Subgroups — #34034626

Order 18688.

876 conjugacy classes of subgroups.

The subgroup lattice has 266 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₈×#10596842	C₂	1	2	yes	maximal
2	#22627080	C₂	1	6	yes	maximal
3	#22627073	C₂	1	3	yes	maximal
4	C₄×C₄⋊#10596842	C₂	1	3	yes	maximal
5	C₄×Q₈×#64408	C₂	1	1	yes	maximal, characteristic
6	Q₈×#10596842	—	2	8	no	—
7	C₂×C₄⋊#10596842	C₂²	1	6	yes	—
8	C₂×C₄⋊#10596842	C₂²	1	6	yes	—
9	C₂×C₄×#10596842	C₂²	1	6	yes	—
10	(C₂×#10596842)⋊C₄	C₂²	1	3	yes	—
11	(C₂×#10596842)⋊C₄	C₂²	1	6	yes	—
12	(C₂×#10596842)⋊C₄	C₂²	1	1	yes	characteristic
13	C₂×Q₈×#64408	C₄	1	1	yes	characteristic
14	C₂×Q₈×#64408	C₂²	1	1	yes	characteristic
15	C₄⋊C₄×#64408	C₄	1	3	yes	—
16	C₄⋊C₄×#64408	C₂²	1	3	yes	—
17	C₄×C₄⋊#64408	C₄	1	3	yes	—
18	C₄²×#64408	C₂²	1	3	yes	—
19	C₄×Q₈×D₇₃	C₄	1	1	yes	characteristic
20	C₂²×#10596842	C₂³	1	6	yes	—
21	C₂²×#10596842	C₂³	1	2	yes	—
22	C₄⋊#10596842	—	2	12	no	—
23	C₄⋊#10596842	—	2	12	no	—
24	C₄×#10596842	—	2	12	no	—
25	Q₈×#64408	—	2	4	no	—
26	Q₈×#64408	D₄	1	4	yes	—
27	Q₈×#64408	Q₈	1	4	yes	—
28	C₂×C₄⋊#64408	C₂×C₄	1	3	yes	—
29	C₂×C₄⋊#64408	C₂×C₄	1	3	yes	—
30	C₂×C₄×#64408	C₂×C₄	1	3	yes	—
31	C₂×C₄×#64408	C₂×C₄	1	3	yes	—
32	C₂×C₄×#64408	C₂×C₄	1	3	yes	—
33	C₂×C₄×#64408	C₂³	1	1	yes	characteristic
34	C₂×C₄×#64408	C₂³	1	3	yes	—
35	C₂×C₄×#64408	C₂³	1	3	yes	—
36	(C₂×#64408)⋊C₄	C₂×C₄	1	6	yes	—
37	C₂×Q₈×D₇₃	C₂×C₄	1	1	yes	characteristic
38	Q₈×Dic₇₃	C₈	1	1	yes	characteristic, complemented
39	C₄×Dic₁₄₆	C₈	1	3	yes	complemented
40	#14556556	C₈	1	3	yes	complemented
41	C₄⋊C₄×D₇₃	C₂×C₄	1	3	yes	—
42	C₄²×D₇₃	C₂×C₄	1	3	yes	—
43	C₄×Q₈×C₇₃	C₈	1	1	yes	characteristic, complemented
44	C₂×#10596842	—	2	2	no	—
45	C₂×#10596842	—	2	2	no	—
46	C₂×#10596842	—	2	6	no	—
47	C₂×#10596842	—	2	6	no	—
48	C₂×#10596842	—	2	6	no	—
49	C₂×#10596842	#41	1	4	yes	—
50	C₂²×#64408	C₂²×C₄	1	3	yes	—
51	C₂²×#64408	C₂²×C₄	1	3	yes	—
52	C₂²×#64408	C₂²×C₄	1	1	yes	characteristic
53	C₂²×#64408	C₂⁴	1	1	yes	characteristic
54	C₄⋊#64408	—	2	6	no	—
55	C₄⋊#64408	C₄⋊C₄	1	12	yes	—
56	C₄×#64408	—	2	6	no	—
57	C₄×#64408	—	2	6	no	—
58	C₄×#64408	—	2	2	no	—
59	C₄×#64408	—	2	6	no	—
