Subgroups — C₂×#14543822 #17960445

Order 4352.

726 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#63216	C₂	1	2	yes	maximal
2	#14543822	C₂	1	4	yes	maximal, direct product factor
3	C₂²×C₄×#3622	C₂	1	1	yes	maximal, characteristic
4	C₂²×#63216	—	2	4	no	—
5	C₂²×#63216	—	2	4	no	—
6	C₂²×#63216	C₄	1	8	yes	complemented
7	C₂²×#63216	C₂²	1	4	yes	—
8	C₂³×#3622	C₄	1	1	yes	characteristic
9	C₂³×#3622	C₂²	1	1	yes	characteristic
10	C₂×C₄×#3622	—	2	4	no	—
11	C₂×C₄×#3622	C₄	1	2	yes	—
12	C₂×C₄×#3622	C₂²	1	2	yes	—
13	C₂²×C₄²×C₁₇	C₄	1	1	yes	characteristic
14	C₂³×C₄×C₁₇	C₈	1	2	yes	—
15	C₂³×C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
16	C₂×C₄²×C₁₇	—	2	2	no	—
17	C₂×C₄²×C₁₇	—	2	2	no	—
18	C₂×C₄²×C₁₇	C₈	1	2	yes	—
19	C₂×C₄²×C₁₇	C₂×C₄	1	2	yes	—
20	C₂²×#3622	—	2	2	no	—
21	C₂²×#3622	—	2	2	no	—
22	C₂²×#3622	C₈	1	4	yes	—
23	C₂²×#3622	C₂×C₄	1	4	yes	—
24	C₂²×#3622	C₂×C₄	1	2	yes	—
25	C₂²×#3622	C₂×C₄	1	1	yes	characteristic
26	C₂²×#3622	D₄	1	4	yes	—
27	C₂²×#3622	D₄	1	2	yes	—
28	C₂²×#3622	Q₈	1	2	yes	—
29	C₂²×#3622	C₂³	1	1	yes	characteristic
30	C₂×#63216	—	2	16	no	—
31	C₂×#63216	—	2	4	no	—
32	C₂×#63216	—	2	4	no	—
33	C₂×#63216	C₂×C₄	1	8	yes	complemented
34	C₄×#3622	—	2	4	no	—
35	C₄×#3622	—	2	4	no	—
36	C₂⁴×C₁₇	C₂×C₈	1	1	yes	characteristic
37	C₂²×C₄×C₁₇	—	2	4	no	—
38	C₂²×C₄×C₁₇	—	2	4	no	—
39	C₂²×C₄×C₁₇	C₁₆	1	8	yes	complemented
40	C₂²×C₄×C₁₇	C₄²	1	2	yes	—
41	C₂²×C₄×C₁₇	C₂²⋊C₄	1	2	yes	—
42	C₂²×C₄×C₁₇	C₄⋊C₄	1	2	yes	—
43	C₂²×C₄×C₁₇	C₂×C₈	1	2	yes	—
44	C₂²×C₄×C₁₇	C₂×C₈	1	4	yes	—
45	C₂²×C₄×C₁₇	C₂×C₈	1	1	yes	characteristic
46	C₂²×C₄×C₁₇	M₁₆	1	2	yes	—
47	C₂²×C₄×C₁₇	M₁₆	1	2	yes	—
48	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
49	C₄²×C₁₇	—	2	4	no	—
50	C₄²×C₁₇	—	2	2	no	—
51	C₄²×C₁₇	—	2	2	no	—
52	C₂×#3622	—	2	8	no	—
53	C₂×#3622	—	2	2	no	—
54	C₂×#3622	—	2	2	no	—
55	C₂×#3622	C₄²	1	2	yes	—
56	C₂×#3622	C₂²⋊C₄	1	2	yes	—
57	C₂×#3622	C₂²⋊C₄	1	8	yes	—
58	C₂×#3622	C₄⋊C₄	1	2	yes	—
59	C₂×#3622	C₄⋊C₄	1	8	yes	—
60	C₂×#3622	C₂×C₈	1	4	yes	—
61	C₂×#3622	C₂²×C₄	1	2	yes	—
62	C₂×#3622	C₂×D₄	1	1	yes	characteristic
63	C₂×#3622	C₂×D₄	1	2	yes	—
64	C₂×#3622	C₂×Q₈	1	1	yes	characteristic
65	#63216	—	2	8	no	—
66	C₂×#1374	—	17	1	no	maximal, Sylow
67	C₂³×C₁₇	C₄×C₈	1	2	yes	—
68	C₂³×C₁₇	C₂²⋊C₈	1	2	yes	—
69	C₂³×C₁₇	C₄⋊C₈	1	2	yes	—
70	C₂³×C₁₇	C₂×C₁₆	1	4	yes	—
71	C₂³×C₁₇	M₃₂	1	4	yes	—
72	C₂³×C₁₇	C₂²×C₈	1	1	yes	characteristic
73	C₂×C₄×C₁₇	—	2	16	no	—
74	C₂×C₄×C₁₇	—	2	4	no	—
