Subgroups — #13268840

Order 1792.

490 conjugacy classes of subgroups.

The subgroup lattice has 254 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#405	C₂	1	1	yes	maximal, characteristic
2	C₂×#89348	C₂	1	2	yes	maximal
3	C₄×#6386	C₂	1	2	yes	maximal, complemented
4	#11725470	C₂	1	2	yes	maximal, complemented
5	C₂²×C₇×M₁₆	C₂²	1	1	yes	characteristic
6	C₂³×C₇×C₈	C₄	1	1	yes	characteristic
7	C₂×C₇×#117	C₄	1	1	yes	characteristic
8	C₇×#405	—	2	2	no	—
9	C₇×#405	—	2	2	no	—
10	C₂×C₇×#97	C₄	1	1	yes	characteristic
11	C₂×C₇×#97	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₇×C₈	C₄	1	1	yes	characteristic
13	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
14	C₂×#6386	C₂²	1	4	yes	—
15	#89348	C₄	1	8	yes	complemented
16	#89256	—	2	2	no	—
17	C₄×C₇⋊C₁₆	—	2	2	no	—
18	#12602	—	7	1	no	maximal, Sylow
19	C₂³×C₄×C₇	C₂×C₄	1	1	yes	characteristic
20	C₂×C₇×M₁₆	—	2	2	no	—
21	C₂×C₇×M₁₆	—	2	2	no	—
22	C₂×C₇×M₁₆	C₂×C₄	1	4	yes	—
23	C₂²×C₇×C₈	—	2	1	no	—
24	C₂²×C₇×C₈	—	2	1	no	—
25	C₂²×C₇×C₈	—	2	2	no	—
26	C₂²×C₇×C₈	C₈	1	4	yes	—
27	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
31	C₇×#117	—	2	2	no	—
32	C₇×#117	—	2	2	no	—
33	C₂×C₇×C₄⋊C₄	C₈	1	2	yes	—
34	C₂×C₇×C₂²⋊C₄	C₈	1	2	yes	—
35	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
37	C₇×#97	—	2	2	no	—
38	C₇×#97	—	2	2	no	—
39	C₄×C₇×C₈	—	2	1	no	—
40	C₄×C₇×C₈	—	2	1	no	—
41	C₄×C₇×C₈	D₄	1	2	yes	—
42	C₄×C₇×C₈	D₄	1	2	yes	—
43	#6386	—	2	4	no	—
44	#6386	C₂×C₄	1	8	yes	—
45	C₂×C₇⋊C₁₆	—	2	4	no	—
46	C₂×#405	D₇	1	1	yes	characteristic
47	C₂×#349	—	7	2	no	—
48	C₄×M₃₂	—	7	2	no	—
49	#1377	—	7	2	no	—
50	C₂⁴×C₇	C₂×C₈	1	1	yes	characteristic
51	C₂²×C₄×C₇	—	2	1	no	—
52	C₂²×C₄×C₇	—	2	1	no	—
53	C₂²×C₄×C₇	—	2	1	no	—
54	C₂²×C₄×C₇	—	2	1	no	—
55	C₂²×C₄×C₇	C₄²	1	2	yes	—
56	C₂²×C₄×C₇	C₂×C₈	1	1	yes	characteristic
57	C₂²×C₄×C₇	C₂×C₈	1	2	yes	—
58	C₂²×C₄×C₇	C₂×C₈	1	2	yes	—
59	C₂²×C₄×C₇	C₂×C₈	1	2	yes	—
60	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
61	C₇×M₁₆	—	2	8	no	—
62	C₂×C₇×C₈	—	2	1	no	—
63	C₂×C₇×C₈	—	2	1	no	—
64	C₂×C₇×C₈	—	2	1	no	—
