Subgroups — C₂×C₇×D₄⋊C₁₆ #13188797

Order 1792.

470 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×C₈×D₄	C₂	1	1	yes	maximal, characteristic
2	C₂×C₇×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₄×C₇×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇×D₄⋊C₁₆	C₂	1	4	yes	maximal, direct product factor
5	C₂³×C₇×C₈	—	2	1	no	—
6	C₂×C₄×C₇×D₄	C₄	1	1	yes	characteristic
7	C₂²×C₇×C₁₆	—	2	1	no	—
8	C₂²×C₇×C₁₆	—	2	1	no	—
9	C₇×C₈×D₄	—	2	1	no	—
10	C₇×C₈×D₄	—	2	1	no	—
11	C₇×C₈×D₄	C₄	1	2	yes	—
12	C₇×C₈×D₄	C₂²	1	2	yes	—
13	C₂×C₇×C₄⋊C₈	C₄	1	1	yes	characteristic
14	C₂×C₇×C₂²⋊C₈	—	2	1	no	—
15	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
16	C₇×C₄⋊C₁₆	—	2	1	no	—
17	C₇×C₄⋊C₁₆	C₂²	1	2	yes	complemented
18	C₄×C₇×C₁₆	—	2	1	no	—
19	C₄×C₇×C₁₆	C₂²	1	2	yes	complemented
20	C₂×D₄⋊C₁₆	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
21	C₂²×C₇×D₄	C₈	1	1	yes	characteristic
22	C₂³×C₄×C₇	—	2	1	no	—
23	C₂²×C₇×C₈	—	2	1	no	—
24	C₂²×C₇×C₈	—	2	2	no	—
25	C₂²×C₇×C₈	—	2	2	no	—
26	C₂²×C₇×C₈	—	4	2	no	—
27	C₂²×C₇×C₈	—	4	2	no	—
28	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
29	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
30	C₄×C₇×D₄	—	2	1	no	—
31	C₄×C₇×D₄	—	2	1	no	—
32	C₄×C₇×D₄	C₈	1	2	yes	—
33	C₄×C₇×D₄	C₂×C₄	1	2	yes	—
34	C₂×C₇×C₄⋊C₄	C₈	1	1	yes	characteristic
35	C₂×C₇×C₂²⋊C₄	—	2	1	no	—
36	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
37	C₂×C₇×C₁₆	—	2	4	no	—
38	C₂×C₇×C₁₆	—	2	2	no	—
39	C₂×C₇×C₁₆	—	2	2	no	—
40	C₂×C₇×C₁₆	—	4	2	no	—
41	C₇×C₄⋊C₈	—	2	1	no	—
42	C₇×C₄⋊C₈	C₂×C₄	1	2	yes	—
43	C₇×C₂²⋊C₈	—	2	2	no	—
44	C₇×C₂²⋊C₈	—	2	2	no	—
45	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
46	C₄×C₇×C₈	D₄	1	1	yes	characteristic
47	C₄×C₇×C₈	D₄	1	1	yes	characteristic
48	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
49	C₂×C₈×D₄	C₂×C₇	1	1	yes	characteristic
50	C₂×C₄⋊C₁₆	C₂×C₇	1	1	yes	characteristic, complemented
51	C₂×C₄×C₁₆	C₂×C₇	1	1	yes	characteristic, complemented
52	D₄⋊C₁₆	C₂×C₇	1	4	yes	direct product factor
53	C₂⁴×C₇	—	2	1	no	—
54	C₂×C₇×D₄	—	2	1	no	—
55	C₂×C₇×D₄	—	2	2	no	—
56	C₂×C₇×D₄	C₁₆	1	4	yes	complemented
57	C₂×C₇×D₄	C₂×C₈	1	2	yes	—
58	C₂²×C₄×C₇	—	2	1	no	—
59	C₂²×C₄×C₇	—	2	2	no	—
60	C₂²×C₄×C₇	—	2	2	no	—
61	C₂²×C₄×C₇	—	4	2	no	—
62	C₂²×C₄×C₇	—	4	2	no	—
63	C₂²×C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
64	C₂²×C₄×C₇	C₂×C₈	1	1	yes	characteristic
65	C₂²×C₄×C₇	M₁₆	1	1	yes	characteristic
66	C₂×C₇×C₈	—	2	2	no	—
67	C₂×C₇×C₈	—	2	2	no	—
