Subgroups — #34030226

Order 18688.

682 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17989645	C₂	1	2	yes	maximal, complemented
2	#22626315	C₂	1	2	yes	maximal, complemented
3	#22626311	C₂	1	4	yes	maximal, complemented
4	D₄×#11924031	C₂	1	2	yes	maximal, complemented
5	C₂×C₄⋊#11924031	C₂	1	1	yes	maximal, characteristic, complemented
6	#22626161	C₂	1	2	yes	maximal, complemented
7	C₄×#14556665	C₂	1	1	yes	maximal, characteristic, complemented
8	#22618643	C₂	1	1	yes	maximal, characteristic
9	#17989645	—	2	8	no	—
10	C₂×#14556665	C₂²	1	4	yes	—
11	C₂×#14556665	C₂²	1	1	yes	characteristic
12	C₂×#14556665	C₂²	1	1	yes	characteristic
13	C₂²×#11924031	C₂²	1	4	yes	—
14	C₂²×#11924031	C₂²	1	2	yes	—
15	C₄⋊#11924031	C₂²	1	1	yes	characteristic
16	C₄⋊#11924031	C₂²	1	2	yes	—
17	C₄⋊#11924031	C₂²	1	1	yes	characteristic
18	C₂²⋊#11924031	C₂²	1	4	yes	—
19	C₂²⋊#11924031	C₂²	1	4	yes	—
20	#17989475	C₂²	1	2	yes	—
21	C₄×#11924031	C₂²	1	2	yes	—
22	C₂×#14556598	C₂²	1	2	yes	—
23	#17988665	C₄	1	2	yes	—
24	#17988665	C₄	1	2	yes	—
25	D₄×#10596831	C₂²	1	2	yes	—
26	D₄×#10596831	C₂²	1	2	yes	—
27	#17988611	C₄	1	1	yes	characteristic
28	#17988554	C₄	1	2	yes	—
29	C₂×C₄×#10596831	C₂²	1	1	yes	characteristic
30	#532×C₇₃	C₄	1	1	yes	characteristic
31	#14556665	—	2	8	no	—
32	#14556665	—	2	2	no	—
33	#14556665	D₄	1	4	yes	complemented
34	C₂×#11924031	—	2	8	no	—
35	C₂×#11924031	—	2	4	no	—
36	C₂×#11924031	C₂³	1	2	yes	—
37	C₂×#11924031	C₂³	1	2	yes	—
38	C₂×#11924031	C₂³	1	2	yes	—
39	C₂×#11924031	C₂³	1	2	yes	—
40	#14556598	—	2	4	no	—
41	#14556598	—	2	4	no	—
42	C₂×#11924012	C₂×C₄	1	2	yes	—
43	C₂×#11924012	C₂×C₄	1	4	yes	—
44	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
45	C₂²×#10596831	C₂³	1	4	yes	—
46	C₂²×#10596831	C₂³	1	1	yes	characteristic
47	C₄⋊#10596831	C₂×C₄	1	2	yes	—
48	C₄⋊#10596831	C₂×C₄	1	2	yes	—
49	C₂²⋊#10596831	C₂×C₄	1	4	yes	—
50	C₂²⋊#10596831	C₂×C₄	1	4	yes	—
51	C₄×#10596831	C₂×C₄	1	1	yes	characteristic
52	C₄×#10596831	C₂×C₄	1	1	yes	characteristic
53	C₄×#10596831	C₂³	1	1	yes	characteristic
54	C₄×#10596831	C₂³	1	1	yes	characteristic
55	C₂×#41×C₇₃	C₂×C₄	1	2	yes	—
56	C₄⋊Q₈×C₇₃	C₈	1	1	yes	characteristic
57	#127×C₇₃	C₈	1	1	yes	characteristic
58	#124×C₇₃	C₈	1	2	yes	—
59	#121×C₇₃	C₈	1	2	yes	—
60	#121×C₇₃	C₈	1	2	yes	—
61	C₄×D₄×C₇₃	C₂×C₄	1	2	yes	—
62	C₄×D₄×C₇₃	C₂×C₄	1	2	yes	—
63	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
64	#11924031	—	2	2	no	—
65	#11924031	—	2	2	no	—
66	#11924031	—	2	2	no	—
