Subgroups — #13265226

Order 1792.

330 conjugacy classes of subgroups.

The subgroup lattice has 212 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	#11725607	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	#11971	—	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	#1364	—	7	4	no	—
46	C₂×C₇×Q₈	—	4	2	no	—
47	C₂×C₇×Q₈	—	4	2	no	—
48	C₂×C₇×Q₈	—	8	2	no	—
49	C₂×C₇×Q₈	—	8	2	no	—
50	C₂²×C₄×C₇	—	4	2	no	—
51	C₂²×C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
52	C₂²×C₄×C₇	D₈	1	1	yes	characteristic
53	C₂²×C₄×C₇	SD₁₆	1	1	yes	characteristic
54	C₂×C₇×C₈	—	2	2	no	—
55	C₂×C₇×C₈	—	2	2	no	—
56	C₂×C₇×C₈	C₂²⋊C₄	1	1	yes	characteristic
57	C₂×C₇×C₈	C₂²⋊C₄	1	1	yes	characteristic
58	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
60	C₇×C₄⋊C₄	—	2	4	no	—
61	C₇×C₄⋊C₄	—	4	2	no	—
62	C₇×C₄⋊C₄	—	4	2	no	—
63	C₇×C₄⋊C₄	—	4	2	no	—
64	C₄²×C₇	C₂²⋊C₄	1	1	yes	characteristic
65	C₄²×C₇	D₈	1	1	yes	characteristic
66	C₄²×C₇	SD₁₆	1	1	yes	characteristic
67	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
68	#1136	—	8	2	no	—
69	C₂×C₇⋊C₈	—	4	2	no	—
70	C₇⋊C₁₆	—	4	2	no	—
71	C₂×C₄⋊Q₈	—	2	1	no	—
72	#500	—	2	1	no	—
73	#500	—	2	1	no	—
74	#500	Dic₇	1	2	yes	—
75	#500	C₂×D₇	1	2	yes	—
76	C₂×#107	—	2	1	no	—
77	C₂×#107	Dic₇	1	2	yes	—
78	#423	—	14	1	no	—
79	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
80	C₄⋊C₁₆	—	7	2	no	—
81	C₂²⋊C₁₆	—	14	1	no	—
82	C₄×C₁₆	—	14	1	no	—
83	#335	—	7	2	no	—
84	#335	—	14	1	no	—
85	C₂³×C₇	#102	1	1	yes	characteristic
86	C₇×Q₈	—	8	4	no	—
87	C₇×Q₈	—	8	2	no	—
88	C₇×Q₈	—	8	2	no	—
89	C₂×C₄×C₇	—	2	2	no	—
90	C₂×C₄×C₇	—	2	1	no	—
91	C₂×C₄×C₇	—	2	1	no	—
92	C₂×C₄×C₇	—	4	4	no	—
93	C₂×C₄×C₇	—	8	4	no	—
94	C₂×C₄×C₇	#102	1	1	yes	characteristic
95	C₂×C₄×C₇	#102	1	1	yes	characteristic
96	C₂×C₄×C₇	#102	1	1	yes	characteristic
97	C₂×C₄×C₇	D₁₆	1	2	yes	—
98	C₂×C₄×C₇	SD₃₂	1	2	yes	—
99	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
100	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
101	C₂×C₄×C₇	C₂×SD₁₆	1	1	yes	characteristic
102	C₇×C₈	—	2	1	no	—
103	C₇×C₈	—	2	1	no	—
104	C₇×C₈	#101	1	2	yes	—
105	C₇×C₈	#137	1	2	yes	—
106	C₇⋊C₈	—	8	2	no	—
107	C₂²×Q₈	—	4	1	no	—
108	C₂×M₁₆	—	28	1	no	—
109	C₂²×C₈	C₂×Dic₇	1	1	yes	characteristic
110	C₂²×C₈	C₂²⋊D₇	1	1	yes	characteristic
111	C₄⋊Q₈	—	2	2	no	—
112	C₄⋊Q₈	—	2	2	no	—
113	C₄⋊Q₈	—	4	1	no	—
114	C₄⋊Q₈	—	4	1	no	—
115	C₂×C₄⋊C₄	—	2	2	no	—
116	C₂×C₄⋊C₄	—	4	1	no	—
117	C₂×C₄²	C₂²⋊D₇	1	1	yes	characteristic
118	C₂×C₁₆	—	14	2	no	—
119	#107	—	2	2	no	—
120	#107	—	2	2	no	—
121	#107	—	2	2	no	—
122	#107	C₂×Dic₇	1	4	yes	—
123	C₄⋊C₈	—	14	2	no	—
124	C₄⋊C₈	—	28	1	no	—
125	C₄×C₈	C₂×Dic₇	1	1	yes	characteristic
126	C₄×C₈	C₂²⋊D₇	1	1	yes	characteristic
127	C₄×C₈	C₂²⋊D₇	1	1	yes	characteristic
128	C₄×C₈	C₂²×D₇	1	1	yes	characteristic
