Subgroups — #17960968

Order 4352.

382 conjugacy classes of subgroups.

The subgroup lattice has 196 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	2	yes	maximal
2	C₂×#11763697	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×#63216	C₂	1	1	yes	maximal, characteristic
4	#14543822	C₂	1	2	yes	maximal
5	#14528471	C₂	1	1	yes	maximal, characteristic
6	C₂²×#63216	C₂²	1	2	yes	—
7	C₂²×#63216	C₂²	1	2	yes	—
8	C₄⋊#63216	C₄	1	8	yes	complemented
9	#11763697	—	2	2	no	—
10	C₄×#63216	—	2	2	no	—
11	C₂×#10587474	C₄	1	1	yes	characteristic
12	C₂×#10587474	C₄	1	1	yes	characteristic
13	C₂×#10587474	C₄	1	1	yes	characteristic
14	C₂×C₄×#3622	C₂²	1	1	yes	characteristic
15	C₂×C₄×#3622	C₂²	1	1	yes	characteristic
16	C₂×C₄×#3622	C₂²	1	1	yes	characteristic
17	C₁₇×#376	C₄	1	1	yes	characteristic
18	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
21	C₂²×#3622	C₂×C₄	1	1	yes	characteristic
22	C₂²×#3622	C₂×C₄	1	1	yes	characteristic
23	C₂²×#3622	C₂×C₄	1	1	yes	characteristic
24	C₂²×#3622	C₂³	1	1	yes	characteristic
25	C₂×#63216	—	2	2	no	—
26	C₂×#63216	—	2	2	no	—
27	C₂×#63216	—	2	2	no	—
28	C₂×#63216	—	2	2	no	—
29	C₂×#63216	C₂×C₄	1	4	yes	—
30	C₂×#63216	C₂×C₄	1	4	yes	—
31	C₁₇×#95	C₈	1	4	yes	—
32	#10587474	—	2	2	no	—
33	#10587474	—	2	2	no	—
34	#10587474	C₈	1	4	yes	—
35	C₄×#3622	—	2	2	no	—
36	C₄×#3622	C₂×C₄	1	4	yes	—
37	C₄×#3622	D₄	1	2	yes	—
38	C₄×#3622	Q₈	1	2	yes	—
39	C₂²×C₄×C₁₇	C₂×C₈	1	2	yes	—
40	C₂²×C₄×C₁₇	C₂×C₈	1	2	yes	—
41	C₂²×C₄×C₁₇	C₂×C₈	1	2	yes	—
42	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
43	C₄²×C₁₇	—	2	1	no	—
44	C₄²×C₁₇	—	2	1	no	—
45	C₄²×C₁₇	C₄²	1	2	yes	—
46	C₄²×C₁₇	C₄⋊C₄	1	2	yes	—
47	C₄²×C₁₇	C₂×C₈	1	2	yes	—
48	C₄²×C₁₇	M₁₆	1	2	yes	—
49	C₂×#3622	—	2	1	no	—
50	C₂×#3622	—	2	1	no	—
51	C₂×#3622	—	2	1	no	—
52	C₂×#3622	—	2	1	no	—
53	C₂×#3622	C₄²	1	2	yes	—
54	C₂×#3622	C₄⋊C₄	1	2	yes	—
55	C₂×#3622	C₂×C₈	1	2	yes	—
56	C₂×#3622	C₂×C₈	1	2	yes	—
57	C₂×#3622	M₁₆	1	2	yes	—
58	C₂×#3622	C₂²×C₄	1	2	yes	—
59	C₂×#3622	C₂×D₄	1	1	yes	characteristic
60	C₂×#3622	C₂×Q₈	1	1	yes	characteristic
61	C₂×#3622	#41	1	2	yes	—
62	#63216	—	2	4	no	—
63	#63216	—	2	4	no	—
64	#12654	—	17	1	no	maximal, Sylow
