Subgroups — #34034859

Order 18688.

490 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22627158	C₂	1	4	yes	maximal, complemented
2	#22627157	C₂	1	4	yes	maximal, complemented
3	#22627123	C₂	1	2	yes	maximal
4	#22627121	C₂	1	1	yes	maximal, characteristic
5	#22627093	C₂	1	2	yes	maximal
6	#22627087	C₂	1	1	yes	maximal, characteristic
7	#22626720	C₂	1	1	yes	maximal, characteristic
8	C₂×#14556688	C₂²	1	2	yes	—
9	C₂×#14556686	C₂²	1	1	yes	characteristic
10	C₂×#14556686	C₂²	1	1	yes	characteristic
11	#17989690	C₂²	1	2	yes	—
12	#17989690	C₂²	1	2	yes	—
13	#17989690	C₂²	1	4	yes	—
14	C₄⋊#11924040	C₂²	1	4	yes	—
15	#17989681	C₂²	1	2	yes	—
16	#17989681	C₂²	1	2	yes	—
17	#17989681	C₂²	1	4	yes	—
18	C₄×#11924040	C₂²	1	4	yes	—
19	#17989638	C₄	1	2	yes	—
20	#17989637	C₄	1	2	yes	—
21	#17989636	C₂²	1	2	yes	—
22	Q₈×#10596833	C₂²	1	2	yes	—
23	#17989594	C₄	1	2	yes	—
24	#17989591	C₄	1	1	yes	characteristic
25	#17989552	C₂²	1	2	yes	—
26	C₄×#11924036	C₂²	1	1	yes	characteristic
27	#17989142	C₄	1	1	yes	characteristic
28	#14556688	—	2	4	no	—
29	#14556686	—	2	2	no	—
30	#14556686	Q₈	1	4	yes	complemented
31	C₂×#11924040	C₂³	1	4	yes	—
32	C₂×#11924040	C₂³	1	2	yes	—
33	C₂×#11924040	C₂³	1	2	yes	—
34	C₂×#11924038	C₂×C₄	1	2	yes	—
35	C₂×#11924036	C₂×C₄	1	1	yes	characteristic
36	C₂×#11924036	C₂³	1	1	yes	characteristic
37	#14556658	C₂×C₄	1	4	yes	—
38	C₄⋊#10596833	C₂×C₄	1	4	yes	—
39	C₄⋊#10596833	C₂×C₄	1	2	yes	—
40	C₄⋊#10596833	C₂×C₄	1	2	yes	—
41	#14556646	C₂×C₄	1	4	yes	—
42	C₄×#10596833	C₂×C₄	1	1	yes	characteristic
43	C₄×#10596833	C₂×C₄	1	1	yes	characteristic
44	C₄×#10596833	C₂³	1	1	yes	characteristic
45	C₄×#10596833	C₂³	1	4	yes	—
46	C₄×#10596833	C₂³	1	1	yes	characteristic
47	#14556591	C₈	1	1	yes	characteristic
48	#14556582	C₈	1	2	yes	—
49	#14556571	C₈	1	2	yes	—
50	#14556567	C₈	1	2	yes	—
51	Q₈×Dic₇₃	C₂×C₄	1	2	yes	—
52	#14556556	C₂×C₄	1	2	yes	—
53	#14556549	C₂×C₄	1	2	yes	—
54	C₄²×D₇₃	C₂×C₄	1	1	yes	characteristic
55	C₄⋊Q₈×C₇₃	C₈	1	1	yes	characteristic
56	#11924040	—	2	2	no	—
57	#11924040	—	2	4	no	—
58	#11924040	C₂×Q₈	1	4	yes	—
59	#11924038	—	2	4	no	—
60	#11924036	—	2	2	no	—
61	#11924036	C₂×Q₈	1	2	yes	—
62	#11924036	#41	1	2	yes	—
63	C₂×#10596833	C₂²×C₄	1	1	yes	characteristic
64	C₂×#10596833	C₂²×C₄	1	1	yes	characteristic
65	C₂×#10596833	C₂²×C₄	1	4	yes	—
66	C₂×#10596833	C₂²×C₄	1	1	yes	characteristic
67	C₂×#10596833	C₂⁴	1	1	yes	characteristic
68	C₂×Dic₁₄₆	C₂×C₈	1	2	yes	—
69	C₂×C₄×D₇₃	C₂×C₈	1	1	yes	characteristic
70	C₂×C₄×D₇₃	C₂×C₈	1	1	yes	characteristic
71	C₂×C₄×D₇₃	C₂²×C₄	1	1	yes	characteristic
72	C₄⋊Dic₇₃	C₂×C₈	1	2	yes	—
73	C₄⋊Dic₇₃	C₂×C₈	1	2	yes	—
