Subgroups — C₂×Q₈×Dic₁₇₂ #25977047

Order 11008.

1830 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₈×Dic₈₆	C₂	1	2	yes	maximal
2	Q₈×Dic₁₇₂	C₂	1	16	yes	maximal, direct product factor
3	C₂×#14583304	C₂	1	3	yes	maximal
4	C₂×#14583195	C₂	1	6	yes	maximal
5	C₂×C₄×Dic₁₇₂	C₂	1	3	yes	maximal
6	C₂×C₈×Q₈×C₄₃	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₁₇₂	C₂²	1	1	yes	characteristic
8	Q₈×Dic₈₆	—	2	8	no	—
9	Q₈×Dic₈₆	C₂²	1	16	yes	—
10	C₂×#13063325	C₂²	1	6	yes	—
11	C₂×#13063324	—	2	6	no	—
12	C₂×Q₈×Dic₄₃	—	2	2	no	—
13	C₂×C₄×Dic₈₆	C₂²	1	6	yes	—
14	#14583304	C₂²	1	24	yes	—
15	#14583195	C₂²	1	48	yes	—
16	C₄×Dic₁₇₂	C₂²	1	24	yes	—
17	C₂×#13063259	C₂²	1	3	yes	—
18	C₂×#13063259	C₂²	1	6	yes	—
19	C₂×#13063250	C₂²	1	6	yes	—
20	C₂×C₄×Q₈×C₄₃	C₂²	1	1	yes	characteristic
21	C₈×Q₈×C₄₃	C₂²	1	8	yes	—
22	C₂×C₄⋊C₈×C₄₃	C₂²	1	3	yes	—
23	C₂×C₄×C₈×C₄₃	C₂²	1	3	yes	—
24	C₂²×Dic₈₆	—	2	6	no	—
25	C₂²×Dic₈₆	C₂³	1	2	yes	—
26	C₂×Dic₁₇₂	Q₈	1	8	yes	direct product factor
27	C₂×Dic₁₇₂	C₂³	1	4	yes	—
28	#13063325	—	2	12	no	—
29	#13063325	C₂³	1	24	yes	—
30	#13063324	—	2	48	no	—
31	Q₈×Dic₄₃	—	2	16	no	—
32	C₄×Dic₈₆	—	2	12	no	—
33	C₄×Dic₈₆	C₂³	1	24	yes	—
34	C₂×C₄⋊Dic₄₃	—	2	6	no	—
35	C₂×C₄⋊Dic₄₃	C₂³	1	6	yes	—
36	C₂×Dic₄₃⋊C₄	—	2	6	no	—
37	C₂×C₄×Dic₄₃	—	2	6	no	—
38	#13063259	C₂³	1	12	yes	—
39	#13063259	C₂³	1	24	yes	—
40	#13063250	C₂³	1	24	yes	—
41	C₂²×Q₈×C₄₃	D₄	1	1	yes	characteristic
42	C₂²×C₈×C₄₃	C₂³	1	3	yes	—
43	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
44	C₄×Q₈×C₄₃	D₄	1	4	yes	—
45	C₄×Q₈×C₄₃	C₂³	1	4	yes	—
46	C₂×C₄⋊C₄×C₄₃	D₄	1	3	yes	—
47	C₂×C₄²×C₄₃	C₂³	1	3	yes	—
48	C₄⋊C₈×C₄₃	C₂³	1	12	yes	—
49	C₄×C₈×C₄₃	C₂³	1	12	yes	—
50	C₂×Q₈×C₄₃	Q₁₆	1	8	yes	complemented
51	C₂×Q₈×C₄₃	C₂×D₄	1	4	yes	—
52	C₂²×C₄×C₄₃	C₂×D₄	1	3	yes	—
53	C₂²×C₄×C₄₃	C₂×D₄	1	3	yes	—
54	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
55	C₂²×Dic₄₃	—	2	2	no	—
56	C₂²×Dic₄₃	—	2	6	no	—
57	C₂×Dic₈₆	—	2	2	no	—
