Subgroups — C₂²×#17989313 #33993028

Order 18688.

7414 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17989313	C₂	1	48	yes	maximal, direct product factor
2	C₂²×#14556590	C₂	1	3	yes	maximal
3	C₂²×#14556583	C₂	1	3	yes	maximal
4	C₂²×#14556582	C₂	1	3	yes	maximal
5	C₂²×Q₈×Dic₇₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×Q₈×Dic₇₃	C₂	1	1	yes	maximal, characteristic
7	C₂²×C₄×Dic₁₄₆	C₂	1	3	yes	maximal
8	C₂²×C₄×Q₈×C₇₃	C₂	1	1	yes	maximal, characteristic
9	C₂³×Dic₁₄₆	C₂²	1	3	yes	—
10	#17989313	C₂²	1	256	yes	direct product factor
11	C₂×#14556590	C₂²	1	72	yes	—
12	C₂×#14556583	C₂²	1	72	yes	—
13	C₂×#14556582	C₂²	1	72	yes	—
14	C₂×Q₈×Dic₇₃	C₂²	1	24	yes	—
15	C₂×Q₈×Dic₇₃	C₂²	1	24	yes	—
16	C₂×C₄×Dic₁₄₆	C₂²	1	72	yes	—
17	C₂²×C₄⋊Dic₇₃	C₂²	1	3	yes	—
18	C₂²×C₄⋊Dic₇₃	C₂²	1	3	yes	—
19	C₂²×C₄⋊Dic₇₃	C₂²	1	3	yes	—
20	C₂²×#11924007	C₂²	1	1	yes	characteristic
21	C₂²×#11924007	C₂²	1	3	yes	—
22	C₂²×#11924007	C₂²	1	6	yes	—
23	C₂²×C₄×Dic₇₃	C₂²	1	3	yes	—
24	C₂²×C₄×Dic₇₃	C₂²	1	3	yes	—
25	C₂³×Q₈×C₇₃	C₂²	1	1	yes	characteristic
26	C₂×C₄×Q₈×C₇₃	C₂²	1	24	yes	—
27	C₂²×C₄⋊C₄×C₇₃	C₂²	1	3	yes	—
28	C₂²×C₄²×C₇₃	C₂²	1	3	yes	—
29	C₂²×Dic₁₄₆	—	2	24	no	—
30	C₂²×Dic₁₄₆	C₂³	1	36	yes	—
31	C₂³×Dic₇₃	C₂³	1	3	yes	—
32	C₂³×Dic₇₃	C₂³	1	1	yes	characteristic
33	C₂³×Dic₇₃	C₂³	1	3	yes	—
34	C₂³×Dic₇₃	C₂³	1	1	yes	characteristic
35	#14556590	C₂³	1	192	yes	—
36	#14556583	C₂³	1	192	yes	—
37	#14556582	C₂³	1	192	yes	—
38	Q₈×Dic₇₃	C₂³	1	64	yes	—
39	Q₈×Dic₇₃	C₂³	1	64	yes	—
40	C₄×Dic₁₄₆	C₂³	1	192	yes	—
41	C₂×C₄⋊Dic₇₃	C₂³	1	36	yes	—
42	C₂×C₄⋊Dic₇₃	C₂³	1	36	yes	—
43	C₂×C₄⋊Dic₇₃	C₂³	1	36	yes	—
44	C₂×#11924007	C₂³	1	72	yes	—
45	C₂×#11924007	C₂³	1	12	yes	—
46	C₂×#11924007	C₂³	1	36	yes	—
47	C₂×C₄×Dic₇₃	C₂³	1	36	yes	—
48	C₂×C₄×Dic₇₃	C₂³	1	36	yes	—
49	C₂²×Q₈×C₇₃	Q₈	1	16	yes	complemented
50	C₂²×Q₈×C₇₃	C₂³	1	12	yes	—
51	C₂³×C₄×C₇₃	C₂³	1	3	yes	—
52	C₂³×C₄×C₇₃	C₂³	1	1	yes	characteristic
53	C₂³×C₄×C₇₃	C₂³	1	3	yes	—
54	C₄×Q₈×C₇₃	C₂³	1	64	yes	—
55	C₂×C₄⋊C₄×C₇₃	C₂³	1	36	yes	—
56	C₂×C₄²×C₇₃	C₂³	1	36	yes	—
57	C₂×Dic₁₄₆	—	2	144	no	—