75	C₂×C₄×C₁₇	—	2	4	no	—
76	C₂×C₄×C₁₇	#95	1	4	yes	—
77	C₂×C₄×C₁₇	C₄×C₈	1	2	yes	—
78	C₂×C₄×C₁₇	#97	1	4	yes	—
79	C₂×C₄×C₁₇	C₂²⋊C₈	1	4	yes	—
80	C₂×C₄×C₁₇	C₂²⋊C₈	1	2	yes	—
81	C₂×C₄×C₁₇	C₄⋊C₈	1	2	yes	—
82	C₂×C₄×C₁₇	C₄⋊C₈	1	4	yes	—
83	C₂×C₄×C₁₇	C₂×C₁₆	1	8	yes	complemented
84	C₂×C₄×C₁₇	C₂×C₄²	1	1	yes	characteristic
85	C₂×C₄×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
86	C₂×C₄×C₁₇	C₂×C₄⋊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×C₄×C₁₇	C₂×M₁₆	1	1	yes	characteristic
90	#3622	—	2	4	no	—
91	#3622	#95	1	4	yes	—
92	#3622	C₂×C₂²⋊C₄	1	2	yes	—
93	#3622	C₂×C₄⋊C₄	1	2	yes	—
94	C₂³×C₁₆	—	17	2	no	—
95	C₂²×C₄×C₈	—	17	1	no	—
96	#1374	—	17	4	no	—
97	C₂²×C₁₇	#336	1	4	yes	—
98	C₂²×C₁₇	C₄×C₁₆	1	4	yes	—
99	C₂²×C₁₇	#346	1	4	yes	—
100	C₂²×C₁₇	C₂²⋊C₁₆	1	8	yes	—
101	C₂²×C₁₇	C₄⋊C₁₆	1	8	yes	—
102	C₂²×C₁₇	C₂×C₄×C₈	1	1	yes	characteristic
103	C₂²×C₁₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
104	C₂²×C₁₇	C₂×C₄⋊C₈	1	1	yes	characteristic
105	C₂²×C₁₇	C₂²×C₁₆	1	2	yes	—
106	C₂²×C₁₇	C₂×M₃₂	1	2	yes	—
107	C₄×C₁₇	—	2	8	no	—
108	C₄×C₁₇	#336	1	4	yes	—
109	C₄×C₁₇	C₂×#95	1	1	yes	characteristic
110	C₄×C₁₇	C₂×#97	1	1	yes	characteristic
111	C₄×C₁₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
112	C₄×C₁₇	C₂×C₄⋊C₈	1	1	yes	characteristic
113	C₂³×C₈	—	17	1	no	—
114	C₂³×C₈	—	17	1	no	—
115	C₂²×C₄²	#595	1	1	yes	characteristic
116	C₂²×C₁₆	—	17	4	no	—
117	C₂²×C₁₆	—	17	8	no	—
118	C₂²×C₁₆	—	34	4	no	—
119	C₂²×C₁₆	—	34	4	no	—
120	C₂×C₄×C₈	—	17	2	no	—
121	C₂×C₄×C₈	—	17	2	no	—
122	C₂×C₄×C₈	—	34	4	no	—
123	C₂×C₁₇	#1374	1	8	yes	complemented
124	C₂×C₁₇	C₂×#336	1	1	yes	characteristic
125	C₂×C₁₇	C₂×C₄×C₁₆	1	1	yes	characteristic
126	C₂×C₁₇	C₂×#346	1	1	yes	characteristic
127	C₂×C₁₇	C₂×C₂²⋊C₁₆	1	2	yes	—
128	C₂×C₁₇	C₂×C₄⋊C₁₆	1	2	yes	—
129	C₂³×C₄	#3624	1	2	yes	—
130	C₂³×C₄	C₂×#595	1	1	yes	characteristic
131	C₂²×C₈	—	17	4	no	—
132	C₂²×C₈	—	17	2	no	—
133	C₂²×C₈	—	17	2	no	—
134	C₂²×C₈	—	17	4	no	—
135	C₂²×C₈	—	17	1	no	—
136	C₂²×C₈	—	17	1	no	—
137	C₂²×C₈	—	17	4	no	—
138	C₂²×C₈	—	17	2	no	—
139	C₂²×C₈	—	34	2	no	—
140	C₂²×C₈	—	34	2	no	—
141	C₂×C₄²	—	2	2	no	—
142	C₂×C₄²	—	2	2	no	—
143	C₂×C₄²	#3624	1	2	yes	—
144	C₂×C₄²	C₂×#595	1	2	yes	—
145	C₂×C₁₆	—	17	8	no	—
146	C₂×C₁₆	—	34	4	no	—
147	C₂×C₁₆	—	34	4	no	—
148	C₂×C₁₆	—	34	16	no	—
149	C₄×C₈	—	34	4	no	—
150	C₄×C₈	—	34	4	no	—
151	C₁₇	C₂×#1374	1	1	yes	characteristic, complemented, Sylow
152	C₂⁴	C₂×#3624	1	1	yes	characteristic
