Subgroups — C₂×#11728670 #13508046

Order 1792.

702 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₇×Q₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂²×#6403	C₂	1	2	yes	maximal
3	C₂×#89471	C₂	1	1	yes	maximal, characteristic
4	#11728670	C₂	1	8	yes	maximal, direct product factor
5	C₂×#89287	C₂	1	2	yes	maximal
6	C₂×C₄⋊(C₇⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
7	C₂²×C₇×Q₁₆	C₂²	1	1	yes	characteristic
8	C₂×C₄×C₇×Q₈	—	2	1	no	—
9	C₂²×Dic₂₈	—	2	1	no	—
10	C₂²×Dic₂₈	C₂²	1	2	yes	—
11	C₂×#6432	—	2	1	no	—
12	C₄×C₇×Q₁₆	—	2	2	no	—
13	C₄×C₇×Q₁₆	C₂²	1	4	yes	—
14	C₂×C₇×#107	C₂²	1	1	yes	characteristic
15	C₂×C₇×#103	—	2	1	no	—
16	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
17	C₂×#6403	—	2	8	no	—
18	C₂×#6403	C₂²	1	8	yes	—
19	C₂²×C₇⋊C₁₆	C₂²	1	2	yes	—
20	#89471	—	2	2	no	—
21	#89471	C₂²	1	4	yes	—
22	#89287	C₂²	1	8	yes	—
23	C₄⋊(C₇⋊C₁₆)	C₂²	1	4	yes	—
24	C₂×#2209	—	7	1	no	maximal, Sylow
25	C₂²×C₇×Q₈	—	2	1	no	—
26	C₂²×Dic₁₄	—	2	2	no	—
27	C₂×C₇×Q₁₆	—	2	1	no	—
28	C₂×C₇×Q₁₆	—	2	2	no	—
29	C₂×C₇×Q₁₆	D₄	1	4	yes	—
30	C₂×C₇×Q₁₆	C₂³	1	2	yes	—
31	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
32	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
33	C₄×C₇×Q₈	—	2	4	no	—
34	C₄×C₇×Q₈	—	4	2	no	—
35	C₂×C₇×C₄⋊C₄	—	2	1	no	—
36	C₂×C₇×C₄⋊C₄	—	4	1	no	—
37	C₂×C₄²×C₇	—	4	1	no	—
38	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
39	C₂×C₄⋊Dic₇	—	4	1	no	—
40	C₂×C₄⋊Dic₇	—	4	1	no	—
41	C₂×Dic₂₈	—	2	4	no	—
42	C₂×Dic₂₈	—	2	4	no	—
43	C₂×Dic₂₈	—	2	4	no	—
44	C₂×Dic₂₈	—	2	2	no	—
45	C₂×Dic₂₈	—	4	4	no	—
46	C₂×Dic₂₈	C₂³	1	4	yes	—
47	#6432	—	2	4	no	—
48	#6432	—	4	2	no	—
49	C₇×#107	—	2	1	no	—
50	C₇×#107	C₂³	1	2	yes	—
51	C₇×#103	—	2	4	no	—
52	C₄×C₇×C₈	D₄	1	2	yes	—
53	C₄×C₇×C₈	C₂³	1	2	yes	—
54	#6403	—	2	16	no	—
55	C₂×C₇⋊C₁₆	—	2	4	no	—
56	C₂×C₇⋊C₁₆	C₂³	1	4	yes	—
57	C₂²×Q₃₂	—	7	2	no	—
58	C₂×#494	—	7	1	no	—
59	C₂×C₄×Q₁₆	D₇	1	1	yes	characteristic
60	#2209	—	7	8	no	—
61	C₂×C₄⋊C₁₆	—	7	1	no	—
62	C₂×#358	—	7	2	no	—
63	C₂×C₇×Q₈	—	2	4	no	—
64	C₂×C₇×Q₈	—	2	2	no	—
65	C₂×C₇×Q₈	—	4	1	no	—
66	C₂×C₇×Q₈	—	4	2	no	—
67	C₂²×C₄×C₇	—	4	1	no	—
68	C₂²×C₄×C₇	—	4	1	no	—
69	C₂²×C₄×C₇	D₈	1	1	yes	characteristic