58	C₂×Dic₈₆	—	2	4	no	—
59	C₂×Dic₈₆	—	2	24	no	—
60	C₂×Dic₈₆	—	8	12	no	—
61	C₂×Dic₈₆	C₂×Q₈	1	8	yes	—
62	C₂×Dic₈₆	C₂⁴	1	4	yes	—
63	C₂×C₈×C₄₃	—	2	6	no	—
64	C₂×C₈×C₄₃	C₂×Q₈	1	4	yes	—
65	C₂×C₈×C₄₃	C₂⁴	1	6	yes	—
66	C₂×C₈×C₄₃	C₂⁴	1	2	yes	—
67	C₄⋊C₄×C₄₃	C₂×D₄	1	12	yes	—
68	C₄²×C₄₃	C₂×D₄	1	6	yes	—
69	C₄²×C₄₃	C₂⁴	1	6	yes	—
70	C₄⋊Dic₄₃	—	2	24	no	—
71	C₄⋊Dic₄₃	—	2	6	no	—
72	C₄⋊Dic₄₃	C₂⁴	1	12	yes	—
73	Dic₄₃⋊C₄	—	2	24	no	—
74	C₄×Dic₄₃	—	2	24	no	—
75	Dic₁₇₂	C₂×Q₈	1	16	yes	direct product factor
76	C₂³×C₄₃	C₂²×D₄	1	1	yes	characteristic
77	Q₈×C₄₃	C₂×Q₁₆	1	16	yes	complemented
78	C₂×C₄×C₄₃	—	2	6	no	—
79	C₂×C₄×C₄₃	C₂×Q₁₆	1	12	yes	—
80	C₂×C₄×C₄₃	C₂²×D₄	1	6	yes	—
81	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
82	C₂×C₄×C₄₃	C₂²×D₄	1	6	yes	—
83	C₂×C₄×C₄₃	C₂²×Q₈	1	2	yes	—
84	C₂×C₄×C₄₃	2₋¹⁺⁴	1	2	yes	—
85	C₂×C₄×C₄₃	C₂⁵	1	1	yes	characteristic
86	C₂×Dic₄₃	—	2	8	no	—
87	C₂×Dic₄₃	—	2	4	no	—
88	C₂×Dic₄₃	—	2	12	no	—
89	C₂×Dic₄₃	—	4	12	no	—
90	Dic₈₆	—	2	8	no	—
91	Dic₈₆	—	2	4	no	—
92	Dic₈₆	—	8	24	no	—
93	Dic₈₆	C₂²×Q₈	1	8	yes	—
94	C₈×C₄₃	—	2	6	no	—
95	C₈×C₄₃	C₂²×Q₈	1	4	yes	—
96	C₂×Q₈×Q₁₆	—	43	1	no	maximal, Sylow
97	C₂²×C₄₃	D₄×Q₈	1	2	yes	—
98	C₂²×C₄₃	C₂²×Q₁₆	1	2	yes	—
99	C₂²×C₄₃	#576	1	2	yes	—
100	C₂²×C₄₃	C₂³×D₄	1	1	yes	characteristic
101	C₄×C₄₃	—	2	6	no	—
102	C₄×C₄₃	D₄×Q₈	1	2	yes	—
103	C₄×C₄₃	C₂²×Q₁₆	1	12	yes	—
104	C₄×C₄₃	C₂³×Q₈	1	1	yes	characteristic
105	C₄×C₄₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
106	Dic₄₃	—	2	8	no	—
107	Dic₄₃	—	4	12	no	—
108	C₂×Q₈²	—	43	2	no	—
109	Q₈×Q₁₆	—	43	16	no	—
110	C₂×#500	—	43	3	no	—
111	C₂×#477	—	43	6	no	—
112	C₂×C₈×Q₈	D₄₃	1	1	yes	characteristic
113	C₂×C₄×Q₁₆	—	43	3	no	—
114	C₂×C₄₃	Q₈×Q₁₆	1	4	yes	complemented
115	C₂×C₄₃	C₂×D₄×Q₈	1	1	yes	characteristic
116	C₂×C₄₃	C₂³×Q₁₆	1	1	yes	characteristic
117	C₂×C₄₃	C₂×#576	1	1	yes	characteristic
118	C₂²×Q₁₆	—	43	1	no	—
119	Q₈²	—	43	16	no	—
120	Q₈²	—	86	8	no	—
121	C₂×C₄⋊Q₈	—	43	6	no	—
122	C₂×C₄⋊Q₈	—	86	6	no	—
