Subgroups — C₄×Q₈×#3624 #17968853

Order 4352.

1070 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₈×#3624	C₂	1	2	yes	maximal
2	C₄⋊C₄×#3624	C₂	1	6	yes	maximal
3	C₄×C₄⋊#3624	C₂	1	3	yes	maximal
4	C₄²×#3624	C₂	1	3	yes	maximal
5	C₄×Q₈×Dic₁₇	C₂	1	1	yes	maximal, characteristic
6	Q₈×#3624	C₄	1	16	yes	direct product factor
7	C₂×C₄⋊#3624	C₂²	1	6	yes	—
8	C₂×C₄×#3624	C₂²	1	3	yes	—
9	C₂×C₄×#3624	C₂²	1	6	yes	—
10	C₂×C₄×#3624	C₂²	1	6	yes	—
11	C₂×C₄×#3624	C₂²	1	1	yes	characteristic
12	#11763675	C₂²	1	6	yes	—
13	C₂×Q₈×Dic₁₇	C₄	1	1	yes	characteristic
14	C₂×Q₈×Dic₁₇	C₂²	1	1	yes	characteristic
15	C₄⋊C₄×Dic₁₇	C₄	1	3	yes	—
16	C₄⋊C₄×Dic₁₇	C₂²	1	3	yes	—
17	C₄×C₄⋊Dic₁₇	C₄	1	3	yes	—
18	C₄²×Dic₁₇	C₂²	1	3	yes	—
19	C₂×C₄×Q₈×C₁₇	C₄	1	1	yes	characteristic
20	C₂²×Q₈×C₁₇	C₂×C₄	1	1	yes	characteristic
21	C₂²×#3624	C₂³	1	6	yes	—
22	C₂²×#3624	C₂³	1	2	yes	—
23	C₄×Q₈×C₁₇	C₈	1	8	yes	complemented
24	C₂×C₄⋊C₄×C₁₇	C₂×C₄	1	3	yes	—
25	C₂×C₄²×C₁₇	C₂×C₄	1	3	yes	—
26	Q₈×Dic₁₇	C₈	1	8	yes	complemented
27	Q₈×Dic₁₇	C₂×C₄	1	8	yes	—
28	C₂×C₄⋊Dic₁₇	C₂×C₄	1	3	yes	—
29	C₂×C₄⋊Dic₁₇	C₂×C₄	1	3	yes	—
30	C₂×C₄×Dic₁₇	C₂×C₄	1	3	yes	—
31	C₂×C₄×Dic₁₇	C₂×C₄	1	3	yes	—
32	C₂×C₄×Dic₁₇	C₂×C₄	1	3	yes	—
33	C₂×C₄×Dic₁₇	C₂³	1	3	yes	—
34	C₂×C₄×Dic₁₇	C₂³	1	3	yes	—
35	C₂×C₄×Dic₁₇	C₂³	1	1	yes	characteristic
36	C₄⋊#3624	C₂×C₄	1	24	yes	—
37	C₄×#3624	—	2	6	no	—
38	C₄×#3624	—	2	12	no	—
39	C₄×#3624	C₂×C₄	1	24	yes	—
40	C₄×#3624	Q₈	1	4	yes	direct product factor
41	#10587502	C₂×C₄	1	6	yes	—
42	C₂×Q₈×C₁₇	C₄²	1	4	yes	—
43	C₂×Q₈×C₁₇	C₂×C₈	1	4	yes	—
44	C₂×Q₈×C₁₇	C₂×C₈	1	4	yes	—
45	C₂²×C₄×C₁₇	C₂²×C₄	1	3	yes	—
46	C₂²×C₄×C₁₇	C₂²×C₄	1	3	yes	—
47	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
48	C₂²×Dic₁₇	C₂²×C₄	1	3	yes	—
49	C₂²×Dic₁₇	C₂²×C₄	1	3	yes	—
50	C₂²×Dic₁₇	C₂²×C₄	1	1	yes	characteristic
51	C₂²×Dic₁₇	C₂⁴	1	1	yes	characteristic
52	C₂×#3624	—	2	6	no	—
53	C₂×#3624	—	2	6	no	—
54	C₂×#3624	C₂²×C₄	1	12	yes	—
55	C₂×#3624	C₂²×C₄	1	4	yes	—
56	C₂×#3624	C₂×Q₈	1	4	yes	—
57	C₂×#3624	#41	1	4	yes	—
58	C₄⋊C₄×C₁₇	C₂×C₈	1	12	yes	—
59	C₄²×C₁₇	C₂×C₈	1	12	yes	—
