Subgroups — C₂²×#11762638 #17187375

Order 4352.

662 conjugacy classes of subgroups.

The subgroup lattice has 186 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#63214	C₂	1	1	yes	maximal, characteristic
2	C₂²×#10587490	C₂	1	1	yes	maximal, characteristic
3	C₂×#11762638	C₂	1	12	yes	maximal, direct product factor
4	C₂²×C₁₇×#108	C₂	1	1	yes	maximal, characteristic
5	C₂²×#63223	—	2	1	no	—
6	C₂²×#63214	C₄	1	6	yes	—
7	C₂²×#63214	C₄	1	2	yes	—
8	C₂²×#63214	C₂²	1	6	yes	—
9	C₂×#10587490	—	2	3	no	—
10	C₂×#10587490	C₂²	1	6	yes	—
11	#11762638	C₂²	1	16	yes	direct product factor
12	C₂²×M₁₆×C₁₇	—	2	1	no	—
13	C₂³×C₈×C₁₇	C₂²	1	1	yes	characteristic
14	C₂×C₁₇×#108	—	2	3	no	—
15	C₂×C₁₇×#108	C₂²	1	6	yes	—
16	C₂³×C₄×C₁₇	D₄	1	1	yes	characteristic
17	C₂×M₁₆×C₁₇	—	2	6	no	—
18	C₂×M₁₆×C₁₇	—	4	3	no	—
19	C₂²×C₈×C₁₇	—	4	1	no	—
20	C₂²×C₈×C₁₇	C₂×C₄	1	3	yes	—
21	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₈×C₁₇	D₄	1	3	yes	—
23	C₂²×C₈×C₁₇	Q₈	1	3	yes	—
24	C₂²×C₈×C₁₇	Q₈	1	1	yes	characteristic
25	C₂²×C₈×C₁₇	C₂³	1	3	yes	—
26	C₂×#63223	—	2	6	no	—
27	C₂×#63223	—	4	3	no	—
28	C₂²×#3622	—	4	1	no	—
29	C₁₇×#108	—	2	6	no	—
30	C₁₇×#108	C₂³	1	4	yes	—
31	#10587490	—	2	6	no	—
32	#10587490	C₂³	1	4	yes	—
33	C₂×#63214	C₂×C₄	1	12	yes	—
34	C₂×#63214	C₂×C₄	1	12	yes	—
35	C₂×#63214	C₂³	1	4	yes	—
36	C₂⁴×C₁₇	Q₁₆	1	1	yes	characteristic
37	C₂²×C₄×C₁₇	C₄⋊C₄	1	3	yes	—
38	C₂²×C₄×C₁₇	C₄⋊C₄	1	1	yes	characteristic
39	C₂²×C₄×C₁₇	D₈	1	3	yes	—
40	C₂²×C₄×C₁₇	D₈	1	1	yes	characteristic
41	C₂²×C₄×C₁₇	Q₁₆	1	3	yes	—
42	C₂²×C₄×C₁₇	C₂×D₄	1	3	yes	—
43	M₁₆×C₁₇	—	2	4	no	—
44	M₁₆×C₁₇	—	4	6	no	—
45	C₂×C₈×C₁₇	—	4	6	no	—
46	C₂×C₈×C₁₇	C₄⋊C₄	1	6	yes	—
47	C₂×C₈×C₁₇	C₄⋊C₄	1	3	yes	—
48	C₂×C₈×C₁₇	C₄⋊C₄	1	3	yes	—
49	C₂×C₈×C₁₇	C₂²×C₄	1	3	yes	—
50	C₂×C₈×C₁₇	C₂²×C₄	1	3	yes	—
51	C₂×C₈×C₁₇	C₂×D₄	1	3	yes	—
52	C₂×C₈×C₁₇	C₂×Q₈	1	3	yes	—
53	C₂×C₈×C₁₇	C₂×Q₈	1	3	yes	—
54	C₂×C₈×C₁₇	C₂⁴	1	1	yes	characteristic
55	#63223	—	2	4	no	—
56	#63223	—	4	6	no	—
57	C₂×#3622	—	4	6	no	—
58	#63214	C₂²×C₄	1	8	yes	—
59	C₂²×#368	—	17	1	no	maximal, Sylow
60	C₂³×C₁₇	—	2	4	no	—
61	C₂³×C₁₇	#107	1	3	yes	—
62	C₂³×C₁₇	#107	1	1	yes	characteristic
63	C₂³×C₁₇	C₂×Q₁₆	1	3	yes	—
64	C₂×C₄×C₁₇	#107	1	6	yes	—
65	C₂×C₄×C₁₇	#107	1	3	yes	—
66	C₂×C₄×C₁₇	#107	1	3	yes	—