70	C₂²×C₄×C₇	D₈	1	1	yes	characteristic
71	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
72	C₂²×Dic₇	—	4	2	no	—
73	C₂×Dic₁₄	—	2	4	no	—
74	C₂×Dic₁₄	—	4	4	no	—
75	C₂×Dic₁₄	—	4	4	no	—
76	C₂×Dic₁₄	—	4	2	no	—
77	C₇×Q₁₆	—	2	4	no	—
78	C₇×Q₁₆	—	2	2	no	—
79	C₇×Q₁₆	C₂×D₄	1	4	yes	—
80	C₂×C₇×C₈	—	2	2	no	—
81	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
82	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
83	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
84	C₂×C₇×C₈	#41	1	2	yes	—
85	C₂×C₇×C₈	C₂⁴	1	1	yes	characteristic
86	C₇×C₄⋊C₄	—	2	2	no	—
87	C₇×C₄⋊C₄	—	4	1	no	—
88	C₇×C₄⋊C₄	—	4	4	no	—
89	C₄²×C₇	—	4	4	no	—
90	C₄²×C₇	D₈	1	2	yes	—
91	C₄²×C₇	C₂×D₄	1	2	yes	—
92	C₄⋊Dic₇	—	4	4	no	—
93	C₄⋊Dic₇	—	4	4	no	—
94	Dic₂₈	—	2	4	no	—
95	Dic₂₈	—	2	4	no	—
96	Dic₂₈	—	4	8	no	—
97	Dic₂₈	—	4	4	no	—
98	C₇⋊C₁₆	—	2	4	no	—
99	C₂²×Q₁₆	—	7	2	no	—
100	C₂²×Q₁₆	—	14	1	no	—
101	C₂²×Q₁₆	C₂×D₇	1	1	yes	characteristic
102	C₂×C₄⋊Q₈	—	14	1	no	—
103	C₂×C₄×Q₈	—	2	1	no	—
104	C₂×Q₃₂	—	7	8	no	—
105	C₂×Q₃₂	—	14	8	no	—
106	C₂²×C₁₆	—	7	2	no	—
107	#494	—	7	4	no	—
108	#494	—	14	2	no	—
109	C₄×Q₁₆	—	2	2	no	—
110	C₄×Q₁₆	C₂×D₇	1	4	yes	—
111	C₂×#107	C₂×D₇	1	1	yes	characteristic
112	C₂×#103	—	2	1	no	—
113	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
114	C₄⋊C₁₆	—	7	4	no	—
115	#358	—	7	8	no	—
116	C₂³×C₇	C₂×D₈	1	1	yes	characteristic
117	C₇×Q₈	—	2	4	no	—
118	C₇×Q₈	—	4	2	no	—
119	C₇×Q₈	—	4	4	no	—
120	C₂×C₄×C₇	—	2	2	no	—
121	C₂×C₄×C₇	—	4	2	no	—
122	C₂×C₄×C₇	—	4	4	no	—
123	C₂×C₄×C₇	—	4	2	no	—
124	C₂×C₄×C₇	—	8	2	no	—
125	C₂×C₄×C₇	Q₃₂	1	4	yes	complemented
126	C₂×C₄×C₇	#121	1	2	yes	—
127	C₂×C₄×C₇	C₂×D₈	1	2	yes	—
128	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
129	C₂×C₄×C₇	C₂×D₈	1	2	yes	—
130	C₂×C₄×C₇	AGL(1,ℤ/8)	1	2	yes	—
131	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
132	C₂×Dic₇	—	4	4	no	—
133	C₂×Dic₇	—	8	4	no	—
134	Dic₁₄	—	4	4	no	—
135	Dic₁₄	—	4	4	no	—
136	Dic₁₄	—	8	4	no	—
137	C₇×C₈	—	2	2	no	—
138	C₇×C₈	#121	1	2	yes	—
139	C₇×C₈	C₂²×D₄	1	1	yes	characteristic
140	C₇×C₈	C₂×#41	1	1	yes	characteristic
141	C₂²×Q₈	—	2	1	no	—
142	C₂²×Q₈	—	14	2	no	—
143	C₂×Q₁₆	—	2	2	no	—