153	C₂²×C₄	—	2	4	no	—
154	C₂²×C₄	—	2	4	no	—
155	C₂²×C₄	#63216	1	8	yes	complemented
156	C₂²×C₄	#63242	1	2	yes	—
157	C₂²×C₄	#63243	1	2	yes	—
158	C₂²×C₄	C₄×#595	1	2	yes	—
159	C₂²×C₄	C₄⋊#595	1	2	yes	—
160	C₂²×C₄	C₂×#3624	1	4	yes	—
161	C₂²×C₄	C₂×#3624	1	1	yes	characteristic
162	C₂²×C₄	#63247	1	2	yes	—
163	C₂²×C₄	C₂²⋊#595	1	2	yes	—
164	C₂²×C₄	C₂²×#595	1	1	yes	characteristic
165	C₂×C₈	—	17	2	no	—
166	C₂×C₈	—	17	2	no	—
167	C₂×C₈	—	17	1	no	—
168	C₂×C₈	—	17	2	no	—
169	C₂×C₈	—	17	1	no	—
170	C₂×C₈	—	17	2	no	—
171	C₂×C₈	—	17	2	no	—
172	C₂×C₈	—	17	8	no	—
173	C₂×C₈	—	17	8	no	—
174	C₂×C₈	—	17	4	no	—
175	C₂×C₈	—	34	2	no	—
176	C₂×C₈	—	34	2	no	—
177	C₂×C₈	—	34	8	no	—
178	C₄²	—	2	2	no	—
179	C₄²	—	2	2	no	—
180	C₄²	—	2	4	no	—
181	C₁₆	—	34	8	no	—
182	C₂³	C₂×#63216	1	4	yes	—
183	C₂³	#10587538	1	4	yes	—
184	C₂³	C₄×#3624	1	2	yes	—
185	C₂³	C₄⋊#3624	1	2	yes	—
186	C₂³	C₂²⋊#3624	1	2	yes	—
187	C₂³	C₂²×#3624	1	1	yes	characteristic
188	C₂×C₄	—	2	4	no	—
189	C₂×C₄	—	2	4	no	—
190	C₂×C₄	—	2	16	no	—
191	C₂×C₄	C₂×#63216	1	8	yes	complemented
192	C₂×C₄	C₄×#3624	1	2	yes	—
193	C₂×C₄	C₄⋊#3624	1	2	yes	—
194	C₂×C₄	#10587542	1	4	yes	—
195	C₂×C₄	#10587543	1	4	yes	—
196	C₂×C₄	#10587544	1	4	yes	—
197	C₂×C₄	(C₂×#595)⋊C₄	1	4	yes	—
198	C₂×C₄	C₂²⋊#3624	1	2	yes	—
199	C₂×C₄	C₂×#63242	1	1	yes	characteristic
200	C₂×C₄	C₂×#63243	1	1	yes	characteristic
201	C₂×C₄	C₂×C₄×#595	1	1	yes	characteristic
202	C₂×C₄	C₂×C₄⋊#595	1	1	yes	characteristic
203	C₂×C₄	C₂×#63247	1	1	yes	characteristic
204	C₂×C₄	C₂×C₂²⋊#595	1	1	yes	characteristic
205	C₈	—	17	2	no	—
206	C₈	—	17	4	no	—
207	C₈	—	17	2	no	—
208	C₈	—	34	4	no	—
209	C₂²	#11763675	1	4	yes	—
210	C₂²	C₄×#63216	1	4	yes	—
211	C₂²	#11763697	1	4	yes	—
212	C₂²	C₂²⋊#63216	1	8	yes	—
213	C₂²	C₄⋊#63216	1	8	yes	—
214	C₂²	C₂×C₄×#3624	1	1	yes	characteristic
215	C₂²	C₂×C₂²⋊#3624	1	1	yes	characteristic
216	C₂²	C₂×C₄⋊#3624	1	1	yes	characteristic
217	C₂²	C₂²×#63216	1	2	yes	—
218	C₂²	C₂×#10587538	1	2	yes	—
219	C₄	—	2	8	no	—
220	C₄	#11763675	1	4	yes	—
221	C₄	C₂×(C₂×#595)⋊C₄	1	1	yes	characteristic
222	C₄	C₂×#10587542	1	1	yes	characteristic
223	C₄	C₂×#10587543	1	1	yes	characteristic
224	C₄	C₂×#10587544	1	1	yes	characteristic
225	C₂	#14543822	1	8	yes	direct product factor
226	C₂	C₂×#11763675	1	1	yes	characteristic
227	C₂	C₂×C₄×#63216	1	1	yes	characteristic
228	C₂	C₂×#11763697	1	1	yes	characteristic
229	C₂	C₂×C₂²⋊#63216	1	2	yes	—
230	C₂	C₂×C₄⋊#63216	1	2	yes	—

