Subgroups — #13243262

Order 1792.

758 conjugacy classes of subgroups.

The subgroup lattice has 238 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×C₇×C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#89696	C₂	1	2	yes	maximal
3	#11725460	C₂	1	4	yes	maximal, complemented
4	C₂³×C₇×C₈	C₄	1	2	yes	—
5	C₂²×C₄²×C₇	C₂²	1	1	yes	characteristic
6	C₂²×#1136	—	2	2	no	—
7	C₂×C₄×C₇×C₈	—	2	4	no	—
8	C₂×C₄×C₇×C₈	C₄	1	2	yes	—
9	C₂×C₄×C₇×C₈	C₂²	1	2	yes	—
10	#89696	—	2	4	no	—
11	#89696	C₄	1	8	yes	—
12	C₂×C₄⋊(C₇⋊C₈)	—	2	2	no	—
13	C₂×C₄⋊(C₇⋊C₈)	C₂²	1	4	yes	—
14	#7874	—	7	1	no	maximal, Sylow
15	C₂³×C₄×C₇	C₂×C₄	1	1	yes	characteristic
16	C₂³×C₄×C₇	D₄	1	1	yes	characteristic
17	C₂³×C₄×C₇	Q₈	1	1	yes	characteristic
18	C₂²×C₇×C₈	—	2	4	no	—
19	C₂²×C₇×C₈	—	2	8	no	—
20	C₂²×C₇×C₈	C₂×C₄	1	4	yes	—
21	C₂×C₄²×C₇	C₂×C₄	1	4	yes	—
22	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
24	C₂×C₄²×C₇	D₄	1	4	yes	—
25	C₂×C₄²×C₇	Q₈	1	1	yes	characteristic
26	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
27	C₂×#1136	—	2	8	no	—
28	C₂×#1136	—	4	4	no	—
29	C₂×#1136	—	4	4	no	—
30	C₂²×C₇⋊C₈	—	2	4	no	—
31	C₄×C₇×C₈	—	2	4	no	—
32	C₄×C₇×C₈	—	2	4	no	—
33	C₄⋊(C₇⋊C₈)	—	2	8	no	—
34	C₄⋊(C₇⋊C₈)	—	2	4	no	—
35	C₄⋊(C₇⋊C₈)	C₂×C₄	1	8	yes	—
36	C₂×#423	—	7	2	no	—
37	C₂²×C₄×C₈	D₇	1	1	yes	characteristic
38	#1277	—	7	4	no	—
39	C₂⁴×C₇	C₄⋊C₄	1	1	yes	characteristic
40	C₂²×C₄×C₇	—	2	4	no	—
41	C₂²×C₄×C₇	—	2	4	no	—
42	C₂²×C₄×C₇	C₄²	1	2	yes	—
43	C₂²×C₄×C₇	C₂²⋊C₄	1	2	yes	—
44	C₂²×C₄×C₇	C₂²⋊C₄	1	4	yes	—
45	C₂²×C₄×C₇	C₄⋊C₄	1	1	yes	characteristic
46	C₂²×C₄×C₇	C₄⋊C₄	1	4	yes	—
47	C₂²×C₄×C₇	C₄⋊C₄	1	1	yes	characteristic
48	C₂²×C₄×C₇	C₄⋊C₄	1	1	yes	characteristic
49	C₂²×C₄×C₇	D₈	1	2	yes	—
50	C₂²×C₄×C₇	SD₁₆	1	4	yes	—
51	C₂²×C₄×C₇	Q₁₆	1	2	yes	—
52	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
53	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄×C₇	C₂×Q₈	1	1	yes	characteristic
55	C₂×C₇×C₈	—	2	16	no	—
56	C₂×C₇×C₈	—	2	4	no	—
