Subgroups — #41×C₇⋊C₁₆ #13507315

Order 1792.

602 conjugacy classes of subgroups.

The subgroup lattice has 184 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×C₈×#41	C₂	1	1	yes	maximal, characteristic
2	Q₈×C₇⋊C₁₆	C₂	1	2	yes	maximal, complemented
3	D₄×C₇⋊C₁₆	C₂	1	6	yes	maximal, complemented
4	#11727498	C₂	1	3	yes	maximal, complemented
5	C₂×C₄×C₇⋊C₁₆	C₂	1	3	yes	maximal, complemented
6	C₄×C₇×#41	C₄	1	1	yes	characteristic
7	C₇×C₈×Q₈	C₄	1	1	yes	characteristic
8	C₇×C₈×Q₈	C₂²	1	1	yes	characteristic
9	C₇×C₈×D₄	C₄	1	3	yes	—
10	C₇×C₈×D₄	C₂²	1	3	yes	—
11	C₇×#431	C₄	1	3	yes	—
12	C₂×C₄×C₇×C₈	C₂²	1	3	yes	—
13	C₂²×C₇⋊C₁₆	C₂²	1	6	yes	—
14	C₂²⋊(C₇⋊C₁₆)	C₂²	1	6	yes	—
15	C₄⋊(C₇⋊C₁₆)	C₂²	1	6	yes	—
16	C₄×C₇⋊C₁₆	C₂²	1	6	yes	—
17	C₄×C₇⋊C₁₆	C₂²	1	1	yes	characteristic
18	C₄×C₇⋊C₁₆	C₂²	1	3	yes	—
19	C₁₆×#41	—	7	1	no	maximal, Sylow
20	C₂×C₇×#41	C₈	1	1	yes	characteristic
21	C₂²×C₇×C₈	C₂×C₄	1	3	yes	—
22	C₂²×C₇×C₈	C₂³	1	3	yes	—
23	C₄×C₇×Q₈	C₈	1	1	yes	characteristic
24	C₄×C₇×Q₈	C₂×C₄	1	1	yes	characteristic
25	C₄×C₇×D₄	C₈	1	3	yes	—
26	C₄×C₇×D₄	C₂×C₄	1	3	yes	—
27	C₇×#117	C₈	1	3	yes	—
28	C₂×C₄²×C₇	C₂×C₄	1	3	yes	—
29	C₇×C₄⋊C₈	C₂×C₄	1	3	yes	—
30	C₇×C₄⋊C₈	C₂×C₄	1	3	yes	—
31	C₇×C₂²⋊C₈	C₂×C₄	1	6	yes	—
32	C₄×C₇×C₈	C₂×C₄	1	3	yes	—
33	C₄×C₇×C₈	C₂×C₄	1	3	yes	—
34	C₄×C₇×C₈	C₂³	1	3	yes	—
35	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
36	C₂×C₇⋊C₁₆	—	2	12	no	—
37	C₂×C₇⋊C₁₆	C₂³	1	6	yes	—
38	C₂×C₇⋊C₁₆	C₂³	1	2	yes	—
39	C₈×#41	D₇	1	1	yes	characteristic
40	Q₈×C₁₆	—	7	2	no	—
41	D₄×C₁₆	—	7	6	no	—
42	#2162	—	7	3	no	—
43	C₂×C₄×C₁₆	—	7	3	no	—
44	C₇×#41	C₁₆	1	8	yes	complemented
45	C₂×C₇×Q₈	C₂×C₈	1	1	yes	characteristic
46	C₂×C₇×D₄	C₂×C₈	1	3	yes	—
47	C₂²×C₄×C₇	C₂×C₈	1	3	yes	—
48	C₂²×C₄×C₇	C₂×C₈	1	3	yes	—
49	C₂²×C₄×C₇	C₂²×C₄	1	3	yes	—
50	C₂×C₇×C₈	—	2	6	no	—
51	C₂×C₇×C₈	—	2	6	no	—
52	C₂×C₇×C₈	C₂²×C₄	1	3	yes	—
53	C₂×C₇×C₈	C₂²×C₄	1	3	yes	—
54	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₇×C₈	C₂⁴	1	1	yes	characteristic
56	C₇×C₄⋊C₄	C₂×C₈	1	3	yes	—
57	C₇×C₄⋊C₄	C₂×C₈	1	3	yes	—
58	C₇×C₂²⋊C₄	C₂×C₈	1	6	yes	—
59	C₄²×C₇	C₂×C₈	1	3	yes	—
60	C₄²×C₇	C₂×C₈	1	3	yes	—
61	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
62	C₄²×C₇	C₂²×C₄	1	3	yes	—
63	C₇⋊C₁₆	—	2	6	no	—
64	C₇⋊C₁₆	#41	1	4	yes	direct product factor
65	C₄×#41	Dic₇	1	1	yes	characteristic
66	C₂²×C₁₆	—	7	6	no	—
67	C₈×Q₈	Dic₇	1	1	yes	characteristic
68	C₈×Q₈	C₂×D₇	1	1	yes	characteristic
69	C₈×D₄	Dic₇	1	3	yes	—
70	C₈×D₄	C₂×D₇	1	3	yes	—
71	#431	Dic₇	1	3	yes	—
72	C₂×C₄×C₈	C₂×D₇	1	3	yes	—
73	C₄⋊C₁₆	—	7	6	no	—
74	C₂²⋊C₁₆	—	7	6	no	—
75	C₄×C₁₆	—	7	6	no	—
76	C₄×C₁₆	—	7	1	no	—
77	C₄×C₁₆	—	7	3	no	—
78	C₂³×C₇	C₂²×C₈	1	3	yes	—
79	C₇×Q₈	C₂×C₁₆	1	4	yes	complemented
80	C₇×D₄	C₂×C₁₆	1	12	yes	complemented
81	C₂×C₄×C₇	—	2	3	no	—
82	C₂×C₄×C₇	—	2	3	no	—
83	C₂×C₄×C₇	—	2	6	no	—
84	C₂×C₄×C₇	C₂×C₁₆	1	12	yes	complemented
85	C₂×C₄×C₇	C₂²×C₈	1	3	yes	—
86	C₂×C₄×C₇	C₂²×C₈	1	3	yes	—
87	C₂×C₄×C₇	C₂²×C₈	1	1	yes	characteristic
88	C₂×C₄×C₇	C₂²×C₈	1	3	yes	—
89	C₂×C₄×C₇	C₂²×C₈	1	1	yes	characteristic
90	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
91	C₇×C₈	—	2	3	no	—
92	C₇×C₈	—	2	3	no	—
