Subgroups — #14533333

Order 2176.

380 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11762750	C₂	1	2	yes	maximal
2	#11762747	C₂	1	2	yes	maximal
3	#11762707	C₂	1	1	yes	maximal, characteristic
4	#11762707	C₂	1	1	yes	maximal, characteristic
5	#11762707	C₂	1	1	yes	maximal, characteristic
6	#11762707	C₂	1	1	yes	maximal, characteristic
7	#11762704	C₂	1	2	yes	maximal
8	#11762691	C₂	1	2	yes	maximal
9	#11762675	C₂	1	1	yes	maximal, characteristic
10	C₄×C₄⋊Dic₁₇	C₂	1	1	yes	maximal, characteristic
11	C₄×C₄⋊C₄×C₁₇	C₂	1	1	yes	maximal, characteristic
12	C₂×C₄⋊C₄×C₁₇	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊C₄×C₁₇	C₂²	1	1	yes	characteristic
14	C₂×C₄²×C₁₇	C₂²	1	1	yes	characteristic
15	C₂×C₄²×C₁₇	C₂²	1	1	yes	characteristic
16	C₂×C₄²×C₁₇	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊Dic₁₇	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊Dic₁₇	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊Dic₁₇	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊Dic₁₇	C₂²	1	1	yes	characteristic
21	C₂×#63225	C₂²	1	2	yes	—
22	C₂×#63225	C₂²	1	2	yes	—
23	C₂×#63225	C₂²	1	2	yes	—
24	C₂×#63225	C₂²	1	2	yes	—
25	C₂×C₄×Dic₁₇	C₂²	1	2	yes	—
26	C₁₇×#95	C₂²	1	1	yes	characteristic
27	C₁₇×#95	C₂²	1	1	yes	characteristic
28	#10587502	C₂²	1	2	yes	—
29	#10587502	C₂²	1	2	yes	—
30	#10587502	C₂²	1	2	yes	—
31	#10587502	C₂²	1	2	yes	—
32	#10587502	C₂²	1	2	yes	—
33	#10587502	C₂²	1	2	yes	—
34	#10587502	C₂²	1	2	yes	—
35	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
36	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
37	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
38	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
39	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
40	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
41	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
42	C₂²×Dic₁₇	C₂³	1	2	yes	—
43	C₂²×Dic₁₇	C₂³	1	2	yes	—
44	C₂²×Dic₁₇	C₂³	1	2	yes	—
45	C₂²×Dic₁₇	C₂³	1	2	yes	—
46	C₄⋊C₄×C₁₇	—	2	2	no	—
47	C₄⋊C₄×C₁₇	—	2	2	no	—
48	C₄²×C₁₇	—	2	2	no	—
49	C₄²×C₁₇	—	2	2	no	—
50	C₄²×C₁₇	—	2	2	no	—
51	C₄⋊Dic₁₇	—	2	2	no	—
52	C₄⋊Dic₁₇	—	2	2	no	—
53	C₄⋊Dic₁₇	—	2	2	no	—
54	C₄⋊Dic₁₇	Q₈	1	4	yes	—
55	#63225	—	2	4	no	—
56	#63225	—	2	4	no	—
57	#63225	—	4	2	no	—
58	#63225	—	4	2	no	—
59	C₄×Dic₁₇	—	4	2	no	—
60	C₂³×C₁₇	C₂⁴	1	1	yes	characteristic
61	C₂×C₄×C₁₇	—	2	1	no	—
62	C₂×C₄×C₁₇	—	2	1	no	—
63	C₂×C₄×C₁₇	—	2	1	no	—
64	C₂×C₄×C₁₇	—	2	1	no	—
65	C₂×C₄×C₁₇	—	2	1	no	—
66	C₂×C₄×C₁₇	—	2	1	no	—
67	C₂×C₄×C₁₇	—	2	1	no	—
68	C₂×C₄×C₁₇	—	2	1	no	—
69	C₂×C₄×C₁₇	—	2	1	no	—
70	C₂×C₄×C₁₇	—	2	1	no	—
71	C₂×C₄×C₁₇	—	2	1	no	—
