Subgroups — #13246926

Order 1792.

394 conjugacy classes of subgroups.

The subgroup lattice has 266 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	C₇×#1564	C₂	1	1	yes	maximal, characteristic
3	#11728524	C₂	1	2	yes	maximal
4	#11727546	C₂	1	1	yes	maximal, characteristic
5	#11725551	C₂	1	2	yes	maximal
6	C₂×C₄×Dic₁₄	C₂²	1	1	yes	characteristic
7	C₇×#431	C₂²	1	2	yes	—
8	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
9	C₂×#6393	—	2	1	no	—
10	C₂×#6393	—	2	1	no	—
11	C₂×Dic₇⋊C₈	—	2	1	no	—
12	C₂×C₈×Dic₇	—	2	1	no	—
13	#89696	C₂²	1	2	yes	—
14	C₂×C₄⋊(C₇⋊C₈)	C₂²	1	1	yes	characteristic
15	C₈×Dic₁₄	—	2	1	no	—
16	C₈×Dic₁₄	—	2	1	no	—
17	C₈×Dic₁₄	C₄	1	2	yes	—
18	C₈×Dic₁₄	C₄	1	2	yes	—
19	C₇×#335	—	2	1	no	—
20	C₇×#334	—	2	1	no	—
21	#89251	—	2	4	no	—
22	#8659	—	7	1	no	maximal, Sylow
23	C₂²×Dic₁₄	Q₈	1	1	yes	characteristic
24	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
25	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
26	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
27	C₂×C₄⋊Dic₇	Q₈	1	1	yes	characteristic
28	C₂×Dic₇⋊C₄	—	2	1	no	—
29	C₂×C₄×Dic₇	—	2	1	no	—
30	C₂×#1136	—	2	2	no	—
31	C₂²×C₇⋊C₈	—	2	1	no	—
32	C₄×Dic₁₄	—	2	1	no	—
33	C₄×Dic₁₄	—	2	1	no	—
34	C₄×Dic₁₄	C₂×C₄	1	2	yes	—
35	C₄×Dic₁₄	D₄	1	2	yes	—
36	C₇×C₄⋊C₈	—	2	2	no	—
37	C₇×C₂²⋊C₈	—	2	2	no	—
38	C₄×C₇×C₈	—	2	2	no	—
39	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
40	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
41	C₄×C₇×C₈	D₄	1	1	yes	characteristic
42	C₄×C₇×C₈	D₄	1	1	yes	characteristic
43	#6393	—	4	2	no	—
44	#6393	—	4	2	no	—
45	Dic₇⋊C₈	—	2	2	no	—
46	Dic₇⋊C₈	—	2	2	no	—
47	C₈×Dic₇	—	2	2	no	—
48	C₈×Dic₇	—	2	2	no	—
49	C₄⋊(C₇⋊C₈)	—	2	1	no	—
50	C₄⋊(C₇⋊C₈)	—	2	2	no	—
51	C₄⋊(C₇⋊C₈)	—	2	2	no	—
52	C₄⋊(C₇⋊C₈)	C₂×C₄	1	2	yes	—
53	C₂×C₈×Q₈	—	7	1	no	—
54	#1848	—	7	1	no	—
55	#1564	D₇	1	1	yes	characteristic
56	#1487	—	7	2	no	—
57	#1301	—	7	2	no	—
58	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₇	C₂×Q₈	1	1	yes	characteristic
60	C₂²×C₄×C₇	#41	1	1	yes	characteristic
61	C₂²×Dic₇	—	2	1	no	—
62	C₂²×Dic₇	—	2	1	no	—
63	C₂×Dic₁₄	—	2	1	no	—
64	C₂×Dic₁₄	—	2	4	no	—
65	C₂×Dic₁₄	C₄⋊C₄	1	2	yes	—
