Subgroups — #13247456

Order 1792.

422 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#1842	C₂	1	1	yes	maximal, characteristic
2	#11727148	C₂	1	1	yes	maximal, characteristic
3	C₄²×C₇⋊C₈	C₂	1	1	yes	maximal, characteristic
4	#11725509	C₂	1	4	yes	maximal
5	C₂×C₇×C₄⋊C₈	—	2	1	no	—
6	C₂×C₇×C₄⋊C₈	—	2	1	no	—
7	C₂×C₇×C₄⋊C₈	C₂²	1	2	yes	—
8	C₄³×C₇	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊(C₇⋊C₈)	—	2	1	no	—
10	C₂×C₄⋊(C₇⋊C₈)	—	2	1	no	—
11	C₂×C₄⋊(C₇⋊C₈)	C₂²	1	2	yes	—
12	C₂×C₄×C₇⋊C₈	C₄	1	2	yes	—
13	C₂×C₄×C₇⋊C₈	C₄	1	1	yes	characteristic
14	C₂×C₄×C₇⋊C₈	C₄	1	1	yes	characteristic
15	C₂×C₄×C₇⋊C₈	C₂²	1	2	yes	—
16	#8784	—	7	1	no	maximal, Sylow
17	C₂²×C₇×C₈	—	2	2	no	—
18	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
21	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
22	C₂×C₄²×C₇	Q₈	1	1	yes	characteristic
23	C₂×C₄²×C₇	Q₈	1	1	yes	characteristic
24	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
25	C₂²×C₇⋊C₈	—	2	2	no	—
26	C₂²×C₇⋊C₈	C₂×C₄	1	2	yes	—
27	C₂²×C₇⋊C₈	C₂×C₄	1	2	yes	—
28	C₇×C₄⋊C₈	—	2	4	no	—
29	C₇×C₄⋊C₈	—	4	2	no	—
30	C₇×C₄⋊C₈	—	4	2	no	—
31	C₄⋊(C₇⋊C₈)	—	2	4	no	—
32	C₄⋊(C₇⋊C₈)	—	4	2	no	—
33	C₄⋊(C₇⋊C₈)	—	4	2	no	—
34	C₄×C₇⋊C₈	—	2	2	no	—
35	C₄×C₇⋊C₈	—	2	2	no	—
36	C₄×C₇⋊C₈	—	2	2	no	—
37	C₄×C₇⋊C₈	—	2	4	no	—
38	C₄×C₇⋊C₈	C₈	1	4	yes	complemented
39	#1842	—	7	1	no	—
40	#1842	D₇	1	1	yes	characteristic
41	C₄²×C₈	—	7	1	no	—
42	#1277	—	7	4	no	—
43	C₂²×C₄×C₇	C₄⋊C₄	1	1	yes	characteristic
44	C₂²×C₄×C₇	C₄⋊C₄	1	1	yes	characteristic
45	C₂²×C₄×C₇	C₄⋊C₄	1	1	yes	characteristic
46	C₂²×C₄×C₇	C₄⋊C₄	1	1	yes	characteristic
47	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
48	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
49	C₂²×C₄×C₇	C₂×Q₈	1	1	yes	characteristic
50	C₂×C₇×C₈	—	4	2	no	—
51	C₂×C₇×C₈	—	4	2	no	—
52	C₂×C₇×C₈	—	4	2	no	—
53	C₄²×C₇	—	2	2	no	—
54	C₄²×C₇	—	2	2	no	—
55	C₄²×C₇	—	2	2	no	—
56	C₄²×C₇	C₂×C₈	1	2	yes	—
57	C₄²×C₇	M₁₆	1	2	yes	—
58	C₄²×C₇	D₈	1	2	yes	—
59	C₄²×C₇	SD₁₆	1	2	yes	—
60	C₄²×C₇	SD₁₆	1	2	yes	—
61	C₄²×C₇	Q₁₆	1	2	yes	—
62	C₄²×C₇	#41	1	2	yes	—
63	C₄²×C₇	#41	1	2	yes	—