60	C₄⋊Dic₁₇	C₄²	1	12	yes	—
61	C₄⋊Dic₁₇	C₂×C₈	1	12	yes	—
62	C₄×Dic₁₇	—	2	6	no	—
63	C₄×Dic₁₇	—	2	6	no	—
64	C₄×Dic₁₇	—	2	6	no	—
65	C₄×Dic₁₇	C₂×C₈	1	12	yes	—
66	C₄×Dic₁₇	C₂²×C₄	1	12	yes	—
67	C₄×Dic₁₇	C₂×Q₈	1	2	yes	—
68	C₄×Dic₁₇	#41	1	2	yes	—
69	C₄×C₈×Q₈	—	17	1	no	maximal, Sylow
70	C₂³×C₁₇	C₂³×C₄	1	1	yes	characteristic
71	Q₈×C₁₇	C₄×C₈	1	16	yes	complemented
72	C₂×C₄×C₁₇	—	2	3	no	—
73	C₂×C₄×C₁₇	—	2	3	no	—
74	C₂×C₄×C₁₇	C₂×C₄²	1	6	yes	—
75	C₂×C₄×C₁₇	C₄×Q₈	1	2	yes	—
76	C₂×C₄×C₁₇	C₂²×C₈	1	6	yes	—
77	C₂×C₄×C₁₇	C₂²×C₈	1	6	yes	—
78	C₂×C₄×C₁₇	C₂²×C₈	1	6	yes	—
79	C₂×C₄×C₁₇	C₂²×C₈	1	2	yes	—
80	C₂×C₄×C₁₇	#131	1	2	yes	—
81	C₂×Dic₁₇	—	2	3	no	—
82	C₂×Dic₁₇	—	2	3	no	—
83	C₂×Dic₁₇	—	2	3	no	—
84	C₂×Dic₁₇	—	2	3	no	—
85	C₂×Dic₁₇	C₂×C₄²	1	6	yes	—
86	C₂×Dic₁₇	C₄×Q₈	1	2	yes	—
87	C₂×Dic₁₇	C₂²×C₈	1	6	yes	—
88	C₂×Dic₁₇	C₂²×C₈	1	2	yes	—
89	C₂×Dic₁₇	#131	1	2	yes	—
90	C₂×Dic₁₇	C₂³×C₄	1	2	yes	—
91	C₂×Dic₁₇	C₂²×Q₈	1	1	yes	characteristic
92	C₂×Dic₁₇	C₂×#41	1	2	yes	—
93	C₂×Dic₁₇	C₂×#41	1	1	yes	characteristic
94	#3624	—	2	12	no	—
95	#3624	C₄×Q₈	1	8	yes	direct product factor
96	C₂×C₈×Q₈	—	17	2	no	—
97	C₄²×Q₈	—	17	1	no	—
98	C₈×C₄⋊C₄	—	17	6	no	—
99	C₄×C₄⋊C₈	—	17	3	no	—
100	C₄²×C₈	—	17	3	no	—
101	C₂²×C₁₇	C₂²×C₄²	1	1	yes	characteristic
102	C₂²×C₁₇	C₂×C₄×Q₈	1	1	yes	characteristic
103	C₂²×C₁₇	C₄×#41	1	1	yes	characteristic
104	C₂²×C₁₇	C₂³×C₈	1	1	yes	characteristic
105	C₂²×C₁₇	C₂³×C₈	1	1	yes	characteristic
106	C₂²×C₁₇	C₂×#131	1	1	yes	characteristic
107	C₂²×C₁₇	C₂×#131	1	1	yes	characteristic
108	C₄×C₁₇	—	2	6	no	—
109	C₄×C₁₇	C₂×C₄×C₈	1	12	yes	—
110	C₄×C₁₇	C₈×Q₈	1	4	yes	complemented
111	Dic₁₇	—	2	6	no	—
112	Dic₁₇	—	2	6	no	—
113	Dic₁₇	C₈×Q₈	1	4	yes	complemented
114	Dic₁₇	C₂×C₄×Q₈	1	2	yes	—
115	Dic₁₇	C₄×#41	1	2	yes	—
116	C₂×C₄×Q₈	—	17	1	no	—
117	C₂×C₄×Q₈	—	17	1	no	—
118	C₂×C₄×Q₈	#595	1	1	yes	characteristic
119	C₈×Q₈	—	17	16	no	—
120	C₂×C₄⋊C₈	—	17	6	no	—
121	C₂×C₄×C₈	—	17	6	no	—
122	C₂×C₄×C₈	—	17	1	no	—
123	C₂×C₄×C₈	—	17	3	no	—
124	C₂×C₄×C₈	—	17	6	no	—
125	C₄×C₄⋊C₄	—	17	3	no	—
126	C₄×C₄⋊C₄	—	17	3	no	—
127	C₄×C₄⋊C₄	—	17	3	no	—