65	C₂×C₇×C₈	—	2	1	no	—
66	C₂×C₇×C₈	—	2	4	no	—
67	C₂×C₇×C₈	—	2	4	no	—
68	C₂×C₇×C₈	C₄²	1	2	yes	—
69	C₂×C₇×C₈	C₂²⋊C₄	1	1	yes	characteristic
70	C₂×C₇×C₈	C₂²⋊C₄	1	1	yes	characteristic
71	C₂×C₇×C₈	C₂×C₈	1	2	yes	—
72	C₂×C₇×C₈	C₂×C₈	1	2	yes	—
73	C₂×C₇×C₈	M₁₆	1	2	yes	—
74	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
75	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
76	C₂×C₇×C₈	#41	1	2	yes	—
77	C₇×C₄⋊C₄	—	2	4	no	—
78	C₇×C₂²⋊C₄	—	2	4	no	—
79	C₄²×C₇	—	2	1	no	—
80	C₄²×C₇	—	2	1	no	—
81	C₄²×C₇	C₂²⋊C₄	1	2	yes	—
82	C₄²×C₇	M₁₆	1	2	yes	—
83	C₇⋊C₁₆	—	2	8	no	—
84	C₂²×M₁₆	C₂×D₇	1	1	yes	characteristic
85	C₂³×C₈	Dic₇	1	1	yes	characteristic
86	C₂×#117	Dic₇	1	1	yes	characteristic
87	C₂×M₃₂	—	7	4	no	—
88	#405	—	2	2	no	—
89	#405	—	2	2	no	—
90	C₂×#97	Dic₇	1	1	yes	characteristic
91	C₂×#97	C₂×D₇	1	1	yes	characteristic
92	C₂×C₄×C₈	Dic₇	1	1	yes	characteristic
93	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
94	#349	—	7	8	no	—
95	#346	—	14	2	no	—
96	C₄×C₁₆	—	14	2	no	—
97	C₂³×C₇	—	2	1	no	—
98	C₂³×C₇	—	2	1	no	—
99	C₂³×C₇	—	4	2	no	—
100	C₂³×C₇	C₄×C₈	1	2	yes	—
101	C₂³×C₇	C₂²×C₈	1	1	yes	characteristic
102	C₂×C₄×C₇	—	2	1	no	—
103	C₂×C₄×C₇	—	2	1	no	—
104	C₂×C₄×C₇	—	2	2	no	—
105	C₂×C₄×C₇	—	2	2	no	—
106	C₂×C₄×C₇	—	2	2	no	—
107	C₂×C₄×C₇	—	2	2	no	—
108	C₂×C₄×C₇	—	2	2	no	—
109	C₂×C₄×C₇	—	4	2	no	—
110	C₂×C₄×C₇	—	4	2	no	—
111	C₂×C₄×C₇	C₄×C₈	1	2	yes	—
112	C₂×C₄×C₇	C₂²⋊C₈	1	4	yes	—
113	C₂×C₄×C₇	C₂×C₄²	1	1	yes	characteristic
114	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
115	C₂×C₄×C₇	#117	1	1	yes	characteristic
116	C₂×C₄×C₇	C₂²×C₈	1	1	yes	characteristic
117	C₂×C₄×C₇	C₂×M₁₆	1	1	yes	characteristic
118	C₂×C₄×C₇	C₂×M₁₆	1	1	yes	characteristic
119	C₇×C₈	—	2	2	no	—
120	C₇×C₈	—	2	2	no	—
121	C₇×C₈	C₂²⋊C₈	1	2	yes	—
122	C₇×C₈	C₂²⋊C₈	1	2	yes	—
123	C₇×C₈	C₄×D₄	1	4	yes	—
124	C₂³×C₄	C₂×Dic₇	1	1	yes	characteristic
125	C₂×M₁₆	—	2	2	no	—
126	C₂×M₁₆	—	2	2	no	—
127	C₂×M₁₆	C₄×D₇	1	4	yes	—
128	C₂²×C₈	—	2	2	no	—
129	C₂²×C₈	—	2	1	no	—
130	C₂²×C₈	—	2	1	no	—
131	C₂²×C₈	C₇⋊C₈	1	4	yes	—
132	C₂²×C₈	C₂×Dic₇	1	1	yes	characteristic
