Subgroups — #13268675

Order 1792.

606 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#427	C₂	1	1	yes	maximal, characteristic
2	C₂×#89878	C₂	1	1	yes	maximal, characteristic
3	C₂³×C₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic
4	#11725528	C₂	1	4	yes	maximal, complemented
5	C₂³×C₇×C₈	C₂²	1	1	yes	characteristic
6	C₂×C₇×#117	—	2	1	no	—
7	C₂×#6489	—	2	1	no	—
8	C₇×#427	—	2	2	no	—
9	C₇×#427	C₄	1	4	yes	complemented
10	C₂×C₇×#107	C₂²	1	2	yes	—
11	C₂×C₇×#106	—	2	1	no	—
12	#89878	—	2	2	no	—
13	#89878	C₄	1	4	yes	complemented
14	C₂×#6392	C₂²	1	2	yes	—
15	C₂×#6391	—	2	1	no	—
16	C₂²×C₇⋊C₁₆	—	2	2	no	—
17	C₂²×C₇⋊C₁₆	—	2	2	no	—
18	C₂²×C₇⋊C₁₆	C₄	1	4	yes	complemented
19	C₂²×C₇⋊C₁₆	C₂²	1	2	yes	—
20	#12556	—	7	1	no	maximal, Sylow
21	C₂³×C₄×C₇	D₄	1	1	yes	characteristic
22	C₂²×C₇×C₈	C₂×C₄	1	2	yes	—
23	C₂²×C₇×C₈	C₂×C₄	1	2	yes	—
24	C₂²×C₇×C₈	C₂×C₄	1	2	yes	—
25	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
26	C₂²×C₇×C₈	D₄	1	2	yes	—
27	C₂²×C₇×C₈	D₄	1	2	yes	—
28	C₂²×C₇×C₈	Q₈	1	2	yes	—
29	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
30	C₇×#117	—	2	4	no	—
31	C₇×#117	—	4	2	no	—
32	C₂×C₇×C₄⋊C₄	—	2	2	no	—
33	C₂×C₇×C₂²⋊C₄	—	4	1	no	—
34	C₂×C₄²×C₇	—	4	1	no	—
35	C₂×C₂²⋊Dic₇	—	4	1	no	—
36	#6489	—	2	4	no	—
37	#6489	—	4	2	no	—
38	C₂×C₄⋊Dic₇	—	2	2	no	—
39	C₂×C₄×Dic₇	—	4	1	no	—
40	C₇×#107	—	2	2	no	—
41	C₇×#107	C₂×C₄	1	4	yes	—
42	C₇×#106	—	2	4	no	—
43	#6392	—	2	2	no	—
44	#6392	C₂×C₄	1	4	yes	—
45	#6391	—	2	4	no	—
46	C₂×C₇⋊C₁₆	—	2	8	no	—
47	C₂×C₇⋊C₁₆	—	2	2	no	—
48	C₂×C₇⋊C₁₆	—	2	2	no	—
49	C₂×C₇⋊C₁₆	C₂×C₄	1	4	yes	—
50	C₂³×C₁₆	—	7	1	no	—
51	C₂×#427	—	7	1	no	—
52	C₂×#427	D₇	1	1	yes	characteristic
53	#1380	—	7	4	no	—
54	C₂⁴×C₇	D₈	1	1	yes	characteristic
55	C₂²×C₄×C₇	—	4	2	no	—
56	C₂²×C₄×C₇	C₂²⋊C₄	1	2	yes	—
57	C₂²×C₄×C₇	C₂²⋊C₄	1	2	yes	—
58	C₂²×C₄×C₇	C₄⋊C₄	1	2	yes	—
59	C₂²×C₄×C₇	D₈	1	1	yes	characteristic
60	C₂²×C₄×C₇	SD₁₆	1	2	yes	—
61	C₂²×C₄×C₇	SD₁₆	1	2	yes	—
62	C₂²×C₄×C₇	Q₁₆	1	2	yes	—
63	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
64	C₂²×Dic₇	—	4	2	no	—
65	C₂×C₇×C₈	C₄²	1	4	yes	—
66	C₂×C₇×C₈	C₂²⋊C₄	1	2	yes	—
67	C₂×C₇×C₈	C₂²⋊C₄	1	2	yes	—
68	C₂×C₇×C₈	C₂²⋊C₄	1	4	yes	—
