Subgroups — C₂²×#17989317 #33993050

Order 18688.

7846 conjugacy classes of subgroups.

The subgroup lattice has 198 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17989317	C₂	1	48	yes	maximal, direct product factor
2	C₂²×#14556590	C₂	1	4	yes	maximal
3	C₂²×#14556589	C₂	1	4	yes	maximal
4	C₂²×C₄×Dic₁₄₆	C₂	1	2	yes	maximal
5	C₂²×#14556556	C₂	1	4	yes	maximal
6	C₂²×C₄⋊Q₈×C₇₃	C₂	1	1	yes	maximal, characteristic
7	C₂³×Dic₁₄₆	C₂²	1	4	yes	—
8	C₂³×Dic₁₄₆	C₂²	1	2	yes	—
9	#17989317	C₂²	1	256	yes	direct product factor
10	C₂×#14556590	C₂²	1	96	yes	—
11	C₂×#14556589	C₂²	1	96	yes	—
12	C₂×C₄×Dic₁₄₆	C₂²	1	48	yes	—
13	C₂×#14556556	C₂²	1	96	yes	—
14	C₂²×C₄⋊Dic₇₃	C₂²	1	2	yes	—
15	C₂²×#11924007	C₂²	1	8	yes	—
16	C₂²×#11924007	C₂²	1	4	yes	—
17	C₂²×C₄×Dic₇₃	C₂²	1	4	yes	—
18	C₂²×C₄×Dic₇₃	C₂²	1	4	yes	—
19	C₂³×Q₈×C₇₃	C₂²	1	2	yes	—
20	C₂×C₄⋊Q₈×C₇₃	C₂²	1	24	yes	—
21	C₂²×C₄⋊C₄×C₇₃	C₂²	1	4	yes	—
22	C₂²×C₄²×C₇₃	C₂²	1	1	yes	characteristic
23	C₂²×Dic₁₄₆	—	4	16	no	—
24	C₂²×Dic₁₄₆	Q₈	1	32	yes	complemented
25	C₂²×Dic₁₄₆	C₂³	1	48	yes	—
26	C₂²×Dic₁₄₆	C₂³	1	24	yes	—
27	C₂³×Dic₇₃	C₂³	1	4	yes	—
28	C₂³×Dic₇₃	C₂³	1	4	yes	—
29	#14556590	C₂³	1	256	yes	—
30	#14556589	C₂³	1	256	yes	—
31	C₄×Dic₁₄₆	C₂³	1	128	yes	—
32	#14556556	C₂³	1	256	yes	—
33	C₂×C₄⋊Dic₇₃	C₂³	1	24	yes	—
34	C₂×#11924007	C₂³	1	48	yes	—
35	C₂×#11924007	C₂³	1	96	yes	—
36	C₂×C₄×Dic₇₃	C₂³	1	48	yes	—
37	C₂×C₄×Dic₇₃	C₂³	1	48	yes	—
38	C₂²×Q₈×C₇₃	—	4	8	no	—
39	C₂²×Q₈×C₇₃	C₂³	1	24	yes	—
40	C₂³×C₄×C₇₃	C₂³	1	4	yes	—
41	C₂³×C₄×C₇₃	C₂³	1	1	yes	characteristic
42	C₂³×C₄×C₇₃	C₂³	1	2	yes	—
43	C₄⋊Q₈×C₇₃	C₂³	1	64	yes	—
44	C₂×C₄⋊C₄×C₇₃	C₂³	1	48	yes	—
45	C₂×C₄²×C₇₃	C₂³	1	12	yes	—
46	C₂×Dic₁₄₆	—	4	96	no	—
47	C₂×Dic₁₄₆	C₂×Q₈	1	192	yes	complemented
48	C₂×Dic₁₄₆	C₂⁴	1	64	yes	—
49	C₂×Dic₁₄₆	C₂⁴	1	32	yes	—
50	C₂²×Dic₇₃	—	2	16	no	—
51	C₂²×Dic₇₃	C₂×Q₈	1	32	yes	—
52	C₂²×Dic₇₃	C₂⁴	1	24	yes	—
53	C₂²×Dic₇₃	C₂⁴	1	24	yes	—
54	C₄⋊Dic₇₃	C₂⁴	1	32	yes	—
55	#11924007	C₂⁴	1	128	yes	—
56	#11924007	C₂⁴	1	64	yes	—
57	C₄×Dic₇₃	C₂⁴	1	64	yes	—
58	C₄×Dic₇₃	C₂⁴	1	64	yes	—
59	C₂⁴×C₇₃	C₂⁴	1	1	yes	characteristic
60	C₂×Q₈×C₇₃	—	4	48	no	—
61	C₂×Q₈×C₇₃	C₂⁴	1	32	yes	—
62	C₂²×C₄×C₇₃	—	2	4	no	—
63	C₂²×C₄×C₇₃	—	2	16	no	—
64	C₂²×C₄×C₇₃	C₂×Q₈	1	16	yes	—
65	C₂²×C₄×C₇₃	C₂⁴	1	6	yes	—
66	C₂²×C₄×C₇₃	C₂⁴	1	24	yes	—