133	C₂²×C₈	C₂×Dic₇	1	1	yes	characteristic
134	C₂²×C₈	C₂×Dic₇	1	1	yes	characteristic
135	C₂²×C₈	C₂²×D₇	1	1	yes	characteristic
136	#117	—	2	2	no	—
137	#117	—	2	2	no	—
138	C₂×C₄⋊C₄	C₇⋊C₈	1	2	yes	—
139	C₂×C₂²⋊C₄	C₇⋊C₈	1	2	yes	—
140	C₂×C₄²	C₂×Dic₇	1	1	yes	characteristic
141	C₂×C₄²	C₂×Dic₇	1	1	yes	characteristic
142	M₃₂	—	7	8	no	—
143	M₃₂	—	14	4	no	—
144	C₂×C₁₆	—	14	4	no	—
145	#97	—	2	2	no	—
146	#97	—	2	2	no	—
147	C₄×C₈	—	2	1	no	—
148	C₄×C₈	—	2	1	no	—
149	C₄×C₈	C₂²⋊D₇	1	2	yes	—
150	C₄×C₈	C₂²⋊D₇	1	2	yes	—
151	C₂²×C₇	—	2	2	no	—
152	C₂²×C₇	—	2	2	no	—
153	C₂²×C₇	—	4	2	no	—
154	C₂²×C₇	—	4	2	no	—
155	C₂²×C₇	—	4	2	no	—
156	C₂²×C₇	C₂×C₄×C₈	1	1	yes	characteristic
157	C₂²×C₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
158	C₂²×C₇	#431	1	1	yes	characteristic
159	C₄×C₇	—	2	4	no	—
160	C₄×C₇	—	2	2	no	—
161	C₄×C₇	—	4	2	no	—
162	C₄×C₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
163	C₄×C₇	C₄×M₁₆	1	1	yes	characteristic
164	C₄×C₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
165	C₄×C₇	#431	1	1	yes	characteristic
166	C₂⁴	C₂×C₇⋊C₈	1	1	yes	characteristic
167	C₂²×C₄	—	2	1	no	—
168	C₂²×C₄	—	2	1	no	—
169	C₂²×C₄	—	2	1	no	—
170	C₂²×C₄	—	2	1	no	—
171	C₂²×C₄	C₂×C₇⋊C₈	1	2	yes	—
172	C₂²×C₄	C₂×C₇⋊C₈	1	1	yes	characteristic
173	C₂²×C₄	C₂×C₇⋊C₈	1	2	yes	—
174	C₂²×C₄	C₂×C₇⋊C₈	1	2	yes	—
175	C₂²×C₄	C₄×Dic₇	1	2	yes	—
176	C₂²×C₄	C₂²×Dic₇	1	1	yes	characteristic
177	M₁₆	—	2	8	no	—
178	C₂×C₈	—	2	1	no	—
179	C₂×C₈	—	2	1	no	—
180	C₂×C₈	—	2	1	no	—
181	C₂×C₈	—	2	1	no	—
182	C₂×C₈	—	2	4	no	—
183	C₂×C₈	—	2	4	no	—
184	C₂×C₈	C₂×C₇⋊C₈	1	2	yes	—
185	C₂×C₈	C₂×C₇⋊C₈	1	2	yes	—
186	C₂×C₈	#1136	1	2	yes	—
187	C₂×C₈	C₄×Dic₇	1	2	yes	—
188	C₂×C₈	C₂²⋊Dic₇	1	1	yes	characteristic
189	C₂×C₈	C₂²⋊Dic₇	1	1	yes	characteristic
190	C₂×C₈	C₂×C₄×D₇	1	2	yes	—
191	C₂×C₈	#1157	1	2	yes	—
192	C₂×C₈	C₂×C₂²⋊D₇	1	2	yes	—
193	C₄⋊C₄	—	2	4	no	—
194	C₂²⋊C₄	—	2	4	no	—
195	C₄²	—	2	1	no	—
196	C₄²	—	2	1	no	—
197	C₄²	#1136	1	2	yes	—
198	C₄²	C₂²⋊Dic₇	1	2	yes	—
199	C₁₆	—	14	8	no	—
200	C₂×C₇	—	2	1	no	—
201	C₂×C₇	—	2	1	no	—