128	C₄³	—	17	3	no	—
129	#336	—	17	6	no	—
130	C₂×C₁₇	C₄²×Q₈	1	1	yes	characteristic
131	C₂×C₁₇	C₂²×C₄×C₈	1	1	yes	characteristic
132	C₂×C₁₇	C₄×#131	1	1	yes	characteristic
133	C₂×C₁₇	C₂×C₈×Q₈	1	1	yes	characteristic
134	C₂×C₁₇	C₂×C₈×Q₈	1	1	yes	characteristic
135	C₂×C₁₇	C₈×#41	1	1	yes	characteristic
136	C₂×C₁₇	C₈×#41	1	1	yes	characteristic
137	C₂²×Q₈	C₂×#595	1	1	yes	characteristic
138	C₂²×C₈	—	17	2	no	—
139	C₂²×C₈	—	17	6	no	—
140	C₄×Q₈	—	17	8	no	—
141	C₄×Q₈	—	17	8	no	—
142	C₄×Q₈	#3624	1	8	yes	direct product factor
143	C₂×C₄⋊C₄	—	17	3	no	—
144	C₂×C₄⋊C₄	—	17	3	no	—
145	C₂×C₄⋊C₄	C₂×#595	1	3	yes	—
146	C₂×C₄²	—	17	3	no	—
147	C₂×C₄²	—	17	3	no	—
148	C₂×C₄²	—	17	1	no	—
149	C₂×C₄²	—	17	3	no	—
150	C₂×C₄²	—	17	3	no	—
151	C₂×C₄²	—	17	3	no	—
152	C₂×C₄²	C₂×#595	1	3	yes	—
153	C₄⋊C₈	—	17	24	no	—
154	C₄×C₈	—	17	24	no	—
155	C₄×C₈	—	17	4	no	—
156	C₄×C₈	—	34	6	no	—
157	C₄×C₈	—	34	12	no	—
158	#95	—	17	6	no	—
159	C₁₇	C₄×C₈×Q₈	1	1	yes	characteristic, complemented, Sylow
160	C₂×Q₈	#63242	1	4	yes	—
161	C₂×Q₈	C₄×#595	1	4	yes	—
162	C₂×Q₈	C₂×#3624	1	4	yes	—
163	C₂²×C₄	—	17	1	no	—
164	C₂²×C₄	—	17	1	no	—
165	C₂²×C₄	—	17	3	no	—
166	C₂²×C₄	—	17	3	no	—
167	C₂²×C₄	C₂²×#595	1	3	yes	—
168	C₂²×C₄	C₂²×#595	1	3	yes	—
169	C₂²×C₄	C₂²×#595	1	1	yes	characteristic
170	C₂×C₈	—	17	4	no	—
171	C₂×C₈	—	17	4	no	—
172	C₂×C₈	—	17	4	no	—
173	C₂×C₈	—	17	12	no	—
174	C₂×C₈	—	34	6	no	—
175	C₂×C₈	—	34	6	no	—
176	C₄⋊C₄	—	17	12	no	—
177	C₄⋊C₄	—	17	12	no	—
178	C₄⋊C₄	C₂×#3624	1	12	yes	—
179	C₄²	—	17	12	no	—
180	C₄²	—	17	12	no	—
181	C₄²	—	17	2	no	—
182	C₄²	—	17	2	no	—
183	C₄²	—	34	6	no	—
184	C₄²	—	34	6	no	—
185	C₄²	—	34	6	no	—
186	C₄²	C₂×#3624	1	12	yes	—
187	C₂³	C₂³×#595	1	1	yes	characteristic
188	Q₈	C₄×#3624	1	16	yes	direct product factor
189	C₂×C₄	—	2	3	no	—
190	C₂×C₄	—	2	3	no	—
191	C₂×C₄	—	17	2	no	—
192	C₂×C₄	—	17	1	no	—
193	C₂×C₄	—	17	1	no	—
194	C₂×C₄	—	17	2	no	—
195	C₂×C₄	—	17	2	no	—
196	C₂×C₄	—	17	2	no	—
197	C₂×C₄	—	17	2	no	—
198	C₂×C₄	—	17	6	no	—
199	C₂×C₄	—	17	6	no	—
200	C₂×C₄	—	34	3	no	—
201	C₂×C₄	—	34	3	no	—
