Finite Group Databasebeta

Subgroups — #14544849

Order 2176.

186 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 186 conjugacy classes of subgroups — too many to draw. Only its 60 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	#11763816	C₂	1	1	yes	maximal, characteristic
2	#11763774	C₂	1	1	yes	maximal, characteristic
3	#11763685	C₂	1	1	yes	maximal, characteristic
4	#11763679	C₂	1	1	yes	maximal, characteristic
5	#11763679	C₂	1	1	yes	maximal, characteristic
6	#11763674	C₂	1	1	yes	maximal, characteristic
7	#11762846	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄×#595	C₂²	1	1	yes	characteristic
9	C₂×#63243	C₂²	1	1	yes	characteristic
10	C₂×#63243	C₂²	1	1	yes	characteristic
11	C₂×#63242	C₂²	1	1	yes	characteristic
12	C₂×C₈×D₁₇	C₂²	1	1	yes	characteristic
13	#10587578	C₂²	1	1	yes	characteristic
14	#10587578	C₂²	1	1	yes	characteristic
15	#10587544	—	2	1	no	—
16	#10587542	—	2	1	no	—
17	C₄⋊#3624	—	2	1	no	—
18	C₄×#3624	—	2	1	no	—
19	C₁₇×#107	C₄	1	1	yes	characteristic, complemented
20	#10587489	C₄	1	1	yes	characteristic, complemented
21	#10587479	C₄	1	1	yes	characteristic, complemented
22	#10587479	C₄	1	1	yes	characteristic, complemented
23	C₂²×#595	—	2	1	no	—
24	C₂×C₄×D₁₇	C₂³	1	1	yes	characteristic
25	C₂×#3624	—	2	1	no	—
26	C₂×#3624	—	2	1	no	—
27	C₂×#3624	D₄	1	1	yes	characteristic
28	C₂×#3624	D₄	1	1	yes	characteristic
29	C₄×#595	—	2	1	no	—
30	C₄×#595	—	2	1	no	—
31	#63243	—	2	1	no	—
32	#63243	—	2	1	no	—
33	#63243	—	2	1	no	—
34	#63243	Q₈	1	2	yes	—
35	#63242	—	2	1	no	—
36	#63242	—	2	1	no	—
37	C₂×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
38	C₄⋊C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
39	C₄⋊C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
40	#63227	—	2	1	no	—
41	#63227	—	2	1	no	—
42	C₄⋊Dic₁₇	C₂×C₄	1	1	yes	characteristic
43	C₄⋊Dic₁₇	C₂×C₄	1	1	yes	characteristic
44	C₄×Dic₁₇	—	2	1	no	—
45	C₄×Dic₁₇	—	2	1	no	—
46	C₂×#3622	C₂×C₄	1	1	yes	characteristic
47	C₈×D₁₇	—	2	1	no	—
48	C₈×D₁₇	—	2	1	no	—
49	C₂²×D₁₇	#41	1	1	yes	characteristic
50	C₂×#595	—	2	2	no	—
51	C₂×#595	—	4	1	no	—
52	C₂×#595	—	4	1	no	—
53	C₂×C₄×C₁₇	—	2	1	no	—
54	C₂×C₄×C₁₇	—	2	1	no	—
55	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
56	C₂×Dic₁₇	—	2	1	no	—
57	C₂×Dic₁₇	—	2	1	no	—
58	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
59	C₄×D₁₇	C₂×Q₈	1	1	yes	characteristic
60	C₄×D₁₇	#41	1	1	yes	characteristic
61	C₄×D₁₇	#41	1	1	yes	characteristic
62	C₄×D₁₇	#41	1	1	yes	characteristic
63	#3624	—	2	2	no	—
64	#3624	—	2	1	no	—
65	#3624	—	2	1	no	—
66	#3624	—	4	1	no	—
67	C₈×C₁₇	—	2	1	no	—
68	#3622	—	2	1	no	—
69	#1922	—	17	1	no	maximal, Sylow
70	C₂²×C₁₇	C₄×D₄	1	1	yes	characteristic
71	C₂×D₁₇	—	2	1	no	—
72	C₂×D₁₇	—	2	1	no	—
73	C₂×D₁₇	#125	1	1	yes	characteristic
74	C₂×D₁₇	#126	1	1	yes	characteristic
75	#595	—	4	2	no	—
76	C₄×C₁₇	—	4	1	no	—
77	C₄×C₁₇	—	4	1	no	—
78	C₄×C₁₇	#117	1	1	yes	characteristic
79	C₄×C₁₇	C₄×Q₈	1	1	yes	characteristic
80	Dic₁₇	—	4	1	no	—
81	Dic₁₇	—	4	1	no	—
82	Dic₁₇	C₂²⋊Q₈	1	1	yes	characteristic