67	#11924031	C₂×D₄	1	4	yes	—
68	#11924012	—	2	4	no	—
69	#11924012	—	2	8	no	—
70	C₂×#10596831	—	2	2	no	—
71	C₂×#10596831	—	2	4	no	—
72	C₂×#10596831	—	2	4	no	—
73	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
74	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
75	C₂×#10596831	C₂²×C₄	1	4	yes	—
76	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
77	C₂×#10596831	C₂×D₄	1	2	yes	—
78	C₂×#10596831	#41	1	2	yes	—
79	C₂×#10596831	C₂⁴	1	1	yes	characteristic
80	#41×C₇₃	—	2	4	no	—
81	#41×C₇₃	—	2	4	no	—
82	C₂×Q₈×C₇₃	C₂×C₈	1	2	yes	—
83	C₂×D₄×C₇₃	C₂×C₈	1	2	yes	—
84	C₂×D₄×C₇₃	C₂×C₈	1	4	yes	—
85	C₂×D₄×C₇₃	C₂×C₈	1	2	yes	—
86	C₂×D₄×C₇₃	C₂×C₈	1	2	yes	—
87	C₂²×C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
88	C₂²×C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
89	C₂²×C₄×C₇₃	C₂²×C₄	1	4	yes	—
90	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
91	C₄⋊C₄×C₇₃	C₂×C₈	1	2	yes	—
92	C₄⋊C₄×C₇₃	C₂×C₈	1	2	yes	—
93	C₂²⋊C₄×C₇₃	C₂×C₈	1	4	yes	—
94	C₂²⋊C₄×C₇₃	C₂×C₈	1	4	yes	—
95	C₄²×C₇₃	C₂×C₈	1	1	yes	characteristic
96	C₄²×C₇₃	C₂×C₈	1	1	yes	characteristic
97	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
98	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
99	C₂³×C₇₃	C₂²×C₈	1	4	yes	—
100	C₂³×C₇₃	C₂²×C₈	1	1	yes	characteristic
101	Q₈×C₇₃	—	2	4	no	—
102	D₄×C₇₃	—	2	8	no	—
103	D₄×C₇₃	—	2	4	no	—
104	D₄×C₇₃	—	2	4	no	—
105	D₄×C₇₃	—	4	2	no	—
106	C₂×C₄×C₇₃	—	2	4	no	—
107	C₂×C₄×C₇₃	—	2	4	no	—
108	C₂×C₄×C₇₃	—	2	2	no	—
109	C₂×C₄×C₇₃	—	4	1	no	—
110	C₂×C₄×C₇₃	C₄×D₄	1	2	yes	—
111	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
112	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
113	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
114	C₂×C₄×C₇₃	C₂²×C₈	1	4	yes	—
115	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
116	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
117	C₂×C₄×C₇₃	#131	1	2	yes	—
118	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
119	#10596831	—	2	4	no	—
120	#10596831	—	2	1	no	—
121	#10596831	—	2	1	no	—
122	#10596831	C₄×D₄	1	2	yes	—
123	#10596831	C₂²×D₄	1	1	yes	characteristic
124	#10596831	C₂×#41	1	1	yes	characteristic
125	C₂²×C₇₃	—	2	1	no	—
126	C₂²×C₇₃	—	2	1	no	—
127	C₂²×C₇₃	—	2	4	no	—
128	C₂²×C₇₃	—	2	4	no	—
129	C₂²×C₇₃	—	2	4	no	—
130	C₂²×C₇₃	C₈×D₄	1	2	yes	—
131	C₂²×C₇₃	C₂³×C₈	1	1	yes	characteristic
132	C₄×C₇₃	—	2	1	no	—
133	C₄×C₇₃	—	2	1	no	—
134	C₄×C₇₃	—	2	1	no	—
135	C₄×C₇₃	—	2	4	no	—
136	C₄×C₇₃	C₈×D₄	1	2	yes	—