93	C₇×C₈	#131	1	2	yes	—
94	C₇×C₈	C₂×#41	1	2	yes	—
95	C₂×#41	C₇⋊C₈	1	1	yes	characteristic
96	C₂²×C₈	C₂×Dic₇	1	3	yes	—
97	C₂²×C₈	C₂²×D₇	1	3	yes	—
98	C₄×Q₈	C₇⋊C₈	1	1	yes	characteristic
99	C₄×Q₈	C₂×Dic₇	1	1	yes	characteristic
100	C₄×D₄	C₇⋊C₈	1	3	yes	—
101	C₄×D₄	C₂×Dic₇	1	3	yes	—
102	#117	C₇⋊C₈	1	3	yes	—
103	C₂×C₄²	C₂×Dic₇	1	3	yes	—
104	C₂×C₁₆	—	7	2	no	—
105	C₂×C₁₆	—	7	6	no	—
106	C₂×C₁₆	—	14	12	no	—
107	C₄⋊C₈	C₂×Dic₇	1	3	yes	—
108	C₄⋊C₈	C₂×Dic₇	1	3	yes	—
109	C₂²⋊C₈	C₂×Dic₇	1	6	yes	—
110	C₄×C₈	C₂×Dic₇	1	3	yes	—
111	C₄×C₈	C₂×Dic₇	1	3	yes	—
112	C₄×C₈	C₂²×D₇	1	3	yes	—
113	C₄×C₈	C₂²×D₇	1	1	yes	characteristic
114	C₂²×C₇	—	2	3	no	—
115	C₂²×C₇	—	2	3	no	—
116	C₂²×C₇	C₂²×C₁₆	1	6	yes	—
117	C₂²×C₇	C₂³×C₈	1	1	yes	characteristic
118	C₄×C₇	—	2	3	no	—
119	C₄×C₇	—	2	3	no	—
120	C₄×C₇	C₂²×C₁₆	1	2	yes	—
121	C₄×C₇	C₂²×C₁₆	1	6	yes	—
122	C₄×C₇	#504	1	2	yes	—
123	C₄×C₇	C₄×#41	1	1	yes	characteristic
124	C₄×C₇	C₂×#131	1	1	yes	characteristic
125	#41	C₇⋊C₁₆	1	8	yes	direct product factor
126	C₂×Q₈	C₂×C₇⋊C₈	1	1	yes	characteristic
127	C₂×D₄	C₂×C₇⋊C₈	1	3	yes	—
128	C₂²×C₄	C₂×C₇⋊C₈	1	3	yes	—
129	C₂²×C₄	C₂×C₇⋊C₈	1	3	yes	—
130	C₂²×C₄	C₂²×Dic₇	1	3	yes	—
131	C₂×C₈	—	2	6	no	—
132	C₂×C₈	—	2	6	no	—
133	C₂×C₈	C₂²×Dic₇	1	3	yes	—
134	C₂×C₈	C₂²×Dic₇	1	3	yes	—
135	C₂×C₈	C₂²×Dic₇	1	1	yes	characteristic
136	C₂×C₈	C₂³×D₇	1	1	yes	characteristic
137	C₄⋊C₄	C₂×C₇⋊C₈	1	3	yes	—
138	C₄⋊C₄	C₂×C₇⋊C₈	1	3	yes	—
139	C₂²⋊C₄	C₂×C₇⋊C₈	1	6	yes	—
140	C₄²	C₂×C₇⋊C₈	1	3	yes	—
141	C₄²	C₂×C₇⋊C₈	1	3	yes	—
142	C₄²	C₂²×Dic₇	1	1	yes	characteristic
143	C₄²	C₂²×Dic₇	1	3	yes	—
144	C₁₆	—	7	4	no	—
145	C₁₆	—	14	6	no	—
146	C₂×C₇	—	2	6	no	—
147	C₂×C₇	C₈×#41	1	1	yes	characteristic
148	C₂×C₇	C₂³×C₁₆	1	1	yes	characteristic
149	C₂×C₇	C₂×#504	1	1	yes	characteristic
150	C₂³	C₂²×C₇⋊C₈	1	3	yes	—
151	Q₈	C₂×C₇⋊C₁₆	1	4	yes	complemented
152	D₄	C₂×C₇⋊C₁₆	1	12	yes	complemented
153	C₂×C₄	—	2	6	no	—
154	C₂×C₄	—	2	3	no	—
155	C₂×C₄	—	2	3	no	—
156	C₂×C₄	C₂×C₇⋊C₁₆	1	12	yes	complemented
157	C₂×C₄	C₂²×C₇⋊C₈	1	3	yes	—
158	C₂×C₄	C₂²×C₇⋊C₈	1	3	yes	—
159	C₂×C₄	C₂²×C₇⋊C₈	1	1	yes	characteristic
160	C₂×C₄	C₂²×C₇⋊C₈	1	3	yes	—
161	C₂×C₄	C₂²×C₇⋊C₈	1	1	yes	characteristic
162	C₂×C₄	C₂³×Dic₇	1	1	yes	characteristic
163	C₈	—	2	3	no	—
164	C₈	—	2	3	no	—
165	C₈	#6511	1	2	yes	—
166	C₈	D₇×#41	1	2	yes	—
167	C₇	C₁₆×#41	1	1	yes	characteristic, complemented, Sylow
168	C₂²	—	2	3	no	—
169	C₂²	—	2	3	no	—
170	C₂²	C₂²×C₇⋊C₁₆	1	6	yes	—
171	C₂²	C₂³×C₇⋊C₈	1	1	yes	characteristic
172	C₄	—	2	3	no	—
173	C₄	—	2	3	no	—
174	C₄	C₂²×C₇⋊C₁₆	1	2	yes	—
175	C₄	C₂²×C₇⋊C₁₆	1	6	yes	—
176	C₄	#89912	1	2	yes	—
177	C₄	C₂×#6511	1	1	yes	characteristic
178	C₄	#41×Dic₇	1	1	yes	characteristic
179	C₂	—	2	6	no	—
180	C₂	#41×C₇⋊C₈	1	1	yes	characteristic
181	C₂	C₂³×C₇⋊C₁₆	1	1	yes	characteristic
182	C₂	C₂×#89912	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₈	C₂²×D₇
commutator	C₂×C₇	C₂³×C₁₆
Frattini	C₂×C₈	C₂³×D₇
Fitting	C₇×C₈×#41	C₂
socle	C₂²×C₇	C₂³×C₈
prime radical	C₂×C₇×#41	C₈
cosocle	C₂×C₇×C₈	C₂⁴
hypercenter	C₈×#41	D₇
hypocenter	C₇	C₁₆×#41
LCS	G ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₄×C₈ ◁ C₈×#41

