Subgroups — #34030149

Order 18688.

590 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22626296	C₂	1	4	yes	maximal, complemented
2	#22626292	C₂	1	2	yes	maximal, complemented
3	#22626288	C₂	1	2	yes	maximal, complemented
4	#22626263	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₄⋊#11924031	C₂	1	1	yes	maximal, characteristic, complemented
6	#22626161	C₂	1	2	yes	maximal
7	#22626140	C₂	1	2	yes	maximal, complemented
8	C₄×#14556598	C₂	1	1	yes	maximal, characteristic
9	C₂×#14556665	C₂²	1	2	yes	—
10	C₂²×#11924031	C₂²	1	2	yes	—
11	C₄⋊#11924031	C₂²	1	1	yes	characteristic
12	C₄⋊#11924031	C₂²	1	1	yes	characteristic
13	C₄⋊#11924031	C₂²	1	2	yes	—
14	C₄⋊#11924031	C₂²	1	2	yes	—
15	C₄⋊#11924031	C₂²	1	2	yes	—
16	C₂²⋊#11924031	C₂²	1	4	yes	—
17	C₂²⋊#11924031	C₂²	1	4	yes	—
18	#17989475	C₂²	1	4	yes	—
19	C₄×#11924031	C₂²	1	4	yes	—
20	C₂×#14556598	C₄	1	1	yes	characteristic
21	C₂×#14556598	C₂²	1	1	yes	characteristic
22	#17988543	C₄	1	2	yes	—
23	#17988543	C₄	1	1	yes	characteristic
24	#17988543	C₂²	1	1	yes	characteristic
25	#17988543	C₂²	1	2	yes	—
26	C₄×#11924012	C₄	1	2	yes	—
27	C₄×#11924012	C₄	1	1	yes	characteristic
28	C₂×C₄×#10596831	C₂²	1	1	yes	characteristic
29	C₂×C₄×#10596831	C₂²	1	2	yes	—
30	C₄×#41×C₇₃	C₄	1	1	yes	characteristic
31	#14556665	—	2	4	no	—
32	C₂×#11924031	—	2	4	no	—
33	C₂×#11924031	C₂³	1	2	yes	—
34	C₂×#11924031	C₂³	1	2	yes	—
35	C₂×#11924031	C₂³	1	4	yes	—
36	#14556598	—	2	4	no	—
37	#14556598	—	2	2	no	—
38	#14556598	—	2	2	no	—
39	C₂×#11924012	C₂×C₄	1	2	yes	—
40	C₂×#11924012	C₂×C₄	1	1	yes	characteristic
41	C₂×#11924012	C₂×C₄	1	2	yes	—
42	C₂×#11924012	C₂×C₄	1	1	yes	characteristic
43	C₂²×#10596831	C₂×C₄	1	2	yes	—
44	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
45	C₂²×#10596831	C₂³	1	2	yes	—
46	C₂²×#10596831	C₂³	1	1	yes	characteristic
47	#14556502	C₂×C₄	1	2	yes	—
48	#14556502	C₂×C₄	1	2	yes	—
49	#14556502	C₂×C₄	1	2	yes	—
50	C₄×#10596831	C₂×C₄	1	2	yes	—
51	C₄×#10596831	C₂×C₄	1	1	yes	characteristic
52	C₄×#10596831	C₂×C₄	1	2	yes	—
53	C₄×#10596831	C₂×C₄	1	1	yes	characteristic
54	C₄×#10596831	C₂³	1	2	yes	—
55	C₄×#10596831	C₂³	1	1	yes	characteristic
56	C₄×#10596831	C₂³	1	1	yes	characteristic
57	C₂×#41×C₇₃	C₂×C₄	1	1	yes	characteristic
58	C₄×Q₈×C₇₃	C₈	1	1	yes	characteristic
59	C₄×Q₈×C₇₃	C₈	1	1	yes	characteristic
60	C₄×D₄×C₇₃	C₈	1	2	yes	—
61	C₄×D₄×C₇₃	C₈	1	1	yes	characteristic
62	C₄×D₄×C₇₃	C₈	1	2	yes	—
63	C₄×D₄×C₇₃	C₈	1	1	yes	characteristic
64	#117×C₇₃	C₂×C₄	1	2	yes	—
65	#117×C₇₃	C₂×C₄	1	1	yes	characteristic
66	C₂×C₄²×C₇₃	C₂×C₄	1	2	yes	—
67	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
68	#11924031	—	2	2	no	—
69	#11924031	—	2	2	no	—
70	#11924031	—	2	4	no	—
71	#11924012	—	2	2	no	—
72	#11924012	—	2	4	no	—
73	#11924012	—	2	4	no	—
74	#11924012	M₁₆	1	4	yes	—
75	C₂×#10596831	—	2	2	no	—
76	C₂×#10596831	—	2	4	no	—
77	C₂×#10596831	—	2	2	no	—
78	C₂×#10596831	—	2	2	no	—
79	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
