Subgroups — C₂×#14543751 #17958757

Order 4352.

382 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	C₂×C₄⋊#63216	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×#63216	C₂	1	1	yes	maximal, characteristic
3	#14543751	C₂	1	4	yes	maximal, direct product factor
4	C₂×#11762933	C₂	1	1	yes	maximal, characteristic
5	C₂²×#63216	—	2	1	no	—
6	C₂²×#63216	—	2	1	no	—
7	C₄⋊#63216	—	2	1	no	—
8	C₄⋊#63216	C₂²	1	2	yes	—
9	C₄×#63216	—	2	1	no	—
10	C₄×#63216	C₂²	1	2	yes	—
11	C₂×C₄×Dic₃₄	C₄	1	1	yes	characteristic
12	#11762933	—	2	1	no	—
13	#11762933	—	2	1	no	—
14	#11762933	C₄	1	2	yes	—
15	#11762933	C₂²	1	2	yes	—
16	C₂×#10587485	—	2	1	no	—
17	C₂×C₄×#3622	C₂²	1	1	yes	characteristic
18	C₂×C₈×Dic₁₇	—	2	1	no	—
19	C₂×C₁₇×C₄⋊C₈	C₄	1	1	yes	characteristic
20	C₂²×Dic₃₄	C₈	1	1	yes	characteristic
21	C₂²×C₈×C₁₇	—	2	1	no	—
22	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄⋊Dic₁₇	C₈	1	1	yes	characteristic
24	C₂×#63225	—	2	1	no	—
25	C₂×C₄×Dic₁₇	—	2	1	no	—
26	C₂²×#3622	D₄	1	1	yes	characteristic
27	C₂²×#3622	D₄	1	1	yes	characteristic
28	C₄×Dic₃₄	—	2	1	no	—
29	C₄×Dic₃₄	—	2	1	no	—
30	C₄×Dic₃₄	C₈	1	2	yes	—
31	C₄×Dic₃₄	C₂×C₄	1	2	yes	—
32	C₂×#63216	—	2	4	no	—
33	C₂×#63216	—	2	2	no	—
34	C₂×#63216	—	2	2	no	—
35	C₂×#63216	—	4	2	no	—
36	C₁₇×C₄⋊C₈	—	2	1	no	—
37	C₁₇×C₄⋊C₈	C₂×C₄	1	2	yes	—
38	#10587485	—	2	2	no	—
39	#10587485	—	2	2	no	—
40	C₈×Dic₁₇	—	2	2	no	—
41	C₈×Dic₁₇	—	2	2	no	—
42	C₄×#3622	C₂×C₄	1	1	yes	characteristic
43	C₄×#3622	D₄	1	1	yes	characteristic
44	C₄×#3622	D₄	1	1	yes	characteristic
45	C₄×#3622	C₂³	1	1	yes	characteristic
46	C₂²×C₄×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
47	C₂²×C₄×C₁₇	C₂×C₈	1	1	yes	characteristic
48	C₂²×C₄×C₁₇	M₁₆	1	1	yes	characteristic
49	C₂²×Dic₁₇	—	2	1	no	—
50	C₂²×Dic₁₇	—	2	1	no	—
51	C₂×Dic₃₄	—	2	1	no	—
52	C₂×Dic₃₄	—	2	2	no	—
53	C₂×Dic₃₄	C₁₆	1	4	yes	complemented
54	C₂×Dic₃₄	C₂×C₈	1	2	yes	—
55	C₂×C₈×C₁₇	—	2	2	no	—
56	C₂×C₈×C₁₇	—	4	2	no	—
57	C₄²×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
58	C₄²×C₁₇	C₂×C₈	1	1	yes	characteristic
59	C₄²×C₁₇	M₁₆	1	1	yes	characteristic
60	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
61	C₄⋊Dic₁₇	—	2	1	no	—
62	C₄⋊Dic₁₇	C₂×C₈	1	2	yes	—
63	#63225	—	2	2	no	—
64	#63225	—	2	2	no	—
65	C₄×Dic₁₇	—	2	2	no	—
66	C₄×Dic₁₇	—	2	2	no	—
67	C₂×#3622	—	2	2	no	—
68	C₂×#3622	C₂²⋊C₄	1	1	yes	characteristic
69	C₂×#3622	C₂²⋊C₄	1	1	yes	characteristic
70	C₂×#3622	SD₁₆	1	2	yes	—
71	C₂×#3622	Q₁₆	1	2	yes	—
72	C₂×#3622	C₂×D₄	1	1	yes	characteristic
73	C₂×#3622	C₂×D₄	1	1	yes	characteristic
74	#63216	—	2	4	no	—
75	#63216	—	4	2	no	—
76	C₂×Q₈⋊C₁₆	—	17	1	no	maximal, Sylow
77	C₂³×C₁₇	C₂²⋊C₈	1	1	yes	characteristic
78	C₂×C₄×C₁₇	—	2	2	no	—
79	C₂×C₄×C₁₇	C₂²⋊C₈	1	1	yes	characteristic