64	C₂×C₇⋊C₈	—	2	2	no	—
65	C₂×C₇⋊C₈	—	2	2	no	—
66	C₂×C₇⋊C₈	—	2	2	no	—
67	C₂×C₇⋊C₈	—	2	2	no	—
68	C₂×C₇⋊C₈	—	4	2	no	—
69	C₂×C₇⋊C₈	—	4	2	no	—
70	C₂×C₇⋊C₈	—	4	2	no	—
71	C₂×C₇⋊C₈	C₂×C₈	1	4	yes	—
72	C₂×C₇⋊C₈	M₁₆	1	4	yes	—
73	C₂×C₄⋊C₈	—	2	1	no	—
74	C₂×C₄⋊C₈	—	2	1	no	—
75	C₂×C₄⋊C₈	—	7	2	no	—
76	C₂×C₄⋊C₈	—	14	1	no	—
77	C₂×C₄⋊C₈	—	14	1	no	—
78	C₂×C₄⋊C₈	C₂×D₇	1	2	yes	—
79	C₂×C₄×C₈	—	7	1	no	—
80	C₂×C₄×C₈	—	7	2	no	—
81	C₂×C₄×C₈	—	7	1	no	—
82	C₂×C₄×C₈	—	7	2	no	—
83	C₄³	C₂×D₇	1	1	yes	characteristic
84	C₂³×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
85	C₂×C₄×C₇	—	2	1	no	—
86	C₂×C₄×C₇	—	2	1	no	—
87	C₂×C₄×C₇	—	2	1	no	—
88	C₂×C₄×C₇	—	2	1	no	—
89	C₂×C₄×C₇	—	2	1	no	—
90	C₂×C₄×C₇	—	2	1	no	—
91	C₂×C₄×C₇	C₄⋊C₈	1	2	yes	—
92	C₂×C₄×C₇	C₄⋊C₈	1	2	yes	—
93	C₂×C₄×C₇	#106	1	2	yes	—
94	C₂×C₄×C₇	#106	1	2	yes	—
95	C₂×C₄×C₇	#107	1	2	yes	—
96	C₂×C₄×C₇	#107	1	2	yes	—
97	C₂×C₄×C₇	#108	1	2	yes	—
98	C₂×C₄×C₇	#108	1	2	yes	—
99	C₂×C₄×C₇	#117	1	2	yes	—
100	C₂×C₄×C₇	#124	1	1	yes	characteristic
101	C₂×C₄×C₇	#124	1	1	yes	characteristic
102	C₂×C₄×C₇	#125	1	2	yes	—
103	C₂×C₄×C₇	C₂²×C₈	1	1	yes	characteristic
104	C₂×C₄×C₇	C₂×M₁₆	1	1	yes	characteristic
105	C₂×C₄×C₇	C₂×M₁₆	1	2	yes	—
106	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
107	C₂×C₄×C₇	C₂×SD₁₆	1	1	yes	characteristic
108	C₂×C₄×C₇	C₂×SD₁₆	1	1	yes	characteristic
109	C₂×C₄×C₇	C₂×Q₁₆	1	1	yes	characteristic
110	C₇×C₈	—	8	2	no	—
111	C₇⋊C₈	—	2	4	no	—
112	C₇⋊C₈	—	2	4	no	—
113	C₇⋊C₈	—	8	2	no	—
114	C₂²×C₈	—	2	2	no	—
115	C₂²×C₈	—	7	2	no	—
116	C₂²×C₈	—	7	2	no	—
117	C₂²×C₈	—	14	2	no	—
118	C₂×C₄²	Dic₁₄	1	1	yes	characteristic
119	C₂×C₄²	Dic₁₄	1	1	yes	characteristic
120	C₂×C₄²	C₄×D₇	1	1	yes	characteristic
121	C₂×C₄²	C₄×D₇	1	1	yes	characteristic
122	C₂×C₄²	C₂²⋊D₇	1	1	yes	characteristic
123	C₂×C₄²	C₂²⋊D₇	1	1	yes	characteristic
124	C₂×C₄²	C₂²×D₇	1	1	yes	characteristic
125	C₄⋊C₈	—	2	4	no	—
126	C₄⋊C₈	—	4	2	no	—
127	C₄⋊C₈	—	4	2	no	—
128	C₄⋊C₈	—	14	4	no	—
129	C₄⋊C₈	—	28	2	no	—
130	C₄⋊C₈	—	28	2	no	—
131	C₄×C₈	—	7	4	no	—
132	C₄×C₈	—	14	4	no	—
133	C₄×C₈	—	14	2	no	—
134	C₄×C₈	—	14	2	no	—
135	C₄×C₈	—	14	2	no	—