144	C₂×Q₁₆	—	2	1	no	—
145	C₂×Q₁₆	—	7	4	no	—
146	C₂×Q₁₆	—	14	4	no	—
147	C₂×Q₁₆	—	14	4	no	—
148	C₂×Q₁₆	—	14	2	no	—
149	C₂×Q₁₆	—	14	4	no	—
150	C₂×Q₁₆	—	28	4	no	—
151	C₂×Q₁₆	D₂₈	1	4	yes	—
152	C₂×Q₁₆	C₂²×D₇	1	2	yes	—
153	C₂²×C₈	C₂²⋊D₇	1	1	yes	characteristic
154	C₂²×C₈	C₂²×D₇	1	1	yes	characteristic
155	C₄⋊Q₈	—	14	4	no	—
156	C₄⋊Q₈	—	28	2	no	—
157	C₄×Q₈	—	2	4	no	—
158	C₄×Q₈	—	4	2	no	—
159	C₂×C₄⋊C₄	—	2	1	no	—
160	C₂×C₄⋊C₄	—	4	1	no	—
161	C₂×C₄⋊C₄	—	28	1	no	—
162	C₂×C₄⋊C₄	—	28	1	no	—
163	C₂×C₄²	—	4	1	no	—
164	C₂×C₄²	C₂²⋊D₇	1	1	yes	characteristic
165	Q₃₂	—	14	16	no	—
166	C₂×C₁₆	—	7	4	no	—
167	C₂×C₁₆	—	14	4	no	—
168	#107	—	2	1	no	—
169	#107	C₂²×D₇	1	2	yes	—
170	#103	—	2	4	no	—
171	C₄×C₈	C₂²⋊D₇	1	2	yes	—
172	C₄×C₈	C₂²×D₇	1	2	yes	—
173	C₂²×C₇	#459	1	2	yes	—
174	C₂²×C₇	C₂×Q₃₂	1	2	yes	—
175	C₂²×C₇	#510	1	2	yes	—
176	C₂²×C₇	C₂²×D₈	1	1	yes	characteristic
177	C₄×C₇	—	2	2	no	—
178	C₄×C₇	—	4	4	no	—
179	C₄×C₇	—	8	2	no	—
180	C₄×C₇	#459	1	2	yes	—
181	C₄×C₇	C₂×Q₃₂	1	4	yes	complemented
182	C₄×C₇	C₂×#121	1	1	yes	characteristic
183	C₄×C₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
184	Dic₇	—	8	4	no	—
185	C₂×Q₈	—	2	4	no	—
186	C₂×Q₈	—	2	2	no	—
187	C₂×Q₈	—	4	1	no	—
188	C₂×Q₈	—	4	2	no	—
189	C₂×Q₈	—	14	4	no	—
190	C₂×Q₈	—	28	2	no	—
191	C₂×Q₈	—	28	4	no	—
192	C₂×Q₈	—	28	4	no	—
193	C₂²×C₄	—	4	1	no	—
194	C₂²×C₄	—	4	1	no	—
195	C₂²×C₄	—	28	2	no	—
196	C₂²×C₄	#1141	1	1	yes	characteristic
197	C₂²×C₄	#1141	1	1	yes	characteristic
198	C₂²×C₄	C₂×C₂²⋊D₇	1	1	yes	characteristic
199	Q₁₆	—	2	4	no	—
200	Q₁₆	—	2	2	no	—
201	Q₁₆	—	14	4	no	—
202	Q₁₆	—	14	4	no	—
203	Q₁₆	—	28	8	no	—
204	Q₁₆	—	28	4	no	—
205	Q₁₆	C₂×D₂₈	1	4	yes	—
206	C₂×C₈	—	2	2	no	—
207	C₂×C₈	C₂×D₂₈	1	2	yes	—
208	C₂×C₈	#1157	1	2	yes	—
209	C₂×C₈	C₂×C₂²⋊D₇	1	2	yes	—
210	C₂×C₈	C₂×C₂²⋊D₇	1	1	yes	characteristic
211	C₂×C₈	C₂³×D₇	1	1	yes	characteristic
212	C₄⋊C₄	—	2	2	no	—
213	C₄⋊C₄	—	4	1	no	—
214	C₄⋊C₄	—	4	4	no	—
215	C₄⋊C₄	—	28	4	no	—
216	C₄⋊C₄	—	28	4	no	—
217	C₄²	—	4	4	no	—
218	C₄²	#1141	1	2	yes	—
219	C₄²	C₂×C₂²⋊D₇	1	2	yes	—
220	C₁₆	—	14	4	no	—
221	C₂×C₇	#2209	1	4	yes	complemented
222	C₂×C₇	C₂×#459	1	1	yes	characteristic
223	C₂×C₇	C₂²×Q₃₂	1	1	yes	characteristic
