Subgroups — #34030421

Order 18688.

564 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22626315	C₂	1	2	yes	maximal, complemented
2	#22626311	C₂	1	4	yes	maximal, complemented
3	D₄×#11924031	C₂	1	2	yes	maximal, complemented
4	#22626292	C₂	1	1	yes	maximal, characteristic, complemented
5	#22626288	C₂	1	1	yes	maximal, characteristic, complemented
6	#22626161	C₂	1	4	yes	maximal, complemented
7	#22618643	C₂	1	1	yes	maximal, characteristic
8	C₂×#14556665	C₂²	1	4	yes	—
9	C₂²×#11924031	C₂²	1	4	yes	—
10	C₄⋊#11924031	C₂²	1	2	yes	—
11	C₄⋊#11924031	C₂²	1	2	yes	—
12	C₂²⋊#11924031	C₂²	1	4	yes	—
13	C₂²⋊#11924031	C₂²	1	4	yes	—
14	C₂²⋊#11924031	C₂²	1	2	yes	—
15	C₂²⋊#11924031	C₂²	1	2	yes	—
16	#17989475	C₂²	1	2	yes	—
17	C₄×#11924031	C₂²	1	1	yes	characteristic
18	C₄×#11924031	C₂²	1	1	yes	characteristic
19	C₂×#14556598	C₂²	1	2	yes	—
20	#17988665	C₄	1	2	yes	—
21	#17988665	C₄	1	2	yes	—
22	D₄×#10596831	C₂²	1	2	yes	—
23	D₄×#10596831	C₂²	1	2	yes	—
24	#17988611	C₄	1	1	yes	characteristic
25	#17988554	C₄	1	2	yes	—
26	C₂×C₄×#10596831	C₂²	1	1	yes	characteristic
27	#532×C₇₃	C₄	1	1	yes	characteristic
28	#14556665	—	4	4	no	—
29	C₂×#11924031	—	4	4	no	—
30	C₂×#11924031	C₂³	1	2	yes	—
31	C₂×#11924031	C₂³	1	4	yes	—
32	C₂×#11924031	C₂³	1	2	yes	—
33	#14556598	—	4	4	no	—
34	C₂×#11924012	C₂×C₄	1	2	yes	—
35	C₂×#11924012	C₂×C₄	1	4	yes	—
36	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
37	C₂²×#10596831	C₂³	1	4	yes	—
38	C₂²×#10596831	C₂³	1	1	yes	characteristic
39	C₄⋊#10596831	C₂×C₄	1	2	yes	—
40	C₄⋊#10596831	C₂×C₄	1	2	yes	—
41	C₂²⋊#10596831	C₂×C₄	1	4	yes	—
42	C₂²⋊#10596831	C₂×C₄	1	4	yes	—
43	C₄×#10596831	C₂×C₄	1	1	yes	characteristic
44	C₄×#10596831	C₂×C₄	1	1	yes	characteristic
45	C₄×#10596831	C₂³	1	1	yes	characteristic
46	C₄×#10596831	C₂³	1	1	yes	characteristic
47	C₂×#41×C₇₃	C₂×C₄	1	2	yes	—
48	C₄⋊Q₈×C₇₃	C₈	1	1	yes	characteristic
49	#127×C₇₃	C₈	1	1	yes	characteristic
50	#124×C₇₃	C₈	1	2	yes	—
51	#121×C₇₃	C₈	1	2	yes	—
52	#121×C₇₃	C₈	1	2	yes	—
53	C₄×D₄×C₇₃	C₂×C₄	1	2	yes	—
54	C₄×D₄×C₇₃	C₂×C₄	1	2	yes	—
55	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
56	#11924031	—	2	2	no	—
57	#11924031	—	2	4	no	—
58	#11924031	—	2	2	no	—
59	#11924012	—	4	2	no	—
60	#11924012	—	4	4	no	—
61	C₂×#10596831	—	2	2	no	—
62	C₂×#10596831	—	2	4	no	—
63	C₂×#10596831	—	2	4	no	—
