Subgroups — #17978665

Order 4352.

498 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14545199	C₂	1	4	yes	maximal, complemented
2	#14545199	C₂	1	2	yes	maximal, complemented
3	Q₈×#63216	C₂	1	2	yes	maximal, complemented
4	#14545119	C₂	1	2	yes	maximal
5	C₂×C₄⋊#63216	C₂	1	1	yes	maximal, characteristic
6	#14544993	C₂	1	2	yes	maximal
7	C₄×#10587538	C₂	1	1	yes	maximal, characteristic
8	#14537719	C₂	1	1	yes	maximal, characteristic
9	C₂×#10587538	C₂²	1	1	yes	characteristic
10	C₂×#10587538	C₂²	1	1	yes	characteristic
11	C₂²×#63216	C₂²	1	2	yes	—
12	C₄⋊#63216	C₂²	1	1	yes	characteristic
13	C₄⋊#63216	C₂²	1	4	yes	—
14	C₄⋊#63216	C₂²	1	1	yes	characteristic
15	C₄⋊#63216	C₂²	1	2	yes	—
16	C₄⋊#63216	C₂²	1	4	yes	—
17	#11763697	C₂²	1	2	yes	—
18	#11763697	C₂²	1	4	yes	—
19	C₄×#63216	C₂²	1	2	yes	—
20	C₄×#63216	C₂²	1	4	yes	—
21	#11762940	C₄	1	2	yes	—
22	#11762940	C₄	1	2	yes	—
23	Q₈×#3622	C₂²	1	2	yes	—
24	Q₈×#3622	C₂²	1	2	yes	—
25	#11762836	C₄	1	2	yes	—
26	#11762833	C₄	1	1	yes	characteristic
27	#11762765	C₂²	1	2	yes	—
28	C₂×C₄×#3622	C₂²	1	1	yes	characteristic
29	C₁₇×#533	C₄	1	1	yes	characteristic
30	C₁₇×C₄⋊Q₈	C₈	1	1	yes	characteristic
31	C₁₇×C₄⋊Q₈	C₈	1	1	yes	characteristic
32	C₁₇×#125	C₈	1	2	yes	—
33	C₁₇×C₂²⋊Q₈	C₈	1	2	yes	—
34	C₁₇×C₂²⋊Q₈	C₈	1	2	yes	—
35	C₄×Q₈×C₁₇	C₂×C₄	1	2	yes	—
36	C₄×Q₈×C₁₇	C₂×C₄	1	2	yes	—
37	C₁₇×#117	C₂×C₄	1	2	yes	—
38	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
39	C₂×#63223	C₂×C₄	1	2	yes	—
40	C₂²×#3622	C₂×C₄	1	1	yes	characteristic
41	C₂²×#3622	C₂³	1	1	yes	characteristic
42	#10587538	—	2	2	no	—
43	#10587538	Q₈	1	4	yes	complemented
44	C₂×#63216	—	2	4	no	—
45	C₂×#63216	C₂³	1	2	yes	—
46	C₂×#63216	C₂³	1	2	yes	—
47	C₂×#63216	C₂³	1	2	yes	—
48	C₂×#63216	C₂³	1	2	yes	—
49	C₄⋊#3622	C₂×C₄	1	2	yes	—
50	C₄⋊#3622	C₂×C₄	1	2	yes	—
51	C₄⋊#3622	C₂×C₄	1	4	yes	—
52	C₄⋊#3622	C₂×C₄	1	4	yes	—
53	#10587474	C₂×C₄	1	4	yes	—
54	C₄×#3622	C₂×C₄	1	1	yes	characteristic
55	C₄×#3622	C₂×C₄	1	1	yes	characteristic
56	C₄×#3622	C₂³	1	1	yes	characteristic
57	C₄×#3622	C₂³	1	1	yes	characteristic
58	C₄×#3622	C₂³	1	4	yes	—
59	C₂×Q₈×C₁₇	C₂×C₈	1	2	yes	—
60	C₂×Q₈×C₁₇	C₂×C₈	1	2	yes	—