83	Dic₁₇	#123	1	1	yes	characteristic
84	#427	—	17	1	no	—
85	#424	—	17	1	no	—
86	#405	—	17	1	no	—
87	#341	—	17	1	no	—
88	#339	—	17	1	no	—
89	#339	—	17	1	no	—
90	#336	—	17	1	no	—
91	C₂×C₁₇	#387	1	1	yes	characteristic
92	C₂×C₁₇	#443	1	1	yes	characteristic
93	C₂×C₁₇	#444	1	1	yes	characteristic
94	D₁₇	—	2	1	no	—
95	D₁₇	—	2	1	no	—
96	C₂×M₁₆	—	17	1	no	—
97	C₂×M₁₆	—	17	1	no	—
98	C₂²×C₈	—	17	1	no	—
99	C₂²×C₈	—	17	1	no	—
100	#117	—	17	1	no	—
101	#117	—	17	1	no	—
102	C₂×C₄²	—	17	1	no	—
103	#107	—	17	1	no	—
104	#107	#595	1	1	yes	characteristic, complemented
105	#106	—	17	1	no	—
106	#106	—	17	1	no	—
107	C₄⋊C₈	—	34	1	no	—
108	C₄⋊C₈	—	34	1	no	—
109	#97	—	34	1	no	—
110	C₄×C₈	—	34	1	no	—
111	C₁₇	#1922	1	1	yes	characteristic, complemented, Sylow
112	C₂²×C₄	—	17	1	no	—
113	C₂²×C₄	—	34	1	no	—
114	M₁₆	—	17	2	no	—
115	M₁₆	—	34	1	no	—
116	M₁₆	—	34	1	no	—
117	M₁₆	—	34	1	no	—
118	C₂×C₈	—	17	1	no	—
119	C₂×C₈	—	17	1	no	—
120	C₂×C₈	—	17	1	no	—
121	C₂×C₈	—	34	1	no	—
122	C₂×C₈	—	34	1	no	—
123	C₂×C₈	—	34	1	no	—
124	C₂×C₈	—	34	1	no	—
125	C₂×C₈	—	34	1	no	—
126	C₂×C₈	—	34	1	no	—
127	C₂×C₈	C₂×#595	1	1	yes	characteristic
128	C₄⋊C₄	—	17	1	no	—
129	C₄⋊C₄	—	17	1	no	—
130	C₄⋊C₄	C₂×#595	1	1	yes	characteristic
131	C₄⋊C₄	C₂×#595	1	1	yes	characteristic
132	C₂²⋊C₄	—	34	1	no	—
133	C₂²⋊C₄	—	34	1	no	—
134	C₄²	—	34	1	no	—
135	C₄²	—	34	1	no	—
136	C₄²	—	34	1	no	—
137	C₄²	—	34	1	no	—
138	C₂³	—	17	1	no	—
139	C₂×C₄	—	2	1	no	—
140	C₂×C₄	—	2	1	no	—
141	C₂×C₄	—	17	1	no	—
142	C₂×C₄	—	17	1	no	—
143	C₂×C₄	—	17	1	no	—
144	C₂×C₄	—	17	1	no	—
145	C₂×C₄	—	17	1	no	—
146	C₂×C₄	—	34	2	no	—
147	C₂×C₄	—	34	1	no	—
148	C₂×C₄	—	34	1	no	—
149	C₂×C₄	—	68	1	no	—
150	C₂×C₄	—	68	1	no	—
151	C₂×C₄	C₂²×#595	1	1	yes	characteristic
152	C₈	—	2	1	no	—
153	C₈	—	34	1	no	—
154	C₈	—	34	1	no	—
155	C₈	—	34	2	no	—
156	C₈	—	34	1	no	—
157	C₈	—	68	1	no	—
158	C₂²	—	17	1	no	—
159	C₂²	—	17	1	no	—
160	C₂²	—	34	1	no	—
161	C₂²	—	34	1	no	—
162	C₂²	D₄×#595	1	1	yes	characteristic
163	C₄	—	4	1	no	—
164	C₄	—	4	1	no	—
165	C₄	—	17	1	no	—
166	C₄	—	17	1	no	—
167	C₄	—	68	2	no	—
168	C₄	—	68	1	no	—
169	C₄	—	68	1	no	—
170	C₄	#10587677	1	1	yes	characteristic
171	C₄	Q₈×#595	1	1	yes	characteristic
172	C₂	—	34	1	no	—
173	C₂	—	34	1	no	—
174	C₂	#11763758	1	1	yes	characteristic
175	C₂	#11763851	1	1	yes	characteristic
176	C₂	#11763855	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×#595
commutator	C₂×C₄×C₁₇	C₂²×C₄
Frattini	C₂×C₄	C₂²×#595
Fitting	C₁₇×#107	C₄
socle	C₂²×C₁₇	C₄×D₄
prime radical	C₂²×D₁₇	#41
cosocle	C₂×C₄×D₁₇	C₂³
hypercenter	#107	#595
hypocenter	C₁₇	#1922
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #107

Complements

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

N	H ≅ G/N	type
19. C₁₇×#107	C₄	semidirect
20. #10587489	C₄	semidirect
21. #10587479	C₄	semidirect
22. #10587479	C₄	semidirect
104. #107	#595	semidirect
111. C₁₇	#1922	semidirect

Maximal subgroups

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

subgroup	index
#11763816	2
#11763774	2
#11763685	2
#11763679	2
#11763674	2
#11762846	2
#1922	17