68	C₂×C₇×C₈	—	4	2	no	—
69	C₂×C₇×C₈	—	4	4	no	—
70	C₂×C₇×C₈	—	4	2	no	—
71	C₂×C₇×C₈	—	4	2	no	—
72	C₂×C₇×C₈	C₂²⋊C₄	1	1	yes	characteristic
73	C₂×C₇×C₈	C₂²⋊C₄	1	1	yes	characteristic
74	C₂×C₇×C₈	D₈	1	2	yes	—
75	C₂×C₇×C₈	SD₁₆	1	2	yes	—
76	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
77	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
78	C₇×C₄⋊C₄	—	2	1	no	—
79	C₇×C₄⋊C₄	C₂×C₈	1	2	yes	—
80	C₇×C₂²⋊C₄	—	2	2	no	—
81	C₇×C₂²⋊C₄	—	2	2	no	—
82	C₄²×C₇	C₂²⋊C₄	1	1	yes	characteristic
83	C₄²×C₇	C₂×C₈	1	1	yes	characteristic
84	C₄²×C₇	M₁₆	1	1	yes	characteristic
85	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
86	C₇×C₁₆	—	2	4	no	—
87	C₇×C₁₆	—	4	2	no	—
88	C₂³×C₈	—	2	1	no	—
89	C₂×C₄×D₄	C₄×C₇	1	1	yes	characteristic
90	C₂²×C₁₆	—	2	1	no	—
91	C₂²×C₁₆	—	2	1	no	—
92	C₈×D₄	—	2	1	no	—
93	C₈×D₄	—	2	1	no	—
94	C₈×D₄	C₄×C₇	1	2	yes	—
95	C₈×D₄	C₂²×C₇	1	2	yes	—
96	C₂×C₄⋊C₈	C₄×C₇	1	1	yes	characteristic
97	C₂×C₂²⋊C₈	—	2	1	no	—
98	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
99	C₄⋊C₁₆	—	2	1	no	—
100	C₄⋊C₁₆	C₂²×C₇	1	2	yes	complemented
101	C₄×C₁₆	—	2	1	no	—
102	C₄×C₁₆	C₂²×C₇	1	2	yes	complemented
103	C₂³×C₇	—	2	4	no	—
104	C₂³×C₇	—	2	2	no	—
105	C₂³×C₇	—	4	2	no	—
106	C₂³×C₇	—	4	2	no	—
107	C₂³×C₇	C₂²⋊C₈	1	1	yes	characteristic
108	C₇×D₄	—	2	4	no	—
109	C₇×D₄	—	2	2	no	—
110	C₇×D₄	C₂×C₁₆	1	4	yes	complemented
111	C₂×C₄×C₇	—	2	2	no	—
112	C₂×C₄×C₇	—	2	2	no	—
113	C₂×C₄×C₇	—	4	2	no	—
114	C₂×C₄×C₇	—	4	4	no	—
115	C₂×C₄×C₇	—	4	2	no	—
116	C₂×C₄×C₇	—	4	2	no	—
117	C₂×C₄×C₇	C₂²⋊C₈	1	1	yes	characteristic
118	C₂×C₄×C₇	C₂²⋊C₈	1	1	yes	characteristic
119	C₂×C₄×C₇	C₂²⋊C₈	1	1	yes	characteristic
120	C₂×C₄×C₇	#102	1	2	yes	—
121	C₂×C₄×C₇	#104	1	2	yes	—
122	C₂×C₄×C₇	C₂×C₁₆	1	2	yes	—
123	C₂×C₄×C₇	M₃₂	1	2	yes	—
124	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
125	C₂×C₄×C₇	C₂²×C₈	1	1	yes	characteristic
126	C₂×C₄×C₇	C₂×M₁₆	1	1	yes	characteristic
127	C₇×C₈	—	2	1	no	—
128	C₇×C₈	—	2	1	no	—
129	C₇×C₈	—	4	2	no	—
130	C₇×C₈	#102	1	2	yes	—
131	C₇×C₈	C₂×D₈	1	1	yes	characteristic
132	C₇×C₈	C₂×SD₁₆	1	1	yes	characteristic
133	C₂²×D₄	C₇×C₈	1	1	yes	characteristic
134	C₂³×C₄	—	2	1	no	—
135	C₂²×C₈	—	2	1	no	—
136	C₂²×C₈	—	2	2	no	—
137	C₂²×C₈	—	2	2	no	—
138	C₂²×C₈	—	4	2	no	—
139	C₂²×C₈	—	4	2	no	—
140	C₂²×C₈	C₇×D₄	1	1	yes	characteristic
141	C₂²×C₈	C₇×D₄	1	1	yes	characteristic
142	C₄×D₄	—	2	1	no	—
143	C₄×D₄	—	2	1	no	—
144	C₄×D₄	C₇×C₈	1	2	yes	—
145	C₄×D₄	C₂×C₄×C₇	1	2	yes	—
146	C₂×C₄⋊C₄	C₇×C₈	1	1	yes	characteristic
147	C₂×C₂²⋊C₄	—	2	1	no	—
