Subgroups — #13506376

Order 1792.

862 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#89646	C₂	1	2	yes	maximal
2	C₂×#89627	C₂	1	2	yes	maximal
3	#11731666	C₂	1	2	yes	maximal
4	C₇×#1846	C₂	1	1	yes	maximal, characteristic
5	#11728361	C₂	1	2	yes	maximal
6	#11728309	C₂	1	2	yes	maximal
7	#11727520	C₂	1	1	yes	maximal, characteristic
8	C₄×C₈×Dic₇	C₂	1	1	yes	maximal, characteristic
9	#11727303	C₂	1	2	yes	maximal
10	C₂×#6451	C₂²	1	4	yes	—
11	C₂×C₇×#107	C₂²	1	2	yes	—
12	C₂×C₇×#106	C₂²	1	2	yes	—
13	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
14	C₇×#382	C₂²	1	2	yes	—
15	C₂×#6392	C₂²	1	2	yes	—
16	C₂×#6391	C₂²	1	2	yes	—
17	C₂×C₈×Dic₇	C₂²	1	1	yes	characteristic
18	C₂×C₈×Dic₇	C₂²	1	1	yes	characteristic
19	C₂×C₈×Dic₇	C₂²	1	1	yes	characteristic
20	C₂×C₈×Dic₇	C₂²	1	1	yes	characteristic
21	#89749	C₂²	1	4	yes	—
22	C₂×#6382	C₂²	1	4	yes	—
23	C₂×#6381	C₂²	1	4	yes	—
24	#89708	C₂²	1	2	yes	—
25	C₄²×Dic₇	C₂²	1	1	yes	characteristic
26	C₂×C₄×C₇⋊C₈	C₂²	1	1	yes	characteristic
27	#89646	—	2	8	no	—
28	#89627	—	2	8	no	—
29	#89418	—	2	8	no	—
30	#33313	—	7	1	no	maximal, Sylow
31	C₂²×Dic₁₄	—	2	4	no	—
32	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
33	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
34	C₂×C₇×C₄⋊C₄	C₂³	1	4	yes	—
35	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
36	C₂×C₄⋊Dic₇	C₂³	1	4	yes	—
37	C₂×Dic₇⋊C₄	—	2	4	no	—
38	C₂×C₄×Dic₇	D₄	1	4	yes	—
39	C₂×C₄×Dic₇	C₂³	1	1	yes	characteristic
40	C₂×C₄×Dic₇	C₂³	1	1	yes	characteristic
41	C₂²×C₇⋊C₈	C₂³	1	1	yes	characteristic
42	C₂²×C₇⋊C₈	C₂³	1	1	yes	characteristic
43	#6451	—	2	8	no	—
44	#6451	—	2	8	no	—
45	C₇×#107	—	2	4	no	—
46	C₇×#106	—	2	4	no	—
47	C₄×C₇×C₈	Q₈	1	4	yes	complemented
48	C₇×#95	—	2	4	no	—
49	C₇⋊#95	—	2	4	no	—
50	C₇⋊#95	—	2	8	no	—
51	#6392	—	2	4	no	—
52	#6391	—	2	4	no	—
53	C₈×Dic₇	D₄	1	4	yes	—
54	C₈×Dic₇	D₄	1	4	yes	—
55	C₈×Dic₇	Q₈	1	4	yes	complemented
56	C₈×Dic₇	Q₈	1	4	yes	complemented
57	#6382	—	2	8	no	—
58	#6381	—	2	8	no	—
59	C₄×C₇⋊C₈	Q₈	1	4	yes	complemented
60	C₂×#500	—	7	2	no	—
61	C₂×#498	—	7	2	no	—
62	#2862	—	7	2	no	—
63	#1846	—	7	2	no	—
64	#1846	—	7	2	no	—
65	#1846	—	7	1	no	—
