Subgroups — #14537727

Order 2176.

356 conjugacy classes of subgroups.

The subgroup lattice has 182 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11763366	C₂	1	1	yes	maximal, characteristic
2	#11762938	C₂	1	4	yes	maximal, complemented
3	#11762936	C₂	1	2	yes	maximal, complemented
4	#11762933	C₂	1	2	yes	maximal, complemented
5	#11762835	C₂	1	2	yes	maximal
6	#11762764	C₂	1	2	yes	maximal
7	C₄×#63223	C₂	1	1	yes	maximal, characteristic
8	C₂×C₁₇×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
9	C₂²×C₈×C₁₇	C₂²	1	2	yes	—
10	C₂×C₄²×C₁₇	C₂²	1	1	yes	characteristic
11	#10587612	C₂²	1	2	yes	—
12	#10587610	C₄	1	2	yes	—
13	#10587610	C₄	1	2	yes	—
14	C₂×#63223	C₂²	1	1	yes	characteristic
15	C₂×#63223	C₂²	1	1	yes	characteristic
16	#10587556	C₄	1	2	yes	—
17	#10587555	C₄	1	1	yes	characteristic
18	#10587555	C₄	1	1	yes	characteristic
19	C₄×Dic₃₄	C₂²	1	2	yes	—
20	C₄×Dic₃₄	C₂²	1	2	yes	—
21	C₁₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
22	C₁₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
23	C₁₇×C₄⋊C₈	C₂²	1	2	yes	—
24	#10587486	C₂²	1	4	yes	—
25	#10587485	C₂²	1	4	yes	—
26	#10587485	C₂²	1	4	yes	—
27	C₈×Dic₁₇	C₂²	1	4	yes	—
28	#10587474	C₂²	1	2	yes	—
29	C₄×#3622	C₂²	1	2	yes	—
30	C₂²×C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
33	C₂×Dic₃₄	C₂×C₄	1	2	yes	—
34	C₂×Dic₃₄	C₂×C₄	1	2	yes	—
35	C₂×C₈×C₁₇	—	2	4	no	—
36	C₂×C₈×C₁₇	C₂³	1	2	yes	—
37	C₂×C₈×C₁₇	C₂³	1	2	yes	—
38	C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
39	C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
40	C₄²×C₁₇	C₂³	1	1	yes	characteristic
41	C₄²×C₁₇	C₂³	1	1	yes	characteristic
42	C₂²⋊Dic₁₇	C₂×C₄	1	4	yes	—
43	C₄⋊Dic₁₇	C₂×C₄	1	2	yes	—
44	C₄⋊Dic₁₇	C₂×C₄	1	2	yes	—
45	C₄⋊Dic₁₇	C₂×C₄	1	2	yes	—
46	C₄⋊Dic₁₇	C₂×C₄	1	2	yes	—
47	#63225	C₂×C₄	1	4	yes	—
48	#63225	C₂×C₄	1	4	yes	—
49	C₄×Dic₁₇	C₂³	1	4	yes	—
50	#63223	—	2	2	no	—
51	#63223	Q₈	1	4	yes	complemented
52	C₂×#3622	C₂³	1	2	yes	—
53	C₂×#3622	C₂³	1	2	yes	—
54	C₂³×C₁₇	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₄×C₁₇	—	2	2	no	—
56	C₂×C₄×C₁₇	—	4	1	no	—
57	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₄×C₁₇	C₂×Q₈	1	2	yes	—
63	C₂×C₄×C₁₇	#41	1	2	yes	—
64	C₂×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
65	C₂×Dic₁₇	C₂²×C₄	1	4	yes	—
66	C₂×Dic₁₇	C₂²×C₄	1	4	yes	—
67	Dic₃₄	—	2	4	no	—
68	Dic₃₄	—	4	2	no	—
69	C₈×C₁₇	—	2	2	no	—
70	C₈×C₁₇	—	2	2	no	—
71	#3622	—	2	2	no	—
72	#3622	C₂×Q₈	1	4	yes	—
73	#2962	—	17	1	no	maximal, Sylow
74	C₂²×C₁₇	—	2	1	no	—
75	C₂²×C₁₇	—	2	1	no	—
76	C₂²×C₁₇	C₄×Q₈	1	2	yes	—
77	C₂²×C₁₇	C₂³×C₄	1	1	yes	characteristic
78	C₄×C₁₇	—	2	1	no	—
79	C₄×C₁₇	—	2	1	no	—
80	C₄×C₁₇	—	2	1	no	—
81	C₄×C₁₇	C₄×Q₈	1	2	yes	—
82	C₄×C₁₇	#131	1	2	yes	—
83	C₄×C₁₇	C₂²×Q₈	1	1	yes	characteristic
84	C₄×C₁₇	C₂×#41	1	1	yes	characteristic
85	Dic₁₇	—	2	4	no	—
86	Dic₁₇	—	2	4	no	—
87	#533	—	17	1	no	—
88	#446	—	17	2	no	—
89	#446	—	17	4	no	—
90	C₈×Q₈	—	17	2	no	—
91	#424	—	17	2	no	—
92	C₂×C₄⋊C₈	D₁₇	1	1	yes	characteristic
93	#405	—	17	2	no	—
94	C₄×M₁₆	—	17	1	no	—
95	C₂×C₁₇	—	2	2	no	—
96	C₂×C₁₇	C₂×C₄×Q₈	1	1	yes	characteristic