202	C₂×C₇	—	4	2	no	—
203	C₂×C₇	C₄×C₂²⋊C₈	1	1	yes	characteristic
204	C₂×C₇	#2115	1	2	yes	—
205	C₂³	—	2	1	no	—
206	C₂³	—	2	1	no	—
207	C₂³	—	4	2	no	—
208	C₂³	C₄×C₇⋊C₈	1	2	yes	—
209	C₂³	C₂²×C₇⋊C₈	1	1	yes	characteristic
210	C₂×C₄	—	2	1	no	—
211	C₂×C₄	—	2	1	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	4	2	no	—
218	C₂×C₄	—	4	2	no	—
219	C₂×C₄	C₄×C₇⋊C₈	1	2	yes	—
220	C₂×C₄	C₂²⋊(C₇⋊C₈)	1	4	yes	—
221	C₂×C₄	C₂²×C₇⋊C₈	1	1	yes	characteristic
222	C₂×C₄	C₂×#1136	1	1	yes	characteristic
223	C₂×C₄	C₂×#1136	1	1	yes	characteristic
224	C₂×C₄	C₂×C₄×Dic₇	1	1	yes	characteristic
225	C₂×C₄	#6489	1	1	yes	characteristic
226	C₂×C₄	C₂×C₂²⋊Dic₇	1	1	yes	characteristic
227	C₈	—	2	2	no	—
228	C₈	—	2	2	no	—
229	C₈	C₂²⋊(C₇⋊C₈)	1	2	yes	—
230	C₈	C₂²⋊(C₇⋊C₈)	1	2	yes	—
231	C₈	C₄×C₂²⋊D₇	1	4	yes	—
232	C₇	#12602	1	1	yes	characteristic, complemented, Sylow
233	C₂²	—	2	2	no	—
234	C₂²	—	2	2	no	—
235	C₂²	—	4	2	no	—
236	C₂²	—	4	2	no	—
237	C₂²	—	4	2	no	—
238	C₂²	C₂×C₄×C₇⋊C₈	1	1	yes	characteristic
239	C₂²	#89697	1	1	yes	characteristic
240	C₂²	C₂×C₂²⋊(C₇⋊C₈)	1	1	yes	characteristic
241	C₄	—	2	4	no	—
242	C₄	—	2	2	no	—
243	C₄	—	4	2	no	—
244	C₄	C₄×#1136	1	1	yes	characteristic
245	C₄	#89697	1	1	yes	characteristic
246	C₄	C₂×C₂²⋊(C₇⋊C₈)	1	1	yes	characteristic
247	C₄	C₄×C₂²⋊Dic₇	1	1	yes	characteristic
248	C₂	—	2	1	no	—
249	C₂	—	2	1	no	—
250	C₂	—	4	2	no	—
251	C₂	C₄×C₂²⋊(C₇⋊C₈)	1	1	yes	characteristic
252	C₂	#11729761	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	C₂²⋊Dic₇
commutator	C₂²×C₇	C₂×C₄×C₈
Frattini	C₂²×C₈	C₂²×D₇
Fitting	C₂×C₇×#405	C₂
socle	C₂²×C₇	C₂×C₂²⋊C₈
prime radical	C₂⁴×C₇	C₂×C₈
cosocle	C₂²×C₇×C₈	C₂³
hypercenter	C₂×#405	D₇
hypocenter	C₇	#12602
LCS	G ▷ C₂²×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ C₂×#405

Complements

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

N	H ≅ G/N	type
3. C₄×#6386	C₂	semidirect
4. #11725470	C₂	semidirect
15. #89348	C₄	semidirect
232. C₇	#12602	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₇×#405	2
C₂×#89348	2
C₄×#6386	2
#11725470	2
#12602	7