66	#1846	—	7	2	no	—
67	#1846	D₇	1	1	yes	characteristic
68	C₄²×C₈	—	7	1	no	—
69	C₂²×C₄×C₇	—	2	4	no	—
70	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
71	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
72	C₂²×Dic₇	—	2	4	no	—
73	C₂²×Dic₇	C₂×D₄	1	2	yes	—
74	C₂²×Dic₇	C₂×D₄	1	2	yes	—
75	C₂×Dic₁₄	—	4	8	no	—
76	C₂×Dic₁₄	—	4	16	no	—
77	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
78	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
79	C₂×C₇×C₈	C₂×Q₈	1	2	yes	—
80	C₂×C₇×C₈	C₂×Q₈	1	2	yes	—
81	C₂×C₇×C₈	C₂×Q₈	1	2	yes	—
82	C₂×C₇×C₈	C₂×Q₈	1	2	yes	—
83	C₇×C₄⋊C₄	—	2	4	no	—
84	C₇×C₄⋊C₄	—	2	4	no	—
85	C₄²×C₇	C₂×Q₈	1	2	yes	—
86	C₄²×C₇	C₂×Q₈	1	2	yes	—
87	C₄⋊Dic₇	—	2	4	no	—
88	C₄⋊Dic₇	—	2	4	no	—
89	Dic₇⋊C₄	—	4	8	no	—
90	C₄×Dic₇	—	2	8	no	—
91	C₄×Dic₇	C₂×D₄	1	2	yes	—
92	C₄×Dic₇	C₂×D₄	1	2	yes	—
93	C₄×Dic₇	C₂×Q₈	1	2	yes	—
94	C₄×Dic₇	C₂×Q₈	1	2	yes	—
95	C₂×C₇⋊C₈	C₂×D₄	1	2	yes	—
96	C₂×C₇⋊C₈	C₂×D₄	1	2	yes	—
97	C₂×C₇⋊C₈	C₂×Q₈	1	2	yes	—
98	C₂×C₇⋊C₈	C₂×Q₈	1	2	yes	—
99	C₂×C₇⋊C₈	C₂×Q₈	1	2	yes	—
100	C₂×C₇⋊C₈	C₂×Q₈	1	2	yes	—
101	C₂×C₄⋊Q₈	—	7	4	no	—
102	#500	—	14	8	no	—
103	#498	—	14	8	no	—
104	C₂×#107	—	7	2	no	—
105	C₂×#107	—	7	4	no	—
106	C₂×#107	C₂×D₇	1	2	yes	—
107	C₂×#106	—	7	2	no	—
108	C₂×#106	—	7	4	no	—
109	C₂×#106	C₂×D₇	1	2	yes	—
110	C₂×C₄×C₈	—	7	1	no	—
111	C₂×C₄×C₈	—	7	1	no	—
112	C₂×C₄×C₈	—	7	1	no	—
113	C₂×C₄×C₈	—	7	1	no	—
114	C₂×C₄×C₈	—	7	1	no	—
115	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
116	#391	—	14	8	no	—
117	#382	—	7	4	no	—
118	#382	—	7	2	no	—
119	#382	C₂×D₇	1	2	yes	—
120	C₄³	—	7	1	no	—
121	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
122	C₂×C₄×C₇	—	2	4	no	—
123	C₂×C₄×C₇	—	4	4	no	—
124	C₂×C₄×C₇	—	4	8	no	—
125	C₂×C₄×C₇	C₄⋊Q₈	1	4	yes	—
126	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
127	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
128	C₂×C₄×C₇	C₂²×Q₈	1	1	yes	characteristic
129	C₂×C₄×C₇	C₂²×Q₈	1	1	yes	characteristic
130	C₂×C₄×C₇	C₂²×Q₈	1	1	yes	characteristic
131	C₂×C₄×C₇	C₂²×Q₈	1	1	yes	characteristic
132	C₂×Dic₇	—	2	4	no	—
133	C₂×Dic₇	—	4	4	no	—
134	C₂×Dic₇	—	4	8	no	—
135	C₂×Dic₇	#127	1	4	yes	—