80	C₂×C₄×C₁₇	C₂²⋊C₈	1	1	yes	characteristic
81	C₂×C₄×C₁₇	C₂²⋊C₈	1	1	yes	characteristic
82	C₂×C₄×C₁₇	#103	1	2	yes	—
83	C₂×C₄×C₁₇	#104	1	2	yes	—
84	C₂×C₄×C₁₇	C₂×C₁₆	1	2	yes	—
85	C₂×C₄×C₁₇	M₃₂	1	2	yes	—
86	C₂×C₄×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
87	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
88	C₂×C₄×C₁₇	C₂×M₁₆	1	1	yes	characteristic
89	C₂×Dic₁₇	—	2	4	no	—
90	C₂×Dic₁₇	—	2	2	no	—
91	C₂×Dic₁₇	—	2	2	no	—
92	C₂×Dic₁₇	—	4	2	no	—
93	Dic₃₄	—	2	4	no	—
94	Dic₃₄	—	2	2	no	—
95	Dic₃₄	C₂×C₁₆	1	4	yes	complemented
96	C₈×C₁₇	—	4	2	no	—
97	#3622	—	2	1	no	—
98	#3622	—	2	1	no	—
99	#3622	#103	1	2	yes	—
100	#3622	C₂×SD₁₆	1	1	yes	characteristic
101	#3622	C₂×Q₁₆	1	1	yes	characteristic
102	C₂×C₈×Q₈	—	17	1	no	—
103	C₂×C₄⋊C₁₆	—	17	1	no	—
104	C₂×C₄×C₁₆	—	17	1	no	—
105	Q₈⋊C₁₆	—	17	4	no	—
106	C₂²×C₁₇	Q₈⋊C₈	1	2	yes	—
107	C₂²×C₁₇	C₂²⋊C₁₆	1	2	yes	—
108	C₂²×C₁₇	#350	1	2	yes	—
109	C₂²×C₁₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
110	C₄×C₁₇	—	2	1	no	—
111	C₄×C₁₇	—	2	1	no	—
112	C₄×C₁₇	Q₈⋊C₈	1	2	yes	—
113	C₄×C₁₇	C₂²⋊C₁₆	1	2	yes	—
114	C₄×C₁₇	C₂×#103	1	1	yes	characteristic
115	C₄×C₁₇	C₂×#104	1	1	yes	characteristic
116	C₄×C₁₇	C₂²×C₁₆	1	1	yes	characteristic
117	C₄×C₁₇	C₂×M₃₂	1	1	yes	characteristic
118	Dic₁₇	—	2	4	no	—
119	Dic₁₇	—	4	2	no	—
120	C₂×C₄×Q₈	—	17	1	no	—
121	C₂²×C₁₆	—	34	1	no	—
122	C₂²×C₁₆	—	34	1	no	—
123	C₈×Q₈	—	17	2	no	—
124	C₈×Q₈	—	17	2	no	—
125	C₈×Q₈	—	34	1	no	—
126	C₈×Q₈	—	34	1	no	—
127	C₂×C₄⋊C₈	—	34	1	no	—
128	C₂×C₄⋊C₈	#595	1	1	yes	characteristic
129	C₂×C₄×C₈	—	17	1	no	—
130	C₂×C₄×C₈	—	34	1	no	—
131	C₄⋊C₁₆	—	17	2	no	—
132	C₄⋊C₁₆	—	34	1	no	—
133	C₄×C₁₆	—	17	2	no	—
134	C₄×C₁₆	—	34	1	no	—
135	C₂×C₁₇	Q₈⋊C₁₆	1	4	yes	complemented
136	C₂×C₁₇	C₂×Q₈⋊C₈	1	1	yes	characteristic
137	C₂×C₁₇	C₂×C₂²⋊C₁₆	1	1	yes	characteristic
138	C₂×C₁₇	C₂×#350	1	1	yes	characteristic
139	C₂²×Q₈	—	17	1	no	—
140	C₂²×C₈	—	2	1	no	—
141	C₂²×C₈	—	17	1	no	—
142	C₂²×C₈	—	17	1	no	—
143	C₄×Q₈	—	17	2	no	—
144	C₄×Q₈	—	17	2	no	—
145	C₄×Q₈	—	34	1	no	—
146	C₄×Q₈	—	34	1	no	—
147	C₂×C₄⋊C₄	—	17	1	no	—
148	C₂×C₄⋊C₄	—	34	1	no	—
149	C₂×C₄²	—	34	1	no	—
150	C₂×C₄²	C₂×#595	1	1	yes	characteristic
151	C₂×C₁₆	—	34	2	no	—
152	C₂×C₁₆	—	34	4	no	—
153	C₂×C₁₆	—	34	2	no	—
154	C₂×C₁₆	—	68	2	no	—
155	C₄⋊C₈	—	2	1	no	—
156	C₄⋊C₈	—	34	2	no	—
157	C₄⋊C₈	—	34	2	no	—
158	C₄⋊C₈	C₂×#595	1	2	yes	—
159	C₄×C₈	—	17	1	no	—
160	C₄×C₈	—	17	1	no	—
161	C₄×C₈	—	17	1	no	—
162	C₄×C₈	—	17	1	no	—
163	C₄×C₈	—	34	2	no	—
164	C₄×C₈	—	34	2	no	—
165	C₁₇	C₂×Q₈⋊C₁₆	1	1	yes	characteristic, complemented, Sylow
166	C₂×Q₈	—	17	2	no	—
167	C₂×Q₈	—	17	4	no	—
168	C₂×Q₈	—	34	2	no	—
169	C₂×Q₈	—	34	1	no	—
170	C₂²×C₄	—	34	1	no	—
171	C₂²×C₄	—	34	1	no	—