137	C₄×C₇₃	#504	1	2	yes	complemented
138	C₄×C₇₃	C₂×C₄×D₄	1	1	yes	characteristic
139	C₄×C₇₃	C₂×#131	1	1	yes	characteristic
140	#33438	—	73	1	no	maximal, Sylow
141	C₂×C₇₃	—	2	2	no	—
142	C₂×C₇₃	—	4	4	no	—
143	C₂×C₇₃	C₂×C₈×D₄	1	1	yes	characteristic
144	C₂×C₇₃	C₂×#504	1	1	yes	characteristic
145	C₂×C₇₃	#3407	1	1	yes	characteristic
146	C₂×#504	—	73	2	no	—
147	#2931	—	73	1	no	—
148	#2169	—	73	2	no	—
149	#2168	—	73	4	no	—
150	D₄×C₁₆	—	73	2	no	—
151	C₂×C₄⋊C₁₆	—	73	1	no	—
152	#2113	—	73	2	no	—
153	C₄×M₃₂	—	73	1	no	—
154	C₇₃	#33438	1	1	yes	characteristic, complemented, Sylow
155	C₂×#131	—	73	2	no	—
156	#532	#64408	1	1	yes	characteristic
157	#504	—	146	8	no	—
158	C₂×M₃₂	—	73	1	no	—
159	C₂×M₃₂	—	73	1	no	—
160	C₂×M₃₂	—	73	4	no	—
161	C₂²×C₁₆	—	73	2	no	—
162	C₂²×C₁₆	—	73	4	no	—
163	#436	—	73	2	no	—
164	#436	—	73	2	no	—
165	C₈×D₄	—	73	2	no	—
166	C₈×D₄	—	73	2	no	—
167	#423	—	73	1	no	—
168	#408	—	73	2	no	—
169	C₂×C₄×C₈	—	73	1	no	—
170	C₄⋊C₁₆	—	73	1	no	—
171	C₄⋊C₁₆	—	73	2	no	—
172	C₄⋊C₁₆	—	73	1	no	—
173	C₂²⋊C₁₆	—	73	4	no	—
174	C₂²⋊C₁₆	—	73	4	no	—
175	#346	—	73	2	no	—
176	C₄×C₁₆	—	73	2	no	—
177	C₂×#41	C₂×#64408	1	2	yes	—
178	#131	—	146	4	no	—
179	#131	—	146	4	no	—
180	C₂×M₁₆	—	73	2	no	—
181	C₂×M₁₆	—	73	4	no	—
182	C₂²×C₈	—	73	1	no	—
183	C₂²×C₈	—	73	4	no	—
184	C₂²×C₈	—	73	1	no	—
185	C₄⋊Q₈	#10596833	1	1	yes	characteristic
186	#127	#10596833	1	1	yes	characteristic
187	#124	#10596833	1	2	yes	—
188	#121	#10596833	1	2	yes	—
189	#121	#10596833	1	2	yes	—
190	C₄×D₄	C₂×#64408	1	2	yes	—
191	C₄×D₄	C₂×#64408	1	2	yes	—
192	C₂×C₄²	C₂×#64408	1	1	yes	characteristic
193	M₃₂	—	73	4	no	—
194	M₃₂	—	146	8	no	—
195	M₃₂	—	146	2	no	—
196	C₂×C₁₆	—	73	2	no	—
197	C₂×C₁₆	—	73	2	no	—
198	C₂×C₁₆	—	73	2	no	—
199	C₂×C₁₆	—	73	2	no	—
200	C₂×C₁₆	—	146	8	no	—
201	C₂×C₁₆	—	146	4	no	—
202	C₄⋊C₈	—	73	2	no	—
203	C₄⋊C₈	—	73	2	no	—
204	C₂²⋊C₈	—	73	4	no	—
205	C₂²⋊C₈	—	73	4	no	—
206	C₄×C₈	—	73	1	no	—
207	C₄×C₈	—	73	1	no	—
208	C₄×C₈	—	73	1	no	—
209	C₄×C₈	—	73	1	no	—
210	#41	—	2	4	no	—
211	#41	—	2	4	no	—
212	C₂×Q₈	C₂×#10596833	1	2	yes	—
213	C₂×D₄	C₂×#10596833	1	2	yes	—
214	C₂×D₄	C₂×#10596833	1	4	yes	—
215	C₂×D₄	C₂×#10596833	1	2	yes	—
216	C₂×D₄	C₂×#10596833	1	2	yes	—
217	C₂²×C₄	C₂×#10596833	1	1	yes	characteristic
218	C₂²×C₄	C₂×#10596833	1	1	yes	characteristic
219	C₂²×C₄	C₂²×#64408	1	4	yes	—
220	C₂²×C₄	C₂²×#64408	1	1	yes	characteristic
221	M₁₆	—	146	8	no	—
222	M₁₆	—	146	4	no	—
223	C₂×C₈	—	73	1	no	—
224	C₂×C₈	—	73	1	no	—
225	C₂×C₈	—	73	2	no	—
