Subgroups — #13265230

Order 1792.

330 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#500	C₂	1	1	yes	maximal, characteristic
2	#11727498	C₂	1	1	yes	maximal, characteristic
3	#11727485	C₂	1	1	yes	maximal, characteristic
4	#11725608	C₂	1	4	yes	maximal, complemented
5	C₂×C₇×C₄⋊Q₈	—	2	1	no	—
6	C₇×#500	—	2	1	no	—
7	C₇×#500	—	2	1	no	—
8	C₇×#500	C₄	1	2	yes	—
9	C₇×#500	C₂²	1	2	yes	—
10	C₂×C₇×#107	—	2	1	no	—
11	C₂×C₇×#107	C₄	1	2	yes	—
12	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
13	#89696	—	2	1	no	—
14	C₂²⋊(C₇⋊C₁₆)	—	2	1	no	—
15	C₄⋊(C₇⋊C₁₆)	C₂²	1	2	yes	—
16	C₄×C₇⋊C₁₆	—	2	1	no	—
17	#89253	—	2	1	no	—
18	#89253	C₂²	1	2	yes	—
19	#11972	—	7	1	no	maximal, Sylow
20	C₂²×C₇×Q₈	—	4	1	no	—
21	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
23	C₇×C₄⋊Q₈	—	2	2	no	—
24	C₇×C₄⋊Q₈	—	2	2	no	—
25	C₇×C₄⋊Q₈	—	4	1	no	—
26	C₇×C₄⋊Q₈	—	4	1	no	—
27	C₂×C₇×C₄⋊C₄	—	2	2	no	—
28	C₂×C₇×C₄⋊C₄	—	4	1	no	—
29	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
30	C₂×#1136	—	4	1	no	—
31	C₇×#107	—	2	2	no	—
32	C₇×#107	—	2	2	no	—
33	C₇×#107	—	2	2	no	—
34	C₇×#107	C₂×C₄	1	4	yes	—
35	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
36	C₄×C₇×C₈	D₄	1	1	yes	characteristic
37	C₄×C₇×C₈	D₄	1	1	yes	characteristic
38	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
39	C₂×C₇⋊C₁₆	—	2	2	no	—
40	C₄⋊(C₇⋊C₈)	—	2	2	no	—
41	C₄⋊(C₇⋊C₈)	—	4	1	no	—
42	C₂×#500	D₇	1	1	yes	characteristic
43	#2162	—	7	1	no	—
44	#1567	—	7	1	no	—
45	#1365	—	7	4	no	—
46	C₂×C₇×Q₈	—	4	2	no	—
47	C₂×C₇×Q₈	—	4	2	no	—
48	C₂×C₇×Q₈	—	8	4	no	—
49	C₂²×C₄×C₇	—	4	2	no	—
50	C₂²×C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
51	C₂²×C₄×C₇	D₈	1	1	yes	characteristic
52	C₂²×C₄×C₇	SD₁₆	1	1	yes	characteristic
53	C₂×C₇×C₈	—	2	2	no	—
54	C₂×C₇×C₈	—	2	2	no	—
55	C₂×C₇×C₈	C₂²⋊C₄	1	1	yes	characteristic
56	C₂×C₇×C₈	C₂²⋊C₄	1	1	yes	characteristic
57	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
59	C₇×C₄⋊C₄	—	2	4	no	—
60	C₇×C₄⋊C₄	—	4	2	no	—
61	C₇×C₄⋊C₄	—	4	2	no	—
62	C₇×C₄⋊C₄	—	4	2	no	—
63	C₄²×C₇	C₂²⋊C₄	1	1	yes	characteristic
64	C₄²×C₇	D₈	1	1	yes	characteristic
65	C₄²×C₇	SD₁₆	1	1	yes	characteristic
66	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
67	#1136	—	8	2	no	—
68	C₂×C₇⋊C₈	—	4	2	no	—
69	C₇⋊C₁₆	—	4	2	no	—
70	C₂×C₄⋊Q₈	—	2	1	no	—
71	#500	—	2	1	no	—
72	#500	—	2	1	no	—