66	C₂×C₇×C₈	—	2	1	no	—
67	C₂×C₇×C₈	—	2	1	no	—
68	C₂×C₇×C₈	—	4	2	no	—
69	C₂×C₇×C₈	C₂²⋊C₄	1	1	yes	characteristic
70	C₂×C₇×C₈	C₂²⋊C₄	1	1	yes	characteristic
71	C₂×C₇×C₈	C₂²⋊C₄	1	1	yes	characteristic
72	C₂×C₇×C₈	C₂²⋊C₄	1	1	yes	characteristic
73	C₂×C₇×C₈	SD₁₆	1	2	yes	—
74	C₂×C₇×C₈	Q₁₆	1	2	yes	—
75	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
76	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
77	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
78	C₄²×C₇	#41	1	1	yes	characteristic
79	C₄⋊Dic₇	—	2	1	no	—
80	C₄⋊Dic₇	C₄⋊C₄	1	2	yes	—
81	Dic₇⋊C₄	—	2	2	no	—
82	Dic₇⋊C₄	—	2	2	no	—
83	C₄×Dic₇	—	2	2	no	—
84	C₄×Dic₇	—	2	2	no	—
85	#1136	—	4	2	no	—
86	#1136	—	4	2	no	—
87	C₂×C₇⋊C₈	—	2	2	no	—
88	C₂×C₇⋊C₈	—	4	1	no	—
89	C₂×C₇⋊C₈	—	4	1	no	—
90	C₂×C₇⋊C₈	—	4	2	no	—
91	C₂×C₄×Q₈	—	7	1	no	—
92	C₈×Q₈	—	7	2	no	—
93	C₈×Q₈	—	7	2	no	—
94	C₈×Q₈	—	14	1	no	—
95	C₈×Q₈	—	14	1	no	—
96	#431	C₂×D₇	1	2	yes	—
97	C₂×#108	—	14	1	no	—
98	C₂×#108	—	14	1	no	—
99	#423	—	7	2	no	—
100	C₂×C₄⋊C₈	—	7	1	no	—
101	C₂×C₄⋊C₈	—	14	1	no	—
102	C₂×C₄×C₈	—	14	1	no	—
103	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
104	#335	—	2	1	no	—
105	#334	—	2	1	no	—
106	Q₈⋊C₈	—	14	4	no	—
107	C₂³×C₇	C₂²⋊Q₈	1	1	yes	characteristic
108	C₂×C₄×C₇	—	2	2	no	—
109	C₂×C₄×C₇	—	4	1	no	—
110	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
111	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
112	C₂×C₄×C₇	#117	1	1	yes	characteristic
113	C₂×C₄×C₇	#121	1	1	yes	characteristic
114	C₂×C₄×C₇	#121	1	1	yes	characteristic
115	C₂×C₄×C₇	C₂²⋊Q₈	1	1	yes	characteristic
116	C₂×C₄×C₇	C₂×SD₁₆	1	1	yes	characteristic
117	C₂×C₄×C₇	C₂×Q₁₆	1	1	yes	characteristic
118	C₂×C₄×C₇	AGL(1,ℤ/8)	1	1	yes	characteristic
119	C₂×C₄×C₇	#137	1	1	yes	characteristic
120	C₂×Dic₇	—	2	4	no	—
121	C₂×Dic₇	—	2	2	no	—
122	C₂×Dic₇	—	2	2	no	—
123	C₂×Dic₇	—	4	2	no	—
124	Dic₁₄	—	2	4	no	—
125	Dic₁₄	—	2	2	no	—
126	Dic₁₄	#108	1	4	yes	complemented
127	C₇×C₈	—	2	1	no	—
128	C₇×C₈	—	2	1	no	—
129	C₇×C₈	—	8	2	no	—
130	C₇×C₈	#103	1	2	yes	—
131	C₇×C₈	#103	1	2	yes	—
132	C₇⋊C₈	—	4	2	no	—
133	C₇⋊C₈	—	8	2	no	—
134	C₂²×Q₈	—	7	1	no	—
135	C₂×M₁₆	—	14	2	no	—
136	C₂²×C₈	—	14	1	no	—
137	C₂²×C₈	D₂₈	1	1	yes	characteristic