226	C₂×C₈	—	73	2	no	—
227	C₂×C₈	—	73	1	no	—
228	C₂×C₈	—	73	1	no	—
229	C₂×C₈	—	73	4	no	—
230	C₂×C₈	—	146	4	no	—
231	C₂×C₈	—	146	4	no	—
232	C₂×C₈	—	146	2	no	—
233	C₄⋊C₄	C₂×#10596833	1	2	yes	—
234	C₄⋊C₄	C₂×#10596833	1	2	yes	—
235	C₂²⋊C₄	C₂×#10596833	1	4	yes	—
236	C₂²⋊C₄	C₂×#10596833	1	4	yes	—
237	C₄²	C₂×#10596833	1	1	yes	characteristic
238	C₄²	C₂×#10596833	1	1	yes	characteristic
239	C₄²	C₂²×#64408	1	1	yes	characteristic
240	C₄²	C₂²×#64408	1	1	yes	characteristic
241	C₁₆	—	73	4	no	—
242	C₁₆	—	146	2	no	—
243	C₁₆	—	146	2	no	—
244	C₁₆	—	146	2	no	—
245	C₂³	C₂²×#10596833	1	4	yes	—
246	C₂³	C₂²×#10596833	1	1	yes	characteristic
247	Q₈	—	2	4	no	—
248	D₄	—	2	8	no	—
249	D₄	—	2	4	no	—
250	D₄	—	2	4	no	—
251	D₄	—	4	2	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	2	4	no	—
254	C₂×C₄	—	2	4	no	—
255	C₂×C₄	—	4	1	no	—
256	C₂×C₄	D₄×#64408	1	2	yes	—
257	C₂×C₄	C₂²×#10596833	1	1	yes	characteristic
258	C₂×C₄	C₂²×#10596833	1	1	yes	characteristic
259	C₂×C₄	C₂²×#10596833	1	1	yes	characteristic
260	C₂×C₄	C₂²×#10596833	1	4	yes	—
261	C₂×C₄	C₂²×#10596833	1	1	yes	characteristic
262	C₂×C₄	C₂²×#10596833	1	1	yes	characteristic
263	C₂×C₄	#14556678	1	2	yes	—
264	C₂×C₄	C₂³×#64408	1	1	yes	characteristic
265	C₈	—	73	1	no	—
266	C₈	—	73	1	no	—
267	C₈	—	73	2	no	—
268	C₈	—	146	4	no	—
269	C₈	—	146	1	no	—
270	C₈	—	146	1	no	—
271	C₂²	—	2	1	no	—
272	C₂²	—	2	1	no	—
273	C₂²	—	2	4	no	—
274	C₂²	—	2	4	no	—
275	C₂²	—	2	4	no	—
276	C₂²	D₄×#10596833	1	2	yes	—
277	C₂²	C₂³×#10596833	1	1	yes	characteristic
278	C₄	—	2	4	no	—
279	C₄	—	2	1	no	—
280	C₄	—	2	1	no	—
281	C₄	—	2	1	no	—
282	C₄	D₄×#10596833	1	2	yes	—
283	C₄	#17989645	1	2	yes	complemented
284	C₄	C₂×D₄×#64408	1	1	yes	characteristic
285	C₄	C₂×#14556678	1	1	yes	characteristic
286	C₂	—	2	2	no	—
287	C₂	—	4	4	no	—
288	C₂	C₂×D₄×#10596833	1	1	yes	characteristic
289	C₂	C₂×#17989645	1	1	yes	characteristic
290	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	#33438
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. C₂×#17989645	C₂	semidirect
2. #22626315	C₂	semidirect
3. #22626311	C₂	semidirect
4. D₄×#11924031	C₂	semidirect
5. C₂×C₄⋊#11924031	C₂	semidirect
6. #22626161	C₂	semidirect
7. C₄×#14556665	C₂	semidirect
33. #14556665	D₄	semidirect
137. C₄×C₇₃	#504	semidirect
154. C₇₃	#33438	semidirect
283. C₄	#17989645	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17989645	2
#22626315	2
#22626311	2
D₄×#11924031	2
C₂×C₄⋊#11924031	2
#22626161	2
C₄×#14556665	2
#22618643	2
#33438	73