61	C₂²×C₄×C₁₇	C₂×C₈	1	1	yes	characteristic
62	C₂²×C₄×C₁₇	C₂×C₈	1	1	yes	characteristic
63	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
64	C₄⋊C₄×C₁₇	C₂×C₈	1	4	yes	—
65	C₄⋊C₄×C₁₇	C₂×C₈	1	4	yes	—
66	C₄⋊C₄×C₁₇	C₂×C₈	1	2	yes	—
67	C₄⋊C₄×C₁₇	C₂×C₈	1	2	yes	—
68	C₄⋊C₄×C₁₇	C₂×C₈	1	2	yes	—
69	C₄⋊C₄×C₁₇	C₂×C₈	1	2	yes	—
70	C₂²⋊C₄×C₁₇	C₂×C₈	1	4	yes	—
71	C₄²×C₁₇	C₂×C₈	1	1	yes	characteristic
72	C₄²×C₁₇	C₂×C₈	1	1	yes	characteristic
73	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
74	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
75	C₄²×C₁₇	C₂²×C₄	1	4	yes	—
76	#63223	—	2	4	no	—
77	C₂×#3622	—	2	2	no	—
78	C₂×#3622	C₂²×C₄	1	1	yes	characteristic
79	C₂×#3622	C₂²×C₄	1	1	yes	characteristic
80	C₂×#3622	C₂²×C₄	1	4	yes	—
81	C₂×#3622	C₂²×C₄	1	1	yes	characteristic
82	C₂×#3622	C₂×Q₈	1	2	yes	—
83	C₂×#3622	#41	1	2	yes	—
84	C₂×#3622	C₂⁴	1	1	yes	characteristic
85	#63216	—	2	2	no	—
86	#63216	—	2	2	no	—
87	#63216	—	2	2	no	—
88	#63216	C₂×Q₈	1	4	yes	—
89	#33476	—	17	1	no	maximal, Sylow
90	C₂³×C₁₇	C₂²×C₈	1	1	yes	characteristic
91	Q₈×C₁₇	—	2	4	no	—
92	Q₈×C₁₇	—	4	2	no	—
93	C₂×C₄×C₁₇	—	2	2	no	—
94	C₂×C₄×C₁₇	—	4	1	no	—
95	C₂×C₄×C₁₇	C₄×Q₈	1	2	yes	—
96	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
97	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
98	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
99	C₂×C₄×C₁₇	C₂²×C₈	1	4	yes	—
100	C₂×C₄×C₁₇	C₂²×C₈	1	4	yes	—
101	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
102	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
103	C₂×C₄×C₁₇	#131	1	2	yes	—
104	C₂×C₄×C₁₇	C₂³×C₄	1	1	yes	characteristic
105	#3622	—	2	1	no	—
106	#3622	—	2	1	no	—
107	#3622	—	2	4	no	—
108	#3622	C₄×Q₈	1	2	yes	—
109	#3622	C₂²×Q₈	1	1	yes	characteristic
110	#3622	C₂×#41	1	1	yes	characteristic
111	#2960	—	17	1	no	—
112	#2183	—	17	2	no	—
113	#2183	—	17	4	no	—
114	Q₈×C₁₆	—	17	2	no	—
115	#2151	—	17	2	no	—
116	C₂×C₄⋊C₁₆	—	17	1	no	—
117	#2108	—	17	2	no	—
118	C₄×M₃₂	—	17	1	no	—
119	C₂²×C₁₇	—	2	1	no	—
120	C₂²×C₁₇	—	2	1	no	—
121	C₂²×C₁₇	C₈×Q₈	1	2	yes	—
122	C₂²×C₁₇	C₂³×C₈	1	1	yes	characteristic
123	C₄×C₁₇	—	2	1	no	—
124	C₄×C₁₇	—	2	4	no	—
125	C₄×C₁₇	—	2	1	no	—
126	C₄×C₁₇	—	2	1	no	—
