Subgroups — #14537923

Order 2176.

356 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11762940	C₂	1	2	yes	maximal, complemented
2	#11762881	C₂	1	2	yes	maximal, complemented
3	#11762881	C₂	1	2	yes	maximal, complemented
4	D₄×#3622	C₂	1	2	yes	maximal, complemented
5	#11762870	C₂	1	2	yes	maximal, complemented
6	#11762866	C₂	1	1	yes	maximal, characteristic, complemented
7	#11762836	C₂	1	2	yes	maximal, complemented
8	C₂×C₄⋊#3622	C₂	1	1	yes	maximal, characteristic, complemented
9	C₁₇×#529	C₂	1	1	yes	maximal, characteristic
10	C₁₇×#126	C₄	1	2	yes	—
11	C₁₇×#125	C₄	1	1	yes	characteristic
12	C₁₇×#124	C₄	1	1	yes	characteristic
13	C₁₇×#123	C₄	1	2	yes	—
14	C₁₇×C₂²⋊Q₈	C₄	1	1	yes	characteristic
15	C₁₇×#121	C₄	1	1	yes	characteristic
16	C₄×Q₈×C₁₇	C₂²	1	1	yes	characteristic
17	C₄×D₄×C₁₇	C₂²	1	2	yes	—
18	C₄×D₄×C₁₇	C₂²	1	1	yes	characteristic
19	C₁₇×#117	C₂²	1	2	yes	—
20	C₂×C₄²×C₁₇	C₂²	1	1	yes	characteristic
21	C₂×#63223	C₂²	1	2	yes	—
22	C₂²×#3622	C₂²	1	1	yes	characteristic
23	C₂²×#3622	C₂²	1	1	yes	characteristic
24	C₂²×#3622	C₂²	1	2	yes	—
25	C₂²⋊#3622	C₂²	1	2	yes	—
26	C₂²⋊#3622	C₂²	1	2	yes	—
27	C₂²⋊#3622	C₂²	1	2	yes	—
28	C₄⋊#3622	C₂²	1	2	yes	—
29	C₄⋊#3622	C₂²	1	2	yes	—
30	C₄⋊#3622	C₂²	1	1	yes	characteristic
31	C₄⋊#3622	C₂²	1	2	yes	—
32	C₄⋊#3622	C₂²	1	2	yes	—
33	C₄⋊#3622	C₂²	1	1	yes	characteristic
34	#10587474	C₂²	1	2	yes	—
35	#10587474	C₂²	1	2	yes	—
36	C₄×#3622	C₂²	1	2	yes	—
37	C₂×Q₈×C₁₇	C₂×C₄	1	1	yes	characteristic
38	C₂×D₄×C₁₇	C₂×C₄	1	2	yes	—
39	C₂×D₄×C₁₇	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
42	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
43	C₂²×C₄×C₁₇	C₂³	1	2	yes	—
44	C₄⋊C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
45	C₄⋊C₄×C₁₇	C₂×C₄	1	2	yes	—
46	C₄⋊C₄×C₁₇	C₂×C₄	1	2	yes	—
47	C₄⋊C₄×C₁₇	C₂×C₄	1	2	yes	—
48	C₄⋊C₄×C₁₇	C₂×C₄	1	2	yes	—
49	C₄⋊C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
50	C₂²⋊C₄×C₁₇	C₂×C₄	1	2	yes	—
51	C₂²⋊C₄×C₁₇	C₂×C₄	1	2	yes	—
52	C₂²⋊C₄×C₁₇	C₂×C₄	1	2	yes	—
53	C₂²⋊C₄×C₁₇	C₂×C₄	1	2	yes	—
54	C₂²⋊C₄×C₁₇	C₂×C₄	1	2	yes	—
55	C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
56	C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
57	C₄²×C₁₇	C₂³	1	2	yes	—
58	C₄²×C₁₇	C₂³	1	1	yes	characteristic
59	C₄²×C₁₇	C₂³	1	1	yes	characteristic
60	#63223	—	4	2	no	—
61	C₂×#3622	—	4	1	no	—
62	C₂×#3622	—	4	2	no	—
63	C₂×#3622	—	4	1	no	—
64	C₂×#3622	C₂³	1	2	yes	—
65	C₂×#3622	C₂³	1	2	yes	—
66	C₂×#3622	C₂³	1	2	yes	—
67	C₂×#3622	C₂³	1	2	yes	—
68	C₂³×C₁₇	C₂²×C₄	1	2	yes	—
69	C₂³×C₁₇	C₂²×C₄	1	1	yes	characteristic
70	Q₈×C₁₇	—	4	1	no	—
71	D₄×C₁₇	—	2	4	no	—
72	D₄×C₁₇	—	4	1	no	—