58	C₂×Dic₁₄₆	C₂⁴	1	48	yes	—
59	C₂²×Dic₇₃	—	2	12	no	—
60	C₂²×Dic₇₃	—	2	4	no	—
61	C₂²×Dic₇₃	—	2	12	no	—
62	C₂²×Dic₇₃	#41	1	8	yes	—
63	C₂²×Dic₇₃	C₂⁴	1	6	yes	—
64	C₂²×Dic₇₃	C₂⁴	1	18	yes	—
65	C₂²×Dic₇₃	C₂⁴	1	18	yes	—
66	C₂²×Dic₇₃	C₂⁴	1	6	yes	—
67	C₄⋊Dic₇₃	C₂⁴	1	48	yes	—
68	C₄⋊Dic₇₃	C₂⁴	1	48	yes	—
69	C₄⋊Dic₇₃	C₂⁴	1	48	yes	—
70	#11924007	C₂⁴	1	16	yes	—
71	#11924007	C₂⁴	1	48	yes	—
72	#11924007	C₂⁴	1	96	yes	—
73	C₄×Dic₇₃	C₂⁴	1	48	yes	—
74	C₄×Dic₇₃	C₂⁴	1	48	yes	—
75	C₂⁴×C₇₃	C₂⁴	1	1	yes	characteristic
76	C₂×Q₈×C₇₃	C₂×Q₈	1	96	yes	complemented
77	C₂×Q₈×C₇₃	C₂⁴	1	16	yes	—
78	C₂²×C₄×C₇₃	—	2	12	no	—
79	C₂²×C₄×C₇₃	—	2	4	no	—
80	C₂²×C₄×C₇₃	C₂×Q₈	1	24	yes	—
81	C₂²×C₄×C₇₃	C₂⁴	1	6	yes	—
82	C₂²×C₄×C₇₃	C₂⁴	1	18	yes	—
83	C₂²×C₄×C₇₃	C₂⁴	1	18	yes	—
84	C₄⋊C₄×C₇₃	C₂⁴	1	48	yes	—
85	C₄²×C₇₃	C₂⁴	1	48	yes	—
86	C₂³×C₇₃	C₂²×Q₈	1	4	yes	—
87	C₂³×C₇₃	C₂×#41	1	4	yes	—
88	C₂³×C₇₃	2₋¹⁺⁴	1	4	yes	—
89	C₂³×C₇₃	C₂⁵	1	3	yes	—
90	Q₈×C₇₃	C₂²×Q₈	1	64	yes	complemented
91	C₂×C₄×C₇₃	—	2	36	no	—
92	C₂×C₄×C₇₃	—	2	12	no	—
93	C₂×C₄×C₇₃	C₂²×Q₈	1	72	yes	—
94	C₂×C₄×C₇₃	C₂⁵	1	4	yes	—
95	C₂×C₄×C₇₃	C₂⁵	1	12	yes	—
96	C₂×C₄×C₇₃	C₂⁵	1	12	yes	—
97	C₂×Dic₇₃	—	2	36	no	—
98	C₂×Dic₇₃	—	2	12	no	—
99	C₂×Dic₇₃	—	2	36	no	—
100	C₂×Dic₇₃	C₂×#41	1	24	yes	—
101	C₂×Dic₇₃	C₂⁵	1	4	yes	—
102	C₂×Dic₇₃	C₂⁵	1	12	yes	—
103	C₂×Dic₇₃	C₂⁵	1	12	yes	—
104	C₂×Dic₇₃	C₂⁵	1	4	yes	—
105	Dic₁₄₆	—	2	96	no	—
106	C₂²×C₇₃	#557	1	16	yes	complemented
107	C₂²×C₇₃	C₂³×Q₈	1	6	yes	—
108	C₂²×C₇₃	C₂²×#41	1	6	yes	—
109	C₂²×C₇₃	C₂×2₋¹⁺⁴	1	6	yes	—
110	C₂²×C₇₃	C₂⁶	1	1	yes	characteristic
111	C₄×C₇₃	—	2	12	no	—
112	C₄×C₇₃	—	2	4	no	—
113	C₄×C₇₃	C₂³×Q₈	1	24	yes	—
114	Dic₇₃	—	2	12	no	—
115	Dic₇₃	—	2	4	no	—
116	Dic₇₃	—	2	12	no	—
117	Dic₇₃	C₂²×#41	1	8	yes	—
118	C₂²×#557	—	73	1	no	maximal, Sylow
119	C₂×C₇₃	C₂×#557	1	12	yes	complemented
120	C₂×C₇₃	C₂⁴×Q₈	1	1	yes	characteristic
121	C₂×C₇₃	C₂³×#41	1	1	yes	characteristic
122	C₂×C₇₃	C₂²×2₋¹⁺⁴	1	1	yes	characteristic
123	C₂×#557	—	73	48	no	—
124	C₂²×C₄⋊Q₈	—	73	3	no	—
125	C₂²×#125	—	73	3	no	—
126	C₂²×#125	—	73	3	no	—