Complements

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

N	H ≅ G/N	type
2. Q₈×C₇⋊C₁₆	C₂	semidirect
3. D₄×C₇⋊C₁₆	C₂	semidirect
4. #11727498	C₂	semidirect
5. C₂×C₄×C₇⋊C₁₆	C₂	semidirect
44. C₇×#41	C₁₆	semidirect
64. C₇⋊C₁₆	#41	direct (N × H)
79. C₇×Q₈	C₂×C₁₆	semidirect
80. C₇×D₄	C₂×C₁₆	semidirect
84. C₂×C₄×C₇	C₂×C₁₆	semidirect
125. #41	C₇⋊C₁₆	direct (N × H)
151. Q₈	C₂×C₇⋊C₁₆	semidirect
152. D₄	C₂×C₇⋊C₁₆	semidirect
156. C₂×C₄	C₂×C₇⋊C₁₆	semidirect
167. C₇	C₁₆×#41	semidirect

Maximal subgroups

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

subgroup	index
C₇×C₈×#41	2
Q₈×C₇⋊C₁₆	2
D₄×C₇⋊C₁₆	2
#11727498	2
C₂×C₄×C₇⋊C₁₆	2
C₁₆×#41	7

Subgroups — #41×C7⋊C16 #13507315

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #41×C₇⋊C₁₆ #13507315