69	C₂×C₇×C₈	C₄⋊C₄	1	2	yes	—
70	C₂×C₇×C₈	C₄⋊C₄	1	4	yes	—
71	C₂×C₇×C₈	C₄⋊C₄	1	4	yes	—
72	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
73	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
76	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
77	C₂×C₇×C₈	C₂×Q₈	1	1	yes	characteristic
78	C₇×C₄⋊C₄	—	2	4	no	—
79	C₇×C₄⋊C₄	—	4	2	no	—
80	C₇×C₂²⋊C₄	—	4	4	no	—
81	C₄²×C₇	—	4	4	no	—
82	C₂²⋊Dic₇	—	4	4	no	—
83	C₄⋊Dic₇	—	2	4	no	—
84	C₄⋊Dic₇	—	4	2	no	—
85	C₄×Dic₇	—	4	4	no	—
86	C₇⋊C₁₆	—	2	4	no	—
87	C₂³×C₈	C₂×D₇	1	1	yes	characteristic
88	C₂×#117	—	2	1	no	—
89	C₂×#117	—	14	1	no	—
90	C₂²×C₁₆	—	7	4	no	—
91	C₂²×C₁₆	—	7	2	no	—
92	C₂²×C₁₆	—	14	2	no	—
93	C₂²×C₁₆	—	14	2	no	—
94	#427	—	2	2	no	—
95	#427	—	7	4	no	—
96	#427	—	14	2	no	—
97	#427	Dic₇	1	4	yes	complemented
98	C₂×#107	—	7	2	no	—
99	C₂×#107	C₂×D₇	1	2	yes	—
100	C₂×#106	—	2	1	no	—
101	C₂×#106	—	14	1	no	—
102	C₂³×C₇	—	2	4	no	—
103	C₂³×C₇	#102	1	2	yes	—
104	C₂³×C₇	#102	1	2	yes	—
105	C₂³×C₇	#107	1	2	yes	—
106	C₂³×C₇	C₂×D₈	1	1	yes	characteristic
107	C₂×C₄×C₇	—	4	4	no	—
108	C₂×C₄×C₇	—	8	4	no	—
109	C₂×C₄×C₇	#95	1	4	yes	—
110	C₂×C₄×C₇	#102	1	2	yes	—
111	C₂×C₄×C₇	#102	1	2	yes	—
112	C₂×C₄×C₇	#103	1	4	yes	—
113	C₂×C₄×C₇	#103	1	4	yes	—
114	C₂×C₄×C₇	#106	1	4	yes	—
115	C₂×C₄×C₇	#107	1	2	yes	—
116	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
117	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
118	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
119	C₂×C₄×C₇	C₂×SD₁₆	1	1	yes	characteristic
120	C₂×C₄×C₇	C₂×SD₁₆	1	1	yes	characteristic
121	C₂×C₄×C₇	C₂×Q₁₆	1	1	yes	characteristic
122	C₂×Dic₇	—	4	4	no	—
123	C₂×Dic₇	—	8	4	no	—
124	C₇×C₈	#95	1	4	yes	—
125	C₇×C₈	C₂×C₄²	1	1	yes	characteristic
126	C₇×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
127	C₇×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
128	C₇×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
129	C₂³×C₄	C₂²⋊D₇	1	1	yes	characteristic
130	C₂²×C₈	Dic₁₄	1	2	yes	—
131	C₂²×C₈	C₄×D₇	1	2	yes	—
132	C₂²×C₈	C₄×D₇	1	2	yes	—
133	C₂²×C₈	D₂₈	1	2	yes	—
134	C₂²×C₈	C₂×Dic₇	1	2	yes	—
135	C₂²×C₈	C₂²⋊D₇	1	1	yes	characteristic
136	C₂²×C₈	C₂²⋊D₇	1	2	yes	—
137	C₂²×C₈	C₂²×D₇	1	1	yes	characteristic