97	C₂×C₁₇	C₂×#131	1	1	yes	characteristic
98	C₂×C₁₇	#568	1	1	yes	characteristic
99	C₂×M₁₆	—	17	1	no	—
100	C₂×M₁₆	—	17	1	no	—
101	C₂²×C₈	C₂×D₁₇	1	2	yes	—
102	C₄⋊Q₈	—	17	1	no	—
103	C₄⋊Q₈	—	17	1	no	—
104	#125	—	17	2	no	—
105	C₂²⋊Q₈	—	17	2	no	—
106	C₂²⋊Q₈	—	17	2	no	—
107	C₄×Q₈	—	17	2	no	—
108	C₄×Q₈	—	17	2	no	—
109	#117	—	17	2	no	—
110	C₂×C₄²	C₂×D₁₇	1	1	yes	characteristic
111	C₄⋊C₈	—	17	4	no	—
112	C₄⋊C₈	—	17	4	no	—
113	C₄⋊C₈	C₂×D₁₇	1	1	yes	characteristic
114	C₄⋊C₈	C₂×D₁₇	1	1	yes	characteristic
115	C₄⋊C₈	C₂×D₁₇	1	2	yes	—
116	#97	—	17	4	no	—
117	#97	—	17	2	no	—
118	C₄×C₈	—	17	4	no	—
119	C₄×C₈	—	17	2	no	—
120	C₁₇	#2962	1	1	yes	characteristic, complemented, Sylow
121	C₂×Q₈	—	17	2	no	—
122	C₂×Q₈	—	17	2	no	—
123	C₂²×C₄	C₄×D₁₇	1	1	yes	characteristic
124	C₂²×C₄	C₄×D₁₇	1	1	yes	characteristic
125	C₂²×C₄	C₂²×D₁₇	1	1	yes	characteristic
126	M₁₆	—	17	4	no	—
127	M₁₆	—	34	2	no	—
128	C₂×C₈	—	2	4	no	—
129	C₂×C₈	—	17	2	no	—
130	C₂×C₈	—	17	2	no	—
131	C₂×C₈	C₂²×D₁₇	1	2	yes	—
132	C₂×C₈	C₂²×D₁₇	1	2	yes	—
133	C₄⋊C₄	—	17	2	no	—
134	C₄⋊C₄	—	17	2	no	—
135	C₄⋊C₄	—	17	4	no	—
136	C₄⋊C₄	—	17	4	no	—
137	C₄⋊C₄	—	17	2	no	—
138	C₄⋊C₄	—	17	2	no	—
139	C₂²⋊C₄	—	17	4	no	—
140	C₄²	—	17	4	no	—
141	C₄²	C₄×D₁₇	1	1	yes	characteristic
142	C₄²	C₄×D₁₇	1	1	yes	characteristic
143	C₄²	C₂²×D₁₇	1	1	yes	characteristic
144	C₄²	C₂²×D₁₇	1	1	yes	characteristic
145	C₂³	C₂×C₄×D₁₇	1	1	yes	characteristic
146	Q₈	—	34	4	no	—
147	Q₈	—	68	2	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	4	1	no	—
150	C₂×C₄	—	17	4	no	—
151	C₂×C₄	—	17	4	no	—
152	C₂×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
153	C₂×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
154	C₂×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
155	C₂×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
156	C₂×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
157	C₂×C₄	#63254	1	2	yes	—
158	C₂×C₄	Q₈×D₁₇	1	2	yes	—
159	C₂×C₄	C₂³×D₁₇	1	1	yes	characteristic
160	C₈	—	2	2	no	—
161	C₈	—	2	2	no	—
162	C₈	—	17	4	no	—
163	C₈	—	34	2	no	—
164	C₂²	—	2	1	no	—
165	C₂²	—	2	1	no	—
166	C₂²	#10587573	1	2	yes	—
167	C₂²	C₂²×C₄×D₁₇	1	1	yes	characteristic
168	C₄	—	2	1	no	—
169	C₄	—	2	1	no	—
170	C₄	—	2	1	no	—
171	C₄	—	34	4	no	—
172	C₄	—	34	4	no	—
173	C₄	#10587573	1	2	yes	—
174	C₄	#10587587	1	2	yes	—
175	C₄	C₂×#63254	1	1	yes	characteristic
176	C₄	C₂×Q₈×D₁₇	1	1	yes	characteristic
177	C₂	—	2	2	no	—
178	C₂	C₂×#10587573	1	1	yes	characteristic
179	C₂	C₂×#10587587	1	1	yes	characteristic
180	C₂	#11763575	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₇
commutator, socle	C₂²×C₁₇	C₂³×C₄
Fitting	C₂×C₁₇×C₄⋊C₈	C₂
prime radical	C₂³×C₁₇	C₂²×C₄
cosocle	C₂×C₄×C₁₇	C₂⁴
hypercenter	C₂×C₄⋊C₈	D₁₇
hypocenter	C₁₇	#2962
LCS	G ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
2. #11762938	C₂	semidirect
3. #11762936	C₂	semidirect
4. #11762933	C₂	semidirect
51. #63223	Q₈	semidirect
120. C₁₇	#2962	semidirect

Maximal subgroups

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

subgroup	index
#11763366	2
#11762938	2
#11762936	2
#11762933	2
#11762835	2
#11762764	2
C₄×#63223	2
C₂×C₁₇×C₄⋊C₈	2
#2962	17