172	C₂²×C₄	#63242	1	1	yes	characteristic
173	C₂²×C₄	#63243	1	1	yes	characteristic
174	C₂²×C₄	C₂²⋊#595	1	1	yes	characteristic
175	C₂×C₈	—	2	2	no	—
176	C₂×C₈	—	4	2	no	—
177	C₂×C₈	—	17	1	no	—
178	C₂×C₈	—	17	1	no	—
179	C₂×C₈	—	17	2	no	—
180	C₂×C₈	—	17	2	no	—
181	C₂×C₈	—	17	1	no	—
182	C₂×C₈	—	17	1	no	—
183	C₂×C₈	—	34	2	no	—
184	C₄⋊C₄	—	17	2	no	—
185	C₄⋊C₄	—	34	2	no	—
186	C₄⋊C₄	—	34	1	no	—
187	C₄⋊C₄	—	34	2	no	—
188	C₄²	—	34	2	no	—
189	C₄²	—	34	2	no	—
190	C₄²	#63242	1	1	yes	characteristic
191	C₄²	#63243	1	1	yes	characteristic
192	C₄²	C₂²⋊#595	1	1	yes	characteristic
193	C₄²	C₂²×#595	1	1	yes	characteristic
194	C₁₆	—	34	4	no	—
195	C₁₆	—	68	2	no	—
196	C₂³	#10587543	1	1	yes	characteristic
197	Q₈	—	17	4	no	—
198	Q₈	—	34	2	no	—
199	Q₈	—	34	4	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	34	2	no	—
202	C₂×C₄	—	34	4	no	—
203	C₂×C₄	—	34	2	no	—
204	C₂×C₄	—	68	2	no	—
205	C₂×C₄	#10587529	1	2	yes	—
206	C₂×C₄	#10587530	1	2	yes	—
207	C₂×C₄	#10587543	1	1	yes	characteristic
208	C₂×C₄	#10587543	1	1	yes	characteristic
209	C₂×C₄	#10587543	1	1	yes	characteristic
210	C₂×C₄	#10587548	1	2	yes	—
211	C₂×C₄	#10587549	1	2	yes	—
212	C₂×C₄	C₂×#63242	1	1	yes	characteristic
213	C₂×C₄	C₂×#63243	1	1	yes	characteristic
214	C₂×C₄	C₂×C₂²⋊#595	1	1	yes	characteristic
215	C₈	—	4	2	no	—
216	C₈	—	17	1	no	—
217	C₈	—	17	2	no	—
218	C₈	—	17	1	no	—
219	C₈	—	34	1	no	—
220	C₈	—	34	1	no	—
221	C₂²	#11763659	1	2	yes	—
222	C₂²	#11763704	1	2	yes	—
223	C₂²	#11763708	1	2	yes	—
224	C₂²	C₂×#10587543	1	1	yes	characteristic
225	C₄	—	2	1	no	—
226	C₄	—	2	1	no	—
227	C₄	—	34	4	no	—
228	C₄	—	68	2	no	—
229	C₄	#11763659	1	2	yes	—
230	C₄	#11763704	1	2	yes	—
231	C₄	C₂×#10587549	1	1	yes	characteristic
232	C₄	C₂×#10587548	1	1	yes	characteristic
233	C₄	C₂×#10587529	1	1	yes	characteristic
234	C₄	C₂×#10587530	1	1	yes	characteristic
235	C₂	#14543751	1	4	yes	direct product factor
236	C₂	C₂×#11763659	1	1	yes	characteristic
237	C₂	C₂×#11763704	1	1	yes	characteristic
238	C₂	C₂×#11763708	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²⋊#595
commutator	C₄×C₁₇	C₂²×C₁₆
Frattini	C₄²	C₂²×#595
Fitting	C₂×C₁₇×C₄⋊C₈	C₄
socle, prime radical	C₂³×C₁₇	C₂²⋊C₈
cosocle	C₄×#3622	C₂³
hypercenter	C₂×C₄⋊C₈	#595
hypocenter	C₁₇	C₂×Q₈⋊C₁₆
LCS	G ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
3. #14543751	C₂	direct (N × H)
53. C₂×Dic₃₄	C₁₆	semidirect
95. Dic₃₄	C₂×C₁₆	semidirect
135. C₂×C₁₇	Q₈⋊C₁₆	semidirect
165. C₁₇	C₂×Q₈⋊C₁₆	semidirect
235. C₂	#14543751	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₄⋊#63216	2
C₂×C₄×#63216	2
#14543751	2
C₂×#11762933	2
C₂×Q₈⋊C₁₆	17

Subgroups — C2×#14543751 #17958757

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#14543751 #17958757