64	C₂×#10596831	—	4	1	no	—
65	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
66	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
67	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
68	C₂×#10596831	C₂²×C₄	1	4	yes	—
69	C₂×#10596831	C₂⁴	1	1	yes	characteristic
70	#41×C₇₃	—	4	2	no	—
71	#41×C₇₃	—	4	2	no	—
72	C₂×Q₈×C₇₃	C₂×C₈	1	2	yes	—
73	C₂×D₄×C₇₃	C₂×C₈	1	2	yes	—
74	C₂×D₄×C₇₃	C₂×C₈	1	2	yes	—
75	C₂×D₄×C₇₃	C₂×C₈	1	4	yes	—
76	C₂×D₄×C₇₃	C₂×C₈	1	2	yes	—
77	C₂²×C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
78	C₂²×C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
79	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
80	C₂²×C₄×C₇₃	C₂²×C₄	1	4	yes	—
81	C₄⋊C₄×C₇₃	C₂×C₈	1	2	yes	—
82	C₄⋊C₄×C₇₃	C₂×C₈	1	2	yes	—
83	C₂²⋊C₄×C₇₃	C₂×C₈	1	4	yes	—
84	C₂²⋊C₄×C₇₃	C₂×C₈	1	4	yes	—
85	C₄²×C₇₃	C₂×C₈	1	1	yes	characteristic
86	C₄²×C₇₃	C₂×C₈	1	1	yes	characteristic
87	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
88	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
89	C₂³×C₇₃	C₂²×C₈	1	4	yes	—
90	C₂³×C₇₃	C₂²×C₈	1	1	yes	characteristic
91	Q₈×C₇₃	—	4	2	no	—
92	D₄×C₇₃	—	2	4	no	—
93	D₄×C₇₃	—	4	2	no	—
94	D₄×C₇₃	—	4	2	no	—
95	D₄×C₇₃	—	4	4	no	—
96	C₂×C₄×C₇₃	—	2	1	no	—
97	C₂×C₄×C₇₃	—	2	4	no	—
98	C₂×C₄×C₇₃	—	2	4	no	—
99	C₂×C₄×C₇₃	—	2	1	no	—
100	C₂×C₄×C₇₃	—	2	2	no	—
101	C₂×C₄×C₇₃	—	4	1	no	—
102	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
103	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
104	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
105	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
106	C₂×C₄×C₇₃	C₂²×C₈	1	4	yes	—
107	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
108	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
109	#10596831	—	2	4	no	—
110	#10596831	—	2	1	no	—
111	#10596831	—	2	1	no	—
112	#10596831	—	2	1	no	—
113	#10596831	H(5,𝔽₂)	1	2	yes	—
114	C₂²×C₇₃	—	2	1	no	—
115	C₂²×C₇₃	—	2	4	no	—
116	C₂²×C₇₃	—	2	1	no	—
117	C₂²×C₇₃	—	2	4	no	—
118	C₂²×C₇₃	—	2	1	no	—
119	C₂²×C₇₃	—	2	4	no	—
120	C₂²×C₇₃	C₂³×C₈	1	1	yes	characteristic
121	C₄×C₇₃	—	2	1	no	—
122	C₄×C₇₃	—	2	1	no	—
123	C₄×C₇₃	—	2	1	no	—
124	C₄×C₇₃	—	2	1	no	—
125	C₄×C₇₃	—	2	4	no	—
126	C₄×C₇₃	#504	1	2	yes	—
127	C₄×C₇₃	#518	1	1	yes	characteristic
128	C₄×C₇₃	#568	1	1	yes	characteristic
129	#33495	—	73	1	no	maximal, Sylow