148	C₂×C₄²	C₂×C₄×C₇	1	1	yes	characteristic
149	C₂×C₁₆	—	2	4	no	—
150	C₂×C₁₆	—	2	2	no	—
151	C₂×C₁₆	—	2	2	no	—
152	C₂×C₁₆	—	4	2	no	—
153	C₄⋊C₈	—	2	1	no	—
154	C₄⋊C₈	C₂×C₄×C₇	1	2	yes	—
155	C₂²⋊C₈	—	2	2	no	—
156	C₂²⋊C₈	—	2	2	no	—
157	C₄×C₈	C₂×C₄×C₇	1	1	yes	characteristic
158	C₄×C₈	C₇×D₄	1	1	yes	characteristic
159	C₄×C₈	C₇×D₄	1	1	yes	characteristic
160	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
161	C₂²×C₇	—	2	4	no	—
162	C₂²×C₇	—	4	8	no	—
163	C₂²×C₇	—	4	2	no	—
164	C₂²×C₇	—	4	2	no	—
165	C₂²×C₇	D₄⋊C₈	1	2	yes	—
166	C₂²×C₇	C₂²⋊C₁₆	1	2	yes	—
167	C₂²×C₇	#350	1	2	yes	—
168	C₂²×C₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
169	C₄×C₇	—	2	1	no	—
170	C₄×C₇	—	2	1	no	—
171	C₄×C₇	—	4	2	no	—
172	C₄×C₇	D₄⋊C₈	1	2	yes	—
173	C₄×C₇	C₂²⋊C₁₆	1	2	yes	—
174	C₄×C₇	C₂×#102	1	1	yes	characteristic
175	C₄×C₇	C₂×#104	1	1	yes	characteristic
176	C₄×C₇	C₂²×C₁₆	1	1	yes	characteristic
177	C₄×C₇	C₂×M₃₂	1	1	yes	characteristic
178	C₂⁴	—	2	1	no	—
179	C₂×D₄	—	2	1	no	—
180	C₂×D₄	—	2	2	no	—
181	C₂×D₄	C₇×C₁₆	1	4	yes	complemented
182	C₂×D₄	C₂×C₇×C₈	1	2	yes	—
183	C₂²×C₄	—	2	1	no	—
184	C₂²×C₄	—	2	2	no	—
185	C₂²×C₄	—	2	2	no	—
186	C₂²×C₄	—	4	2	no	—
187	C₂²×C₄	—	4	2	no	—
188	C₂²×C₄	C₇×C₂²⋊C₄	1	1	yes	characteristic
189	C₂²×C₄	C₂×C₇×C₈	1	1	yes	characteristic
190	C₂²×C₄	C₇×M₁₆	1	1	yes	characteristic
191	C₂×C₈	—	2	2	no	—
192	C₂×C₈	—	2	2	no	—
193	C₂×C₈	—	4	2	no	—
194	C₂×C₈	—	4	4	no	—
195	C₂×C₈	—	4	2	no	—
196	C₂×C₈	—	4	2	no	—
197	C₂×C₈	C₇×C₂²⋊C₄	1	1	yes	characteristic
198	C₂×C₈	C₇×C₂²⋊C₄	1	1	yes	characteristic
199	C₂×C₈	C₇×D₈	1	2	yes	—
200	C₂×C₈	C₇×SD₁₆	1	2	yes	—
201	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
202	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
203	C₄⋊C₄	—	2	1	no	—
204	C₄⋊C₄	C₂×C₇×C₈	1	2	yes	—
205	C₂²⋊C₄	—	2	2	no	—
206	C₂²⋊C₄	—	2	2	no	—
207	C₄²	C₇×C₂²⋊C₄	1	1	yes	characteristic
208	C₄²	C₂×C₇×C₈	1	1	yes	characteristic
209	C₄²	C₇×M₁₆	1	1	yes	characteristic
210	C₄²	C₂²×C₄×C₇	1	1	yes	characteristic
211	C₁₆	—	2	4	no	—
212	C₁₆	—	4	2	no	—
213	C₂×C₇	—	4	4	no	—
214	C₂×C₇	D₄⋊C₁₆	1	4	yes	direct product factor
215	C₂×C₇	C₂×D₄⋊C₈	1	1	yes	characteristic
216	C₂×C₇	C₂×C₂²⋊C₁₆	1	1	yes	characteristic
217	C₂×C₇	C₂×#350	1	1	yes	characteristic
218	C₂³	—	2	4	no	—
219	C₂³	—	2	2	no	—
220	C₂³	—	4	2	no	—
221	C₂³	—	4	2	no	—
222	C₂³	C₇×C₂²⋊C₈	1	1	yes	characteristic
223	D₄	—	2	2	no	—
224	D₄	—	2	4	no	—
225	D₄	C₂×C₇×C₁₆	1	4	yes	complemented
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	4	2	no	—