73	#500	Dic₇	1	2	yes	—
74	#500	C₂×D₇	1	2	yes	—
75	C₂×#107	—	2	1	no	—
76	C₂×#107	Dic₇	1	2	yes	—
77	#423	—	14	1	no	—
78	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
79	C₄⋊C₁₆	—	7	2	no	—
80	C₂²⋊C₁₆	—	14	1	no	—
81	C₄×C₁₆	—	14	1	no	—
82	#335	—	7	2	no	—
83	#335	—	14	1	no	—
84	C₂³×C₇	#102	1	1	yes	characteristic
85	C₇×Q₈	—	8	4	no	—
86	C₇×Q₈	—	8	4	no	—
87	C₂×C₄×C₇	—	2	2	no	—
88	C₂×C₄×C₇	—	2	1	no	—
89	C₂×C₄×C₇	—	2	1	no	—
90	C₂×C₄×C₇	—	4	4	no	—
91	C₂×C₄×C₇	—	8	4	no	—
92	C₂×C₄×C₇	#102	1	1	yes	characteristic
93	C₂×C₄×C₇	#102	1	1	yes	characteristic
94	C₂×C₄×C₇	#102	1	1	yes	characteristic
95	C₂×C₄×C₇	SD₃₂	1	2	yes	—
96	C₂×C₄×C₇	Q₃₂	1	2	yes	—
97	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
98	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
99	C₂×C₄×C₇	C₂×SD₁₆	1	1	yes	characteristic
100	C₇×C₈	—	2	1	no	—
101	C₇×C₈	—	2	1	no	—
102	C₇×C₈	#101	1	2	yes	—
103	C₇×C₈	#137	1	2	yes	—
104	C₇⋊C₈	—	8	2	no	—
105	C₂²×Q₈	—	4	1	no	—
106	C₂×M₁₆	—	28	1	no	—
107	C₂²×C₈	C₂×Dic₇	1	1	yes	characteristic
108	C₂²×C₈	C₂²⋊D₇	1	1	yes	characteristic
109	C₄⋊Q₈	—	2	2	no	—
110	C₄⋊Q₈	—	2	2	no	—
111	C₄⋊Q₈	—	4	1	no	—
112	C₄⋊Q₈	—	4	1	no	—
113	C₂×C₄⋊C₄	—	2	2	no	—
114	C₂×C₄⋊C₄	—	4	1	no	—
115	C₂×C₄²	C₂²⋊D₇	1	1	yes	characteristic
116	C₂×C₁₆	—	14	2	no	—
117	#107	—	2	2	no	—
118	#107	—	2	2	no	—
119	#107	—	2	2	no	—
120	#107	C₂×Dic₇	1	4	yes	—
121	C₄⋊C₈	—	14	2	no	—
122	C₄⋊C₈	—	28	1	no	—
123	C₄×C₈	C₂×Dic₇	1	1	yes	characteristic
124	C₄×C₈	C₂²⋊D₇	1	1	yes	characteristic
125	C₄×C₈	C₂²⋊D₇	1	1	yes	characteristic
126	C₄×C₈	C₂²×D₇	1	1	yes	characteristic
127	C₂²×C₇	—	2	1	no	—
128	C₂²×C₇	—	2	1	no	—
129	C₂²×C₇	#358	1	2	yes	—
130	C₂²×C₇	C₂×#102	1	1	yes	characteristic
131	C₄×C₇	—	2	1	no	—
132	C₄×C₇	—	2	1	no	—
133	C₄×C₇	—	8	4	no	—
134	C₄×C₇	#358	1	2	yes	—
135	C₄×C₇	#361	1	2	yes	—
136	C₄×C₇	C₂×#101	1	1	yes	characteristic
137	C₄×C₇	#419	1	1	yes	characteristic
138	C₄×C₇	C₂×SD₃₂	1	1	yes	characteristic
139	C₄×C₇	C₂×Q₃₂	1	1	yes	characteristic
140	C₂×Q₈	—	4	2	no	—
141	C₂×Q₈	—	4	2	no	—
142	C₂×Q₈	—	8	4	no	—
143	C₂²×C₄	—	4	2	no	—
144	C₂²×C₄	#1141	1	1	yes	characteristic
145	C₂²×C₄	#1142	1	1	yes	characteristic
146	C₂²×C₄	C₂²⋊Dic₇	1	1	yes	characteristic
147	M₁₆	—	56	2	no	—
148	C₂×C₈	—	2	2	no	—
149	C₂×C₈	—	2	2	no	—
150	C₂×C₈	—	28	2	no	—
151	C₂×C₈	C₂²⋊Dic₇	1	1	yes	characteristic
