Finite Group Databasebeta

Subgroups — #14544900

Order 2176.

200 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 72 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	C₂×#10587536	C₂	1	1	yes	maximal, characteristic
2	C₂×#10587535	C₂	1	1	yes	maximal, characteristic
3	#11763814	C₂	1	1	yes	maximal, characteristic
4	C₄×#63243	C₂	1	1	yes	maximal, characteristic
5	#11763685	C₂	1	1	yes	maximal, characteristic
6	#11763685	C₂	1	1	yes	maximal, characteristic
7	C₄⋊C₈×D₁₇	C₂	1	1	yes	maximal, characteristic
8	C₂×#63243	C₂²	1	1	yes	characteristic
9	C₂×#63243	C₂²	1	1	yes	characteristic
10	C₂×#63243	C₂²	1	1	yes	characteristic
11	C₂×#63243	C₂²	1	1	yes	characteristic
12	C₂×C₈×D₁₇	C₂²	1	1	yes	characteristic
13	C₂×C₈×D₁₇	C₂²	1	1	yes	characteristic
14	C₄²×D₁₇	C₂²	1	1	yes	characteristic
15	#10587544	—	2	1	no	—
16	#10587544	—	2	1	no	—
17	#10587542	—	2	1	no	—
18	C₄×#3624	—	2	1	no	—
19	#10587536	—	2	2	no	—
20	#10587535	—	2	2	no	—
21	C₁₇×C₄⋊C₈	C₄	1	1	yes	characteristic
22	#10587485	C₄	1	1	yes	characteristic
23	#10587485	C₄	1	1	yes	characteristic
24	C₄⋊#3622	C₄	1	1	yes	characteristic
25	C₂×C₄×D₁₇	Q₈	1	1	yes	characteristic
26	C₂×C₄×D₁₇	Q₈	1	1	yes	characteristic
27	C₂×C₄×D₁₇	C₂³	1	1	yes	characteristic
28	C₂×#3624	—	2	1	no	—
29	C₂×#3624	—	2	1	no	—
30	C₂×#3624	—	2	1	no	—
31	C₂×#3624	—	2	1	no	—
32	#63243	—	2	1	no	—
33	#63243	—	2	1	no	—
34	#63243	—	2	1	no	—
35	#63243	—	2	1	no	—
36	#63243	—	2	1	no	—
37	#63243	—	2	1	no	—
38	#63243	D₄	1	2	yes	—
39	#63243	Q₈	1	2	yes	—
40	C₂×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
41	C₂×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
42	C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
43	C₄×Dic₁₇	C₂×C₄	1	1	yes	characteristic
44	C₄×Dic₁₇	D₄	1	1	yes	characteristic
45	C₄×Dic₁₇	D₄	1	1	yes	characteristic
46	C₂×#3622	C₂×C₄	1	1	yes	characteristic
47	C₂×#3622	C₂×C₄	1	1	yes	characteristic
48	C₈×D₁₇	—	2	1	no	—
49	C₈×D₁₇	—	2	1	no	—
50	C₈×D₁₇	—	2	1	no	—
51	C₈×D₁₇	—	2	1	no	—
52	C₂²×D₁₇	C₂×Q₈	1	1	yes	characteristic
53	C₂×C₄×C₁₇	C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₄×C₁₇	C₄⋊C₄	1	1	yes	characteristic
55	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
56	C₂×Dic₁₇	C₄⋊C₄	1	1	yes	characteristic
57	C₂×Dic₁₇	C₄⋊C₄	1	1	yes	characteristic
58	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
59	C₄×D₁₇	—	2	2	no	—
60	C₄×D₁₇	—	4	1	no	—
61	C₄×D₁₇	C₂×D₄	1	1	yes	characteristic
62	C₄×D₁₇	C₂×Q₈	1	1	yes	characteristic
63	C₄×D₁₇	#41	1	1	yes	characteristic
64	C₄×D₁₇	#41	1	1	yes	characteristic
65	#3624	—	2	2	no	—
66	#3624	—	2	2	no	—
67	#3624	—	4	1	no	—
68	#3624	—	4	1	no	—
69	C₈×C₁₇	—	2	1	no	—
70	C₈×C₁₇	—	2	1	no	—
71	#3622	—	2	1	no	—
72	#3622	—	2	1	no	—
73	#1950	—	17	1	no	maximal, Sylow
74	C₂²×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
75	C₂×D₁₇	—	2	1	no	—
76	C₂×D₁₇	—	2	1	no	—
77	C₂×D₁₇	#125	1	1	yes	characteristic
78	C₂×D₁₇	C₄⋊Q₈	1	1	yes	characteristic
79	C₄×C₁₇	—	2	1	no	—
80	C₄×C₁₇	—	2	1	no	—
81	C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
82	C₄×C₁₇	C₄×Q₈	1	1	yes	characteristic
83	Dic₁₇	—	2	1	no	—
84	Dic₁₇	#108	1	2	yes	complemented