229	C₂×C₄	—	4	4	no	—
230	C₂×C₄	—	4	2	no	—
231	C₂×C₄	—	4	2	no	—
232	C₂×C₄	C₇×C₂²⋊C₈	1	1	yes	characteristic
233	C₂×C₄	C₇×C₂²⋊C₈	1	1	yes	characteristic
234	C₂×C₄	C₇×C₂²⋊C₈	1	1	yes	characteristic
235	C₂×C₄	C₇×#102	1	2	yes	—
236	C₂×C₄	C₇×#104	1	2	yes	—
237	C₂×C₄	C₂×C₇×C₁₆	1	2	yes	—
238	C₂×C₄	C₇×M₃₂	1	2	yes	—
239	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	1	yes	characteristic
240	C₂×C₄	C₂²×C₇×C₈	1	1	yes	characteristic
241	C₂×C₄	C₂×C₇×M₁₆	1	1	yes	characteristic
242	C₈	—	2	1	no	—
243	C₈	—	2	1	no	—
244	C₈	—	4	2	no	—
245	C₈	C₇×#102	1	2	yes	—
246	C₈	C₂×C₇×D₈	1	1	yes	characteristic
247	C₈	C₂×C₇×SD₁₆	1	1	yes	characteristic
248	C₇	C₂×D₄⋊C₁₆	1	1	yes	characteristic, direct product factor, Sylow
249	C₂²	—	2	4	no	—
250	C₂²	—	4	2	no	—
251	C₂²	—	4	8	no	—
252	C₂²	—	4	2	no	—
253	C₂²	C₇×D₄⋊C₈	1	2	yes	—
254	C₂²	C₇×C₂²⋊C₁₆	1	2	yes	—
255	C₂²	C₇×#350	1	2	yes	—
256	C₂²	C₂×C₇×C₂²⋊C₈	1	1	yes	characteristic
257	C₄	—	2	1	no	—
258	C₄	—	2	1	no	—
259	C₄	—	4	2	no	—
260	C₄	C₇×D₄⋊C₈	1	2	yes	—
261	C₄	C₇×C₂²⋊C₁₆	1	2	yes	—
262	C₄	C₂×C₇×#102	1	1	yes	characteristic
263	C₄	C₂×C₇×#104	1	1	yes	characteristic
264	C₄	C₂²×C₇×C₁₆	1	1	yes	characteristic
265	C₄	C₂×C₇×M₃₂	1	1	yes	characteristic
266	C₂	—	4	4	no	—
267	C₂	C₇×D₄⋊C₁₆	1	4	yes	direct product factor
268	C₂	C₂×C₇×D₄⋊C₈	1	1	yes	characteristic
269	C₂	C₂×C₇×C₂²⋊C₁₆	1	1	yes	characteristic
270	C₂	C₂×C₇×#350	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₇×C₈	D₄
commutator	C₄	C₂²×C₇×C₁₆
Frattini, cosocle	C₄×C₈	C₂³×C₇
socle	C₂³×C₇	C₂²⋊C₈
prime radical	C₂²×C₇×D₄	C₈
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇×C₈ ◁ C₂×C₄×C₇×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₇×C₄⋊C₁₆	C₂	semidirect
3. C₂×C₄×C₇×C₁₆	C₂	semidirect
4. C₇×D₄⋊C₁₆	C₂	direct (N × H)
17. C₇×C₄⋊C₁₆	C₂²	semidirect
19. C₄×C₇×C₁₆	C₂²	semidirect
20. C₂×D₄⋊C₁₆	C₇	direct (N × H)
50. C₂×C₄⋊C₁₆	C₂×C₇	semidirect
51. C₂×C₄×C₁₆	C₂×C₇	semidirect
52. D₄⋊C₁₆	C₂×C₇	direct (N × H)
56. C₂×C₇×D₄	C₁₆	semidirect
100. C₄⋊C₁₆	C₂²×C₇	semidirect
102. C₄×C₁₆	C₂²×C₇	semidirect
110. C₇×D₄	C₂×C₁₆	semidirect
181. C₂×D₄	C₇×C₁₆	semidirect
214. C₂×C₇	D₄⋊C₁₆	direct (N × H)
225. D₄	C₂×C₇×C₁₆	semidirect
248. C₇	C₂×D₄⋊C₁₆	direct (N × H)
267. C₂	C₇×D₄⋊C₁₆	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×C₇×C₈×D₄	2
C₂×C₇×C₄⋊C₁₆	2
C₂×C₄×C₇×C₁₆	2
C₇×D₄⋊C₁₆	2
C₂×D₄⋊C₁₆	7

Subgroups — C2×C7×D4⋊C16 #13188797

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₇×D₄⋊C₁₆ #13188797