152	C₂×C₈	C₂²⋊Dic₇	1	1	yes	characteristic
153	C₂×C₈	C₂²×Dic₇	1	1	yes	characteristic
154	C₂×C₈	C₂×C₂²⋊D₇	1	1	yes	characteristic
155	C₄⋊C₄	—	2	4	no	—
156	C₄⋊C₄	—	4	2	no	—
157	C₄⋊C₄	—	4	2	no	—
158	C₄⋊C₄	—	4	2	no	—
159	C₄²	#1141	1	1	yes	characteristic
160	C₄²	#1142	1	1	yes	characteristic
161	C₄²	C₂²⋊Dic₇	1	1	yes	characteristic
162	C₄²	C₂×C₂²⋊D₇	1	1	yes	characteristic
163	C₁₆	—	28	2	no	—
164	C₂×C₇	—	2	2	no	—
165	C₂×C₇	#1546	1	1	yes	characteristic
166	C₂×C₇	C₂×#358	1	1	yes	characteristic
167	C₂×C₇	C₂×#361	1	1	yes	characteristic
168	C₂³	#6406	1	1	yes	characteristic
169	Q₈	—	8	4	no	—
170	Q₈	—	8	4	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	2	1	no	—
173	C₂×C₄	—	2	1	no	—
174	C₂×C₄	—	4	4	no	—
175	C₂×C₄	—	8	4	no	—
176	C₂×C₄	#6402	1	2	yes	—
177	C₂×C₄	#6403	1	2	yes	—
178	C₂×C₄	#6406	1	1	yes	characteristic
179	C₂×C₄	#6406	1	1	yes	characteristic
180	C₂×C₄	#6406	1	1	yes	characteristic
181	C₂×C₄	C₂×#1141	1	1	yes	characteristic
182	C₂×C₄	C₂×#1142	1	1	yes	characteristic
183	C₂×C₄	C₂×C₂²⋊Dic₇	1	1	yes	characteristic
184	C₈	—	2	1	no	—
185	C₈	—	2	1	no	—
186	C₈	—	56	2	no	—
187	C₈	#6410	1	2	yes	—
188	C₈	#6505	1	2	yes	—
189	C₇	#11972	1	1	yes	characteristic, complemented, Sylow
190	C₂²	—	2	1	no	—
191	C₂²	—	2	1	no	—
192	C₂²	#89359	1	2	yes	—
193	C₂²	C₂×#6406	1	1	yes	characteristic
194	C₄	—	2	1	no	—
195	C₄	—	2	1	no	—
196	C₄	—	8	4	no	—
197	C₄	#89358	1	2	yes	—
198	C₄	#89359	1	2	yes	—
199	C₄	C₂×#6402	1	1	yes	characteristic
200	C₄	C₂×#6403	1	1	yes	characteristic
201	C₄	#89997	1	1	yes	characteristic
202	C₄	C₂×#6410	1	1	yes	characteristic
203	C₂	—	2	2	no	—
204	C₂	#11729527	1	1	yes	characteristic
205	C₂	C₂×#89358	1	1	yes	characteristic
206	C₂	C₂×#89359	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#6406
commutator	C₂×C₇×C₈	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₇
Fitting	C₂×C₇×#500	C₂
socle	C₂²×C₇	C₂×#102
prime radical	C₂³×C₇	#102
cosocle	C₄×C₇×C₈	C₂³
hypercenter	C₂×#500	D₇
hypocenter	C₇	#11972
LCS	G ▷ C₂×C₇×C₈ ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#500

Complements

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

N	H ≅ G/N	type
4. #11725608	C₂	semidirect
189. C₇	#11972	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₇×#500	2
#11727498	2
#11727485	2
#11725608	2
#11972	7