67	C₂²×C₄×C₇₃	C₂⁴	1	12	yes	—
68	C₄⋊C₄×C₇₃	C₂⁴	1	64	yes	—
69	C₄²×C₇₃	C₂⁴	1	16	yes	—
70	C₂³×C₇₃	C₂²×Q₈	1	8	yes	—
71	C₂³×C₇₃	H(5,𝔽₂)	1	4	yes	—
72	C₂³×C₇₃	C₂⁵	1	3	yes	—
73	Q₈×C₇₃	—	4	32	no	—
74	C₂×C₄×C₇₃	—	2	12	no	—
75	C₂×C₄×C₇₃	—	2	48	no	—
76	C₂×C₄×C₇₃	C₂²×Q₈	1	48	yes	—
77	C₂×C₄×C₇₃	C₂⁵	1	4	yes	—
78	C₂×C₄×C₇₃	C₂⁵	1	16	yes	—
79	C₂×C₄×C₇₃	C₂⁵	1	8	yes	—
80	C₂×Dic₇₃	—	2	48	no	—
81	C₂×Dic₇₃	C₂²×Q₈	1	96	yes	—
82	C₂×Dic₇₃	C₂⁵	1	16	yes	—
83	C₂×Dic₇₃	C₂⁵	1	16	yes	—
84	Dic₁₄₆	—	4	64	no	—
85	Dic₁₄₆	C₂²×Q₈	1	128	yes	complemented
86	C₂²×C₇₃	Q₈²	1	16	yes	complemented
87	C₂²×C₇₃	C₂³×Q₈	1	12	yes	—
88	C₂²×C₇₃	C₂×H(5,𝔽₂)	1	6	yes	—
89	C₂²×C₇₃	C₂⁶	1	1	yes	characteristic
90	C₄×C₇₃	—	2	16	no	—
91	C₄×C₇₃	—	2	4	no	—
92	C₄×C₇₃	C₂³×Q₈	1	16	yes	—
93	Dic₇₃	—	2	16	no	—
94	Dic₇₃	C₂³×Q₈	1	32	yes	—
95	C₂²×Q₈²	—	73	1	no	maximal, Sylow
96	C₂×C₇₃	C₂×Q₈²	1	12	yes	complemented
97	C₂×C₇₃	C₂⁴×Q₈	1	2	yes	—
98	C₂×C₇₃	C₂²×H(5,𝔽₂)	1	1	yes	characteristic
99	C₂×Q₈²	—	73	48	no	—
100	C₂²×C₄⋊Q₈	—	73	4	no	—
101	C₂²×C₄⋊Q₈	—	73	4	no	—
102	C₂²×C₄⋊Q₈	D₇₃	1	1	yes	characteristic
103	C₂²×C₄×Q₈	—	73	2	no	—
104	C₂²×C₄×Q₈	—	73	4	no	—
105	C₇₃	C₂²×Q₈²	1	1	yes	characteristic, complemented, Sylow
106	C₂³×Q₈	—	73	4	no	—
107	C₂³×Q₈	—	73	2	no	—
108	C₂³×Q₈	C₂×D₇₃	1	2	yes	—
109	Q₈²	—	73	256	no	—
110	C₂×C₄⋊Q₈	—	73	96	no	—
111	C₂×C₄⋊Q₈	—	73	96	no	—
112	C₂×C₄⋊Q₈	C₂×D₇₃	1	24	yes	—
113	C₂×C₄×Q₈	—	73	96	no	—
114	C₂×C₄×Q₈	—	73	48	no	—
115	C₂²×C₄⋊C₄	—	73	4	no	—
116	C₂²×C₄⋊C₄	—	73	8	no	—
117	C₂²×C₄⋊C₄	—	73	2	no	—
118	C₂²×C₄⋊C₄	C₂×D₇₃	1	4	yes	—
119	C₂²×C₄²	—	73	4	no	—
120	C₂²×C₄²	—	73	4	no	—
121	C₂²×C₄²	C₂×D₇₃	1	1	yes	characteristic
122	C₂²×Q₈	—	4	8	no	—
123	C₂²×Q₈	—	292	16	no	—
124	C₂²×Q₈	—	73	32	no	—
125	C₂²×Q₈	—	73	48	no	—
126	C₂²×Q₈	—	73	24	no	—
127	C₂²×Q₈	C₂²×D₇₃	1	24	yes	—
128	C₂³×C₄	—	73	4	no	—
129	C₂³×C₄	—	73	4	no	—
130	C₂³×C₄	C₂²×D₇₃	1	1	yes	characteristic
131	C₂³×C₄	C₂²×D₇₃	1	4	yes	—
132	C₂³×C₄	C₂²×D₇₃	1	2	yes	—
133	C₄⋊Q₈	—	73	256	no	—
134	C₄⋊Q₈	—	73	256	no	—
135	C₄⋊Q₈	C₂²×D₇₃	1	64	yes	—
136	C₄×Q₈	—	73	256	no	—
137	C₄×Q₈	—	73	128	no	—
138	C₂×C₄⋊C₄	—	73	96	no	—
139	C₂×C₄⋊C₄	—	73	48	no	—
140	C₂×C₄⋊C₄	—	73	24	no	—
141	C₂×C₄⋊C₄	C₂²×D₇₃	1	48	yes	—
142	C₂×C₄²	—	73	48	no	—
143	C₂×C₄²	—	73	48	no	—
144	C₂×C₄²	C₂²×D₇₃	1	12	yes	—