138	C₂²×C₈	D₂₈	1	1	yes	characteristic
139	C₄×Q₈	—	7	2	no	—
140	C₄×Q₈	—	7	2	no	—
141	C₄×Q₈	—	14	1	no	—
142	C₄×Q₈	—	14	1	no	—
143	C₂×C₄⋊C₄	—	7	1	no	—
144	C₂×C₄⋊C₄	—	14	1	no	—
145	C₂×C₄²	—	14	1	no	—
146	C₂×C₄²	C₂²×D₇	1	1	yes	characteristic
147	#108	—	28	2	no	—
148	#108	—	28	2	no	—
149	C₄⋊C₈	—	2	2	no	—
150	C₄⋊C₈	—	7	2	no	—
151	C₄⋊C₈	—	14	2	no	—
152	C₄⋊C₈	—	14	2	no	—
153	C₄⋊C₈	—	14	2	no	—
154	C₄⋊C₈	—	14	2	no	—
155	C₄⋊C₈	—	14	1	no	—
156	C₂²⋊C₈	—	2	2	no	—
157	C₄×C₈	—	2	2	no	—
158	C₄×C₈	—	14	2	no	—
159	C₄×C₈	—	14	2	no	—
160	C₄×C₈	C₄×D₇	1	1	yes	characteristic
161	C₄×C₈	C₄×D₇	1	1	yes	characteristic
162	C₄×C₈	C₂²⋊D₇	1	1	yes	characteristic
163	C₄×C₈	C₂²⋊D₇	1	1	yes	characteristic
164	C₂²×C₇	—	2	1	no	—
165	C₂²×C₇	—	2	1	no	—
166	C₂²×C₇	#380	1	1	yes	characteristic
167	C₂²×C₇	#475	1	1	yes	characteristic
168	C₂²×C₇	#477	1	1	yes	characteristic
169	C₄×C₇	—	2	1	no	—
170	C₄×C₇	—	2	1	no	—
171	C₄×C₇	—	2	1	no	—
172	C₄×C₇	C₂×#103	1	1	yes	characteristic
173	C₄×C₇	#417	1	1	yes	characteristic
174	C₄×C₇	C₂×#108	1	2	yes	—
175	C₄×C₇	#460	1	1	yes	characteristic
176	C₄×C₇	#462	1	1	yes	characteristic
177	Dic₇	—	2	4	no	—
178	Dic₇	—	4	2	no	—
179	C₂×Q₈	—	7	2	no	—
180	C₂×Q₈	—	14	4	no	—
181	C₂×Q₈	—	14	1	no	—
182	C₂²×C₄	—	14	1	no	—
183	C₂²×C₄	—	14	1	no	—
184	C₂²×C₄	C₂×D₂₈	1	1	yes	characteristic
185	C₂²×C₄	#1159	1	1	yes	characteristic
186	C₂²×C₄	Q₈×D₇	1	1	yes	characteristic
187	M₁₆	—	28	2	no	—
188	M₁₆	—	28	2	no	—
189	C₂×C₈	—	2	1	no	—
190	C₂×C₈	—	2	1	no	—
191	C₂×C₈	—	4	2	no	—
192	C₂×C₈	—	14	2	no	—
193	C₂×C₈	—	28	1	no	—
194	C₂×C₈	—	28	1	no	—
195	C₂×C₈	—	28	2	no	—
196	C₂×C₈	#1132	1	2	yes	—
197	C₂×C₈	Dic₂₈	1	2	yes	—
198	C₂×C₈	(C₂×D₇)⋊C₄	1	1	yes	characteristic
199	C₂×C₈	(C₂×D₇)⋊C₄	1	1	yes	characteristic
200	C₂×C₈	(C₂×D₇)⋊C₄	1	1	yes	characteristic
201	C₂×C₈	(C₂×D₇)⋊C₄	1	1	yes	characteristic
202	C₄⋊C₄	—	7	2	no	—
203	C₄⋊C₄	—	14	2	no	—
204	C₄⋊C₄	—	14	2	no	—
205	C₄⋊C₄	—	14	1	no	—
206	C₄²	—	14	2	no	—
207	C₄²	—	14	2	no	—
208	C₄²	C₂×C₄×D₇	1	1	yes	characteristic
209	C₄²	D₄×D₇	1	1	yes	characteristic
210	C₄²	#1161	1	1	yes	characteristic
211	C₄²	C₂×C₂²⋊D₇	1	1	yes	characteristic
212	C₂×C₇	—	2	1	no	—