73	C₂×C₄×C₁₇	—	2	2	no	—
74	C₂×C₄×C₁₇	—	2	1	no	—
75	C₂×C₄×C₁₇	—	2	2	no	—
76	C₂×C₄×C₁₇	—	2	1	no	—
77	C₂×C₄×C₁₇	—	2	2	no	—
78	C₂×C₄×C₁₇	—	4	1	no	—
79	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
84	C₂×C₄×C₁₇	C₂²×C₄	1	2	yes	—
85	C₂×C₄×C₁₇	C₂²×C₄	1	2	yes	—
86	C₂×C₄×C₁₇	C₂²×C₄	1	2	yes	—
87	C₂×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
88	#3622	—	2	2	no	—
89	#3622	—	2	2	no	—
90	#3622	—	2	2	no	—
91	#3622	—	2	2	no	—
92	#2992	—	17	1	no	maximal, Sylow
93	C₂²×C₁₇	—	2	1	no	—
94	C₂²×C₁₇	—	2	1	no	—
95	C₂²×C₁₇	—	2	2	no	—
96	C₂²×C₁₇	—	2	2	no	—
97	C₂²×C₁₇	—	2	2	no	—
98	C₂²×C₁₇	#131	1	2	yes	—
99	C₂²×C₁₇	C₂³×C₄	1	1	yes	characteristic
100	C₄×C₁₇	—	2	1	no	—
101	C₄×C₁₇	—	2	2	no	—
102	C₄×C₁₇	—	2	1	no	—
103	C₄×C₁₇	—	2	1	no	—
104	C₄×C₁₇	—	2	2	no	—
105	C₄×C₁₇	—	2	1	no	—
106	C₄×C₁₇	—	2	1	no	—
107	C₄×C₁₇	—	2	2	no	—
108	C₄×C₁₇	H(5,𝔽₂)	1	1	yes	characteristic
109	C₄×C₁₇	2₋¹⁺⁴	1	1	yes	characteristic
110	#529	D₁₇	1	1	yes	characteristic
111	#446	—	17	2	no	—
112	#435	—	17	2	no	—
113	#435	—	17	2	no	—
114	C₈×D₄	—	17	2	no	—
115	#433	—	17	2	no	—
116	#432	—	17	1	no	—
117	#424	—	17	2	no	—
118	C₂×C₄⋊C₈	—	17	1	no	—
119	C₂×C₁₇	—	2	2	no	—
120	C₂×C₁₇	—	4	2	no	—
121	C₂×C₁₇	#520	1	1	yes	characteristic
122	C₂×C₁₇	C₂×#131	1	1	yes	characteristic
123	C₂×C₁₇	#568	1	1	yes	characteristic
124	C₂×M₁₆	—	17	2	no	—
125	C₂²×C₈	—	17	1	no	—
126	C₂²×C₈	—	17	1	no	—
127	C₂²×C₈	—	17	2	no	—
128	#126	Dic₁₇	1	2	yes	—
129	#125	Dic₁₇	1	1	yes	characteristic
130	#124	Dic₁₇	1	1	yes	characteristic
131	#123	Dic₁₇	1	2	yes	—
132	C₂²⋊Q₈	Dic₁₇	1	1	yes	characteristic
133	#121	Dic₁₇	1	1	yes	characteristic
134	C₄×Q₈	C₂×D₁₇	1	1	yes	characteristic
135	C₄×D₄	C₂×D₁₇	1	2	yes	—
136	C₄×D₄	C₂×D₁₇	1	1	yes	characteristic
137	#117	C₂×D₁₇	1	2	yes	—
138	C₂×C₄²	C₂×D₁₇	1	1	yes	characteristic
139	C₄⋊C₈	—	17	2	no	—
140	C₄⋊C₈	—	17	2	no	—
141	C₄⋊C₈	—	17	1	no	—
142	C₄⋊C₈	—	17	2	no	—
143	C₄⋊C₈	—	17	2	no	—
144	C₄⋊C₈	—	17	1	no	—
145	C₂²⋊C₈	—	17	2	no	—
146	C₂²⋊C₈	—	17	2	no	—
147	C₂²⋊C₈	—	17	2	no	—
148	#97	—	17	2	no	—
149	#97	—	17	2	no	—
150	C₄×C₈	—	17	2	no	—
151	C₁₇	#2992	1	1	yes	characteristic, complemented, Sylow
152	C₂×Q₈	C₂×Dic₁₇	1	1	yes	characteristic
153	C₂×D₄	C₂×Dic₁₇	1	2	yes	—
154	C₂×D₄	C₂×Dic₁₇	1	1	yes	characteristic
155	C₂²×C₄	C₂×Dic₁₇	1	1	yes	characteristic
156	C₂²×C₄	C₂×Dic₁₇	1	1	yes	characteristic
157	C₂²×C₄	C₂²×D₁₇	1	1	yes	characteristic
158	C₂²×C₄	C₂²×D₁₇	1	2	yes	—
159	M₁₆	—	68	2	no	—
160	C₂×C₈	—	17	2	no	—
161	C₂×C₈	—	17	2	no	—
162	C₂×C₈	—	17	2	no	—
163	C₂×C₈	—	17	2	no	—
164	C₂×C₈	—	68	1	no	—
165	C₂×C₈	—	68	2	no	—
166	C₂×C₈	—	68	1	no	—
167	C₄⋊C₄	C₂×Dic₁₇	1	2	yes	—
