Subgroups — C₂×#11726822 #13507281

Order 1792.

542 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×Dic₁₄	C₂	1	1	yes	maximal, characteristic
2	C₂×C₇×#346	C₂	1	1	yes	maximal, characteristic
3	C₂×Dic₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic
4	C₂×Dic₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic
5	C₂×#89297	C₂	1	1	yes	maximal, characteristic
6	C₂×C₁₆×Dic₇	C₂	1	1	yes	maximal, characteristic
7	C₂×C₄⋊(C₇⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
8	#11726822	C₂	1	8	yes	maximal, direct product factor
9	C₂×C₄×Dic₁₄	C₄	1	1	yes	characteristic
10	C₂²×C₇×C₁₆	C₂²	1	1	yes	characteristic
11	C₂²×C₇×C₁₆	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
13	C₂×Dic₇⋊C₈	C₄	1	1	yes	characteristic
14	C₂×Dic₇⋊C₈	C₄	1	1	yes	characteristic
15	C₂×C₈×Dic₇	C₂²	1	1	yes	characteristic
16	C₂×C₈×Dic₇	C₂²	1	1	yes	characteristic
17	C₂²×C₇⋊C₁₆	C₂²	1	1	yes	characteristic
18	C₂²×C₇⋊C₁₆	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊(C₇⋊C₈)	C₄	1	1	yes	characteristic
20	C₈×Dic₁₄	C₄	1	4	yes	—
21	C₈×Dic₁₄	C₂²	1	4	yes	—
22	C₇×#346	C₂²	1	4	yes	—
23	#89297	C₂²	1	4	yes	—
24	Dic₇⋊C₁₆	C₂²	1	4	yes	—
25	Dic₇⋊C₁₆	C₂²	1	4	yes	—
26	C₁₆×Dic₇	C₂²	1	4	yes	—
27	C₄⋊(C₇⋊C₁₆)	C₂²	1	4	yes	—
28	C₂×#2183	—	7	1	no	maximal, Sylow
29	C₂²×Dic₁₄	C₈	1	1	yes	characteristic
30	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
32	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄⋊Dic₇	C₈	1	1	yes	characteristic
34	C₂×Dic₇⋊C₄	C₈	1	1	yes	characteristic
35	C₂×Dic₇⋊C₄	C₈	1	1	yes	characteristic
36	C₂×C₄×Dic₇	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄×Dic₇	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₇⋊C₈	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₇⋊C₈	C₂×C₄	1	1	yes	characteristic
40	C₄×Dic₁₄	C₈	1	4	yes	—
41	C₄×Dic₁₄	C₂×C₄	1	4	yes	—
42	C₂×C₇×C₁₆	—	2	2	no	—
43	C₂×C₇×C₁₆	Q₈	1	4	yes	complemented
44	C₂×C₇×C₁₆	C₂³	1	2	yes	—
45	C₂×C₇×C₁₆	C₂³	1	2	yes	—
46	C₄×C₇×C₈	C₂×C₄	1	2	yes	—
47	C₄×C₇×C₈	C₂³	1	2	yes	—
48	Dic₇⋊C₈	C₂×C₄	1	4	yes	—
49	Dic₇⋊C₈	C₂×C₄	1	4	yes	—
50	C₈×Dic₇	C₂×C₄	1	2	yes	—
51	C₈×Dic₇	C₂×C₄	1	2	yes	—
52	C₈×Dic₇	C₂³	1	2	yes	—
53	C₈×Dic₇	C₂³	1	2	yes	—
54	C₂×C₇⋊C₁₆	—	2	2	no	—
55	C₂×C₇⋊C₁₆	—	2	2	no	—
56	C₂×C₇⋊C₁₆	C₂³	1	2	yes	—
57	C₂×C₇⋊C₁₆	C₂³	1	2	yes	—
