Subgroups — #13509326

Order 1792.

482 conjugacy classes of subgroups.

The subgroup lattice has 214 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11735887	C₂	1	1	yes	maximal, characteristic
2	C₂×#89455	C₂	1	1	yes	maximal, characteristic
3	C₄×C₇×M₃₂	C₂	1	1	yes	maximal, characteristic
4	#11729182	C₂	1	4	yes	maximal
5	#11726823	C₂	1	4	yes	maximal, complemented
6	#11726799	C₂	1	4	yes	maximal, complemented
7	#90161	—	2	1	no	—
8	C₂×#6433	—	2	1	no	—
9	C₂×C₇×M₃₂	C₂²	1	2	yes	—
10	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
11	#89878	—	2	1	no	—
12	#89878	C₂²	1	2	yes	—
13	C₂×#6392	C₂²	1	2	yes	—
14	C₂×#6391	—	2	1	no	—
15	#89455	—	2	1	no	—
16	#89455	—	2	1	no	—
17	#89455	—	2	1	no	—
18	#89455	C₂²	1	2	yes	—
19	#89455	C₂²	1	2	yes	—
20	#89455	C₂²	1	2	yes	—
21	#89454	C₂²	1	2	yes	—
22	#89452	—	2	1	no	—
23	C₇×#346	C₂²	1	2	yes	—
24	C₄×C₇×C₁₆	C₂²	1	2	yes	—
25	#89300	C₂²	1	8	yes	—
26	#89299	C₂²	1	8	yes	—
27	#33732	—	7	1	no	maximal, Sylow
28	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
29	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
30	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
31	#6489	—	2	2	no	—
32	C₂×C₄⋊Dic₇	—	2	2	no	—
33	#6487	—	4	2	no	—
34	C₂×Dic₇⋊C₄	—	4	2	no	—
35	#6433	—	2	2	no	—
36	#6433	—	2	2	no	—
37	#6433	—	2	2	no	—
38	#6433	—	4	2	no	—
39	#6432	—	2	2	no	—
40	C₄×Dic₁₄	—	4	2	no	—
41	C₇×M₃₂	Q₈	1	8	yes	complemented
42	C₂×C₇×C₁₆	C₂³	1	4	yes	—
43	C₄×C₇×C₈	D₄	1	1	yes	characteristic
44	C₄×C₇×C₈	D₄	1	1	yes	characteristic
45	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
46	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
47	#6392	—	2	2	no	—
48	#6392	—	2	2	no	—
49	#6392	C₂³	1	4	yes	—
50	#6392	C₂³	1	4	yes	—
51	#6391	—	2	2	no	—
52	#6391	—	2	2	no	—
53	#6391	—	2	2	no	—
54	#6391	—	2	2	no	—
55	#3160	—	7	1	no	—
56	C₂×#499	—	7	1	no	—
57	#2255	—	7	4	no	—
58	#2253	—	7	4	no	—
59	#2159	—	7	4	no	—
60	C₄×M₃₂	D₇	1	1	yes	characteristic
61	C₂²×C₄×C₇	D₈	1	2	yes	—
62	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
63	C₂²×Dic₇	—	4	2	no	—
64	C₂×Dic₁₄	—	4	2	no	—
65	C₂×C₇×C₈	Q₁₆	1	4	yes	—
66	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
67	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
68	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
69	C₂×C₇×C₈	C₂×Q₈	1	4	yes	—
70	C₂×C₇×C₈	C₂⁴	1	1	yes	characteristic
