1	yes	characteristic
261	C₈	—	31	4	no	—
262	C₈	—	62	2	no	—
263	C₂²	—	2	1	no	—
264	C₂²	—	2	1	no	—
265	C₂²	—	62	2	no	—
266	C₂²	—	124	1	no	—
267	C₂²	—	124	1	no	—
268	C₂²	C₄×C₂²⋊D₃₁	1	2	yes	—
269	C₂²	C₂²×C₂²⋊D₃₁	1	1	yes	characteristic
270	C₄	—	2	4	no	—
271	C₄	—	2	1	no	—
272	C₄	—	2	1	no	—
273	C₄	—	2	1	no	—
274	C₄	—	4	2	no	—
275	C₄	—	62	2	no	—
276	C₄	—	124	1	no	—
277	C₄	—	124	1	no	—
278	C₄	C₄×C₂²⋊D₃₁	1	2	yes	—
279	C₄	#11758519	1	2	yes	direct product factor
280	C₄	C₂²×C₄×D₃₁	1	1	yes	characteristic
281	C₄	C₂×#92495	1	1	yes	characteristic
282	C₂	—	2	2	no	—
283	C₂	—	124	2	no	—
284	C₂	C₂×C₄×C₂²⋊D₃₁	1	1	yes	characteristic
285	C₂	C₂×#11758519	1	1	yes	characteristic
286	C₂	#14521847	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₃₁
commutator	C₄×C₃₁	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₃₁
Fitting	C₂×C₄×Q₈×C₃₁	C₂
socle	C₂²×C₃₁	C₂²×D₄
prime radical	C₂×#92495	C₂²
cosocle	C₂×C₄×C₃₁	C₂⁴
hypercenter	C₂×C₄×Q₈	D₃₁
hypocenter	C₃₁	C₄×#137
LCS	G ▷ C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×Q₈

Complements

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

N	H ≅ G/N	type
3. #14521154	C₂	semidirect
4. #14521147	C₂	semidirect
5. C₄×#92482	C₂	semidirect
6. C₄×#92481	C₂	semidirect
7. #14521089	C₂	semidirect
8. #14521058	C₂	semidirect
10. C₄×#92474	C₂	semidirect
11. C₂×C₄×Q₈×C₃₁	C₂	semidirect
23. #11758519	C₄	direct (N × H)
46. C₂×Q₈×C₃₁	C₂×C₄	semidirect
72. #92482	C₂×C₄	semidirect
73. #92481	C₂×C₄	semidirect
84. #92474	C₂×C₄	semidirect
133. C₄×C₃₁	#137	semidirect
141. C₂×C₄×Q₈	D₃₁	semidirect
187. C₃₁	C₄×#137	semidirect
193. C₂×Q₈	C₄×D₃₁	semidirect
279. C₄	#11758519	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×#11758519	2
C₄×#92495	2
#14521154	2
#14521147	2
C₄×#92482	2
C₄×#92481	2
#14521089	2
#14521058	2
#14521048	2
C₄×#92474	2
C₂×C₄×Q₈×C₃₁	2
C₄×#137	31

Subgroups — C4×#11758519 #16854530

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#11758519 #16854530