57	C₂×C₇×C₈	—	2	4	no	—
58	C₂×C₇×C₈	M₁₆	1	8	yes	complemented
59	C₄²×C₇	C₄²	1	2	yes	—
60	C₄²×C₇	C₂²⋊C₄	1	4	yes	—
61	C₄²×C₇	C₂²⋊C₄	1	2	yes	—
62	C₄²×C₇	C₄⋊C₄	1	4	yes	—
63	C₄²×C₇	C₂²×C₄	1	2	yes	—
64	C₄²×C₇	C₂×D₄	1	2	yes	—
65	#1136	—	4	16	no	—
66	C₂×C₇⋊C₈	—	2	8	no	—
67	C₂×C₇⋊C₈	—	4	4	no	—
68	C₂×C₇⋊C₈	—	4	4	no	—
69	C₂²×M₁₆	—	14	2	no	—
70	C₂³×C₈	Dic₇	1	2	yes	—
71	C₂²×C₄²	C₂×D₇	1	1	yes	characteristic
72	#423	—	7	8	no	—
73	#423	—	14	4	no	—
74	C₂×C₄⋊C₈	—	7	4	no	—
75	C₂×C₄⋊C₈	—	14	2	no	—
76	C₂×C₄×C₈	—	2	4	no	—
77	C₂×C₄×C₈	Dic₇	1	2	yes	—
78	C₂×C₄×C₈	C₂×D₇	1	2	yes	—
79	C₂³×C₇	—	2	2	no	—
80	C₂³×C₇	—	2	2	no	—
81	C₂³×C₇	#95	1	2	yes	—
82	C₂³×C₇	#106	1	2	yes	—
83	C₂³×C₇	#107	1	2	yes	—
84	C₂³×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
85	C₂×C₄×C₇	—	2	2	no	—
86	C₂×C₄×C₇	—	2	2	no	—
87	C₂×C₄×C₇	—	2	8	no	—
88	C₂×C₄×C₇	—	2	4	no	—
89	C₂×C₄×C₇	—	2	4	no	—
90	C₂×C₄×C₇	#95	1	2	yes	—
91	C₂×C₄×C₇	#95	1	2	yes	—
92	C₂×C₄×C₇	#95	1	2	yes	—
93	C₂×C₄×C₇	#102	1	8	yes	—
94	C₂×C₄×C₇	#103	1	8	yes	—
95	C₂×C₄×C₇	#106	1	2	yes	—
96	C₂×C₄×C₇	#107	1	2	yes	—
97	C₂×C₄×C₇	#108	1	4	yes	—
98	C₂×C₄×C₇	C₂×C₄²	1	1	yes	characteristic
99	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	2	yes	—
100	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
101	C₂×C₄×C₇	C₂×C₄⋊C₄	1	2	yes	—
102	C₂×C₄×C₇	C₂×M₁₆	1	4	yes	—
103	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
104	C₂×C₄×C₇	C₂×SD₁₆	1	2	yes	—
105	C₂×C₄×C₇	C₂×Q₁₆	1	1	yes	characteristic
106	C₇×C₈	—	2	8	no	—
107	C₇⋊C₈	—	4	8	no	—
108	C₂³×C₄	Dic₁₄	1	1	yes	characteristic
109	C₂³×C₄	D₂₈	1	1	yes	characteristic
110	C₂³×C₄	C₂×Dic₇	1	1	yes	characteristic
111	C₂×M₁₆	—	14	8	no	—
112	C₂×M₁₆	—	28	4	no	—
113	C₂×M₁₆	—	28	4	no	—
114	C₂²×C₈	—	2	4	no	—
115	C₂²×C₈	—	2	8	no	—
116	C₂²×C₈	—	14	4	no	—
117	C₂²×C₈	C₂×Dic₇	1	4	yes	—
118	C₂×C₄²	Dic₁₄	1	1	yes	characteristic
119	C₂×C₄²	C₄×D₇	1	4	yes	—
120	C₂×C₄²	D₂₈	1	1	yes	characteristic