129	C₂²×C₇	—	2	1	no	—
130	C₂²×C₇	—	2	1	no	—
131	C₂²×C₇	#357	1	2	yes	—
132	C₂²×C₇	C₂×#102	1	1	yes	characteristic
133	C₄×C₇	—	2	1	no	—
134	C₄×C₇	—	2	1	no	—
135	C₄×C₇	—	8	4	no	—
136	C₄×C₇	#357	1	2	yes	—
137	C₄×C₇	#362	1	2	yes	—
138	C₄×C₇	C₂×#101	1	1	yes	characteristic
139	C₄×C₇	#419	1	1	yes	characteristic
140	C₄×C₇	C₂×D₁₆	1	1	yes	characteristic
141	C₄×C₇	C₂×SD₃₂	1	1	yes	characteristic
142	C₂×Q₈	—	4	2	no	—
143	C₂×Q₈	—	4	2	no	—
144	C₂×Q₈	—	8	2	no	—
145	C₂×Q₈	—	8	2	no	—
146	C₂²×C₄	—	4	2	no	—
147	C₂²×C₄	#1141	1	1	yes	characteristic
148	C₂²×C₄	#1142	1	1	yes	characteristic
149	C₂²×C₄	C₂²⋊Dic₇	1	1	yes	characteristic
150	M₁₆	—	56	2	no	—
151	C₂×C₈	—	2	2	no	—
152	C₂×C₈	—	2	2	no	—
153	C₂×C₈	—	28	2	no	—
154	C₂×C₈	C₂²⋊Dic₇	1	1	yes	characteristic
155	C₂×C₈	C₂²⋊Dic₇	1	1	yes	characteristic
156	C₂×C₈	C₂²×Dic₇	1	1	yes	characteristic
157	C₂×C₈	C₂×C₂²⋊D₇	1	1	yes	characteristic
158	C₄⋊C₄	—	2	4	no	—
159	C₄⋊C₄	—	4	2	no	—
160	C₄⋊C₄	—	4	2	no	—
161	C₄⋊C₄	—	4	2	no	—
162	C₄²	#1141	1	1	yes	characteristic
163	C₄²	#1142	1	1	yes	characteristic
164	C₄²	C₂²⋊Dic₇	1	1	yes	characteristic
165	C₄²	C₂×C₂²⋊D₇	1	1	yes	characteristic
166	C₁₆	—	28	2	no	—
167	C₂×C₇	—	2	2	no	—
168	C₂×C₇	#1546	1	1	yes	characteristic
169	C₂×C₇	C₂×#357	1	1	yes	characteristic
170	C₂×C₇	C₂×#362	1	1	yes	characteristic
171	C₂³	#6406	1	1	yes	characteristic
172	Q₈	—	8	4	no	—
173	Q₈	—	8	2	no	—
174	Q₈	—	8	2	no	—
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	2	1	no	—
177	C₂×C₄	—	2	1	no	—
178	C₂×C₄	—	4	4	no	—
179	C₂×C₄	—	8	4	no	—
180	C₂×C₄	#6400	1	2	yes	—
181	C₂×C₄	#6401	1	2	yes	—
182	C₂×C₄	#6406	1	1	yes	characteristic
183	C₂×C₄	#6406	1	1	yes	characteristic
184	C₂×C₄	#6406	1	1	yes	characteristic
185	C₂×C₄	C₂×#1141	1	1	yes	characteristic
186	C₂×C₄	C₂×#1142	1	1	yes	characteristic
187	C₂×C₄	C₂×C₂²⋊Dic₇	1	1	yes	characteristic
188	C₈	—	2	1	no	—
189	C₈	—	2	1	no	—
190	C₈	—	56	2	no	—
191	C₈	#6410	1	2	yes	—
192	C₈	#6505	1	2	yes	—
193	C₇	#11971	1	1	yes	characteristic, complemented, Sylow
194	C₂²	—	2	1	no	—
195	C₂²	—	2	1	no	—
196	C₂²	#89357	1	2	yes	—
197	C₂²	C₂×#6406	1	1	yes	characteristic
198	C₄	—	2	1	no	—
199	C₄	—	2	1	no	—
200	C₄	—	8	4	no	—
201	C₄	#89357	1	2	yes	—
202	C₄	#89360	1	2	yes	—
203	C₄	C₂×#6400	1	1	yes	characteristic
204	C₄	C₂×#6401	1	1	yes	characteristic
205	C₄	#89997	1	1	yes	characteristic
206	C₄	C₂×#6410	1	1	yes	characteristic
207	C₂	—	2	2	no	—
208	C₂	#11729527	1	1	yes	characteristic
209	C₂	C₂×#89357	1	1	yes	characteristic
210	C₂	C₂×#89360	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	#11971
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. #11725607	C₂	semidirect
193. C₇	#11971	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
#11725607	2
#11971	7