Finite Group Databasebeta

Subgroups — #13268593

Order 1792.

238 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 83 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	#11726819	C₂	1	1	yes	maximal, characteristic
2	#11726819	C₂	1	1	yes	maximal, characteristic
3	#11726800	C₂	1	1	yes	maximal, characteristic
4	#11726798	C₂	1	1	yes	maximal, characteristic
5	#11725818	C₂	1	1	yes	maximal, characteristic
6	#11725817	C₂	1	1	yes	maximal, characteristic
7	C₇×#1312	C₂	1	1	yes	maximal, characteristic
8	#89471	C₂²	1	1	yes	characteristic
9	#89454	C₂²	1	1	yes	characteristic
10	#89454	C₂²	1	1	yes	characteristic
11	#89454	C₂²	1	1	yes	characteristic
12	#89453	—	2	1	no	—
13	#89453	—	2	1	no	—
14	#89453	—	2	1	no	—
15	#89452	—	2	1	no	—
16	C₄×C₇×C₁₆	C₂²	1	1	yes	characteristic
17	C₄×C₇×C₁₆	C₂²	1	1	yes	characteristic
18	C₇×C₈²	C₂²	1	1	yes	characteristic
19	#89300	—	2	2	no	—
20	#89299	—	2	2	no	—
21	#89298	—	2	2	no	—
22	#89298	—	2	2	no	—
23	#12533	—	7	1	no	maximal, Sylow
24	C₂×Dic₂₈	—	2	1	no	—
25	C₂×Dic₂₈	—	2	1	no	—
26	#6432	—	2	1	no	—
27	#6432	—	2	1	no	—
28	#6432	—	2	1	no	—
29	#6432	—	2	1	no	—
30	C₂×C₇×C₁₆	D₄	1	2	yes	—
31	C₂×C₇×C₁₆	D₄	1	2	yes	—
32	C₄×C₇×C₈	D₄	1	1	yes	characteristic
33	C₄×C₇×C₈	D₄	1	1	yes	characteristic
34	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
35	#6392	—	2	1	no	—
36	#6392	—	2	1	no	—
37	#6392	—	2	1	no	—
38	#6392	—	2	1	no	—
39	#6392	—	2	1	no	—
40	#6392	—	2	1	no	—
41	#6391	—	4	2	no	—
42	#6390	—	4	2	no	—
43	#6390	—	4	2	no	—
44	#6390	—	4	2	no	—
45	#2254	—	7	1	no	—
46	#2252	—	7	1	no	—
47	#2241	—	7	1	no	—
48	#2241	—	7	1	no	—
49	#1710	—	7	1	no	—
50	#1707	—	7	1	no	—
51	#1312	D₇	1	1	yes	characteristic
52	C₂×Dic₁₄	—	4	1	no	—
53	C₂×Dic₁₄	—	4	1	no	—
54	C₂×Dic₁₄	—	4	1	no	—
55	C₂×Dic₁₄	—	4	1	no	—
56	C₂×C₇×C₈	D₈	1	2	yes	—
57	C₂×C₇×C₈	D₈	1	2	yes	—
58	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
59	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
60	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
61	C₄⋊Dic₇	—	4	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	C₄⋊Dic₇	—	4	1	no	—
67	C₄⋊Dic₇	—	4	1	no	—
68	C₄⋊Dic₇	—	4	1	no	—
69	Dic₂₈	—	4	2	no	—
70	Dic₂₈	—	4	2	no	—
71	C₇×C₁₆	—	2	2	no	—
72	C₇×C₁₆	—	2	2	no	—
73	#500	—	7	1	no	—
74	#500	—	7	1	no	—
75	#500	—	7	1	no	—
76	#498	—	14	1	no	—
77	#494	—	7	1	no	—
78	#487	—	14	1	no	—
79	#487	—	14	1	no	—
80	#487	—	14	1	no	—
81	#367	—	14	2	no	—
82	#366	—	14	2	no	—
83	#358	—	14	2	no	—
84	#358	—	14	2	no	—
85	C₄×C₁₆	C₂×D₇	1	1	yes	characteristic
86	C₄×C₁₆	C₂×D₇	1	1	yes	characteristic
87	C₈²	C₂×D₇	1	1	yes	characteristic
88	C₂×C₄×C₇	#127	1	1	yes	characteristic
89	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
90	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
91	C₂×Dic₇	—	8	1	no	—
92	C₂×Dic₇	—	8	1	no	—
93	C₂×Dic₇	—	8	1	no	—
94	C₂×Dic₇	—	8	1	no	—
95	Dic₁₄	—	8	1	no	—
96	Dic₁₄	—	8	1	no	—
97	Dic₁₄	—	8	2	no	—
98	Dic₁₄	—	8	2	no	—