65	C₂³×C₁₇	C₂²×C₈	1	1	yes	characteristic
66	C₂×C₄×C₁₇	—	2	4	no	—
67	C₂×C₄×C₁₇	—	2	4	no	—
68	C₂×C₄×C₁₇	—	2	2	no	—
69	C₂×C₄×C₁₇	—	2	4	no	—
70	C₂×C₄×C₁₇	C₄×C₈	1	4	yes	—
71	C₂×C₄×C₁₇	C₄⋊C₈	1	4	yes	—
72	C₂×C₄×C₁₇	C₂×C₄²	1	1	yes	characteristic
73	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
74	C₂×C₄×C₁₇	#117	1	1	yes	characteristic
75	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
76	C₂×C₄×C₁₇	C₂×M₁₆	1	1	yes	characteristic
77	C₂×C₄×C₁₇	C₂×M₁₆	1	1	yes	characteristic
78	#3622	—	2	2	no	—
79	#3622	—	2	2	no	—
80	#3622	C₄⋊C₈	1	4	yes	—
81	#3622	C₄×D₄	1	2	yes	—
82	#3622	C₄×Q₈	1	2	yes	—
83	C₂×C₄⋊C₁₆	—	17	2	no	—
84	C₂×#346	—	17	1	no	—
85	C₂×C₄×C₁₆	—	17	1	no	—
86	#1726	—	17	1	no	—
87	#1374	—	17	2	no	—
88	C₂²×C₁₇	C₂×C₄×C₈	1	1	yes	characteristic
89	C₂²×C₁₇	C₂×C₄⋊C₈	1	1	yes	characteristic
90	C₂²×C₁₇	#431	1	1	yes	characteristic
91	C₂²×C₁₇	#504	1	4	yes	—
92	C₄×C₁₇	—	2	4	no	—
93	C₄×C₁₇	—	2	4	no	—
94	C₄×C₁₇	—	4	2	no	—
95	C₄×C₁₇	C₄×C₄⋊C₄	1	1	yes	characteristic
96	C₄×C₁₇	C₄×M₁₆	1	1	yes	characteristic
97	C₄×C₁₇	C₂×C₄⋊C₈	1	1	yes	characteristic
98	C₄×C₁₇	#431	1	1	yes	characteristic
99	C₂²×C₁₆	—	17	2	no	—
100	C₂²×C₁₆	—	17	2	no	—
101	C₂×#97	—	17	1	no	—
102	C₂×#97	—	17	1	no	—
103	C₂×#97	—	17	1	no	—
104	C₂×C₄×C₈	—	17	1	no	—
105	C₂×C₄×C₈	—	17	1	no	—
106	C₂×C₄×C₈	—	17	1	no	—
107	#376	#595	1	1	yes	characteristic
108	C₄⋊C₁₆	—	17	8	no	—
109	#346	—	34	2	no	—
110	C₄×C₁₆	—	34	2	no	—
111	C₂×C₁₇	C₄×C₄⋊C₈	1	1	yes	characteristic
112	C₂×C₁₇	#2108	1	2	yes	—
113	C₂×C₁₇	#2151	1	2	yes	—
114	C₂×C₁₇	#2164	1	2	yes	—
115	C₂²×C₈	—	17	1	no	—
116	C₂²×C₈	—	17	1	no	—
117	C₂²×C₈	—	17	1	no	—
118	C₂²×C₈	—	17	1	no	—
119	C₂×C₄²	C₂×#595	1	1	yes	characteristic
120	C₂×C₄²	C₂×#595	1	1	yes	characteristic
121	C₂×C₄²	C₂×#595	1	1	yes	characteristic
122	C₂×C₁₆	—	17	4	no	—
123	C₂×C₁₆	—	17	4	no	—
124	C₂×C₁₆	—	34	2	no	—
125	C₂×C₁₆	—	34	2	no	—
126	C₂×C₁₆	—	34	2	no	—
127	C₂×C₁₆	—	34	2	no	—
128	#97	—	17	4	no	—
129	#97	—	34	2	no	—
130	#97	—	34	2	no	—
131	C₄×C₈	—	17	4	no	—
132	C₄×C₈	—	17	2	no	—
133	C₄×C₈	—	17	2	no	—
134	C₄×C₈	—	34	2	no	—
135	#95	#3624	1	4	yes	—
136	C₁₇	#12654	1	1	yes	characteristic, complemented, Sylow
137	C₂²×C₄	C₂×#3624	1	2	yes	—