72	C₂×C₄×C₁₇	—	2	1	no	—
73	C₂×C₄×C₁₇	—	2	1	no	—
74	C₂×C₄×C₁₇	—	2	1	no	—
75	C₂×C₄×C₁₇	—	2	1	no	—
76	C₂×C₄×C₁₇	—	2	1	no	—
77	C₂×C₄×C₁₇	C₂×Q₈	1	2	yes	—
78	C₂×C₄×C₁₇	#41	1	2	yes	—
79	C₂×C₄×C₁₇	#41	1	2	yes	—
80	C₂×C₄×C₁₇	#41	1	2	yes	—
81	C₂×C₄×C₁₇	#41	1	2	yes	—
82	C₂×Dic₁₇	—	2	2	no	—
83	C₂×Dic₁₇	—	2	2	no	—
84	C₂×Dic₁₇	—	2	2	no	—
85	C₂×Dic₁₇	—	2	2	no	—
86	C₂×Dic₁₇	—	2	2	no	—
87	C₂×Dic₁₇	—	2	2	no	—
88	C₂×Dic₁₇	—	2	2	no	—
89	C₂×Dic₁₇	—	2	2	no	—
90	C₂×Dic₁₇	—	2	2	no	—
91	C₂×Dic₁₇	—	2	2	no	—
92	C₂×Dic₁₇	—	2	2	no	—
93	C₂×Dic₁₇	C₂×Q₈	1	4	yes	—
94	#2509	—	17	1	no	maximal, Sylow
95	C₂²×C₁₇	C₂²×Q₈	1	1	yes	characteristic
96	C₂²×C₁₇	C₂×#41	1	1	yes	characteristic
97	C₂²×C₁₇	C₂×#41	1	1	yes	characteristic
98	C₂²×C₁₇	C₂×#41	1	1	yes	characteristic
99	C₂²×C₁₇	C₂×#41	1	1	yes	characteristic
100	C₂²×C₁₇	H(5,𝔽₂)	1	1	yes	characteristic
101	C₂²×C₁₇	2₋¹⁺⁴	1	1	yes	characteristic
102	C₄×C₁₇	—	2	2	no	—
103	C₄×C₁₇	—	2	2	no	—
104	C₄×C₁₇	—	4	1	no	—
105	C₄×C₁₇	—	4	1	no	—
106	C₄×C₁₇	—	4	1	no	—
107	C₄×C₁₇	—	4	1	no	—
108	C₄×C₁₇	#125	1	4	yes	—
109	Dic₁₇	—	2	4	no	—
110	Dic₁₇	—	4	2	no	—
111	Dic₁₇	—	4	2	no	—
112	Dic₁₇	—	4	2	no	—
113	#400	—	17	2	no	—
114	#400	—	17	2	no	—
115	#389	—	17	2	no	—
116	#389	—	17	1	no	—
117	#389	—	17	1	no	—
118	#389	—	17	1	no	—
119	#389	—	17	1	no	—
120	#387	—	17	2	no	—
121	#382	—	17	1	no	—
122	C₄×C₄⋊C₄	—	17	1	no	—
123	C₄×C₄⋊C₄	D₁₇	1	1	yes	characteristic
124	C₂×C₁₇	C₂×#125	1	1	yes	characteristic
125	C₂×C₁₇	#529	1	1	yes	characteristic
126	C₂×C₁₇	#542	1	1	yes	characteristic
127	C₂×C₁₇	#553	1	1	yes	characteristic
128	C₂×C₁₇	#554	1	1	yes	characteristic
129	C₂×C₁₇	#556	1	1	yes	characteristic
130	C₂×C₁₇	#557	1	1	yes	characteristic
131	C₂×C₄⋊C₄	—	17	1	no	—
132	C₂×C₄⋊C₄	—	17	1	no	—
133	C₂×C₄⋊C₄	—	17	1	no	—
134	C₂×C₄⋊C₄	—	17	2	no	—
135	C₂×C₄⋊C₄	—	17	1	no	—
136	C₂×C₄⋊C₄	—	17	2	no	—
137	C₂×C₄⋊C₄	—	17	2	no	—
138	C₂×C₄⋊C₄	—	17	2	no	—
139	C₂×C₄⋊C₄	C₂×D₁₇	1	1	yes	characteristic
140	C₂×C₄⋊C₄	C₂×D₁₇	1	1	yes	characteristic
141	C₂×C₄²	—	17	2	no	—
142	C₂×C₄²	C₂×D₁₇	1	1	yes	characteristic
143	C₂×C₄²	C₂×D₁₇	1	1	yes	characteristic
144	C₂×C₄²	C₂×D₁₇	1	1	yes	characteristic
145	#95	—	17	2	no	—
146	#95	—	17	2	no	—
147	#95	—	17	2	no	—
148	#95	—	17	2	no	—
149	#95	—	17	2	no	—
150	#95	—	17	2	no	—
151	#95	—	17	2	no	—
152	#95	C₂×D₁₇	1	1	yes	characteristic
153	#95	C₂×D₁₇	1	1	yes	characteristic
154	C₁₇	#2509	1	1	yes	characteristic, complemented, Sylow
155	C₂²×C₄	—	17	2	no	—
156	C₂²×C₄	—	17	2	no	—
157	C₂²×C₄	—	17	2	no	—
158	C₂²×C₄	—	17	2	no	—
159	C₂²×C₄	C₂²×D₁₇	1	1	yes	characteristic
160	C₂²×C₄	C₂²×D₁₇	1	1	yes	characteristic
161	C₂²×C₄	C₂²×D₁₇	1	1	yes	characteristic