58	C₄⋊(C₇⋊C₈)	C₂×C₄	1	4	yes	—
59	C₂×C₈×Q₈	—	7	1	no	—
60	#2183	—	7	8	no	—
61	C₂×C₄⋊C₁₆	—	7	1	no	—
62	C₂×C₄⋊C₁₆	—	7	1	no	—
63	C₂×C₄⋊C₁₆	—	7	1	no	—
64	C₂×#346	—	7	1	no	—
65	C₂×#346	D₇	1	1	yes	characteristic
66	C₂×C₄×C₁₆	—	7	1	no	—
67	C₂²×C₄×C₇	C₂×C₈	1	1	yes	characteristic
68	C₂²×C₄×C₇	C₂×C₈	1	1	yes	characteristic
69	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
70	C₂²×Dic₇	C₂×C₈	1	1	yes	characteristic
71	C₂²×Dic₇	C₂×C₈	1	1	yes	characteristic
72	C₂²×Dic₇	C₂×C₈	1	1	yes	characteristic
73	C₂²×Dic₇	C₂×C₈	1	1	yes	characteristic
74	C₂×Dic₁₄	—	2	4	no	—
75	C₂×Dic₁₄	C₂×C₈	1	4	yes	—
76	C₂×C₇×C₈	—	2	2	no	—
77	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
78	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₇×C₈	C₂×Q₈	1	2	yes	—
80	C₂×C₇×C₈	#41	1	2	yes	—
81	C₂×C₇×C₈	C₂⁴	1	1	yes	characteristic
82	C₄²×C₇	C₂×C₈	1	2	yes	—
83	C₄²×C₇	C₂²×C₄	1	2	yes	—
84	C₄⋊Dic₇	C₂×C₈	1	4	yes	—
85	Dic₇⋊C₄	C₂×C₈	1	4	yes	—
86	Dic₇⋊C₄	C₂×C₈	1	4	yes	—
87	C₄×Dic₇	C₂×C₈	1	2	yes	—
88	C₄×Dic₇	C₂×C₈	1	2	yes	—
89	C₄×Dic₇	C₂²×C₄	1	2	yes	—
90	C₄×Dic₇	C₂²×C₄	1	2	yes	—
91	C₂×C₇⋊C₈	—	2	2	no	—
92	C₂×C₇⋊C₈	—	2	2	no	—
93	C₂×C₇⋊C₈	C₂²×C₄	1	2	yes	—
94	C₂×C₇⋊C₈	C₂²×C₄	1	2	yes	—
95	C₇×C₁₆	—	2	2	no	—
96	C₇×C₁₆	C₂×Q₈	1	4	yes	complemented
97	C₇⋊C₁₆	—	2	2	no	—
98	C₇⋊C₁₆	—	2	2	no	—
99	C₂×C₄×Q₈	—	7	1	no	—
100	C₂²×C₁₆	—	7	1	no	—
101	C₂²×C₁₆	—	7	1	no	—
102	C₂²×C₁₆	C₂×D₇	1	1	yes	characteristic
103	C₂²×C₁₆	C₂×D₇	1	1	yes	characteristic
104	C₈×Q₈	—	7	4	no	—
105	C₈×Q₈	—	7	4	no	—
106	C₂×C₄⋊C₈	—	7	1	no	—
107	C₂×C₄⋊C₈	—	7	1	no	—
108	C₂×C₄⋊C₈	—	7	1	no	—
109	C₂×C₄×C₈	—	7	1	no	—
110	C₂×C₄×C₈	—	7	1	no	—
111	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
112	C₄⋊C₁₆	—	7	4	no	—
113	C₄⋊C₁₆	—	7	4	no	—
114	C₄⋊C₁₆	—	7	4	no	—
115	#346	—	7	4	no	—
116	#346	C₂×D₇	1	4	yes	—
117	C₄×C₁₆	—	7	4	no	—
118	C₂³×C₇	C₂²×C₈	1	1	yes	characteristic
119	C₂×C₄×C₇	—	2	2	no	—
120	C₂×C₄×C₇	—	2	2	no	—
121	C₂×C₄×C₇	C₄×Q₈	1	2	yes	—
122	C₂×C₄×C₇	C₂²×C₈	1	2	yes	—
123	C₂×C₄×C₇	C₂²×C₈	1	2	yes	—
124	C₂×C₄×C₇	C₂²×C₈	1	1	yes	characteristic
125	C₂×C₄×C₇	#131	1	2	yes	—
126	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
127	C₂×Dic₇	—	2	2	no	—
128	C₂×Dic₇	—	2	2	no	—
129	C₂×Dic₇	—	2	2	no	—
130	C₂×Dic₇	M₃₂	1	4	yes	—
131	C₂×Dic₇	C₂²×C₈	1	2	yes	—