127	C₂²×C₄×Q₈	—	73	3	no	—
128	C₂²×C₄×Q₈	—	73	1	no	—
129	C₂²×C₄×Q₈	—	73	1	no	—
130	C₂²×C₄×Q₈	D₇₃	1	1	yes	characteristic
131	C₇₃	C₂²×#557	1	1	yes	characteristic, complemented, Sylow
132	C₂³×Q₈	—	73	3	no	—
133	C₂³×Q₈	C₂×D₇₃	1	1	yes	characteristic
134	#557	—	73	256	no	—
135	C₂×C₄⋊Q₈	—	73	72	no	—
136	C₂×#125	—	73	72	no	—
137	C₂×#125	—	73	72	no	—
138	C₂×C₄×Q₈	—	73	24	no	—
139	C₂×C₄×Q₈	—	73	24	no	—
140	C₂×C₄×Q₈	—	73	72	no	—
141	C₂×C₄×Q₈	C₂×D₇₃	1	24	yes	—
142	C₂²×C₄⋊C₄	—	73	1	no	—
143	C₂²×C₄⋊C₄	—	73	3	no	—
144	C₂²×C₄⋊C₄	—	73	3	no	—
145	C₂²×C₄⋊C₄	—	73	3	no	—
146	C₂²×C₄⋊C₄	—	73	3	no	—
147	C₂²×C₄⋊C₄	—	73	6	no	—
148	C₂²×C₄⋊C₄	C₂×D₇₃	1	3	yes	—
149	C₂²×C₄²	—	73	3	no	—
150	C₂²×C₄²	—	73	3	no	—
151	C₂²×C₄²	C₂×D₇₃	1	3	yes	—
152	C₂²×Q₈	—	73	36	no	—
153	C₂²×Q₈	—	146	24	no	—
154	C₂²×Q₈	Dic₁₄₆	1	16	yes	complemented
155	C₂²×Q₈	C₂²×D₇₃	1	12	yes	—
156	C₂³×C₄	—	73	1	no	—
157	C₂³×C₄	—	73	3	no	—
158	C₂³×C₄	—	73	3	no	—
159	C₂³×C₄	—	73	1	no	—
160	C₂³×C₄	C₂²×D₇₃	1	1	yes	characteristic
161	C₂³×C₄	C₂²×D₇₃	1	3	yes	—
162	C₂³×C₄	C₂²×D₇₃	1	3	yes	—
163	C₄⋊Q₈	—	73	192	no	—
164	#125	—	73	192	no	—
165	#125	—	73	192	no	—
166	C₄×Q₈	—	73	64	no	—
167	C₄×Q₈	—	73	64	no	—
168	C₄×Q₈	—	73	192	no	—
169	C₄×Q₈	C₂²×D₇₃	1	64	yes	—
170	C₂×C₄⋊C₄	—	73	12	no	—
171	C₂×C₄⋊C₄	—	73	36	no	—
172	C₂×C₄⋊C₄	—	73	36	no	—
173	C₂×C₄⋊C₄	—	73	36	no	—
174	C₂×C₄⋊C₄	—	73	36	no	—
175	C₂×C₄⋊C₄	—	73	72	no	—
176	C₂×C₄⋊C₄	C₂²×D₇₃	1	36	yes	—
177	C₂×C₄²	—	73	36	no	—
178	C₂×C₄²	—	73	36	no	—
179	C₂×C₄²	C₂²×D₇₃	1	36	yes	—
180	C₂⁴	C₂³×D₇₃	1	1	yes	characteristic
181	C₂×Q₈	—	73	48	no	—
182	C₂×Q₈	—	146	144	no	—
183	C₂×Q₈	C₂×Dic₁₄₆	1	96	yes	complemented
184	C₂×Q₈	C₂³×D₇₃	1	16	yes	—
185	C₂²×C₄	—	2	12	no	—
186	C₂²×C₄	—	2	4	no	—
187	C₂²×C₄	—	73	8	no	—
188	C₂²×C₄	—	73	6	no	—
189	C₂²×C₄	—	73	6	no	—
190	C₂²×C₄	—	73	18	no	—
191	C₂²×C₄	—	73	18	no	—
192	C₂²×C₄	—	146	12	no	—
193	C₂²×C₄	—	146	12	no	—
194	C₂²×C₄	—	146	4	no	—
195	C₂²×C₄	C₂×Dic₁₄₆	1	24	yes	—
196	C₂²×C₄	C₂³×D₇₃	1	6	yes	—
197	C₂²×C₄	C₂³×D₇₃	1	18	yes	—
198	C₂²×C₄	C₂³×D₇₃	1	18	yes	—
199	C₄⋊C₄	—	73	16	no	—