224	C₂×C₇	C₂×#510	1	1	yes	characteristic
225	C₂³	C₂×#1141	1	1	yes	characteristic
226	Q₈	—	2	4	no	—
227	Q₈	—	4	2	no	—
228	Q₈	—	4	4	no	—
229	Q₈	—	28	4	no	—
230	Q₈	—	28	4	no	—
231	Q₈	—	56	4	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	4	2	no	—
234	C₂×C₄	—	4	4	no	—
235	C₂×C₄	—	4	2	no	—
236	C₂×C₄	—	8	2	no	—
237	C₂×C₄	—	28	4	no	—
238	C₂×C₄	—	56	4	no	—
239	C₂×C₄	#6403	1	4	yes	complemented
240	C₂×C₄	#6493	1	2	yes	—
241	C₂×C₄	C₂×#1141	1	2	yes	—
242	C₂×C₄	C₂×#1141	1	2	yes	—
243	C₂×C₄	C₂×#1141	1	1	yes	characteristic
244	C₂×C₄	#6512	1	2	yes	—
245	C₂×C₄	C₂²×C₂²⋊D₇	1	1	yes	characteristic
246	C₈	—	2	2	no	—
247	C₈	#6493	1	2	yes	—
248	C₈	C₂²×D₂₈	1	1	yes	characteristic
249	C₈	C₂×#1157	1	1	yes	characteristic
250	C₇	C₂×#2209	1	1	yes	characteristic, complemented, Sylow
251	C₂²	#89787	1	2	yes	—
252	C₂²	C₂×#6403	1	2	yes	—
253	C₂²	#89967	1	2	yes	—
254	C₂²	C₂²×#1141	1	1	yes	characteristic
255	C₄	—	2	2	no	—
256	C₄	—	4	4	no	—
257	C₄	—	8	2	no	—
258	C₄	—	56	4	no	—
259	C₄	#89787	1	2	yes	—
260	C₄	C₂×#6403	1	4	yes	complemented
261	C₄	C₂×#6493	1	1	yes	characteristic
262	C₄	C₂×#6512	1	1	yes	characteristic
263	C₂	#11728670	1	4	yes	direct product factor
264	C₂	C₂×#89787	1	1	yes	characteristic
265	C₂	C₂²×#6403	1	1	yes	characteristic
266	C₂	C₂×#89967	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#1141
commutator, cosocle	C₂×C₇×C₈	C₂⁴
Frattini	C₂×C₈	C₂³×D₇
Fitting	C₂×C₄×C₇×Q₁₆	C₂
socle, prime radical	C₂³×C₇	C₂×D₈
hypercenter	C₂×C₄×Q₁₆	D₇
hypocenter	C₇	C₂×#2209
LCS	G ▷ C₂×C₇×C₈ ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×Q₁₆

Complements

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

N	H ≅ G/N	type
4. #11728670	C₂	direct (N × H)
125. C₂×C₄×C₇	Q₃₂	semidirect
181. C₄×C₇	C₂×Q₃₂	semidirect
221. C₂×C₇	#2209	semidirect
239. C₂×C₄	#6403	semidirect
250. C₇	C₂×#2209	semidirect
260. C₄	C₂×#6403	semidirect
263. C₂	#11728670	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×C₄×C₇×Q₁₆	2
C₂²×#6403	2
C₂×#89471	2
#11728670	2
C₂×#89287	2
C₂×C₄⋊(C₇⋊C₁₆)	2
C₂×#2209	7

Subgroups — C2×#11728670 #13508046

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11728670 #13508046