74	C₄⋊Dic₇₃	C₂×C₈	1	4	yes	—
75	#11924007	C₂×C₈	1	4	yes	—
76	#11924005	C₂×C₈	1	4	yes	—
77	C₄×Dic₇₃	C₂×C₈	1	1	yes	characteristic
78	C₄×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
79	C₄×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
80	C₄×Dic₇₃	C₂²×C₄	1	4	yes	—
81	C₂×Q₈×C₇₃	C₂×C₈	1	2	yes	—
82	C₄⋊C₄×C₇₃	C₂×C₈	1	2	yes	—
83	C₄⋊C₄×C₇₃	C₂×C₈	1	2	yes	—
84	C₄²×C₇₃	C₂×C₈	1	1	yes	characteristic
85	C₂²×D₇₃	C₂²×C₈	1	1	yes	characteristic
86	Q₈×C₇₃	—	4	2	no	—
87	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
88	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
89	C₂×C₄×C₇₃	C₂²×C₈	1	4	yes	—
90	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
91	C₂×Dic₇₃	C₂²×C₈	1	1	yes	characteristic
92	C₂×Dic₇₃	C₂²×C₈	1	1	yes	characteristic
93	C₂×Dic₇₃	C₂²×C₈	1	4	yes	—
94	C₂×Dic₇₃	C₂³×C₄	1	1	yes	characteristic
95	C₄×D₇₃	—	4	1	no	—
96	C₄×D₇₃	—	4	1	no	—
97	C₄×D₇₃	C₄×Q₈	1	2	yes	—
98	C₄×D₇₃	#131	1	2	yes	—
99	Dic₁₄₆	—	4	2	no	—
100	#10596833	—	2	1	no	—
101	#10596833	—	2	1	no	—
102	#10596833	—	2	4	no	—
103	#10596833	C₄×Q₈	1	2	yes	—
104	#10596833	C₂²×Q₈	1	1	yes	characteristic
105	#10596833	C₂×#41	1	1	yes	characteristic
106	C₂²×C₇₃	C₂³×C₈	1	1	yes	characteristic
107	C₂×D₇₃	—	2	1	no	—
108	C₂×D₇₃	—	2	1	no	—
109	C₂×D₇₃	C₈×Q₈	1	2	yes	—
110	C₄×C₇₃	—	2	1	no	—
111	C₄×C₇₃	—	2	1	no	—
112	C₄×C₇₃	—	2	4	no	—
113	C₄×C₇₃	C₈×Q₈	1	2	yes	—
114	Dic₇₃	—	2	4	no	—
115	Dic₇₃	—	2	1	no	—
116	Dic₇₃	—	2	1	no	—
117	Dic₇₃	C₂×C₄×Q₈	1	1	yes	characteristic
118	Dic₇₃	C₂×#131	1	1	yes	characteristic
119	#33475	—	73	1	no	maximal, Sylow
120	C₂×C₇₃	C₂×C₈×Q₈	1	1	yes	characteristic
121	C₂×C₇₃	#3407	1	1	yes	characteristic
122	C₂×C₇₃	#3407	1	1	yes	characteristic
123	D₇₃	—	2	2	no	—
124	#2960	—	73	1	no	—
125	#2183	—	73	4	no	—
126	#2183	—	73	4	no	—
127	#2151	—	73	2	no	—
128	#2150	—	73	1	no	—
129	#2108	—	73	2	no	—
130	C₂×#346	—	73	1	no	—
131	C₇₃	#33475	1	1	yes	characteristic, complemented, Sylow
132	#533	—	73	1	no	—
133	C₂×M₃₂	—	73	2	no	—
134	C₂²×C₁₆	—	73	1	no	—
135	C₂²×C₁₆	—	73	1	no	—
136	#446	—	73	2	no	—
137	#446	—	73	2	no	—
138	C₈×Q₈	—	73	2	no	—
139	C₈×Q₈	—	73	2	no	—
140	#424	—	73	2	no	—
141	#423	—	73	1	no	—
142	#405	—	73	2	no	—
143	C₂×C₄×C₈	—	73	1	no	—
144	C₄⋊C₁₆	—	73	4	no	—
145	C₄⋊C₁₆	—	73	2	no	—
146	C₄⋊C₁₆	—	73	2	no	—
147	C₄⋊C₁₆	—	73	4	no	—
148	#346	—	73	2	no	—
149	#346	—	73	2	no	—
150	#346	—	73	4	no	—
151	C₄×C₁₆	—	73	4	no	—
152	C₂×M₁₆	—	73	2	no	—
153	C₂²×C₈	—	73	1	no	—
154	C₂²×C₈	—	73	1	no	—
155	C₄⋊Q₈	—	73	1	no	—
156	C₄⋊Q₈	#10596842	1	1	yes	characteristic
157	#125	—	73	2	no	—
158	C₂²⋊Q₈	—	73	2	no	—
159	C₂²⋊Q₈	—	73	2	no	—
160	C₄×Q₈	—	73	2	no	—