121	C₂×C₄²	C₂×Dic₇	1	1	yes	characteristic
122	C₂×C₄²	C₂²⋊D₇	1	4	yes	—
123	C₂×C₄²	C₂²×D₇	1	1	yes	characteristic
124	C₄⋊C₈	—	7	8	no	—
125	C₄⋊C₈	—	14	8	no	—
126	C₄⋊C₈	—	14	4	no	—
127	C₄×C₈	—	2	4	no	—
128	C₄×C₈	—	2	4	no	—
129	C₂²×C₇	—	2	4	no	—
130	C₂²×C₇	—	2	4	no	—
131	C₂²×C₇	—	2	2	no	—
132	C₂²×C₇	—	2	2	no	—
133	C₂²×C₇	#340	1	4	yes	—
134	C₂²×C₇	C₂×#95	1	1	yes	characteristic
135	C₂²×C₇	#423	1	2	yes	—
136	C₂²×C₇	C₂×#106	1	1	yes	characteristic
137	C₂²×C₇	C₂×#107	1	1	yes	characteristic
138	C₂²×C₇	C₂×#108	1	2	yes	—
139	C₄×C₇	—	2	2	no	—
140	C₄×C₇	—	2	2	no	—
141	C₄×C₇	#340	1	4	yes	—
142	C₄×C₇	#341	1	4	yes	—
143	C₄×C₇	C₄×M₁₆	1	2	yes	—
144	C₄×C₇	#407	1	2	yes	—
145	C₄×C₇	C₂×#102	1	2	yes	—
146	C₄×C₇	C₂×#103	1	2	yes	—
147	C₂⁴	C₄⋊Dic₇	1	1	yes	characteristic
148	C₂²×C₄	—	2	4	no	—
149	C₂²×C₄	—	2	4	no	—
150	C₂²×C₄	#1132	1	4	yes	—
151	C₂²×C₄	D₅₆	1	2	yes	—
152	C₂²×C₄	Dic₂₈	1	2	yes	—
153	C₂²×C₄	C₄×Dic₇	1	2	yes	—
154	C₂²×C₄	Dic₇⋊C₄	1	4	yes	—
155	C₂²×C₄	C₄⋊Dic₇	1	1	yes	characteristic
156	C₂²×C₄	C₄⋊Dic₇	1	1	yes	characteristic
157	C₂²×C₄	C₄⋊Dic₇	1	1	yes	characteristic
158	C₂²×C₄	(C₂×D₇)⋊C₄	1	4	yes	—
159	C₂²×C₄	C₂²⋊Dic₇	1	2	yes	—
160	C₂²×C₄	C₂×Dic₁₄	1	1	yes	characteristic
161	C₂²×C₄	C₂×D₂₈	1	1	yes	characteristic
162	C₂²×C₄	C₂²×Dic₇	1	1	yes	characteristic
163	M₁₆	—	28	16	no	—
164	C₂×C₈	—	2	16	no	—
165	C₂×C₈	—	2	4	no	—
166	C₂×C₈	—	2	4	no	—
167	C₂×C₈	—	14	8	no	—
168	C₂×C₈	—	28	4	no	—
169	C₂×C₈	—	28	4	no	—
170	C₂×C₈	#1136	1	8	yes	complemented
171	C₄²	C₄×Dic₇	1	2	yes	—
172	C₄²	Dic₇⋊C₄	1	4	yes	—
173	C₄²	(C₂×D₇)⋊C₄	1	4	yes	—
174	C₄²	C₂²⋊Dic₇	1	2	yes	—
175	C₄²	C₂×C₄×D₇	1	2	yes	—
176	C₄²	C₂×C₂²⋊D₇	1	2	yes	—
177	C₂×C₇	—	2	4	no	—
178	C₂×C₇	#1566	1	2	yes	—
179	C₂×C₇	#1567	1	2	yes	—
180	C₂×C₇	#1728	1	1	yes	characteristic
181	C₂×C₇	C₂×#340	1	1	yes	characteristic
182	C₂×C₇	C₂×#341	1	1	yes	characteristic
183	C₂³	—	2	2	no	—
184	C₂³	—	2	2	no	—
185	C₂³	#6391	1	2	yes	—
186	C₂³	#6392	1	2	yes	—
187	C₂³	C₇⋊#95	1	2	yes	—