85	Dic₁₇	#121	1	1	yes	characteristic
86	Dic₁₇	C₂²⋊Q₈	1	1	yes	characteristic
87	C₂×#108	—	17	1	no	—
88	C₂×#108	—	17	1	no	—
89	#423	—	17	1	no	—
90	C₂×C₄⋊C₈	—	17	1	no	—
91	C₄×M₁₆	—	17	1	no	—
92	#341	—	17	1	no	—
93	#341	—	17	1	no	—
94	C₂×C₁₇	#389	1	1	yes	characteristic
95	C₂×C₁₇	C₂×#108	1	1	yes	characteristic
96	C₂×C₁₇	#430	1	1	yes	characteristic
97	D₁₇	—	2	1	no	—
98	D₁₇	—	2	1	no	—
99	C₂×M₁₆	—	17	1	no	—
100	C₂×M₁₆	—	17	1	no	—
101	C₂×M₁₆	—	17	1	no	—
102	C₂×M₁₆	—	17	1	no	—
103	C₂²×C₈	—	17	1	no	—
104	C₂²×C₈	—	17	1	no	—
105	C₂×C₄²	—	17	1	no	—
106	#108	—	34	2	no	—
107	#108	—	34	2	no	—
108	C₄⋊C₈	—	17	1	no	—
109	C₄⋊C₈	—	17	1	no	—
110	C₄⋊C₈	—	17	1	no	—
111	C₄⋊C₈	—	34	1	no	—
112	C₄⋊C₈	—	34	1	no	—
113	C₄⋊C₈	#595	1	1	yes	characteristic
114	#97	—	34	1	no	—
115	C₄×C₈	—	34	1	no	—
116	C₁₇	#1950	1	1	yes	characteristic, complemented, Sylow
117	C₂²×C₄	—	17	1	no	—
118	C₂²×C₄	—	17	1	no	—
119	C₂²×C₄	—	17	1	no	—
120	M₁₆	—	17	2	no	—
121	M₁₆	—	17	2	no	—
122	M₁₆	—	34	1	no	—
123	M₁₆	—	34	1	no	—
124	M₁₆	—	34	1	no	—
125	M₁₆	—	34	1	no	—
126	M₁₆	—	34	1	no	—
127	M₁₆	—	34	1	no	—
128	C₂×C₈	—	17	1	no	—
129	C₂×C₈	—	17	1	no	—
130	C₂×C₈	—	34	1	no	—
131	C₂×C₈	—	34	1	no	—
132	C₂×C₈	—	34	1	no	—
133	C₂×C₈	—	34	1	no	—
134	C₂×C₈	—	34	1	no	—
135	C₂×C₈	—	34	1	no	—
136	C₂×C₈	—	34	1	no	—
137	C₂×C₈	—	34	1	no	—
138	C₂×C₈	C₂×#595	1	1	yes	characteristic
139	C₂×C₈	C₂×#595	1	1	yes	characteristic
140	C₄²	—	17	1	no	—
141	C₄²	—	17	1	no	—
142	C₄²	—	17	1	no	—
143	C₄²	C₂×#595	1	1	yes	characteristic
144	C₂³	—	17	1	no	—
145	C₂×C₄	—	17	1	no	—
146	C₂×C₄	—	17	1	no	—
147	C₂×C₄	—	17	1	no	—
148	C₂×C₄	—	17	1	no	—
149	C₂×C₄	—	17	1	no	—
150	C₂×C₄	—	17	1	no	—
151	C₂×C₄	—	17	1	no	—
152	C₂×C₄	—	34	2	no	—
153	C₂×C₄	—	68	1	no	—
154	C₂×C₄	C₄⋊#595	1	1	yes	characteristic
155	C₂×C₄	C₄⋊#595	1	1	yes	characteristic
156	C₂×C₄	C₂²×#595	1	1	yes	characteristic
157	C₈	—	2	1	no	—
158	C₈	—	2	1	no	—
159	C₈	—	34	2	no	—
160	C₈	—	34	2	no	—
161	C₈	—	34	1	no	—
162	C₈	—	34	1	no	—
163	C₈	—	68	1	no	—
164	C₈	—	68	1	no	—
165	C₂²	—	17	1	no	—
166	C₂²	—	17	1	no	—
167	C₂²	—	34	1	no	—
168	C₂²	—	34	1	no	—
169	C₂²	C₂×C₄⋊#595	1	1	yes	characteristic
170	C₄	—	2	1	no	—
171	C₄	—	2	1	no	—
172	C₄	—	17	1	no	—
173	C₄	—	17	1	no	—
174	C₄	—	17	2	no	—
175	C₄	—	34	1	no	—
176	C₄	D₄×#595	1	1	yes	characteristic
177	C₄	Q₈×#595	1	1	yes	characteristic
178	C₂	—	34	1	no	—
179	C₂	—	34	1	no	—
180	C₂	#11763760	1	1	yes	characteristic
181	C₂	D₁₇⋊#108	1	1	yes	characteristic
182	C₂	#11763824	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₄⋊#595
commutator	C₂×C₄×C₁₇	C₂²×C₄
Frattini	C₂×C₄	C₂²×#595
Fitting	C₁₇×C₄⋊C₈	C₄
socle	C₂²×C₁₇	C₂×C₄⋊C₄
prime radical	C₂²×D₁₇	C₂×Q₈
cosocle	C₂×C₄×D₁₇	C₂³
hypercenter	C₄⋊C₈	#595
hypocenter	C₁₇	#1950
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊C₈

Complements

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

N	H ≅ G/N	type
84. Dic₁₇	#108	semidirect
116. C₁₇	#1950	semidirect

Maximal subgroups

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

subgroup	index
C₂×#10587536	2
C₂×#10587535	2
#11763814	2
C₄×#63243	2
#11763685	2
C₄⋊C₈×D₁₇	2
#1950	17