60	C₄×#64408	—	4	3	no	—
61	C₄×#64408	C₂×D₄	1	6	yes	—
62	C₄×#64408	C₂×Q₈	1	6	yes	—
63	C₂×Dic₁₄₆	C₂×C₈	1	3	yes	—
64	Q₈×D₇₃	C₄⋊C₄	1	4	yes	—
65	Q₈×D₇₃	M₁₆	1	4	yes	—
66	C₂×C₄×D₇₃	C₂²×C₄	1	1	yes	characteristic
67	C₂×C₄×D₇₃	C₂²×C₄	1	3	yes	—
68	C₂×C₄×D₇₃	C₂²×C₄	1	3	yes	—
69	C₄⋊Dic₇₃	C₂×C₈	1	3	yes	—
70	#11924007	C₂×C₈	1	6	yes	—
71	C₄×Dic₇₃	C₂×C₈	1	3	yes	—
72	C₄×Dic₇₃	C₂×C₈	1	3	yes	—
73	C₄×Dic₇₃	C₂×C₈	1	3	yes	—
74	C₂×Q₈×C₇₃	C₂×C₈	1	1	yes	characteristic
75	C₄⋊C₄×C₇₃	C₂×C₈	1	3	yes	—
76	C₄²×C₇₃	C₂×C₈	1	3	yes	—
77	C₂²×D₇₃	C₂³×C₄	1	1	yes	characteristic
78	C₂×#64408	—	2	1	no	—
79	C₂×#64408	—	2	1	no	—
80	C₂×#64408	—	2	3	no	—
81	C₂×#64408	—	2	3	no	—
82	C₂×#64408	—	2	3	no	—
83	C₂×#64408	—	2	3	no	—
84	C₂×#64408	—	2	3	no	—
85	C₂×#64408	C₂×C₄⋊C₄	1	6	yes	—
86	C₂×#64408	#131	1	2	yes	—
87	C₂×#64408	C₂²×D₄	1	1	yes	characteristic
88	C₂×#64408	C₂²×Q₈	1	1	yes	characteristic
89	C₂×#64408	C₂×#41	1	2	yes	—
90	C₂×#64408	2₋¹⁺⁴	1	2	yes	—
91	#10596842	—	2	4	no	—
92	#10596842	—	4	6	no	—
93	Q₈×C₇₃	C₄⋊C₈	1	4	yes	complemented
94	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
95	C₂×C₄×C₇₃	C₂²×C₈	1	3	yes	—
96	C₂×C₄×C₇₃	C₂²×C₈	1	3	yes	—
97	C₂×Dic₇₃	C₂²×C₈	1	1	yes	characteristic
98	C₂×Dic₇₃	C₂²×C₈	1	3	yes	—
99	C₂×Dic₇₃	C₂²×C₈	1	3	yes	—
100	C₄×D₇₃	—	2	3	no	—
101	C₄×D₇₃	—	2	1	no	—
102	C₄×D₇₃	—	2	3	no	—
103	C₄×D₇₃	—	2	1	no	—
104	C₄×D₇₃	C₂×C₄⋊C₄	1	6	yes	—
105	C₄×D₇₃	C₂×M₁₆	1	6	yes	—
106	Dic₁₄₆	C₄⋊C₈	1	12	yes	complemented
107	C₂²×C₇₃	C₂³×C₈	1	1	yes	characteristic
108	C₂×D₇₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
109	C₂×D₇₃	C₄×#41	1	1	yes	characteristic
110	C₂×D₇₃	#519	1	1	yes	characteristic
111	C₂×D₇₃	C₂²×M₁₆	1	1	yes	characteristic
112	C₂×D₇₃	C₂×#131	1	1	yes	characteristic
113	C₂×D₇₃	#568	1	1	yes	characteristic
114	#64408	—	2	6	no	—
115	#64408	—	2	2	no	—
116	#64408	—	4	3	no	—
117	#64408	#548	1	2	yes	—
118	#64408	#557	1	2	yes	—
119	C₄×C₇₃	—	2	3	no	—
120	C₄×C₇₃	—	2	1	no	—
121	C₄×C₇₃	C₂×C₄⋊C₈	1	6	yes	—
122	Dic₇₃	—	2	3	no	—
123	Dic₇₃	—	2	1	no	—
124	Dic₇₃	C₂×C₄⋊C₈	1	6	yes	—
125	#22734	—	73	1	no	maximal, Sylow
126	C₂×C₇₃	C₂²×C₄⋊C₈	1	1	yes	characteristic
127	C₂×C₇₃	C₈×#41	1	1	yes	characteristic