132	C₂×Dic₇	C₂²×C₈	1	2	yes	—
133	C₂×Dic₇	C₂²×C₈	1	2	yes	—
134	C₂×Dic₇	C₂²×C₈	1	2	yes	—
135	Dic₁₄	—	2	8	no	—
136	C₇×C₈	—	2	2	no	—
137	C₇×C₈	C₄×Q₈	1	2	yes	—
138	C₇×C₈	C₂²×Q₈	1	1	yes	characteristic
139	C₇×C₈	C₂×#41	1	1	yes	characteristic
140	C₇⋊C₈	—	2	2	no	—
141	C₇⋊C₈	—	2	2	no	—
142	C₂²×Q₈	—	7	1	no	—
143	C₂²×C₈	—	7	1	no	—
144	C₂²×C₈	—	7	1	no	—
145	C₂²×C₈	C₄×D₇	1	1	yes	characteristic
146	C₂²×C₈	C₂²×D₇	1	1	yes	characteristic
147	C₄×Q₈	—	7	4	no	—
148	C₄×Q₈	—	7	4	no	—
149	C₂×C₄⋊C₄	—	7	1	no	—
150	C₂×C₄⋊C₄	—	7	1	no	—
151	C₂×C₄⋊C₄	—	7	1	no	—
152	C₂×C₄²	—	7	1	no	—
153	C₂×C₄²	—	7	1	no	—
154	C₂×C₄²	C₄×D₇	1	1	yes	characteristic
155	C₂×C₁₆	—	2	2	no	—
156	C₂×C₁₆	—	7	2	no	—
157	C₂×C₁₆	—	7	2	no	—
158	C₂×C₁₆	—	14	2	no	—
159	C₂×C₁₆	—	14	2	no	—
160	C₂×C₁₆	Dic₁₄	1	4	yes	complemented
161	C₂×C₁₆	C₂²×D₇	1	2	yes	—
162	C₂×C₁₆	C₂²×D₇	1	2	yes	—
163	C₄⋊C₈	—	7	4	no	—
164	C₄⋊C₈	—	7	4	no	—
165	C₄⋊C₈	—	7	4	no	—
166	C₄×C₈	—	7	2	no	—
167	C₄×C₈	—	7	2	no	—
168	C₄×C₈	—	7	2	no	—
169	C₄×C₈	—	7	2	no	—
170	C₄×C₈	C₄×D₇	1	2	yes	—
171	C₄×C₈	C₂²×D₇	1	2	yes	—
172	C₂²×C₇	C₈×Q₈	1	2	yes	—
173	C₂²×C₇	C₂×M₃₂	1	2	yes	—
174	C₂²×C₇	#504	1	2	yes	—
175	C₂²×C₇	C₂³×C₈	1	1	yes	characteristic
176	C₄×C₇	—	2	2	no	—
177	C₄×C₇	—	2	2	no	—
178	C₄×C₇	C₈×Q₈	1	2	yes	—
179	C₄×C₇	C₂×C₄×Q₈	1	1	yes	characteristic
180	C₄×C₇	C₂×#131	1	1	yes	characteristic
181	Dic₇	—	2	2	no	—
182	Dic₇	—	2	2	no	—
183	Dic₇	—	2	2	no	—
184	Dic₇	C₂×M₃₂	1	4	yes	—
185	C₂×Q₈	—	7	4	no	—
186	C₂×Q₈	—	14	4	no	—
187	C₂²×C₄	—	7	1	no	—
188	C₂²×C₄	—	7	1	no	—
189	C₂²×C₄	—	7	1	no	—
190	C₂²×C₄	—	7	1	no	—
191	C₂²×C₄	C₈×D₇	1	1	yes	characteristic
192	C₂²×C₄	C₈×D₇	1	1	yes	characteristic
193	C₂²×C₄	C₂×C₄×D₇	1	1	yes	characteristic
194	C₂×C₈	—	2	2	no	—
195	C₂×C₈	—	7	2	no	—
196	C₂×C₈	—	7	2	no	—
197	C₂×C₈	—	14	2	no	—
198	C₂×C₈	—	14	2	no	—
199	C₂×C₈	C₂×Dic₁₄	1	2	yes	—
200	C₂×C₈	C₂×C₄×D₇	1	2	yes	—
201	C₂×C₈	C₂×C₄×D₇	1	1	yes	characteristic
202	C₂×C₈	#1157	1	2	yes	—
203	C₂×C₈	C₂³×D₇	1	1	yes	characteristic
204	C₄⋊C₄	—	7	4	no	—
205	C₄⋊C₄	—	7	4	no	—
206	C₄⋊C₄	—	7	4	no	—
207	C₄²	—	7	2	no	—
208	C₄²	—	7	2	no	—
209	C₄²	—	7	2	no	—
210	C₄²	—	7	2	no	—
211	C₄²	C₈×D₇	1	2	yes	—
212	C₄²	C₂×C₄×D₇	1	2	yes	—
213	C₁₆	—	2	2	no	—
214	C₁₆	—	14	2	no	—
215	C₁₆	—	14	2	no	—
216	C₁₆	C₂×Dic₁₄	1	4	yes	complemented