71	C₄²×C₇	D₈	1	2	yes	—
72	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
73	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
74	C₂²⋊Dic₇	—	4	2	no	—
75	C₄⋊Dic₇	—	2	4	no	—
76	C₄⋊Dic₇	—	2	4	no	—
77	C₄⋊Dic₇	—	4	2	no	—
78	C₄⋊Dic₇	—	4	2	no	—
79	Dic₇⋊C₄	—	4	4	no	—
80	Dic₇⋊C₄	—	4	4	no	—
81	Dic₇⋊C₄	—	4	4	no	—
82	C₄×Dic₇	—	4	2	no	—
83	C₇×C₁₆	C₂×Q₈	1	8	yes	—
84	#533	—	14	1	no	—
85	C₂×#125	—	14	1	no	—
86	C₂×M₃₂	C₂×D₇	1	2	yes	—
87	#500	—	7	2	no	—
88	#499	—	7	2	no	—
89	#499	—	7	2	no	—
90	#499	—	7	2	no	—
91	#499	—	14	1	no	—
92	#499	—	14	1	no	—
93	#499	—	14	1	no	—
94	#498	—	14	1	no	—
95	#427	—	7	2	no	—
96	#427	—	14	1	no	—
97	C₂×#107	—	7	2	no	—
98	C₂×#106	—	14	1	no	—
99	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
100	#367	—	7	8	no	—
101	#366	—	7	8	no	—
102	#346	C₂×D₇	1	2	yes	—
103	C₄×C₁₆	C₂×D₇	1	2	yes	—
104	C₂³×C₇	C₂×D₈	1	1	yes	characteristic
105	C₂×C₄×C₇	—	4	2	no	—
106	C₂×C₄×C₇	C₄⋊Q₈	1	2	yes	—
107	C₂×C₄×C₇	C₂×D₈	1	2	yes	—
108	C₂×C₄×C₇	C₂×D₈	1	2	yes	—
109	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
110	C₂×C₄×C₇	C₂×Q₁₆	1	2	yes	—
111	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
112	C₂×Dic₇	—	4	4	no	—
113	C₂×Dic₇	—	4	4	no	—
114	C₂×Dic₇	—	8	4	no	—
115	Dic₁₄	—	8	4	no	—
116	C₇×C₈	C₄⋊Q₈	1	2	yes	—
117	C₇×C₈	C₂×Q₁₆	1	2	yes	—
118	C₇×C₈	C₂×Q₁₆	1	2	yes	—
119	C₇×C₈	C₂²×Q₈	1	2	yes	—
120	C₂²×C₈	D₂₈	1	1	yes	characteristic
121	C₂²×C₈	C₂²×D₇	1	1	yes	characteristic
122	C₄⋊Q₈	—	14	2	no	—
123	#125	—	14	2	no	—
124	#125	—	14	2	no	—
125	#125	—	14	2	no	—
126	#125	—	28	2	no	—
127	C₂²⋊Q₈	—	28	2	no	—
128	C₄×Q₈	—	28	2	no	—
129	#117	—	14	2	no	—
130	C₂×C₄⋊C₄	—	14	2	no	—
131	C₂×C₄⋊C₄	—	28	2	no	—
132	C₂×C₄²	D₂₈	1	1	yes	characteristic
133	M₃₂	Dic₁₄	1	8	yes	complemented
134	C₂×C₁₆	C₂²×D₇	1	4	yes	—
135	#107	—	7	4	no	—
136	#107	—	7	4	no	—
137	#107	—	14	2	no	—
138	#107	—	14	2	no	—
139	#106	—	14	2	no	—
140	#106	—	14	2	no	—
141	#106	—	14	2	no	—
142	#106	—	14	2	no	—
143	C₄×C₈	D₂₈	1	1	yes	characteristic
144	C₄×C₈	D₂₈	1	1	yes	characteristic
145	C₄×C₈	C₂²×D₇	1	1	yes	characteristic
146	C₄×C₈	C₂²×D₇	1	1	yes	characteristic
147	C₂²×C₇	—	2	2	no	—
148	C₂²×C₇	#500	1	2	yes	—
149	C₂²×C₇	C₂²×D₈	1	1	yes	characteristic
150	C₄×C₇	—	2	2	no	—
151	C₄×C₇	—	2	2	no	—
152	C₄×C₇	#500	1	2	yes	—
153	C₄×C₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
154	C₄×C₇	C₂²×Q₁₆	1	1	yes	characteristic