128	C₂×C₇₃	#2982	1	1	yes	characteristic
129	D₇₃	#2351	1	1	yes	characteristic
130	D₇₃	#2906	1	1	yes	characteristic
131	D₇₃	#2971	1	1	yes	characteristic
132	D₇₃	#2984	1	1	yes	characteristic
133	C₂×C₈×Q₈	—	73	2	no	—
134	C₄²×Q₈	—	73	1	no	—
135	#1939	—	73	6	no	—
136	#1831	—	73	3	no	—
137	C₄×C₄⋊C₈	—	73	3	no	—
138	C₇₃	#22734	1	1	yes	characteristic, complemented, Sylow
139	C₂×C₄×Q₈	—	73	1	no	—
140	C₂×C₄×Q₈	—	73	1	no	—
141	C₂×C₄×Q₈	—	73	1	no	—
142	C₈×Q₈	—	146	8	no	—
143	C₂×C₄⋊C₈	—	73	6	no	—
144	C₂×C₄⋊C₈	—	73	6	no	—
145	C₂×C₄×C₈	—	73	6	no	—
146	C₄×C₄⋊C₄	—	73	3	no	—
147	C₄×C₄⋊C₄	—	73	3	no	—
148	C₄×C₄⋊C₄	—	73	3	no	—
149	C₄³	—	73	3	no	—
150	#336	—	73	3	no	—
151	#336	—	73	6	no	—
152	#336	—	73	1	no	—
153	C₂²×Q₈	—	73	1	no	—
154	C₂²×C₈	—	73	2	no	—
155	C₂²×C₈	—	73	6	no	—
156	C₄×Q₈	—	73	4	no	—
157	C₄×Q₈	—	73	4	no	—
158	C₄×Q₈	—	73	1	no	—
159	C₄×Q₈	—	73	3	no	—
160	C₄×Q₈	—	73	3	no	—
161	C₄×Q₈	—	146	4	no	—
162	C₄×Q₈	#10596842	1	1	yes	characteristic, complemented
163	C₂×C₄⋊C₄	—	73	3	no	—
164	C₂×C₄⋊C₄	—	73	3	no	—
165	C₂×C₄⋊C₄	—	73	3	no	—
166	C₂×C₄²	—	73	3	no	—
167	C₂×C₄²	—	73	3	no	—
168	C₂×C₄²	—	73	3	no	—
169	C₂×C₄²	—	73	3	no	—
170	C₂×C₄²	—	73	1	no	—
171	C₂×C₄²	—	73	3	no	—
172	C₂×C₄²	—	73	3	no	—
173	C₄⋊C₈	—	146	12	no	—
174	C₄⋊C₈	—	146	12	no	—
175	C₄×C₈	—	146	12	no	—
176	#95	—	73	6	no	—
177	C₂×Q₈	—	73	4	no	—
178	C₂×Q₈	—	73	4	no	—
179	C₂×Q₈	—	73	3	no	—
180	C₂×Q₈	C₂×#10596842	1	1	yes	characteristic
181	C₂²×C₄	—	73	1	no	—
182	C₂²×C₄	—	73	3	no	—
183	C₂²×C₄	—	73	3	no	—
184	C₂²×C₄	—	73	1	no	—
185	C₂²×C₄	—	73	3	no	—
186	C₂²×C₄	—	73	1	no	—
187	C₂²×C₄	—	73	3	no	—
188	C₂×C₈	—	73	4	no	—
189	C₂×C₈	—	146	6	no	—
190	C₂×C₈	—	146	6	no	—
191	C₂×C₈	—	146	6	no	—
192	C₂×C₈	—	146	2	no	—
193	C₂×C₈	—	146	2	no	—
194	C₄⋊C₄	—	73	12	no	—
195	C₄⋊C₄	—	73	6	no	—
196	C₄⋊C₄	—	73	3	no	—
197	C₄⋊C₄	—	146	6	no	—
198	C₄⋊C₄	C₂×#10596842	1	3	yes	—
199	C₄²	—	292	3	no	—
200	C₄²	—	73	6	no	—
201	C₄²	—	73	6	no	—
202	C₄²	—	73	3	no	—
203	C₄²	—	73	3	no	—
204	C₄²	—	73	3	no	—
205	C₄²	—	146	6	no	—
206	C₄²	—	146	2	no	—
207	C₄²	—	146	6	no	—
208	C₄²	—	146	6	no	—
209	C₄²	C₂×#10596842	1	3	yes	—
210	C₂³	—	73	1	no	—