136	C₂×Dic₇	C₄⋊Q₈	1	4	yes	—
137	C₂×Dic₇	#135	1	8	yes	—
138	Dic₁₄	—	8	16	no	—
139	C₇×C₈	C₄⋊Q₈	1	4	yes	—
140	C₇×C₈	C₄⋊Q₈	1	4	yes	—
141	C₇⋊C₈	C₄⋊Q₈	1	4	yes	—
142	C₇⋊C₈	C₄⋊Q₈	1	4	yes	—
143	C₂²×Q₈	—	14	4	no	—
144	C₂²×C₈	—	7	1	no	—
145	C₂²×C₈	—	7	1	no	—
146	C₂²×C₈	C₂²×D₇	1	1	yes	characteristic
147	C₂²×C₈	C₂²×D₇	1	1	yes	characteristic
148	C₄⋊Q₈	—	14	8	no	—
149	C₄⋊Q₈	—	14	8	no	—
150	C₂×C₄⋊C₄	—	7	4	no	—
151	C₂×C₄⋊C₄	—	14	4	no	—
152	C₂×C₄⋊C₄	C₂²×D₇	1	4	yes	—
153	C₂×C₄²	—	7	1	no	—
154	C₂×C₄²	—	7	1	no	—
155	C₂×C₄²	—	7	4	no	—
156	C₂×C₄²	C₂²×D₇	1	1	yes	characteristic
157	#107	—	2	4	no	—
158	#107	—	14	8	no	—
159	#107	—	14	4	no	—
160	#106	—	2	4	no	—
161	#106	—	14	8	no	—
162	#106	—	14	4	no	—
163	C₄×C₈	—	7	4	no	—
164	C₄×C₈	—	7	4	no	—
165	C₄×C₈	—	7	4	no	—
166	C₄×C₈	—	7	4	no	—
167	C₄×C₈	—	7	4	no	—
168	C₄×C₈	Dic₁₄	1	4	yes	complemented
169	#95	—	2	4	no	—
170	#95	—	14	8	no	—
171	#95	—	14	4	no	—
172	C₂²×C₇	C₂×#127	1	1	yes	characteristic
173	C₂²×C₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
174	C₂²×C₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
175	C₂²×C₇	C₂×#135	1	4	yes	—
176	C₄×C₇	—	2	4	no	—
177	C₄×C₇	—	8	4	no	—
178	C₄×C₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
179	C₄×C₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
180	C₄×C₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
181	C₄×C₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
182	Dic₇	—	2	4	no	—
183	Dic₇	—	8	4	no	—
184	Dic₇	#495	1	8	yes	—
185	C₂×Q₈	—	28	8	no	—
186	C₂×Q₈	—	28	16	no	—
187	C₂²×C₄	—	2	4	no	—
188	C₂²×C₄	—	7	2	no	—
189	C₂²×C₄	—	7	2	no	—
190	C₂²×C₄	—	14	4	no	—
191	C₂²×C₄	D₄×D₇	1	2	yes	—
192	C₂²×C₄	C₂³×D₇	1	1	yes	characteristic
193	C₂×C₈	—	7	2	no	—
194	C₂×C₈	—	7	2	no	—
195	C₂×C₈	—	7	2	no	—
196	C₂×C₈	—	7	2	no	—
197	C₂×C₈	—	7	2	no	—
198	C₂×C₈	—	7	2	no	—
199	C₂×C₈	C₂×Dic₁₄	1	2	yes	—
200	C₂×C₈	C₂×Dic₁₄	1	2	yes	—
201	C₂×C₈	D₄×D₇	1	2	yes	—
202	C₂×C₈	D₄×D₇	1	2	yes	—
203	C₂×C₈	Q₈×D₇	1	2	yes	—
204	C₂×C₈	Q₈×D₇	1	2	yes	—
205	C₄⋊C₄	—	2	4	no	—
206	C₄⋊C₄	—	2	4	no	—
207	C₄⋊C₄	—	14	4	no	—
208	C₄⋊C₄	—	14	4	no	—
209	C₄⋊C₄	—	28	8	no	—
210	C₄²	—	7	2	no	—
211	C₄²	—	7	2	no	—