200	C₄⋊C₄	—	73	48	no	—
201	C₄⋊C₄	—	73	96	no	—
202	C₄⋊C₄	—	73	48	no	—
203	C₄⋊C₄	—	73	48	no	—
204	C₄⋊C₄	—	73	48	no	—
205	C₄⋊C₄	C₂³×D₇₃	1	48	yes	—
206	C₄²	—	73	48	no	—
207	C₄²	—	73	48	no	—
208	C₄²	C₂³×D₇₃	1	48	yes	—
209	C₂³	C₂²×Dic₁₄₆	1	4	yes	—
210	C₂³	#41×D₇₃	1	4	yes	—
211	C₂³	#14556639	1	4	yes	—
212	C₂³	C₂⁴×D₇₃	1	3	yes	—
213	Q₈	—	146	96	no	—
214	Q₈	C₂²×Dic₁₄₆	1	64	yes	complemented
215	C₂×C₄	—	2	36	no	—
216	C₂×C₄	—	2	12	no	—
217	C₂×C₄	—	73	24	no	—
218	C₂×C₄	—	73	4	no	—
219	C₂×C₄	—	73	4	no	—
220	C₂×C₄	—	73	12	no	—
221	C₂×C₄	—	73	12	no	—
222	C₂×C₄	—	146	36	no	—
223	C₂×C₄	—	146	12	no	—
224	C₂×C₄	—	146	36	no	—
225	C₂×C₄	C₂²×Dic₁₄₆	1	72	yes	—
226	C₂×C₄	C₂⁴×D₇₃	1	4	yes	—
227	C₂×C₄	C₂⁴×D₇₃	1	12	yes	—
228	C₂×C₄	C₂⁴×D₇₃	1	12	yes	—
229	C₂²	#17989313	1	16	yes	direct product factor
230	C₂²	C₂³×Dic₁₄₆	1	6	yes	—
231	C₂²	C₂×#41×D₇₃	1	6	yes	—
232	C₂²	C₂×#14556639	1	6	yes	—
233	C₂²	C₂⁵×D₇₃	1	1	yes	characteristic
234	C₄	—	2	12	no	—
235	C₄	—	2	4	no	—
236	C₄	—	73	8	no	—
237	C₄	—	146	12	no	—
238	C₄	—	146	4	no	—
239	C₄	—	146	12	no	—
240	C₄	C₂³×Dic₁₄₆	1	24	yes	—
241	C₂	C₂×#17989313	1	12	yes	direct product factor
242	C₂	C₂⁴×Dic₁₄₆	1	1	yes	characteristic
243	C₂	C₂²×#41×D₇₃	1	1	yes	characteristic
244	C₂	C₂²×#14556639	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×#557
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×#17989313	C₂	direct (N × H)
10. #17989313	C₂²	direct (N × H)
49. C₂²×Q₈×C₇₃	Q₈	semidirect
76. C₂×Q₈×C₇₃	C₂×Q₈	semidirect
90. Q₈×C₇₃	C₂²×Q₈	semidirect
106. C₂²×C₇₃	#557	semidirect
119. C₂×C₇₃	C₂×#557	semidirect
131. C₇₃	C₂²×#557	semidirect
154. C₂²×Q₈	Dic₁₄₆	semidirect
183. C₂×Q₈	C₂×Dic₁₄₆	semidirect
214. Q₈	C₂²×Dic₁₄₆	semidirect
229. C₂²	#17989313	direct (N × H)
241. C₂	C₂×#17989313	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×#17989313	2
C₂²×#14556590	2
C₂²×#14556583	2
C₂²×#14556582	2
C₂²×Q₈×Dic₇₃	2
C₂²×C₄×Dic₁₄₆	2
C₂²×C₄×Q₈×C₇₃	2
C₂²×#557	73

Subgroups — C22×#17989313 #33993028

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#17989313 #33993028