130	C₂×C₇₃	—	4	1	no	—
131	C₂×C₇₃	—	4	4	no	—
132	C₂×C₇₃	#2972	1	1	yes	characteristic
133	C₂×C₇₃	C₂×#504	1	1	yes	characteristic
134	C₂×C₇₃	#3407	1	1	yes	characteristic
135	#2931	—	73	1	no	—
136	#2169	—	73	2	no	—
137	#2168	—	73	4	no	—
138	D₄×C₁₆	—	73	2	no	—
139	#2163	—	73	1	no	—
140	#2162	—	73	1	no	—
141	#2113	—	73	4	no	—
142	C₇₃	#33495	1	1	yes	characteristic, complemented, Sylow
143	C₂×#131	—	73	2	no	—
144	#532	#64408	1	1	yes	characteristic
145	C₂×M₃₂	—	73	4	no	—
146	C₂²×C₁₆	—	73	4	no	—
147	#436	—	73	2	no	—
148	#436	—	73	2	no	—
149	C₈×D₄	—	73	2	no	—
150	C₈×D₄	—	73	2	no	—
151	#423	—	73	1	no	—
152	#408	—	73	2	no	—
153	C₂×C₄×C₈	—	73	1	no	—
154	C₄⋊C₁₆	—	73	2	no	—
155	C₄⋊C₁₆	—	73	2	no	—
156	C₂²⋊C₁₆	—	73	2	no	—
157	C₂²⋊C₁₆	—	73	2	no	—
158	C₂²⋊C₁₆	—	73	4	no	—
159	C₂²⋊C₁₆	—	73	4	no	—
160	#346	—	73	2	no	—
161	C₄×C₁₆	—	73	1	no	—
162	C₄×C₁₆	—	73	1	no	—
163	C₂×#41	C₂×#64408	1	2	yes	—
164	#131	—	292	4	no	—
165	C₂×M₁₆	—	73	2	no	—
166	C₂×M₁₆	—	73	4	no	—
167	C₂²×C₈	—	73	1	no	—
168	C₂²×C₈	—	73	1	no	—
169	C₂²×C₈	—	73	4	no	—
170	C₄⋊Q₈	#10596833	1	1	yes	characteristic
171	#127	#10596833	1	1	yes	characteristic
172	#124	#10596833	1	2	yes	—
173	#121	#10596833	1	2	yes	—
174	#121	#10596833	1	2	yes	—
175	C₄×D₄	C₂×#64408	1	2	yes	—
176	C₄×D₄	C₂×#64408	1	2	yes	—
177	C₂×C₄²	C₂×#64408	1	1	yes	characteristic
178	M₃₂	—	292	4	no	—
179	C₂×C₁₆	—	292	4	no	—
180	C₂×C₁₆	—	73	2	no	—
181	C₂×C₁₆	—	73	2	no	—
182	C₂×C₁₆	—	73	4	no	—
183	C₄⋊C₈	—	73	2	no	—
184	C₄⋊C₈	—	73	2	no	—
185	C₂²⋊C₈	—	73	4	no	—
186	C₂²⋊C₈	—	73	4	no	—
187	C₄×C₈	—	73	1	no	—
188	C₄×C₈	—	73	1	no	—
189	C₄×C₈	—	73	1	no	—
190	C₄×C₈	—	73	1	no	—
191	#41	—	4	2	no	—
192	#41	—	4	2	no	—
193	C₂×Q₈	C₂×#10596833	1	2	yes	—
194	C₂×D₄	C₂×#10596833	1	2	yes	—
195	C₂×D₄	C₂×#10596833	1	2	yes	—
196	C₂×D₄	C₂×#10596833	1	4	yes	—
197	C₂×D₄	C₂×#10596833	1	2	yes	—
198	C₂²×C₄	C₂×#10596833	1	1	yes	characteristic
199	C₂²×C₄	C₂×#10596833	1	1	yes	characteristic
200	C₂²×C₄	C₂²×#64408	1	4	yes	—
201	C₂²×C₄	C₂²×#64408	1	1	yes	characteristic
202	M₁₆	—	292	4	no	—
203	M₁₆	—	292	2	no	—
204	C₂×C₈	—	292	1	no	—
205	C₂×C₈	—	73	1	no	—
206	C₂×C₈	—	73	1	no	—
207	C₂×C₈	—	73	1	no	—
208	C₂×C₈	—	73	1	no	—
209	C₂×C₈	—	73	4	no	—
210	C₂×C₈	—	146	4	no	—
211	C₂×C₈	—	146	4	no	—
212	C₂×C₈	—	146	2	no	—
213	C₄⋊C₄	C₂×#10596833	1	2	yes	—