136	C₂²×C₇	#382	1	1	yes	characteristic
137	C₂²×C₇	C₂×C₄⋊C₈	1	1	yes	characteristic
138	C₂²×C₇	#423	1	1	yes	characteristic
139	C₂²×C₇	C₂×#106	1	1	yes	characteristic
140	C₂²×C₇	C₂×#107	1	1	yes	characteristic
141	C₂²×C₇	C₂×#108	1	1	yes	characteristic
142	C₂²×C₇	C₂×#108	1	1	yes	characteristic
143	C₄×C₇	—	2	2	no	—
144	C₄×C₇	—	2	2	no	—
145	C₄×C₇	—	2	2	no	—
146	C₄×C₇	#334	1	4	yes	—
147	C₄×C₇	#335	1	4	yes	—
148	C₄×C₇	#431	1	2	yes	—
149	C₄×C₇	#432	1	2	yes	—
150	C₄×C₇	#486	1	1	yes	characteristic
151	C₄×C₇	#486	1	1	yes	characteristic
152	C₄×C₇	#487	1	1	yes	characteristic
153	C₄×C₇	#487	1	1	yes	characteristic
154	C₂²×C₄	Dic₇⋊C₄	1	1	yes	characteristic
155	C₂²×C₄	Dic₇⋊C₄	1	1	yes	characteristic
156	C₂²×C₄	Dic₇⋊C₄	1	1	yes	characteristic
157	C₂²×C₄	Dic₇⋊C₄	1	1	yes	characteristic
158	C₂²×C₄	C₂×Dic₁₄	1	1	yes	characteristic
159	C₂²×C₄	C₂×C₄×D₇	1	1	yes	characteristic
160	C₂²×C₄	C₂×C₂²⋊D₇	1	1	yes	characteristic
161	C₂×C₈	—	4	2	no	—
162	C₂×C₈	—	4	2	no	—
163	C₂×C₈	—	4	2	no	—
164	C₂×C₈	—	7	4	no	—
165	C₂×C₈	—	7	4	no	—
166	C₂×C₈	—	14	2	no	—
167	C₂×C₈	—	14	2	no	—
168	C₂×C₈	—	14	2	no	—
169	C₂×C₈	—	14	2	no	—
170	C₂×C₈	—	28	2	no	—
171	C₂×C₈	—	28	2	no	—
172	C₂×C₈	—	28	2	no	—
173	C₄²	—	2	2	no	—
174	C₄²	—	2	2	no	—
175	C₄²	—	2	2	no	—
176	C₄²	C₈×D₇	1	2	yes	—
177	C₄²	#1131	1	2	yes	—
178	C₄²	#1141	1	2	yes	—
179	C₄²	#1142	1	2	yes	—
180	C₄²	#1143	1	2	yes	—
181	C₄²	#1144	1	2	yes	—
182	C₄²	#1159	1	2	yes	—
183	C₄²	#1161	1	2	yes	—
184	C₂×C₇	C₂×#334	1	1	yes	characteristic
185	C₂×C₇	C₂×#335	1	1	yes	characteristic
186	C₂×C₇	#1566	1	1	yes	characteristic
187	C₂×C₇	#1567	1	1	yes	characteristic
188	C₂×C₇	#1831	1	1	yes	characteristic
189	C₂×C₇	#1834	1	1	yes	characteristic
190	C₂×C₇	#1837	1	1	yes	characteristic
191	C₂³	C₂×Dic₇⋊C₄	1	1	yes	characteristic
192	C₂×C₄	—	2	1	no	—
193	C₂×C₄	—	2	1	no	—
194	C₂×C₄	—	2	1	no	—
195	C₂×C₄	—	2	1	no	—
196	C₂×C₄	—	2	1	no	—
197	C₂×C₄	—	2	1	no	—
198	C₂×C₄	#6381	1	2	yes	—
199	C₂×C₄	#6381	1	2	yes	—
200	C₂×C₄	#6382	1	2	yes	—
201	C₂×C₄	#6382	1	2	yes	—
202	C₂×C₄	Dic₇⋊C₈	1	2	yes	—
203	C₂×C₄	Dic₇⋊C₈	1	2	yes	—