145	C₂⁴	C₂³×D₇₃	1	1	yes	characteristic
146	C₂×Q₈	—	4	48	no	—
147	C₂×Q₈	—	292	96	no	—
148	C₂×Q₈	—	73	192	no	—
149	C₂×Q₈	—	73	64	no	—
150	C₂×Q₈	—	73	32	no	—
151	C₂×Q₈	C₂³×D₇₃	1	32	yes	—
152	C₂²×C₄	—	2	4	no	—
153	C₂²×C₄	—	2	16	no	—
154	C₂²×C₄	—	73	32	no	—
155	C₂²×C₄	—	73	24	no	—
156	C₂²×C₄	—	73	24	no	—
157	C₂²×C₄	—	146	16	no	—
158	C₂²×C₄	Q₈×D₇₃	1	16	yes	—
159	C₂²×C₄	C₂³×D₇₃	1	24	yes	—
160	C₂²×C₄	C₂³×D₇₃	1	6	yes	—
161	C₂²×C₄	C₂³×D₇₃	1	12	yes	—
162	C₄⋊C₄	—	73	128	no	—
163	C₄⋊C₄	—	73	64	no	—
164	C₄⋊C₄	—	73	32	no	—
165	C₄⋊C₄	C₂³×D₇₃	1	64	yes	—
166	C₄²	—	73	64	no	—
167	C₄²	—	73	64	no	—
168	C₄²	C₂³×D₇₃	1	16	yes	—
169	C₂³	C₂×Q₈×D₇₃	1	8	yes	—
170	C₂³	#14556636	1	4	yes	—
171	C₂³	C₂⁴×D₇₃	1	3	yes	—
172	Q₈	—	4	32	no	—
173	Q₈	—	292	64	no	—
174	Q₈	—	73	128	no	—
175	C₂×C₄	—	2	12	no	—
176	C₂×C₄	—	2	48	no	—
177	C₂×C₄	—	73	96	no	—
178	C₂×C₄	—	73	16	no	—
179	C₂×C₄	—	73	16	no	—
180	C₂×C₄	—	146	48	no	—
181	C₂×C₄	C₂×Q₈×D₇₃	1	48	yes	—
182	C₂×C₄	C₂⁴×D₇₃	1	16	yes	—
183	C₂×C₄	C₂⁴×D₇₃	1	4	yes	—
184	C₂×C₄	C₂⁴×D₇₃	1	8	yes	—
185	C₂²	#17989317	1	16	yes	direct product factor
186	C₂²	C₂²×Q₈×D₇₃	1	12	yes	—
187	C₂²	C₂×#14556636	1	6	yes	—
188	C₂²	C₂⁵×D₇₃	1	1	yes	characteristic
189	C₄	—	2	4	no	—
190	C₄	—	2	16	no	—
191	C₄	—	73	32	no	—
192	C₄	—	146	16	no	—
193	C₄	C₂²×Q₈×D₇₃	1	16	yes	—
194	C₂	C₂×#17989317	1	12	yes	direct product factor
195	C₂	C₂³×Q₈×D₇₃	1	2	yes	—
196	C₂	C₂²×#14556636	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂³×D₇₃
commutator, cosocle	C₂²×C₇₃	C₂⁶
Frattini	C₂²	C₂⁵×D₇₃
Fitting	C₂²×C₄⋊Q₈×C₇₃	C₂
socle, prime radical	C₂⁴×C₇₃	C₂⁴
hypercenter	C₂²×C₄⋊Q₈	D₇₃
hypocenter	C₇₃	C₂²×Q₈²
LCS	G ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
1. C₂×#17989317	C₂	direct (N × H)
9. #17989317	C₂²	direct (N × H)
24. C₂²×Dic₁₄₆	Q₈	semidirect
47. C₂×Dic₁₄₆	C₂×Q₈	semidirect
85. Dic₁₄₆	C₂²×Q₈	semidirect
86. C₂²×C₇₃	Q₈²	semidirect
96. C₂×C₇₃	C₂×Q₈²	semidirect
105. C₇₃	C₂²×Q₈²	semidirect
185. C₂²	#17989317	direct (N × H)
194. C₂	C₂×#17989317	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×#17989317	2
C₂²×#14556590	2
C₂²×#14556589	2
C₂²×C₄×Dic₁₄₆	2
C₂²×#14556556	2
C₂²×C₄⋊Q₈×C₇₃	2
C₂²×Q₈²	73

Subgroups — C22×#17989317 #33993050

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#17989317 #33993050