214	C₄⋊C₄	C₂×#10596833	1	2	yes	—
215	C₂²⋊C₄	C₂×#10596833	1	4	yes	—
216	C₂²⋊C₄	C₂×#10596833	1	4	yes	—
217	C₄²	C₂×#10596833	1	1	yes	characteristic
218	C₄²	C₂×#10596833	1	1	yes	characteristic
219	C₄²	C₂²×#64408	1	1	yes	characteristic
220	C₄²	C₂²×#64408	1	1	yes	characteristic
221	C₁₆	—	146	2	no	—
222	C₁₆	—	146	4	no	—
223	C₁₆	—	146	2	no	—
224	C₂³	C₂²×#10596833	1	4	yes	—
225	C₂³	C₂²×#10596833	1	1	yes	characteristic
226	Q₈	—	4	2	no	—
227	D₄	—	2	4	no	—
228	D₄	—	4	2	no	—
229	D₄	—	4	4	no	—
230	D₄	—	4	2	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	4	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	4	no	—
236	C₂×C₄	—	4	1	no	—
237	C₂×C₄	C₂²×#10596833	1	1	yes	characteristic
238	C₂×C₄	C₂²×#10596833	1	1	yes	characteristic
239	C₂×C₄	C₂²×#10596833	1	1	yes	characteristic
240	C₂×C₄	C₂²×#10596833	1	1	yes	characteristic
241	C₂×C₄	C₂²×#10596833	1	4	yes	—
242	C₂×C₄	C₂²×#10596833	1	1	yes	characteristic
243	C₂×C₄	C₂³×#64408	1	1	yes	characteristic
244	C₈	—	73	2	no	—
245	C₈	—	146	1	no	—
246	C₈	—	146	4	no	—
247	C₈	—	146	1	no	—
248	C₈	—	146	1	no	—
249	C₂²	—	2	1	no	—
250	C₂²	—	2	4	no	—
251	C₂²	—	2	1	no	—
252	C₂²	—	2	4	no	—
253	C₂²	—	2	1	no	—
254	C₂²	—	2	4	no	—
255	C₂²	C₂³×#10596833	1	1	yes	characteristic
256	C₄	—	2	1	no	—
257	C₄	—	2	1	no	—
258	C₄	—	2	1	no	—
259	C₄	—	2	1	no	—
260	C₄	—	2	4	no	—
261	C₄	#17989645	1	2	yes	—
262	C₄	#17989654	1	1	yes	characteristic
263	C₄	#17989665	1	1	yes	characteristic
264	C₂	—	4	1	no	—
265	C₂	—	4	4	no	—
266	C₂	#22626778	1	1	yes	characteristic
267	C₂	C₂×#17989645	1	1	yes	characteristic
268	C₂	#22626898	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×#64408
commutator, socle	C₂²×C₇₃	C₂³×C₈
Fitting, prime radical	#532×C₇₃	C₄
cosocle	C₂×#10596831	C₂⁴
hypercenter	#532	#64408
hypocenter	C₇₃	#33495
LCS	G ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ #532

Complements

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

N	H ≅ G/N	type
1. #22626315	C₂	semidirect
2. #22626311	C₂	semidirect
3. D₄×#11924031	C₂	semidirect
4. #22626292	C₂	semidirect
5. #22626288	C₂	semidirect
6. #22626161	C₂	semidirect
142. C₇₃	#33495	semidirect

Maximal subgroups

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

subgroup	index
#22626315	2
#22626311	2
D₄×#11924031	2
#22626292	2
#22626288	2
#22626161	2
#22618643	2
#33495	73