213	C₂×C₇	—	2	1	no	—
214	C₂×C₇	#1803	1	1	yes	characteristic
215	C₂×C₇	#1810	1	1	yes	characteristic
216	C₂×C₇	#1814	1	1	yes	characteristic
217	C₂³	#6460	1	1	yes	characteristic
218	Q₈	—	7	4	no	—
219	Q₈	—	14	4	no	—
220	Q₈	—	14	2	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	4	1	no	—
223	C₂×C₄	—	14	4	no	—
224	C₂×C₄	—	14	2	no	—
225	C₂×C₄	—	14	2	no	—
226	C₂×C₄	—	28	2	no	—
227	C₂×C₄	D₇×C₄⋊C₄	1	1	yes	characteristic
228	C₂×C₄	#6455	1	1	yes	characteristic
229	C₂×C₄	#6458	1	1	yes	characteristic
230	C₂×C₄	C₂×#1132	1	1	yes	characteristic
231	C₂×C₄	C₂×Dic₂₈	1	1	yes	characteristic
232	C₂×C₄	#6471	1	1	yes	characteristic
233	C₂×C₄	#6472	1	1	yes	characteristic
234	C₂×C₄	C₂×(C₂×D₇)⋊C₄	1	1	yes	characteristic
235	C₂×C₄	#6501	1	1	yes	characteristic
236	C₂×C₄	#6509	1	1	yes	characteristic
237	C₈	—	2	1	no	—
238	C₈	—	2	1	no	—
239	C₈	—	8	2	no	—
240	C₈	—	28	2	no	—
241	C₈	—	56	2	no	—
242	C₈	#6390	1	2	yes	—
243	C₈	#6390	1	2	yes	—
244	C₇	#8659	1	1	yes	characteristic, complemented, Sylow
245	C₂²	—	2	1	no	—
246	C₂²	—	2	1	no	—
247	C₂²	#89622	1	1	yes	characteristic
248	C₂²	#89623	1	1	yes	characteristic
249	C₂²	#89759	1	1	yes	characteristic
250	C₄	—	2	1	no	—
251	C₄	—	2	1	no	—
252	C₄	—	2	1	no	—
253	C₄	—	14	4	no	—
254	C₄	—	28	2	no	—
255	C₄	#89620	1	1	yes	characteristic
256	C₄	#89621	1	1	yes	characteristic
257	C₄	D₇×#108	1	2	yes	—
258	C₄	C₂×#6390	1	1	yes	characteristic
259	C₄	#89905	1	1	yes	characteristic
260	C₂	—	2	1	no	—
261	C₂	—	2	1	no	—
262	C₂	#11728239	1	1	yes	characteristic
263	C₂	#11728419	1	1	yes	characteristic
264	C₂	#11728465	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#6458
commutator	C₄²×C₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇
Fitting	C₇×#1564	C₂
socle	C₂²×C₇	#380
prime radical	C₂³×C₇	C₂²⋊Q₈
cosocle	C₂×C₄²×C₇	C₂³
hypercenter	#1564	D₇
hypocenter	C₇	#8659
LCS	G ▷ C₄²×C₇ ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1564

Complements

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

N	H ≅ G/N	type
126. Dic₁₄	#108	semidirect
244. C₇	#8659	semidirect

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
C₇×#1564	2
#11728524	2
#11727546	2
#11725551	2
#8659	7