67	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	3	yes	—
68	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	3	yes	—
69	C₂×C₄×C₁₇	C₂×D₈	1	3	yes	—
70	C₂×C₄×C₁₇	C₂×D₈	1	3	yes	—
71	C₂×C₄×C₁₇	C₂×Q₁₆	1	3	yes	—
72	C₂×C₄×C₁₇	C₂²×D₄	1	1	yes	characteristic
73	C₈×C₁₇	—	4	4	no	—
74	C₈×C₁₇	C₂×C₄⋊C₄	1	3	yes	—
75	C₈×C₁₇	C₂×C₄⋊C₄	1	3	yes	—
76	C₈×C₁₇	C₂³×C₄	1	1	yes	characteristic
77	C₈×C₁₇	C₂²×Q₈	1	1	yes	characteristic
78	#3622	—	4	4	no	—
79	C₂³×C₁₆	—	17	1	no	—
80	C₂²×#108	—	17	1	no	—
81	C₂²×#108	D₁₇	1	1	yes	characteristic
82	C₂×#368	—	17	12	no	—
83	C₂²×C₁₇	—	2	6	no	—
84	C₂²×C₁₇	—	2	6	no	—
85	C₂²×C₁₇	#368	1	4	yes	complemented
86	C₂²×C₁₇	C₂×#107	1	3	yes	—
87	C₂²×C₁₇	C₂×#107	1	3	yes	—
88	C₂²×C₁₇	C₂²×Q₁₆	1	1	yes	characteristic
89	C₄×C₁₇	C₂×#107	1	3	yes	—
90	C₄×C₁₇	C₂×#107	1	3	yes	—
91	C₄×C₁₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
92	C₄×C₁₇	C₂²×D₈	1	1	yes	characteristic
93	C₂²×M₁₆	—	2	1	no	—
94	C₂²×M₁₆	—	34	1	no	—
95	C₂³×C₈	C₂×D₁₇	1	1	yes	characteristic
96	C₂²×C₁₆	—	17	2	no	—
97	C₂²×C₁₆	—	17	6	no	—
98	C₂²×C₁₆	—	17	6	no	—
99	C₂×#108	—	2	3	no	—
100	C₂×#108	—	17	6	no	—
101	C₂×#108	—	34	3	no	—
102	C₂×#108	C₂×D₁₇	1	6	yes	—
103	#368	—	17	16	no	—
104	C₂×C₁₇	—	2	1	no	—
105	C₂×C₁₇	—	2	3	no	—
106	C₂×C₁₇	C₂×#368	1	6	yes	complemented
107	C₂×C₁₇	C₂²×#107	1	1	yes	characteristic
108	C₂³×C₄	C₂²⋊D₁₇	1	1	yes	characteristic
109	C₂×M₁₆	—	2	6	no	—
110	C₂×M₁₆	—	4	3	no	—
111	C₂×M₁₆	—	34	6	no	—
112	C₂×M₁₆	—	68	3	no	—
113	C₂²×C₈	—	4	1	no	—
114	C₂²×C₈	—	68	1	no	—
115	C₂²×C₈	Dic₃₄	1	1	yes	characteristic
116	C₂²×C₈	Dic₃₄	1	3	yes	—
117	C₂²×C₈	C₄×D₁₇	1	1	yes	characteristic
118	C₂²×C₈	C₄×D₁₇	1	3	yes	—
119	C₂²×C₈	C₂²⋊D₁₇	1	3	yes	—
120	C₂²×C₈	C₂²×D₁₇	1	3	yes	—
121	C₂×C₁₆	—	17	12	no	—
122	C₂×C₁₆	—	17	12	no	—
123	C₂×C₁₆	—	17	4	no	—
124	#108	—	2	6	no	—
125	#108	—	17	4	no	—
126	#108	—	34	6	no	—
127	#108	C₂²×D₁₇	1	4	yes	—
128	C₁₇	C₂²×#368	1	1	yes	characteristic, complemented, Sylow
129	C₂⁴	#63231	1	1	yes	characteristic
130	C₂²×C₄	#63225	1	1	yes	characteristic
131	C₂²×C₄	#63225	1	3	yes	—
132	C₂²×C₄	#63228	1	3	yes	—
133	C₂²×C₄	#63228	1	1	yes	characteristic
134	C₂²×C₄	#63231	1	3	yes	—
135	C₂²×C₄	C₂×C₂²⋊D₁₇	1	3	yes	—
136	M₁₆	—	2	4	no	—
137	M₁₆	—	4	6	no	—
138	M₁₆	—	34	4	no	—
139	M₁₆	—	68	6	no	—
140	C₂×C₈	—	4	6	no	—