127	C₄×C₁₇	—	2	4	no	—
128	C₄×C₁₇	C₈×Q₈	1	2	yes	—
129	C₄×C₁₇	#504	1	2	yes	—
130	C₄×C₁₇	C₂×C₄×Q₈	1	1	yes	characteristic
131	C₄×C₁₇	C₂×#131	1	1	yes	characteristic
132	#533	#595	1	1	yes	characteristic
133	C₂×M₃₂	—	17	1	no	—
134	C₂×M₃₂	—	17	1	no	—
135	C₂²×C₁₆	—	17	2	no	—
136	#446	—	17	2	no	—
137	#446	—	17	2	no	—
138	C₈×Q₈	—	17	2	no	—
139	C₈×Q₈	—	17	2	no	—
140	#424	—	17	2	no	—
141	#423	—	17	1	no	—
142	#405	—	17	2	no	—
143	C₂×C₄×C₈	—	17	1	no	—
144	C₄⋊C₁₆	—	17	1	no	—
145	C₄⋊C₁₆	—	17	1	no	—
146	C₄⋊C₁₆	—	17	2	no	—
147	C₄⋊C₁₆	—	17	4	no	—
148	C₄⋊C₁₆	—	17	4	no	—
149	#346	—	17	2	no	—
150	#346	—	17	4	no	—
151	C₄×C₁₆	—	17	2	no	—
152	C₄×C₁₆	—	17	4	no	—
153	C₂×C₁₇	—	2	2	no	—
154	C₂×C₁₇	C₂×C₈×Q₈	1	1	yes	characteristic
155	C₂×C₁₇	C₂×#504	1	1	yes	characteristic
156	C₂×C₁₇	#3407	1	1	yes	characteristic
157	C₂×M₁₆	—	17	2	no	—
158	C₂²×C₈	—	17	1	no	—
159	C₂²×C₈	—	17	1	no	—
160	C₄⋊Q₈	#3624	1	1	yes	characteristic
161	C₄⋊Q₈	#3624	1	1	yes	characteristic
162	#125	#3624	1	2	yes	—
163	C₂²⋊Q₈	#3624	1	2	yes	—
164	C₂²⋊Q₈	#3624	1	2	yes	—
165	C₄×Q₈	C₂×#595	1	2	yes	—
166	C₄×Q₈	C₂×#595	1	2	yes	—
167	#117	C₂×#595	1	2	yes	—
168	C₂×C₄²	C₂×#595	1	1	yes	characteristic
169	M₃₂	—	17	4	no	—
170	M₃₂	—	34	2	no	—
171	C₂×C₁₆	—	17	2	no	—
172	C₂×C₁₆	—	17	2	no	—
173	C₂×C₁₆	—	17	2	no	—
174	C₂×C₁₆	—	17	2	no	—
175	C₂×C₁₆	—	34	4	no	—
176	C₄⋊C₈	—	17	2	no	—
177	C₄⋊C₈	—	17	2	no	—
178	C₄⋊C₈	—	17	4	no	—
179	C₄⋊C₈	—	17	4	no	—
180	#97	—	17	4	no	—
181	C₄×C₈	—	17	1	no	—
182	C₄×C₈	—	17	1	no	—
183	C₄×C₈	—	17	1	no	—
184	C₄×C₈	—	17	4	no	—
185	C₄×C₈	—	17	1	no	—
186	C₁₇	#33476	1	1	yes	characteristic, complemented, Sylow
187	C₂×Q₈	C₂×#3624	1	2	yes	—
188	C₂×Q₈	C₂×#3624	1	2	yes	—
189	C₂²×C₄	C₂×#3624	1	1	yes	characteristic
190	C₂²×C₄	C₂×#3624	1	1	yes	characteristic
191	C₂²×C₄	C₂²×#595	1	1	yes	characteristic
192	M₁₆	—	34	4	no	—
193	C₂×C₈	—	17	2	no	—
194	C₂×C₈	—	17	2	no	—
195	C₂×C₈	—	17	1	no	—
196	C₂×C₈	—	17	1	no	—
197	C₂×C₈	—	17	1	no	—
198	C₂×C₈	—	17	1	no	—
199	C₂×C₈	—	17	4	no	—
200	C₂×C₈	—	34	2	no	—
201	C₄⋊C₄	C₂×#3624	1	2	yes	—
202	C₄⋊C₄	C₂×#3624	1	2	yes	—
203	C₄⋊C₄	C₂×#3624	1	4	yes	—
204	C₄⋊C₄	C₂×#3624	1	4	yes	—