123	C₂×C₄×Q₈	—	43	6	no	—
124	C₂×C₄×Q₈	—	86	2	no	—
125	C₂×C₄×Q₈	C₂×D₄₃	1	1	yes	characteristic
126	#500	—	43	24	no	—
127	#477	—	43	48	no	—
128	C₈×Q₈	C₂×D₄₃	1	8	yes	—
129	C₄×Q₁₆	—	43	24	no	—
130	C₂×#107	—	43	6	no	—
131	C₂×#107	—	43	3	no	—
132	C₂×C₄⋊C₈	C₂×D₄₃	1	3	yes	—
133	C₂×#103	—	43	6	no	—
134	C₂×C₄×C₈	C₂×D₄₃	1	3	yes	—
135	C₄₃	C₂×Q₈×Q₁₆	1	1	yes	characteristic, complemented, Sylow
136	C₂²×Q₈	—	43	2	no	—
137	C₂²×Q₈	—	86	6	no	—
138	C₂²×Q₈	D₁₇₂	1	1	yes	characteristic
139	C₂×Q₁₆	—	43	8	no	—
140	C₂×Q₁₆	—	43	4	no	—
141	C₂²×C₈	C₂²×D₄₃	1	3	yes	—
142	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
143	C₄⋊Q₈	—	43	24	no	—
144	C₄⋊Q₈	—	86	48	no	—
145	C₄⋊Q₈	—	86	12	no	—
146	C₄×Q₈	—	43	24	no	—
147	C₄×Q₈	—	86	16	no	—
148	C₄×Q₈	—	86	12	no	—
149	C₄×Q₈	D₁₇₂	1	4	yes	—
150	C₄×Q₈	C₂²×D₄₃	1	4	yes	—
151	C₂×C₄⋊C₄	—	43	6	no	—
152	C₂×C₄⋊C₄	—	86	6	no	—
153	C₂×C₄⋊C₄	—	86	6	no	—
154	C₂×C₄⋊C₄	D₁₇₂	1	3	yes	—
155	C₂×C₄²	—	86	6	no	—
156	C₂×C₄²	C₂²×D₄₃	1	3	yes	—
157	#107	—	43	12	no	—
158	#107	—	43	24	no	—
159	C₄⋊C₈	C₂²×D₄₃	1	12	yes	—
160	#103	—	43	24	no	—
161	C₄×C₈	C₂²×D₄₃	1	12	yes	—
162	C₂×Q₈	—	43	8	no	—
163	C₂×Q₈	—	43	4	no	—
164	C₂×Q₈	—	86	2	no	—
165	C₂×Q₈	—	86	24	no	—
166	C₂×Q₈	—	86	4	no	—
167	C₂×Q₈	—	344	12	no	—
168	C₂×Q₈	Dic₁₇₂	1	8	yes	direct product factor
169	C₂×Q₈	C₂×D₁₇₂	1	4	yes	—
170	C₂²×C₄	—	86	2	no	—
171	C₂²×C₄	—	86	6	no	—
172	C₂²×C₄	C₂×D₁₇₂	1	3	yes	—
173	C₂²×C₄	C₂×D₁₇₂	1	3	yes	—
174	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
175	Q₁₆	—	43	16	no	—
176	C₂×C₈	—	2	6	no	—
177	C₂×C₈	Q₈×D₄₃	1	4	yes	—
178	C₂×C₈	C₂³×D₄₃	1	6	yes	—
179	C₂×C₈	C₂³×D₄₃	1	2	yes	—
180	C₄⋊C₄	—	43	12	no	—
181	C₄⋊C₄	—	86	6	no	—
182	C₄⋊C₄	—	86	24	no	—
183	C₄⋊C₄	—	86	24	no	—
184	C₄⋊C₄	C₂×D₁₇₂	1	12	yes	—
185	C₄²	—	86	24	no	—
186	C₄²	C₂×D₁₇₂	1	6	yes	—
187	C₄²	C₂³×D₄₃	1	6	yes	—
188	C₂³	C₂²×D₁₇₂	1	1	yes	characteristic
189	Q₈	—	43	8	no	—
190	Q₈	—	86	8	no	—
191	Q₈	—	86	4	no	—
192	Q₈	—	344	24	no	—