155	Dic₇	—	8	4	no	—
156	Dic₇	—	8	4	no	—
157	C₂×Q₈	—	28	2	no	—
158	C₂²×C₄	—	28	2	no	—
159	C₂²×C₄	D₅₆	1	2	yes	—
160	C₂²×C₄	C₂×D₂₈	1	1	yes	characteristic
161	C₂×C₈	Dic₂₈	1	4	yes	—
162	C₂×C₈	C₂×Dic₁₄	1	4	yes	—
163	C₂×C₈	C₂×D₂₈	1	1	yes	characteristic
164	C₂×C₈	C₂×D₂₈	1	1	yes	characteristic
165	C₂×C₈	C₂×D₂₈	1	1	yes	characteristic
166	C₂×C₈	C₂³×D₇	1	1	yes	characteristic
167	C₄⋊C₄	—	14	4	no	—
168	C₄⋊C₄	—	14	4	no	—
169	C₄⋊C₄	—	28	4	no	—
170	C₄⋊C₄	—	28	4	no	—
171	C₄⋊C₄	—	28	4	no	—
172	C₄⋊C₄	—	28	2	no	—
173	C₄⋊C₄	—	28	2	no	—
174	C₂²⋊C₄	—	28	2	no	—
175	C₄²	—	28	2	no	—
176	C₄²	D₅₆	1	2	yes	—
177	C₄²	C₂×D₂₈	1	1	yes	characteristic
178	C₄²	C₂×D₂₈	1	1	yes	characteristic
179	C₁₆	C₂×Dic₁₄	1	8	yes	—
180	C₂×C₇	—	2	2	no	—
181	C₂×C₇	C₂×#500	1	1	yes	characteristic
182	C₂×C₇	#3414	1	2	yes	—
183	C₂³	C₂×D₅₆	1	1	yes	characteristic
184	Q₈	—	56	4	no	—
185	C₂×C₄	—	4	2	no	—
186	C₂×C₄	—	28	4	no	—
187	C₂×C₄	—	28	4	no	—
188	C₂×C₄	—	56	4	no	—
189	C₂×C₄	#6432	1	2	yes	—
190	C₂×C₄	C₂×D₅₆	1	2	yes	—
191	C₂×C₄	C₂×D₅₆	1	2	yes	—
192	C₂×C₄	C₂×D₅₆	1	1	yes	characteristic
193	C₂×C₄	C₂×Dic₂₈	1	2	yes	—
194	C₂×C₄	C₂²×D₂₈	1	1	yes	characteristic
195	C₈	#6432	1	2	yes	—
196	C₈	C₂×Dic₂₈	1	2	yes	—
197	C₈	C₂×Dic₂₈	1	2	yes	—
198	C₈	C₂²×Dic₁₄	1	2	yes	—
199	C₇	#33732	1	1	yes	characteristic, complemented, Sylow
200	C₂²	—	2	2	no	—
201	C₂²	#89454	1	2	yes	—
202	C₂²	C₂²×D₅₆	1	1	yes	characteristic
203	C₄	—	2	2	no	—
204	C₄	—	2	2	no	—
205	C₄	—	56	4	no	—
206	C₄	—	56	4	no	—
207	C₄	#89454	1	2	yes	—
208	C₄	C₂×#6432	1	1	yes	characteristic
209	C₄	C₂²×Dic₂₈	1	1	yes	characteristic
210	C₂	—	2	2	no	—
211	C₂	C₂×#89454	1	1	yes	characteristic
212	C₂	#11739454	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂²×D₅₆
commutator, cosocle	C₂×C₇×C₈	C₂⁴
Frattini	C₂×C₈	C₂³×D₇
Fitting	C₄×C₇×M₃₂	C₂
socle	C₂²×C₇	C₂²×D₈
prime radical	C₂³×C₇	C₂×D₈
hypercenter	C₄×M₃₂	D₇
hypocenter	C₇	#33732
LCS	G ▷ C₂×C₇×C₈ ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×M₃₂

Complements

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

N	H ≅ G/N	type
5. #11726823	C₂	semidirect
6. #11726799	C₂	semidirect
41. C₇×M₃₂	Q₈	semidirect
133. M₃₂	Dic₁₄	semidirect
199. C₇	#33732	semidirect

Maximal subgroups

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

subgroup	index
#11735887	2
C₂×#89455	2
C₄×C₇×M₃₂	2
#11729182	2
#11726823	2
#11726799	2
#33732	7