	subgroup	G/H
center	C₂³×C₄	C₂×#595
commutator	C₂×C₁₇	C₂×C₄×C₁₆
Frattini	C₂²×C₄	C₂²×#595
Fitting	C₂²×C₄²×C₁₇	C₄
socle, prime radical	C₂⁴×C₁₇	C₂×C₈
cosocle	C₂²×#3622	C₂³
hypercenter	C₂²×C₄²	#595
hypocenter	C₁₇	C₂×#1374
LCS	G ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂³×C₄ ◁ C₂²×C₄²

Complements

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

N	H ≅ G/N	type
2. #14543822	C₂	direct (N × H)
6. C₂²×#63216	C₄	semidirect
33. C₂×#63216	C₂×C₄	semidirect
39. C₂²×C₄×C₁₇	C₁₆	semidirect
83. C₂×C₄×C₁₇	C₂×C₁₆	semidirect
123. C₂×C₁₇	#1374	semidirect
151. C₁₇	C₂×#1374	semidirect
155. C₂²×C₄	#63216	semidirect
191. C₂×C₄	C₂×#63216	semidirect
225. C₂	#14543822	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×#63216	2
#14543822	2
C₂²×C₄×#3622	2
C₂×#1374	17

Subgroups — C2×#14543822 #17960445

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#14543822 #17960445