Finite Group Databasebeta

Subgroups — #13268115

Order 1792.

206 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 67 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11727050	C₂	1	1	yes	maximal, characteristic
2	#11727049	C₂	1	1	yes	maximal, characteristic
3	#11725818	C₂	1	1	yes	maximal, characteristic
4	C₈×Dic₂₈	C₂	1	1	yes	maximal, characteristic
5	C₇×#1370	C₂	1	1	yes	maximal, characteristic
6	#11725371	C₂	1	1	yes	maximal, characteristic
7	#11725370	C₂	1	1	yes	maximal, characteristic
8	C₄×Dic₂₈	C₂²	1	1	yes	characteristic
9	#89454	C₂²	1	1	yes	characteristic
10	#89454	C₂²	1	1	yes	characteristic
11	#89453	—	2	1	no	—
12	#89453	—	2	1	no	—
13	C₈×Dic₁₄	—	2	1	no	—
14	C₇×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
15	C₇×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
16	C₇×C₈²	C₂²	1	1	yes	characteristic
17	#89300	—	2	1	no	—
18	#89299	—	2	1	no	—
19	#89298	—	2	1	no	—
20	#89298	—	2	1	no	—
21	#89253	C₂²	1	1	yes	characteristic
22	#89251	—	2	1	no	—
23	#12450	—	7	1	no	maximal, Sylow
24	C₂×Dic₂₈	D₄	1	1	yes	characteristic
25	#6432	—	2	1	no	—
26	#6432	—	2	1	no	—
27	C₄×Dic₁₄	—	2	1	no	—
28	C₂×C₇×C₁₆	—	2	1	no	—
29	C₂×C₇×C₁₆	—	2	1	no	—
30	C₄×C₇×C₈	D₄	1	1	yes	characteristic
31	C₄×C₇×C₈	D₄	1	1	yes	characteristic
32	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
33	#6392	—	2	1	no	—
34	#6392	—	2	1	no	—
35	#6392	—	2	1	no	—
36	#6392	—	2	1	no	—
37	#6392	D₄	1	1	yes	characteristic
38	#6390	—	2	1	no	—
39	#6390	—	4	2	no	—
40	#6390	—	4	1	no	—
41	#6390	—	4	1	no	—
42	Dic₇⋊C₈	—	4	1	no	—
43	C₈×Dic₇	—	4	1	no	—
44	C₄⋊(C₇⋊C₈)	—	2	1	no	—
45	#2227	—	7	1	no	—
46	#2225	—	7	1	no	—
47	#1710	—	7	1	no	—
48	C₈×Q₁₆	—	7	1	no	—
49	#1370	D₇	1	1	yes	characteristic
50	#1365	—	7	1	no	—
51	#1364	—	7	1	no	—
52	C₂×Dic₁₄	—	2	1	no	—
53	C₂×Dic₁₄	—	4	1	no	—
54	C₂×Dic₁₄	—	4	1	no	—
55	C₂×C₇×C₈	—	2	1	no	—
56	C₂×C₇×C₈	D₈	1	1	yes	characteristic
57	C₂×C₇×C₈	D₈	1	1	yes	characteristic
58	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
59	C₂×C₇×C₈	#41	1	1	yes	characteristic
60	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
61	C₄⋊Dic₇	—	2	1	no	—
62	C₄⋊Dic₇	—	4	1	no	—
63	C₄⋊Dic₇	—	4	1	no	—
64	C₄⋊Dic₇	—	4	1	no	—
65	C₄⋊Dic₇	—	4	1	no	—
66	Dic₇⋊C₄	—	4	1	no	—
67	C₄×Dic₇	—	4	1	no	—
68	C₂×C₇⋊C₈	—	4	1	no	—
69	Dic₂₈	—	2	1	no	—
70	Dic₂₈	Q₁₆	1	2	yes	—
71	C₇×C₁₆	—	4	1	no	—
72	C₇×C₁₆	—	4	1	no	—
73	#500	—	7	1	no	—
74	#500	—	7	1	no	—
75	#487	—	14	1	no	—
76	#487	—	14	1	no	—
77	C₈×Q₈	—	14	1	no	—
78	C₄×Q₁₆	—	7	1	no	—
79	#367	—	14	1	no	—
80	#366	—	14	1	no	—
81	C₄⋊C₁₆	C₂×D₇	1	1	yes	characteristic
82	C₄⋊C₁₆	C₂×D₇	1	1	yes	characteristic
83	#358	—	14	1	no	—
84	#358	—	14	1	no	—
85	#335	—	7	1	no	—
86	Q₈⋊C₈	—	14	1	no	—
87	C₈²	C₂×D₇	1	1	yes	characteristic
88	C₂×C₄×C₇	#121	1	1	yes	characteristic
89	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
90	C₂×C₄×C₇	AGL(1,ℤ/8)	1	1	yes	characteristic
91	C₂×Dic₇	—	4	1	no	—
92	C₂×Dic₇	—	4	1	no	—
93	C₂×Dic₇	—	8	1	no	—
94	C₂×Dic₇	—	8	1	no	—
95	Dic₁₄	—	2	2	no	—
96	Dic₁₄	—	4	1	no	—
97	Dic₁₄	—	8	2	no	—
98	Dic₁₄	—	8	1	no	—
99	Dic₁₄	—	8	1	no	—
100	C₇×C₈	—	2	1	no	—
101	C₇×C₈	—	2	1	no	—
102	C₇×C₈	—	2	1	no	—