205	C₄⋊C₄	C₂×#3624	1	2	yes	—
206	C₄⋊C₄	C₂×#3624	1	2	yes	—
207	C₂²⋊C₄	C₂×#3624	1	4	yes	—
208	C₄²	C₂×#3624	1	1	yes	characteristic
209	C₄²	C₂×#3624	1	1	yes	characteristic
210	C₄²	C₂²×#595	1	1	yes	characteristic
211	C₄²	C₂²×#595	1	1	yes	characteristic
212	C₄²	C₂²×#595	1	4	yes	—
213	C₁₆	—	17	4	no	—
214	C₁₆	—	34	2	no	—
215	C₁₆	—	34	2	no	—
216	C₁₆	—	34	2	no	—
217	C₂³	C₂²×#3624	1	1	yes	characteristic
218	Q₈	—	2	4	no	—
219	Q₈	—	4	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	4	1	no	—
222	C₂×C₄	#10587680	1	2	yes	—
223	C₂×C₄	Q₈×#595	1	2	yes	—
224	C₂×C₄	C₂²×#3624	1	1	yes	characteristic
225	C₂×C₄	C₂²×#3624	1	1	yes	characteristic
226	C₂×C₄	C₂²×#3624	1	1	yes	characteristic
227	C₂×C₄	C₂²×#3624	1	4	yes	—
228	C₂×C₄	C₂²×#3624	1	4	yes	—
229	C₂×C₄	C₂²×#3624	1	1	yes	characteristic
230	C₂×C₄	C₂²×#3624	1	1	yes	characteristic
231	C₂×C₄	C₂³×#595	1	1	yes	characteristic
232	C₈	—	17	1	no	—
233	C₈	—	17	1	no	—
234	C₈	—	17	2	no	—
235	C₈	—	34	1	no	—
236	C₈	—	34	1	no	—
237	C₈	—	34	4	no	—
238	C₂²	—	2	1	no	—
239	C₂²	—	2	1	no	—
240	C₂²	Q₈×#3624	1	2	yes	—
241	C₂²	C₂³×#3624	1	1	yes	characteristic
242	C₄	—	2	1	no	—
243	C₄	—	2	4	no	—
244	C₄	—	2	1	no	—
245	C₄	—	2	1	no	—
246	C₄	—	2	4	no	—
247	C₄	Q₈×#3624	1	2	yes	—
248	C₄	#11763867	1	2	yes	—
249	C₄	C₂×Q₈×#595	1	1	yes	characteristic
250	C₄	C₂×#10587680	1	1	yes	characteristic
251	C₂	—	2	2	no	—
252	C₂	C₂×Q₈×#3624	1	1	yes	characteristic
253	C₂	C₂×#11763867	1	1	yes	characteristic
254	C₂	#14545751	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×#595
commutator, socle	C₂²×C₁₇	C₂³×C₈
Fitting	C₁₇×#533	C₄
prime radical	C₂³×C₁₇	C₂²×C₈
cosocle	C₂×#3622	C₂⁴
hypercenter	#533	#595
hypocenter	C₁₇	#33476
LCS	G ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ #533

Complements

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

N	H ≅ G/N	type
1. #14545199	C₂	semidirect
2. #14545199	C₂	semidirect
3. Q₈×#63216	C₂	semidirect
43. #10587538	Q₈	semidirect
186. C₁₇	#33476	semidirect

Maximal subgroups

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

subgroup	index
#14545199	2
Q₈×#63216	2
#14545119	2
C₂×C₄⋊#63216	2
#14544993	2
C₄×#10587538	2
#14537719	2
#33476	17