161	C₄×Q₈	—	73	2	no	—
162	#117	—	73	2	no	—
163	C₂×C₄²	—	73	1	no	—
164	M₃₂	—	146	4	no	—
165	C₂×C₁₆	—	73	2	no	—
166	C₂×C₁₆	—	73	4	no	—
167	C₂×C₁₆	—	73	4	no	—
168	C₂×C₁₆	—	73	2	no	—
169	C₂×C₁₆	—	146	2	no	—
170	C₄⋊C₈	—	73	2	no	—
171	C₄⋊C₈	—	73	2	no	—
172	C₄⋊C₈	—	73	4	no	—
173	C₄⋊C₈	—	73	4	no	—
174	#97	—	73	4	no	—
175	C₄×C₈	—	73	4	no	—
176	C₄×C₈	—	73	1	no	—
177	C₄×C₈	—	73	1	no	—
178	C₄×C₈	—	73	1	no	—
179	C₄×C₈	—	73	1	no	—
180	C₂×Q₈	—	73	2	no	—
181	C₂×Q₈	C₂×#10596842	1	2	yes	—
182	C₂²×C₄	—	73	1	no	—
183	C₂²×C₄	—	73	1	no	—
184	C₂²×C₄	—	73	1	no	—
185	M₁₆	—	146	4	no	—
186	C₂×C₈	—	73	1	no	—
187	C₂×C₈	—	73	1	no	—
188	C₂×C₈	—	73	2	no	—
189	C₂×C₈	—	73	2	no	—
190	C₂×C₈	—	73	1	no	—
191	C₂×C₈	—	73	1	no	—
192	C₂×C₈	—	73	4	no	—
193	C₂×C₈	—	146	2	no	—
194	C₄⋊C₄	—	73	4	no	—
195	C₄⋊C₄	—	73	2	no	—
196	C₄⋊C₄	—	73	4	no	—
197	C₄⋊C₄	—	73	2	no	—
198	C₄⋊C₄	C₂×#10596842	1	2	yes	—
199	C₄⋊C₄	C₂×#10596842	1	2	yes	—
200	C₂²⋊C₄	—	73	4	no	—
201	C₄²	—	73	1	no	—
202	C₄²	—	73	1	no	—
203	C₄²	—	73	1	no	—
204	C₄²	—	73	4	no	—
205	C₄²	C₂×#10596842	1	1	yes	characteristic
206	C₁₆	—	73	4	no	—
207	C₁₆	—	146	2	no	—
208	C₁₆	—	146	4	no	—
209	C₂³	—	73	1	no	—
210	Q₈	—	4	2	no	—
211	Q₈	—	292	2	no	—
212	C₂×C₄	—	292	1	no	—
213	C₂×C₄	—	292	1	no	—
214	C₂×C₄	—	73	1	no	—
215	C₂×C₄	—	73	2	no	—
216	C₂×C₄	—	73	2	no	—
217	C₂×C₄	—	73	1	no	—
218	C₂×C₄	—	73	1	no	—
219	C₂×C₄	—	73	4	no	—
220	C₂×C₄	C₂²×#10596842	1	1	yes	characteristic
221	C₂×C₄	C₂²×#10596842	1	1	yes	characteristic
222	C₂×C₄	C₂²×#10596842	1	4	yes	—
223	C₂×C₄	C₂²×#10596842	1	1	yes	characteristic
224	C₈	—	73	1	no	—
225	C₈	—	73	1	no	—
226	C₈	—	73	2	no	—
227	C₈	—	146	1	no	—
228	C₈	—	146	1	no	—
229	C₈	—	146	4	no	—
230	C₂²	—	73	2	no	—
231	C₂²	—	146	1	no	—
232	C₂²	—	146	1	no	—
233	C₂²	C₂³×#10596842	1	1	yes	characteristic
234	C₄	—	2	1	no	—
235	C₄	—	2	4	no	—
236	C₄	—	2	1	no	—
237	C₄	—	73	1	no	—
238	C₄	—	73	1	no	—
239	C₄	—	146	1	no	—
240	C₄	—	146	1	no	—
241	C₄	—	146	4	no	—
242	C₄	Q₈×#10596842	1	2	yes	—
243	C₂	—	146	2	no	—
244	C₂	C₂×Q₈×#10596842	1	1	yes	characteristic
245	C₂	#22627184	1	1	yes	characteristic
246	C₂	#22627186	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	C₂²×C₈
cosocle	C₂×#10596833	C₂⁴
hypercenter	C₄⋊Q₈	#10596842
hypocenter	C₇₃	#33475
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
1. #22627158	C₂	semidirect
2. #22627157	C₂	semidirect
30. #14556686	Q₈	semidirect
131. C₇₃	#33475	semidirect

Maximal subgroups

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

subgroup	index
#22627158	2
#22627157	2
#22627123	2
#22627121	2
#22627093	2
#22627087	2
#22626720	2
#33475	73