138	C₂²×C₄	C₂×#3624	1	2	yes	—
139	C₂²×C₄	C₂×#3624	1	2	yes	—
140	C₂²×C₄	C₂²×#595	1	1	yes	characteristic
141	C₂×C₈	—	17	2	no	—
142	C₂×C₈	—	17	2	no	—
143	C₂×C₈	—	17	2	no	—
144	C₂×C₈	—	17	1	no	—
145	C₂×C₈	—	17	2	no	—
146	C₂×C₈	—	17	1	no	—
147	C₂×C₈	—	17	2	no	—
148	C₂×C₈	—	17	2	no	—
149	C₂×C₈	—	17	2	no	—
150	C₂×C₈	—	34	1	no	—
151	C₂×C₈	—	34	1	no	—
152	C₂×C₈	—	34	1	no	—
153	C₂×C₈	—	34	1	no	—
154	C₄²	—	2	1	no	—
155	C₄²	—	2	1	no	—
156	C₄²	#63242	1	2	yes	—
157	C₄²	#63243	1	2	yes	—
158	C₄²	C₄×#595	1	2	yes	—
159	C₄²	C₄⋊#595	1	2	yes	—
160	C₁₆	—	34	4	no	—
161	C₁₆	—	34	4	no	—
162	C₂³	C₂²×#3624	1	1	yes	characteristic
163	C₂×C₄	—	2	4	no	—
164	C₂×C₄	—	2	4	no	—
165	C₂×C₄	—	2	2	no	—
166	C₂×C₄	—	2	4	no	—
167	C₂×C₄	C₄×#3624	1	4	yes	—
168	C₂×C₄	C₄⋊#3624	1	4	yes	—
169	C₂×C₄	C₂×#63242	1	1	yes	characteristic
170	C₂×C₄	C₂×#63243	1	1	yes	characteristic
171	C₂×C₄	D₁₇⋊M₁₆	1	1	yes	characteristic
172	C₂×C₄	C₂×C₄×#595	1	1	yes	characteristic
173	C₂×C₄	C₂×C₄⋊#595	1	1	yes	characteristic
174	C₂×C₄	#10587677	1	1	yes	characteristic
175	C₈	—	17	2	no	—
176	C₈	—	17	2	no	—
177	C₈	—	17	4	no	—
178	C₈	—	34	2	no	—
179	C₈	—	34	2	no	—
180	C₂²	C₂×C₄×#3624	1	1	yes	characteristic
181	C₂²	C₂×C₄⋊#3624	1	1	yes	characteristic
182	C₂²	#11763825	1	1	yes	characteristic
183	C₂²	#11763867	1	4	yes	—
184	C₄	—	2	4	no	—
185	C₄	—	2	4	no	—
186	C₄	—	4	2	no	—
187	C₄	C₄×C₄⋊#595	1	1	yes	characteristic
188	C₄	C₄×#63243	1	1	yes	characteristic
189	C₄	#11763810	1	1	yes	characteristic
190	C₄	#11763828	1	1	yes	characteristic
191	C₂	C₄×C₄⋊#3624	1	1	yes	characteristic
192	C₂	#14544993	1	2	yes	—
193	C₂	#14545119	1	2	yes	—
194	C₂	#14545149	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×#595
commutator	C₂²×C₁₇	C₂×C₄×C₈
Fitting	C₁₇×#376	C₄
socle, prime radical	C₂³×C₁₇	C₂²×C₈
cosocle	C₂²×#3622	C₂³
hypercenter	#376	#595
hypocenter	C₁₇	#12654
LCS	G ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂²×C₄ ◁ #376

Complements

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

N	H ≅ G/N	type
8. C₄⋊#63216	C₄	semidirect
136. C₁₇	#12654	semidirect

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×#11763697	2
C₂×C₄×#63216	2
#14543822	2
#14528471	2
#12654	17