103	C₇×C₈	—	4	1	no	—
104	C₇×C₈	C₂×Q₁₆	1	1	yes	characteristic
105	C₇×C₈	#135	1	1	yes	characteristic
106	C₇⋊C₈	—	8	1	no	—
107	C₂×Q₁₆	—	7	1	no	—
108	C₄⋊Q₈	—	14	1	no	—
109	C₄⋊Q₈	—	14	1	no	—
110	C₄×Q₈	—	14	1	no	—
111	C₂×C₁₆	—	2	1	no	—
112	C₂×C₁₆	—	2	1	no	—
113	#107	—	7	1	no	—
114	#107	—	14	1	no	—
115	#107	—	14	1	no	—
116	#107	—	14	1	no	—
117	#107	—	14	1	no	—
118	C₄⋊C₈	—	14	1	no	—
119	C₄⋊C₈	—	28	1	no	—
120	#103	—	14	1	no	—
121	#103	—	28	2	no	—
122	#103	—	28	1	no	—
123	#103	—	28	1	no	—
124	C₄×C₈	—	28	1	no	—
125	C₄×C₈	D₂₈	1	1	yes	characteristic
126	C₄×C₈	D₂₈	1	1	yes	characteristic
127	C₄×C₈	C₂²×D₇	1	1	yes	characteristic
128	C₂²×C₇	#459	1	1	yes	characteristic
129	C₄×C₇	—	2	1	no	—
130	C₄×C₇	#467	1	1	yes	characteristic
131	C₄×C₇	#473	1	1	yes	characteristic
132	C₄×C₇	#508	1	1	yes	characteristic
133	C₄×C₇	#510	1	1	yes	characteristic
134	Dic₇	—	4	2	no	—
135	Dic₇	—	8	1	no	—
136	Dic₇	—	16	1	no	—
137	Dic₇	—	16	1	no	—
138	C₂×Q₈	—	14	1	no	—
139	C₂×Q₈	—	28	1	no	—
140	C₂×Q₈	—	28	1	no	—
141	Q₁₆	—	7	2	no	—
142	Q₁₆	—	14	1	no	—
143	C₂×C₈	—	2	1	no	—
144	C₂×C₈	—	28	1	no	—
145	C₂×C₈	D₅₆	1	1	yes	characteristic
146	C₂×C₈	D₅₆	1	1	yes	characteristic
147	C₂×C₈	D₄×D₇	1	1	yes	characteristic
148	C₂×C₈	#1161	1	1	yes	characteristic
149	C₄⋊C₄	—	14	1	no	—
150	C₄⋊C₄	—	28	1	no	—
151	C₄⋊C₄	—	28	1	no	—
152	C₄⋊C₄	—	28	1	no	—
153	C₄⋊C₄	—	28	1	no	—
154	C₄⋊C₄	—	28	1	no	—
155	C₄²	—	28	1	no	—
156	C₄²	C₂×D₂₈	1	1	yes	characteristic
157	C₁₆	—	4	1	no	—
158	C₁₆	—	4	1	no	—
159	C₂×C₇	#1669	1	1	yes	characteristic
160	C₂×C₇	#2211	1	1	yes	characteristic
161	C₂×C₇	#2212	1	1	yes	characteristic
162	Q₈	—	14	2	no	—
163	Q₈	—	28	1	no	—
164	Q₈	—	56	2	no	—
165	Q₈	—	56	1	no	—
166	Q₈	—	56	1	no	—
167	C₂×C₄	—	28	1	no	—
168	C₂×C₄	—	28	1	no	—
169	C₂×C₄	—	56	1	no	—
170	C₂×C₄	—	56	1	no	—
171	C₂×C₄	#6458	1	1	yes	characteristic
172	C₂×C₄	C₂×D₅₆	1	1	yes	characteristic
173	C₂×C₄	#6471	1	1	yes	characteristic
174	C₈	—	2	1	no	—
175	C₈	—	2	1	no	—
176	C₈	—	2	1	no	—
177	C₈	—	4	1	no	—
178	C₈	—	56	1	no	—
179	C₈	#6475	1	1	yes	characteristic
180	C₈	D₇×Q₁₆	1	1	yes	characteristic
181	C₇	#12450	1	1	yes	characteristic, complemented, Sylow
182	C₂²	#89615	1	1	yes	characteristic
183	C₄	—	2	1	no	—
184	C₄	—	28	2	no	—
185	C₄	—	56	1	no	—
186	C₄	—	112	1	no	—
187	C₄	—	112	1	no	—
188	C₄	#89659	1	1	yes	characteristic
189	C₄	#89668	1	1	yes	characteristic
190	C₄	#89676	1	1	yes	characteristic
191	C₄	#89681	1	1	yes	characteristic
192	C₂	#11726331	1	1	yes	characteristic
193	C₂	#11727048	1	1	yes	characteristic
194	C₂	#11727058	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#89615
commutator, cosocle	C₄×C₇×C₈	C₂³
Frattini	C₄×C₈	C₂²×D₇
Fitting	C₇×#1370	C₂
socle, prime radical	C₂²×C₇	#459
hypercenter	#1370	D₇
hypocenter	C₇	#12450
LCS	G ▷ C₄×C₇×C₈ ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1370

Complements

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

N	H ≅ G/N	type
181. C₇	#12450	semidirect

Maximal subgroups

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

subgroup	index
#11727050	2
#11727049	2
#11725818	2
C₈×Dic₂₈	2
C₇×#1370	2
#11725371	2
#11725370	2
#12450	7