141	C₂×C₈	—	68	6	no	—
142	C₂×C₈	#63225	1	6	yes	—
143	C₂×C₈	#63225	1	3	yes	—
144	C₂×C₈	#63225	1	3	yes	—
145	C₂×C₈	C₂×Dic₃₄	1	3	yes	—
146	C₂×C₈	C₂×Dic₃₄	1	3	yes	—
147	C₂×C₈	C₂×C₄×D₁₇	1	3	yes	—
148	C₂×C₈	C₂×C₄×D₁₇	1	3	yes	—
149	C₂×C₈	C₂×C₂²⋊D₁₇	1	3	yes	—
150	C₂×C₈	C₂³×D₁₇	1	1	yes	characteristic
151	C₁₆	—	17	8	no	—
152	C₂³	—	2	4	no	—
153	C₂³	#10587478	1	3	yes	—
154	C₂³	#10587478	1	1	yes	characteristic
155	C₂³	C₂×#63231	1	3	yes	—
156	C₂×C₄	#10587478	1	6	yes	—
157	C₂×C₄	#10587478	1	3	yes	—
158	C₂×C₄	#10587478	1	3	yes	—
159	C₂×C₄	C₂×#63225	1	3	yes	—
160	C₂×C₄	C₂×#63225	1	3	yes	—
161	C₂×C₄	C₂×#63228	1	3	yes	—
162	C₂×C₄	C₂×#63228	1	3	yes	—
163	C₂×C₄	C₂×#63231	1	3	yes	—
164	C₂×C₄	C₂²×C₂²⋊D₁₇	1	1	yes	characteristic
165	C₈	—	4	4	no	—
166	C₈	—	68	4	no	—
167	C₈	C₂×#63225	1	3	yes	—
168	C₈	C₂×#63225	1	3	yes	—
169	C₈	C₂²×Dic₃₄	1	1	yes	characteristic
170	C₈	C₂²×C₄×D₁₇	1	1	yes	characteristic
171	C₂²	—	2	6	no	—
172	C₂²	—	2	6	no	—
173	C₂²	#11762638	1	4	yes	direct product factor
174	C₂²	C₂×#10587478	1	3	yes	—
175	C₂²	C₂×#10587478	1	3	yes	—
176	C₂²	C₂²×#63231	1	1	yes	characteristic
177	C₄	C₂×#10587478	1	3	yes	—
178	C₄	C₂×#10587478	1	3	yes	—
179	C₄	C₂²×#63225	1	1	yes	characteristic
180	C₄	C₂²×#63228	1	1	yes	characteristic
181	C₂	—	2	1	no	—
182	C₂	—	2	3	no	—
183	C₂	C₂×#11762638	1	6	yes	direct product factor
184	C₂	C₂²×#10587478	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	#63228
commutator	C₈×C₁₇	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₁₇
Fitting	C₂²×C₁₇×#108	C₂
socle	C₂³×C₁₇	#107
prime radical	C₂⁴×C₁₇	Q₁₆
cosocle	C₂×C₈×C₁₇	C₂⁴
hypercenter	C₂²×#108	D₁₇
hypocenter	C₁₇	C₂²×#368
LCS	G ▷ C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂²×#108

Complements

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

N	H ≅ G/N	type
3. C₂×#11762638	C₂	direct (N × H)
11. #11762638	C₂²	direct (N × H)
85. C₂²×C₁₇	#368	semidirect
106. C₂×C₁₇	C₂×#368	semidirect
128. C₁₇	C₂²×#368	semidirect
173. C₂²	#11762638	direct (N × H)
183. C₂	C₂×#11762638	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×#63214	2
C₂²×#10587490	2
C₂×#11762638	2
C₂²×C₁₇×#108	2
C₂²×#368	17

Subgroups — C22×#11762638 #17187375

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#11762638 #17187375