99	Dic₁₄	—	8	2	no	—
100	C₇×C₈	—	2	2	no	—
101	C₇×C₈	—	2	2	no	—
102	C₇×C₈	#137	1	2	yes	—
103	C₇×C₈	#137	1	2	yes	—
104	C₂×Q₁₆	—	14	1	no	—
105	C₂×Q₁₆	—	14	1	no	—
106	C₄⋊Q₈	—	14	1	no	—
107	C₄⋊Q₈	—	14	1	no	—
108	C₄⋊Q₈	—	14	1	no	—
109	C₄⋊Q₈	—	14	1	no	—
110	C₂×C₁₆	D₂₈	1	2	yes	—
111	C₂×C₁₆	D₂₈	1	2	yes	—
112	#107	—	14	1	no	—
113	#107	—	14	1	no	—
114	#107	—	14	1	no	—
115	#107	—	14	1	no	—
116	#107	—	14	1	no	—
117	#107	—	14	1	no	—
118	#106	—	28	2	no	—
119	#103	—	28	2	no	—
120	#103	—	28	2	no	—
121	#103	—	28	2	no	—
122	C₄×C₈	D₂₈	1	1	yes	characteristic
123	C₄×C₈	D₂₈	1	1	yes	characteristic
124	C₄×C₈	C₂²×D₇	1	1	yes	characteristic
125	C₂²×C₇	#493	1	1	yes	characteristic
126	C₄×C₇	#497	1	1	yes	characteristic
127	C₄×C₇	#497	1	1	yes	characteristic
128	C₄×C₇	#508	1	2	yes	—
129	C₄×C₇	#510	1	2	yes	—
130	Dic₇	—	16	1	no	—
131	Dic₇	—	16	1	no	—
132	Dic₇	—	16	1	no	—
133	Dic₇	—	16	1	no	—
134	C₂×Q₈	—	28	1	no	—
135	C₂×Q₈	—	28	1	no	—
136	C₂×Q₈	—	28	1	no	—
137	C₂×Q₈	—	28	1	no	—
138	Q₁₆	—	28	2	no	—
139	Q₁₆	—	28	2	no	—
140	C₂×C₈	D₅₆	1	2	yes	—
141	C₂×C₈	D₅₆	1	2	yes	—
142	C₂×C₈	C₂×D₂₈	1	1	yes	characteristic
143	C₂×C₈	C₂×D₂₈	1	1	yes	characteristic
144	C₄⋊C₄	—	28	1	no	—
145	C₄⋊C₄	—	28	1	no	—
146	C₄⋊C₄	—	28	1	no	—
147	C₄⋊C₄	—	28	1	no	—
148	C₄⋊C₄	—	28	1	no	—
149	C₄⋊C₄	—	28	1	no	—
150	C₄⋊C₄	—	28	1	no	—
151	C₄⋊C₄	—	28	1	no	—
152	C₄²	C₂×D₂₈	1	1	yes	characteristic
153	C₁₆	—	2	2	no	—
154	C₁₆	—	2	2	no	—
155	C₂×C₇	#1722	1	1	yes	characteristic
156	C₂×C₇	#2249	1	1	yes	characteristic
157	C₂×C₇	#2251	1	1	yes	characteristic
158	Q₈	—	56	2	no	—
159	Q₈	—	56	2	no	—
160	Q₈	—	56	2	no	—
161	Q₈	—	56	1	no	—
162	Q₈	—	56	1	no	—
163	C₂×C₄	—	56	1	no	—
164	C₂×C₄	—	56	1	no	—
165	C₂×C₄	—	56	1	no	—
166	C₂×C₄	—	56	1	no	—
167	C₂×C₄	#6437	1	1	yes	characteristic
168	C₂×C₄	C₂×D₅₆	1	1	yes	characteristic
169	C₂×C₄	C₂×D₅₆	1	1	yes	characteristic
170	C₈	—	2	2	no	—
171	C₈	—	2	2	no	—
172	C₈	#6472	1	2	yes	—
173	C₈	#6472	1	2	yes	—
174	C₇	#12533	1	1	yes	characteristic, complemented, Sylow
175	C₂²	#89467	1	1	yes	characteristic
176	C₄	—	112	1	no	—
177	C₄	—	112	1	no	—
178	C₄	—	112	1	no	—
179	C₄	—	112	1	no	—
180	C₄	#89487	1	1	yes	characteristic
181	C₄	#89487	1	1	yes	characteristic
182	C₄	#89676	1	2	yes	—
183	C₄	#89681	1	2	yes	—
184	C₂	#11725852	1	1	yes	characteristic
185	C₂	#11726816	1	1	yes	characteristic
186	C₂	#11726838	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#89467
commutator, cosocle	C₄×C₇×C₈	C₂³
Frattini	C₄×C₈	C₂²×D₇
Fitting	C₇×#1312	C₂
socle, prime radical	C₂²×C₇	#493
hypercenter	#1312	D₇
hypocenter	C₇	#12533
LCS	G ▷ C₄×C₇×C₈ ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1312

Complements

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

N	H ≅ G/N	type
174. C₇	#12533	semidirect

Maximal subgroups

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

subgroup	index
#11726819	2
#11726800	2
#11726798	2
#11725818	2
#11725817	2
C₇×#1312	2
#12533	7