188	C₂³	C₂×C₄⋊Dic₇	1	1	yes	characteristic
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	2	8	no	—
192	C₂×C₄	—	2	4	no	—
193	C₂×C₄	—	2	4	no	—
194	C₂×C₄	#6390	1	8	yes	—
195	C₂×C₄	#6391	1	2	yes	—
196	C₂×C₄	#6392	1	2	yes	—
197	C₂×C₄	#6393	1	4	yes	—
198	C₂×C₄	#6395	1	8	yes	—
199	C₂×C₄	C₇⋊#95	1	2	yes	—
200	C₂×C₄	C₇⋊#95	1	2	yes	—
201	C₂×C₄	C₇⋊#95	1	2	yes	—
202	C₂×C₄	C₂×#1132	1	2	yes	—
203	C₂×C₄	C₂×D₅₆	1	1	yes	characteristic
204	C₂×C₄	C₂×Dic₂₈	1	1	yes	characteristic
205	C₂×C₄	C₂×#1136	1	4	yes	—
206	C₂×C₄	C₂×C₄×Dic₇	1	1	yes	characteristic
207	C₂×C₄	C₂×Dic₇⋊C₄	1	2	yes	—
208	C₂×C₄	C₂×(C₂×D₇)⋊C₄	1	2	yes	—
209	C₂×C₄	C₂×C₂²⋊Dic₇	1	1	yes	characteristic
210	C₈	—	2	8	no	—
211	C₈	—	28	8	no	—
212	C₇	#7874	1	1	yes	characteristic, complemented, Sylow
213	C₂²	—	2	2	no	—
214	C₂²	—	2	4	no	—
215	C₂²	—	2	4	no	—
216	C₂²	—	2	2	no	—
217	C₂²	#89346	1	4	yes	—
218	C₂²	#89696	1	2	yes	—
219	C₂²	C₂×#6391	1	1	yes	characteristic
220	C₂²	C₂×#6392	1	1	yes	characteristic
221	C₂²	C₂×#6393	1	2	yes	—
222	C₂²	C₂×C₇⋊#95	1	1	yes	characteristic
223	C₄	—	2	2	no	—
224	C₄	—	2	2	no	—
225	C₄	#89346	1	4	yes	—
226	C₄	#89347	1	4	yes	—
227	C₄	C₄×#1136	1	2	yes	—
228	C₄	C₂×#6390	1	2	yes	—
229	C₄	C₂×#6395	1	2	yes	—
230	C₄	#89979	1	2	yes	—
231	C₂	—	2	4	no	—
232	C₂	#11727484	1	2	yes	—
233	C₂	#11727485	1	2	yes	—
234	C₂	#11729281	1	1	yes	characteristic
235	C₂	C₂×#89346	1	1	yes	characteristic
236	C₂	C₂×#89347	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₈
commutator	C₂×C₄×C₇	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₇
Fitting	C₂²×C₄×C₇×C₈	C₂
socle	C₂³×C₇	C₂×C₄⋊C₄
prime radical	C₂⁴×C₇	C₄⋊C₄
cosocle	C₂×C₄²×C₇	C₂³
hypercenter	C₂²×C₄×C₈	D₇
hypocenter	C₇	#7874
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
3. #11725460	C₂	semidirect
58. C₂×C₇×C₈	M₁₆	semidirect
170. C₂×C₈	#1136	semidirect
212. C₇	#7874	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₄×C₇×C₈	2
C₂×#89696	2
#11725460	2
#7874	7