217	C₂×C₇	#2183	1	4	yes	complemented
218	C₂×C₇	C₂×C₈×Q₈	1	1	yes	characteristic
219	C₂×C₇	C₂²×M₃₂	1	1	yes	characteristic
220	C₂×C₇	C₂×#504	1	1	yes	characteristic
221	C₂³	C₂×C₈×D₇	1	1	yes	characteristic
222	Q₈	—	14	8	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	7	2	no	—
226	C₂×C₄	—	7	4	no	—
227	C₂×C₄	—	7	2	no	—
228	C₂×C₄	—	7	2	no	—
229	C₂×C₄	—	7	2	no	—
230	C₂×C₄	—	14	2	no	—
231	C₂×C₄	—	14	2	no	—
232	C₂×C₄	—	14	2	no	—
233	C₂×C₄	C₄×Dic₁₄	1	2	yes	—
234	C₂×C₄	C₂×C₈×D₇	1	2	yes	—
235	C₂×C₄	C₂×C₈×D₇	1	2	yes	—
236	C₂×C₄	C₂×C₈×D₇	1	1	yes	characteristic
237	C₂×C₄	#6464	1	2	yes	—
238	C₂×C₄	C₂²×C₄×D₇	1	1	yes	characteristic
239	C₈	—	2	2	no	—
240	C₈	—	14	2	no	—
241	C₈	—	14	2	no	—
242	C₈	C₄×Dic₁₄	1	2	yes	—
243	C₈	C₂²×Dic₁₄	1	1	yes	characteristic
244	C₈	C₂×#1157	1	1	yes	characteristic
245	C₇	C₂×#2183	1	1	yes	characteristic, complemented, Sylow
246	C₂²	C₈×Dic₁₄	1	2	yes	—
247	C₂²	D₇×M₃₂	1	2	yes	—
248	C₂²	#89679	1	2	yes	—
249	C₂²	C₂²×C₈×D₇	1	1	yes	characteristic
250	C₄	—	2	2	no	—
251	C₄	—	2	2	no	—
252	C₄	—	7	4	no	—
253	C₄	—	14	2	no	—
254	C₄	—	14	2	no	—
255	C₄	—	14	2	no	—
256	C₄	C₈×Dic₁₄	1	2	yes	—
257	C₄	C₂×C₄×Dic₁₄	1	1	yes	characteristic
258	C₄	C₂×#6464	1	1	yes	characteristic
259	C₂	#11726822	1	4	yes	direct product factor
260	C₂	C₂×C₈×Dic₁₄	1	1	yes	characteristic
261	C₂	C₂×D₇×M₃₂	1	1	yes	characteristic
262	C₂	C₂×#89679	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₈	C₂²×D₇
commutator	C₂²×C₇	C₂³×C₈
Frattini	C₂×C₈	C₂³×D₇
Fitting	C₂×C₇×#346	C₂
socle, prime radical	C₂³×C₇	C₂²×C₈
cosocle	C₂×C₇×C₈	C₂⁴
hypercenter	C₂×#346	D₇
hypocenter	C₇	C₂×#2183
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×#346

Complements

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

N	H ≅ G/N	type
8. #11726822	C₂	direct (N × H)
43. C₂×C₇×C₁₆	Q₈	semidirect
96. C₇×C₁₆	C₂×Q₈	semidirect
160. C₂×C₁₆	Dic₁₄	semidirect
216. C₁₆	C₂×Dic₁₄	semidirect
217. C₂×C₇	#2183	semidirect
245. C₇	C₂×#2183	semidirect
259. C₂	#11726822	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×C₈×Dic₁₄	2
C₂×C₇×#346	2
C₂×Dic₇⋊C₁₆	2
C₂×#89297	2
C₂×C₁₆×Dic₇	2
C₂×C₄⋊(C₇⋊C₁₆)	2
#11726822	2
C₂×#2183	7

Subgroups — C2×#11726822 #13507281

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11726822 #13507281