212	C₄²	—	7	2	no	—
213	C₄²	—	7	2	no	—
214	C₄²	—	14	8	no	—
215	C₄²	C₂×Dic₁₄	1	2	yes	—
216	C₄²	Q₈×D₇	1	2	yes	—
217	C₂×C₇	#2868	1	1	yes	characteristic
218	C₂×C₇	C₂×#495	1	2	yes	—
219	C₂×C₇	#3160	1	2	yes	—
220	C₂×C₇	#3160	1	2	yes	—
221	C₂³	C₂×D₄×D₇	1	1	yes	characteristic
222	Q₈	—	56	16	no	—
223	C₂×C₄	—	2	4	no	—
224	C₂×C₄	—	4	4	no	—
225	C₂×C₄	—	4	8	no	—
226	C₂×C₄	—	7	4	no	—
227	C₂×C₄	—	7	8	no	—
228	C₂×C₄	—	7	4	no	—
229	C₂×C₄	—	14	4	no	—
230	C₂×C₄	—	28	4	no	—
231	C₂×C₄	—	28	8	no	—
232	C₂×C₄	#6451	1	4	yes	—
233	C₂×C₄	C₂²×Dic₁₄	1	1	yes	characteristic
234	C₂×C₄	C₂×D₄×D₇	1	1	yes	characteristic
235	C₂×C₄	C₂×D₄×D₇	1	1	yes	characteristic
236	C₂×C₄	C₂×Q₈×D₇	1	1	yes	characteristic
237	C₂×C₄	C₂×Q₈×D₇	1	1	yes	characteristic
238	C₂×C₄	C₂×Q₈×D₇	1	1	yes	characteristic
239	C₈	—	7	4	no	—
240	C₈	—	7	4	no	—
241	C₈	#6451	1	4	yes	—
242	C₈	#6451	1	4	yes	—
243	C₇	#33313	1	1	yes	characteristic, complemented, Sylow
244	C₂²	C₂×#6451	1	1	yes	characteristic
245	C₂²	D₇×#127	1	1	yes	characteristic
246	C₂²	D₇×C₄⋊Q₈	1	1	yes	characteristic
247	C₂²	D₇×#135	1	4	yes	—
248	C₄	—	2	4	no	—
249	C₄	—	7	8	no	—
250	C₄	—	8	4	no	—
251	C₄	—	14	4	no	—
252	C₄	—	56	4	no	—
253	C₄	C₂×#6451	1	1	yes	characteristic
254	C₄	C₂×#6451	1	1	yes	characteristic
255	C₄	D₇×C₄⋊Q₈	1	1	yes	characteristic
256	C₄	D₇×C₄⋊Q₈	1	1	yes	characteristic
257	C₂	#11731801	1	1	yes	characteristic
258	C₂	#11736615	1	2	yes	—
259	C₂	D₇×#495	1	2	yes	—
260	C₂	#11739361	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄×D₇
commutator, cosocle	C₂²×C₄×C₇	C₂⁴
Frattini	C₂²×C₄	C₂³×D₇
Fitting	C₇×#1846	C₂
socle, prime radical	C₂³×C₇	C₂²×D₄
hypercenter	#1846	D₇
hypocenter	C₇	#33313
LCS	G ▷ C₂²×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1846

Complements

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

N	H ≅ G/N	type
47. C₄×C₇×C₈	Q₈	semidirect
55. C₈×Dic₇	Q₈	semidirect
56. C₈×Dic₇	Q₈	semidirect
59. C₄×C₇⋊C₈	Q₈	semidirect
168. C₄×C₈	Dic₁₄	semidirect
243. C₇	#33313	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#89646	2
C₂×#89627	2
#11731666	2
C₇×#1846	2
#11728361	2
#11728309	2
#11727520	2
C₄×C₈×Dic₇	2
#11727303	2
#33313	7