138	#117	—	2	4	no	—
139	#117	—	4	2	no	—
140	#117	—	14	4	no	—
141	#117	—	28	2	no	—
142	C₂×C₄⋊C₄	—	2	2	no	—
143	C₂×C₄⋊C₄	—	14	2	no	—
144	C₂×C₂²⋊C₄	—	4	1	no	—
145	C₂×C₂²⋊C₄	—	28	1	no	—
146	C₂×C₄²	—	4	1	no	—
147	C₂×C₄²	—	28	1	no	—
148	C₂×C₁₆	—	7	4	no	—
149	C₂×C₁₆	—	14	8	no	—
150	C₂×C₁₆	—	14	2	no	—
151	C₂×C₁₆	—	14	2	no	—
152	#107	—	2	2	no	—
153	#107	—	7	4	no	—
154	#107	—	14	2	no	—
155	#107	C₂×Dic₇	1	4	yes	—
156	#106	—	2	4	no	—
157	#106	—	14	4	no	—
158	C₂²×C₇	—	2	4	no	—
159	C₂²×C₇	—	2	4	no	—
160	C₂²×C₇	—	2	4	no	—
161	C₂²×C₇	#340	1	4	yes	—
162	C₂²×C₇	C₂×#102	1	1	yes	characteristic
163	C₂²×C₇	C₂×#102	1	1	yes	characteristic
164	C₂²×C₇	C₂×#107	1	1	yes	characteristic
165	C₂²×C₇	#508	1	4	yes	—
166	C₄×C₇	—	8	4	no	—
167	C₄×C₇	#340	1	4	yes	—
168	C₄×C₇	C₂×#95	1	1	yes	characteristic
169	C₄×C₇	C₂×#103	1	1	yes	characteristic
170	C₄×C₇	C₂×#103	1	1	yes	characteristic
171	C₄×C₇	C₂×#106	1	1	yes	characteristic
172	Dic₇	—	8	4	no	—
173	C₂⁴	#1141	1	1	yes	characteristic
174	C₂²×C₄	—	4	2	no	—
175	C₂²×C₄	—	28	2	no	—
176	C₂²×C₄	Dic₇⋊C₄	1	2	yes	—
177	C₂²×C₄	(C₂×D₇)⋊C₄	1	2	yes	—
178	C₂²×C₄	#1141	1	1	yes	characteristic
179	C₂²×C₄	#1142	1	2	yes	—
180	C₂²×C₄	#1143	1	2	yes	—
181	C₂²×C₄	#1144	1	2	yes	—
182	C₂²×C₄	C₂²⋊Dic₇	1	2	yes	—
183	C₂²×C₄	C₂×C₂²⋊D₇	1	1	yes	characteristic
184	C₂×C₈	C₄×Dic₇	1	4	yes	—
185	C₂×C₈	Dic₇⋊C₄	1	2	yes	—
186	C₂×C₈	Dic₇⋊C₄	1	4	yes	—
187	C₂×C₈	C₄⋊Dic₇	1	4	yes	—
188	C₂×C₈	(C₂×D₇)⋊C₄	1	2	yes	—
189	C₂×C₈	(C₂×D₇)⋊C₄	1	4	yes	—
190	C₂×C₈	C₂²⋊Dic₇	1	2	yes	—
191	C₂×C₈	C₂×Dic₁₄	1	1	yes	characteristic
192	C₂×C₈	C₂×C₄×D₇	1	1	yes	characteristic
193	C₂×C₈	C₂×C₄×D₇	1	1	yes	characteristic
194	C₂×C₈	C₂×D₂₈	1	1	yes	characteristic
195	C₂×C₈	C₂²×Dic₇	1	1	yes	characteristic
196	C₂×C₈	C₂×C₂²⋊D₇	1	1	yes	characteristic
197	C₄⋊C₄	—	2	4	no	—
198	C₄⋊C₄	—	4	2	no	—
199	C₄⋊C₄	—	14	4	no	—
200	C₄⋊C₄	—	28	2	no	—
201	C₂²⋊C₄	—	4	4	no	—
202	C₂²⋊C₄	—	28	4	no	—
203	C₄²	—	4	4	no	—
204	C₄²	—	28	4	no	—
205	C₁₆	—	14	4	no	—
206	C₂×C₇	—	2	4	no	—
207	C₂×C₇	C₂×#340	1	1	yes	characteristic
208	C₂×C₇	#2138	1	2	yes	—
209	C₂×C₇	#2138	1	2	yes	—
210	C₂×C₇	#2158	1	2	yes	—
211	C₂³	—	2	4	no	—
212	C₂³	#6381	1	2	yes	—