204	C₂×C₄	#6396	1	2	yes	—
205	C₂×C₄	#6396	1	2	yes	—
206	C₂×C₄	#6453	1	2	yes	—
207	C₂×C₄	#6455	1	2	yes	—
208	C₂×C₄	C₂×C₈×D₇	1	1	yes	characteristic
209	C₂×C₄	C₂×#1131	1	1	yes	characteristic
210	C₂×C₄	D₇×M₁₆	1	2	yes	—
211	C₂×C₄	C₂×#1141	1	1	yes	characteristic
212	C₂×C₄	C₂×#1142	1	1	yes	characteristic
213	C₂×C₄	#6499	1	1	yes	characteristic
214	C₂×C₄	C₂×#1143	1	1	yes	characteristic
215	C₂×C₄	C₂×#1144	1	1	yes	characteristic
216	C₂×C₄	#6510	1	1	yes	characteristic
217	C₈	—	8	2	no	—
218	C₈	—	14	4	no	—
219	C₈	—	14	4	no	—
220	C₈	—	56	2	no	—
221	C₇	#8784	1	1	yes	characteristic, complemented, Sylow
222	C₂²	C₂×#6381	1	1	yes	characteristic
223	C₂²	C₂×#6382	1	1	yes	characteristic
224	C₂²	#89749	1	1	yes	characteristic
225	C₂²	C₂×Dic₇⋊C₈	1	1	yes	characteristic
226	C₂²	#89890	1	1	yes	characteristic
227	C₂²	C₂×#6396	1	1	yes	characteristic
228	C₂²	C₂×#6396	1	1	yes	characteristic
229	C₄	—	2	2	no	—
230	C₄	—	2	2	no	—
231	C₄	—	2	2	no	—
232	C₄	#89274	1	4	yes	—
233	C₄	#89275	1	4	yes	—
234	C₄	#89605	1	2	yes	—
235	C₄	#89607	1	2	yes	—
236	C₄	#89842	1	1	yes	characteristic
237	C₄	#89853	1	1	yes	characteristic
238	C₄	#89860	1	1	yes	characteristic
239	C₄	#89865	1	1	yes	characteristic
240	C₂	#11727328	1	1	yes	characteristic
241	C₂	C₂×#89275	1	1	yes	characteristic
242	C₂	C₂×#89274	1	1	yes	characteristic
243	C₂	#11728197	1	1	yes	characteristic
244	C₂	#11728244	1	1	yes	characteristic
245	C₂	#11728245	1	1	yes	characteristic
246	C₂	#11728270	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₇
commutator	C₂×C₄×C₇	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₇
Fitting	C₇×#1842	C₂
socle, prime radical	C₂³×C₇	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₇	C₂³
hypercenter	#1842	D₇
hypocenter	C₇	#8784
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1842

Complements

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

N	H ≅ G/N	type
38. C₄×C₇⋊C₈	C₈	semidirect
221. C₇	#8784	semidirect

Maximal subgroups

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

subgroup	index
C₇×#1842	2
#11727148	2
C₄²×C₇⋊C₈	2
#11725509	2
#8784	7