211	Q₈	—	73	12	no	—
212	Q₈	C₄⋊#10596842	1	4	yes	complemented
213	C₂×C₄	—	73	1	no	—
214	C₂×C₄	—	73	1	no	—
215	C₂×C₄	—	73	2	no	—
216	C₂×C₄	—	73	2	no	—
217	C₂×C₄	—	73	3	no	—
218	C₂×C₄	—	73	6	no	—
219	C₂×C₄	—	73	6	no	—
220	C₂×C₄	—	73	6	no	—
221	C₂×C₄	—	73	2	no	—
222	C₂×C₄	—	73	1	no	—
223	C₂×C₄	—	73	3	no	—
224	C₂×C₄	—	146	1	no	—
225	C₂×C₄	—	146	3	no	—
226	C₂×C₄	—	146	3	no	—
227	C₂×C₄	—	146	1	no	—
228	C₂×C₄	—	146	3	no	—
229	C₂×C₄	—	146	3	no	—
230	C₂×C₄	—	146	3	no	—
231	C₂×C₄	—	146	3	no	—
232	C₂×C₄	—	146	1	no	—
233	C₂×C₄	—	146	3	no	—
234	C₂×C₄	—	146	1	no	—
235	C₂×C₄	C₂²×#10596842	1	1	yes	characteristic
236	C₂×C₄	C₂²×#10596842	1	3	yes	—
237	C₂×C₄	C₂²×#10596842	1	3	yes	—
238	C₈	—	292	6	no	—
239	C₈	—	146	4	no	—
240	C₂²	—	73	1	no	—
241	C₂²	—	73	1	no	—
242	C₂²	—	73	1	no	—
243	C₂²	—	73	1	no	—
244	C₂²	—	73	1	no	—
245	C₂²	—	73	1	no	—
246	C₂²	C₂³×#10596842	1	1	yes	characteristic
247	C₄	—	2	3	no	—
248	C₄	—	2	1	no	—
249	C₄	—	292	3	no	—
250	C₄	—	73	2	no	—
251	C₄	—	73	2	no	—
252	C₄	—	73	6	no	—
253	C₄	—	146	6	no	—
254	C₄	—	146	2	no	—
255	C₄	—	146	3	no	—
256	C₄	—	146	1	no	—
257	C₄	C₂×C₄⋊#10596842	1	6	yes	—
258	C₂	—	73	1	no	—
259	C₂	—	73	1	no	—
260	C₂	—	73	1	no	—
261	C₂	—	73	1	no	—
262	C₂	C₂²×C₄⋊#10596842	1	1	yes	characteristic
263	C₂	#41×#10596842	1	1	yes	characteristic
264	C₂	#22627174	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×#10596842
commutator, socle	C₂²×C₇₃	C₂³×C₈
Fitting	C₄×Q₈×C₇₃	C₈
prime radical	C₂²×D₇₃	C₂³×C₄
cosocle	C₂²×#64408	C₂⁴
hypercenter	C₄×Q₈	#10596842
hypocenter	C₇₃	#22734
LCS	G ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄×Q₈

Complements

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

N	H ≅ G/N	type
38. Q₈×Dic₇₃	C₈	semidirect
39. C₄×Dic₁₄₆	C₈	semidirect
40. #14556556	C₈	semidirect
43. C₄×Q₈×C₇₃	C₈	semidirect
93. Q₈×C₇₃	C₄⋊C₈	semidirect
106. Dic₁₄₆	C₄⋊C₈	semidirect
138. C₇₃	#22734	semidirect
162. C₄×Q₈	#10596842	semidirect
212. Q₈	C₄⋊#10596842	semidirect

Maximal subgroups

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

subgroup	index
C₂×Q₈×#10596842	2
#22627080	2
#22627073	2
C₄×C₄⋊#10596842	2
C₄×Q₈×#64408	2
#22734	73