193	Q₈	C₂×Dic₁₇₂	1	16	yes	direct product factor
194	C₂×C₄	—	2	6	no	—
195	C₂×C₄	—	86	8	no	—
196	C₂×C₄	—	86	4	no	—
197	C₂×C₄	—	86	12	no	—
198	C₂×C₄	—	172	12	no	—
199	C₂×C₄	C₂×Dic₁₇₂	1	12	yes	—
200	C₂×C₄	C₂²×D₁₇₂	1	6	yes	—
201	C₂×C₄	C₂²×D₁₇₂	1	1	yes	characteristic
202	C₂×C₄	C₂²×D₁₇₂	1	6	yes	—
203	C₂×C₄	C₂×Q₈×D₄₃	1	2	yes	—
204	C₂×C₄	#13063372	1	2	yes	—
205	C₂×C₄	C₂⁴×D₄₃	1	1	yes	characteristic
206	C₈	—	2	6	no	—
207	C₈	C₂×Q₈×D₄₃	1	4	yes	—
208	C₂²	Q₈×D₁₇₂	1	2	yes	—
209	C₂²	C₂²×Dic₁₇₂	1	2	yes	—
210	C₂²	#14583704	1	2	yes	—
211	C₂²	C₂³×D₁₇₂	1	1	yes	characteristic
212	C₄	—	2	6	no	—
213	C₄	—	86	8	no	—
214	C₄	—	172	12	no	—
215	C₄	Q₈×D₁₇₂	1	2	yes	—
216	C₄	C₂²×Dic₁₇₂	1	12	yes	—
217	C₄	C₂²×Q₈×D₄₃	1	1	yes	characteristic
218	C₄	C₂×#13063372	1	1	yes	characteristic
219	C₂	Q₈×Dic₁₇₂	1	4	yes	direct product factor
220	C₂	C₂×Q₈×D₁₇₂	1	1	yes	characteristic
221	C₂	C₂³×Dic₁₇₂	1	1	yes	characteristic
222	C₂	C₂×#14583704	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²×D₁₇₂
commutator, cosocle	C₂×C₄×C₄₃	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₄₃
Fitting	C₂×C₈×Q₈×C₄₃	C₂
socle, prime radical	C₂³×C₄₃	C₂²×D₄
hypercenter	C₂×C₈×Q₈	D₄₃
hypocenter	C₄₃	C₂×Q₈×Q₁₆
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×Q₈ ◁ C₂×C₈×Q₈

Complements

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

N	H ≅ G/N	type
2. Q₈×Dic₁₇₂	C₂	direct (N × H)
26. C₂×Dic₁₇₂	Q₈	direct (N × H)
50. C₂×Q₈×C₄₃	Q₁₆	semidirect
75. Dic₁₇₂	C₂×Q₈	direct (N × H)
77. Q₈×C₄₃	C₂×Q₁₆	semidirect
114. C₂×C₄₃	Q₈×Q₁₆	semidirect
135. C₄₃	C₂×Q₈×Q₁₆	semidirect
168. C₂×Q₈	Dic₁₇₂	direct (N × H)
193. Q₈	C₂×Dic₁₇₂	direct (N × H)
219. C₂	Q₈×Dic₁₇₂	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×Q₈×Dic₈₆	2
Q₈×Dic₁₇₂	2
C₂×#14583304	2
C₂×#14583195	2
C₂×C₄×Dic₁₇₂	2
C₂×C₈×Q₈×C₄₃	2
C₂×Q₈×Q₁₆	43

Subgroups — C2×Q8×Dic172 #25977047

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×Q₈×Dic₁₇₂ #25977047