168	C₄⋊C₄	C₂×Dic₁₇	1	2	yes	—
169	C₄⋊C₄	C₂×Dic₁₇	1	1	yes	characteristic
170	C₄⋊C₄	C₂×Dic₁₇	1	2	yes	—
171	C₄⋊C₄	C₂×Dic₁₇	1	1	yes	characteristic
172	C₄⋊C₄	C₂×Dic₁₇	1	2	yes	—
173	C₂²⋊C₄	C₂×Dic₁₇	1	2	yes	—
174	C₂²⋊C₄	C₂×Dic₁₇	1	2	yes	—
175	C₂²⋊C₄	C₂×Dic₁₇	1	2	yes	—
176	C₂²⋊C₄	C₂×Dic₁₇	1	2	yes	—
177	C₂²⋊C₄	C₂×Dic₁₇	1	2	yes	—
178	C₄²	C₂×Dic₁₇	1	1	yes	characteristic
179	C₄²	C₂×Dic₁₇	1	1	yes	characteristic
180	C₄²	C₂²×D₁₇	1	2	yes	—
181	C₄²	C₂²×D₁₇	1	1	yes	characteristic
182	C₄²	C₂²×D₁₇	1	1	yes	characteristic
183	C₂³	C₂²×Dic₁₇	1	2	yes	—
184	C₂³	C₂²×Dic₁₇	1	1	yes	characteristic
185	Q₈	—	4	1	no	—
186	D₄	—	2	4	no	—
187	D₄	—	4	1	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	2	1	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	2	1	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	4	1	no	—
194	C₂×C₄	C₂²×Dic₁₇	1	1	yes	characteristic
195	C₂×C₄	C₂²×Dic₁₇	1	1	yes	characteristic
196	C₂×C₄	C₂²×Dic₁₇	1	1	yes	characteristic
197	C₂×C₄	C₂²×Dic₁₇	1	1	yes	characteristic
198	C₂×C₄	C₂²×Dic₁₇	1	1	yes	characteristic
199	C₂×C₄	C₂²×Dic₁₇	1	2	yes	—
200	C₂×C₄	C₂²×Dic₁₇	1	2	yes	—
201	C₂×C₄	C₂²×Dic₁₇	1	2	yes	—
202	C₂×C₄	C₂³×D₁₇	1	1	yes	characteristic
203	C₈	—	34	2	no	—
204	C₈	—	34	2	no	—
205	C₈	—	34	2	no	—
206	C₈	—	34	2	no	—
207	C₂²	—	2	1	no	—
208	C₂²	—	2	1	no	—
209	C₂²	—	2	2	no	—
210	C₂²	—	2	2	no	—
211	C₂²	—	2	2	no	—
212	C₂²	#10587634	1	2	yes	—
213	C₂²	C₂³×Dic₁₇	1	1	yes	characteristic
214	C₄	—	2	1	no	—
215	C₄	—	2	2	no	—
216	C₄	—	2	1	no	—
217	C₄	—	2	1	no	—
218	C₄	—	2	2	no	—
219	C₄	—	2	1	no	—
220	C₄	—	2	1	no	—
221	C₄	—	2	2	no	—
222	C₄	#10587696	1	1	yes	characteristic
223	C₄	#10587697	1	1	yes	characteristic
224	C₂	—	2	2	no	—
225	C₂	—	4	2	no	—
226	C₂	#11763292	1	1	yes	characteristic
227	C₂	C₂×#10587634	1	1	yes	characteristic
228	C₂	#11763576	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₇
commutator, socle	C₂²×C₁₇	C₂³×C₄
Fitting	C₁₇×#529	C₂
prime radical	C₁₇×#121	C₄
cosocle	C₂×C₄×C₁₇	C₂⁴
hypercenter	#529	D₁₇
hypocenter	C₁₇	#2992
LCS	G ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ #529

Complements

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

N	H ≅ G/N	type
1. #11762940	C₂	semidirect
2. #11762881	C₂	semidirect
3. #11762881	C₂	semidirect
4. D₄×#3622	C₂	semidirect
5. #11762870	C₂	semidirect
6. #11762866	C₂	semidirect
7. #11762836	C₂	semidirect
8. C₂×C₄⋊#3622	C₂	semidirect
151. C₁₇	#2992	semidirect

Maximal subgroups

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

subgroup	index
#11762940	2
#11762881	2
D₄×#3622	2
#11762870	2
#11762866	2
#11762836	2
C₂×C₄⋊#3622	2
C₁₇×#529	2
#2992	17