202	C₂×C₄	—	34	3	no	—
203	C₂×C₄	—	34	3	no	—
204	C₂×C₄	C₂×#63242	1	6	yes	—
205	C₂×C₄	C₂×C₄×#595	1	6	yes	—
206	C₂×C₄	#10587680	1	2	yes	—
207	C₂×C₄	Q₈×#595	1	2	yes	—
208	C₂×C₄	C₂²×#3624	1	6	yes	—
209	C₂×C₄	C₂²×#3624	1	6	yes	—
210	C₂×C₄	C₂²×#3624	1	2	yes	—
211	C₈	—	17	8	no	—
212	C₈	—	34	12	no	—
213	C₂²	C₂²×C₄×#595	1	1	yes	characteristic
214	C₂²	C₂×Q₈×#595	1	1	yes	characteristic
215	C₂²	#41×#595	1	1	yes	characteristic
216	C₂²	C₂³×#3624	1	1	yes	characteristic
217	C₂²	C₂²×#63242	1	1	yes	characteristic
218	C₂²	D₁₇⋊#131	1	1	yes	characteristic
219	C₂²	C₂×#10587680	1	1	yes	characteristic
220	C₄	—	2	6	no	—
221	C₄	—	17	2	no	—
222	C₄	—	17	2	no	—
223	C₄	—	17	4	no	—
224	C₄	—	34	6	no	—
225	C₄	—	34	6	no	—
226	C₄	C₂×C₄×#3624	1	12	yes	—
227	C₄	Q₈×#3624	1	4	yes	direct product factor
228	C₂	C₄×Q₈×#595	1	1	yes	characteristic
229	C₂	C₂²×C₄×#3624	1	1	yes	characteristic
230	C₂	C₄×#10587680	1	1	yes	characteristic
231	C₂	C₂×Q₈×#3624	1	1	yes	characteristic
232	C₂	Q₈×#63242	1	1	yes	characteristic
233	C₂	#41×#3624	1	1	yes	characteristic
234	C₂	#14545590	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²×#595
commutator	C₂×C₁₇	C₂²×C₄×C₈
Frattini	C₂³	C₂³×#595
Fitting	C₂×C₄×Q₈×C₁₇	C₄
socle, prime radical	C₂³×C₁₇	C₂³×C₄
cosocle	C₂²×Dic₁₇	C₂⁴
hypercenter	C₂×C₄×Q₈	#595
hypocenter	C₁₇	C₄×C₈×Q₈
LCS	G ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×Q₈

Complements

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

N	H ≅ G/N	type
6. Q₈×#3624	C₄	direct (N × H)
23. C₄×Q₈×C₁₇	C₈	semidirect
26. Q₈×Dic₁₇	C₈	semidirect
40. C₄×#3624	Q₈	direct (N × H)
71. Q₈×C₁₇	C₄×C₈	semidirect
95. #3624	C₄×Q₈	direct (N × H)
110. C₄×C₁₇	C₈×Q₈	semidirect
113. Dic₁₇	C₈×Q₈	semidirect
142. C₄×Q₈	#3624	direct (N × H)
159. C₁₇	C₄×C₈×Q₈	semidirect
188. Q₈	C₄×#3624	direct (N × H)
227. C₄	Q₈×#3624	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×#3624	2
C₄⋊C₄×#3624	2
C₄×C₄⋊#3624	2
C₄²×#3624	2
C₄×Q₈×Dic₁₇	2
C₄×C₈×Q₈	17

Subgroups — C4×Q8×#3624 #17968853

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×Q₈×#3624 #17968853