162	C₂²×C₄	C₂²×D₁₇	1	1	yes	characteristic
163	C₂²×C₄	C₂²×D₁₇	1	1	yes	characteristic
164	C₂²×C₄	C₂²×D₁₇	1	1	yes	characteristic
165	C₂²×C₄	C₂²×D₁₇	1	1	yes	characteristic
166	C₄⋊C₄	—	2	2	no	—
167	C₄⋊C₄	—	2	2	no	—
168	C₄⋊C₄	—	17	4	no	—
169	C₄⋊C₄	—	34	4	no	—
170	C₄⋊C₄	—	34	4	no	—
171	C₄⋊C₄	—	34	2	no	—
172	C₄⋊C₄	—	34	2	no	—
173	C₄⋊C₄	—	34	2	no	—
174	C₄⋊C₄	—	68	2	no	—
175	C₄⋊C₄	—	68	2	no	—
176	C₄²	—	2	2	no	—
177	C₄²	—	2	2	no	—
178	C₄²	—	2	2	no	—
179	C₄²	—	68	2	no	—
180	C₂³	C₂³×D₁₇	1	1	yes	characteristic
181	C₂×C₄	—	2	1	no	—
182	C₂×C₄	—	2	1	no	—
183	C₂×C₄	—	2	1	no	—
184	C₂×C₄	—	2	1	no	—
185	C₂×C₄	—	2	1	no	—
186	C₂×C₄	—	2	1	no	—
187	C₂×C₄	—	2	1	no	—
188	C₂×C₄	—	2	1	no	—
189	C₂×C₄	—	2	1	no	—
190	C₂×C₄	—	2	1	no	—
191	C₂×C₄	—	2	1	no	—
192	C₂×C₄	—	2	1	no	—
193	C₂×C₄	—	2	1	no	—
194	C₂×C₄	—	2	1	no	—
195	C₂×C₄	—	2	1	no	—
196	C₂×C₄	—	2	1	no	—
197	C₂×C₄	—	17	4	no	—
198	C₂×C₄	—	34	2	no	—
199	C₂×C₄	—	34	2	no	—
200	C₂×C₄	—	34	2	no	—
201	C₂×C₄	—	34	2	no	—
202	C₂×C₄	—	34	2	no	—
203	C₂×C₄	—	34	2	no	—
204	C₂×C₄	—	34	2	no	—
205	C₂×C₄	—	34	2	no	—
206	C₂×C₄	—	34	2	no	—
207	C₂×C₄	—	34	2	no	—
208	C₂×C₄	—	34	2	no	—
209	C₂×C₄	#63252	1	2	yes	—
210	C₂×C₄	#63252	1	2	yes	—
211	C₂×C₄	#63252	1	2	yes	—
212	C₂×C₄	#63254	1	2	yes	—
213	C₂×C₄	Q₈×D₁₇	1	2	yes	—
214	C₂²	C₂×#63252	1	1	yes	characteristic
215	C₂²	C₂×#63252	1	1	yes	characteristic
216	C₂²	C₂×#63252	1	1	yes	characteristic
217	C₂²	C₂×#63254	1	1	yes	characteristic
218	C₂²	C₂×Q₈×D₁₇	1	1	yes	characteristic
219	C₂²	#10587696	1	1	yes	characteristic
220	C₂²	#10587697	1	1	yes	characteristic
221	C₄	—	2	2	no	—
222	C₄	—	2	2	no	—
223	C₄	—	4	1	no	—
224	C₄	—	4	1	no	—
225	C₄	—	4	1	no	—
226	C₄	—	4	1	no	—
227	C₄	—	34	4	no	—
228	C₄	—	68	2	no	—
229	C₄	—	68	2	no	—
230	C₄	—	68	2	no	—
231	C₄	#10587575	1	4	yes	—
232	C₂	C₂×#10587575	1	1	yes	characteristic
233	C₂	#11763343	1	1	yes	characteristic
234	C₂	#11763430	1	1	yes	characteristic
235	C₂	#11763505	1	1	yes	characteristic
236	C₂	#11763519	1	1	yes	characteristic
237	C₂	#11763529	1	1	yes	characteristic
238	C₂	#11763537	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₇
commutator, socle, prime radical, cosocle	C₂³×C₁₇	C₂⁴
Fitting	C₄×C₄⋊C₄×C₁₇	C₂
hypercenter	C₄×C₄⋊C₄	D₁₇
hypocenter	C₁₇	#2509
LCS	G ▷ C₂³×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂³ ◁ C₄×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
154. C₁₇	#2509	semidirect

Maximal subgroups

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

subgroup	index
#11762750	2
#11762747	2
#11762707	2
#11762704	2
#11762691	2
#11762675	2
C₄×C₄⋊Dic₁₇	2
C₄×C₄⋊C₄×C₁₇	2
#2509	17