213	C₂³	#6383	1	2	yes	—
214	C₂³	#6406	1	2	yes	—
215	C₂³	C₂×#1141	1	1	yes	characteristic
216	C₂×C₄	—	4	4	no	—
217	C₂×C₄	—	8	4	no	—
218	C₂×C₄	—	28	4	no	—
219	C₂×C₄	—	56	4	no	—
220	C₂×C₄	#6381	1	2	yes	—
221	C₂×C₄	#6382	1	4	yes	—
222	C₂×C₄	#6383	1	2	yes	—
223	C₂×C₄	#6384	1	4	yes	—
224	C₂×C₄	C₇⋊#95	1	4	yes	—
225	C₂×C₄	#6406	1	2	yes	—
226	C₂×C₄	#6409	1	4	yes	—
227	C₂×C₄	C₂×Dic₇⋊C₄	1	1	yes	characteristic
228	C₂×C₄	C₂×(C₂×D₇)⋊C₄	1	1	yes	characteristic
229	C₂×C₄	C₂×#1142	1	1	yes	characteristic
230	C₂×C₄	C₂×#1143	1	1	yes	characteristic
231	C₂×C₄	C₂×#1144	1	1	yes	characteristic
232	C₂×C₄	C₂×C₂²⋊Dic₇	1	1	yes	characteristic
233	C₈	C₇⋊#95	1	4	yes	—
234	C₈	C₂×C₄×Dic₇	1	1	yes	characteristic
235	C₈	C₂×Dic₇⋊C₄	1	1	yes	characteristic
236	C₈	C₂×C₄⋊Dic₇	1	1	yes	characteristic
237	C₈	C₂×(C₂×D₇)⋊C₄	1	1	yes	characteristic
238	C₇	#12556	1	1	yes	characteristic, complemented, Sylow
239	C₂²	—	2	4	no	—
240	C₂²	—	2	4	no	—
241	C₂²	—	2	4	no	—
242	C₂²	#89324	1	4	yes	—
243	C₂²	C₂×#6381	1	1	yes	characteristic
244	C₂²	C₂×#6383	1	1	yes	characteristic
245	C₂²	#89966	1	4	yes	—
246	C₂²	C₂×#6406	1	1	yes	characteristic
247	C₄	—	8	4	no	—
248	C₄	—	56	4	no	—
249	C₄	#89324	1	4	yes	—
250	C₄	C₂×#6382	1	1	yes	characteristic
251	C₄	C₂×#6384	1	1	yes	characteristic
252	C₄	C₂×C₇⋊#95	1	1	yes	characteristic
253	C₄	C₂×#6409	1	1	yes	characteristic
254	C₂	—	2	4	no	—
255	C₂	#11728402	1	2	yes	—
256	C₂	#11728405	1	2	yes	—
257	C₂	C₂×#89324	1	1	yes	characteristic
258	C₂	#11729831	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	#1141
commutator	C₇×C₈	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₇
Fitting	C₂×C₇×#427	C₂
socle	C₂³×C₇	C₂×D₈
prime radical	C₂⁴×C₇	D₈
cosocle	C₂²×C₇×C₈	C₂³
hypercenter	C₂×#427	D₇
hypocenter	C₇	#12556
LCS	G ▷ C₇×C₈ ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂×#427

Complements

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

N	H ≅ G/N	type
4. #11725528	C₂	semidirect
9. C₇×#427	C₄	semidirect
13. #89878	C₄	semidirect
18. C₂²×C₇⋊C₁₆	C₄	semidirect
97. #427	Dic₇	semidirect
238. C₇	#12556	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₇×#427	2
C₂×#89878	2
C₂³×C₇⋊C₁₆	2
#11725528	2
#12556	7