80	C₂×#10596831	C₂²×C₄	1	2	yes	—
81	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
82	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
83	C₂×#10596831	C₂²×C₄	1	2	yes	—
84	C₂×#10596831	#41	1	2	yes	—
85	C₂×#10596831	#41	1	2	yes	—
86	C₂×#10596831	C₂⁴	1	1	yes	characteristic
87	#41×C₇₃	—	2	2	no	—
88	#41×C₇₃	—	2	2	no	—
89	C₂×Q₈×C₇₃	C₂×C₈	1	1	yes	characteristic
90	C₂×D₄×C₇₃	C₂×C₈	1	2	yes	—
91	C₂×D₄×C₇₃	C₂×C₈	1	1	yes	characteristic
92	C₂²×C₄×C₇₃	C₂×C₈	1	2	yes	—
93	C₂²×C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
94	C₂²×C₄×C₇₃	C₂×C₈	1	2	yes	—
95	C₂²×C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
96	C₂²×C₄×C₇₃	C₂²×C₄	1	2	yes	—
97	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
98	C₄⋊C₄×C₇₃	C₂×C₈	1	2	yes	—
99	C₄⋊C₄×C₇₃	C₂×C₈	1	2	yes	—
100	C₄⋊C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
101	C₄⋊C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
102	C₂²⋊C₄×C₇₃	C₂×C₈	1	2	yes	—
103	C₂²⋊C₄×C₇₃	C₂×C₈	1	2	yes	—
104	C₂²⋊C₄×C₇₃	C₂×C₈	1	2	yes	—
105	C₄²×C₇₃	C₂×C₈	1	2	yes	—
106	C₄²×C₇₃	C₂×C₈	1	1	yes	characteristic
107	C₄²×C₇₃	C₂×C₈	1	2	yes	—
108	C₄²×C₇₃	C₂×C₈	1	1	yes	characteristic
109	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
110	C₄²×C₇₃	C₂²×C₄	1	2	yes	—
111	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
112	C₂³×C₇₃	C₂²×C₈	1	2	yes	—
113	C₂³×C₇₃	C₂²×C₈	1	1	yes	characteristic
114	Q₈×C₇₃	—	2	2	no	—
115	D₄×C₇₃	—	2	4	no	—
116	D₄×C₇₃	—	2	2	no	—
117	C₂×C₄×C₇₃	—	2	2	no	—
118	C₂×C₄×C₇₃	—	2	2	no	—
119	C₂×C₄×C₇₃	—	2	2	no	—
120	C₂×C₄×C₇₃	—	2	4	no	—
121	C₂×C₄×C₇₃	—	4	1	no	—
122	C₂×C₄×C₇₃	—	4	2	no	—
123	C₂×C₄×C₇₃	#117	1	2	yes	—
124	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
125	C₂×C₄×C₇₃	C₂²×C₈	1	2	yes	—
126	C₂×C₄×C₇₃	C₂²×C₈	1	2	yes	—
127	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
128	C₂×C₄×C₇₃	C₂²×C₈	1	2	yes	—
129	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
130	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
131	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
132	C₂×C₄×C₇₃	C₂×M₁₆	1	2	yes	—
133	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
134	#10596831	—	2	2	no	—
135	#10596831	—	2	1	no	—
136	#10596831	—	2	1	no	—
137	#10596831	#117	1	2	yes	—
138	#10596831	C₂×M₁₆	1	4	yes	—
139	#10596831	C₂×#41	1	1	yes	characteristic
140	#10596831	C₂×#41	1	1	yes	characteristic
141	C₂²×C₇₃	—	2	1	no	—
142	C₂²×C₇₃	—	2	1	no	—
143	C₂²×C₇₃	—	2	2	no	—
144	C₂²×C₇₃	—	2	2	no	—
145	C₂²×C₇₃	—	2	2	no	—
146	C₂²×C₇₃	#431	1	2	yes	—
147	C₂²×C₇₃	C₂³×C₈	1	1	yes	characteristic
148	C₄×C₇₃	—	2	1	no	—
149	C₄×C₇₃	—	2	2	no	—
150	C₄×C₇₃	—	2	1	no	—
151	C₄×C₇₃	—	2	1	no	—
152	C₄×C₇₃	—	2	2	no	—
153	C₄×C₇₃	—	2	2	no	—
154	C₄×C₇₃	#431	1	2	yes	—
155	C₄×C₇₃	#504	1	2	yes	—
156	C₄×C₇₃	C₂×#117	1	1	yes	characteristic
157	C₄×C₇₃	C₂²×M₁₆	1	1	yes	characteristic
158	#33423	—	73	1	no	maximal, Sylow
159	C₂×C₇₃	—	2	2	no	—
160	C₂×C₇₃	—	4	2	no	—
161	C₂×C₇₃	C₂×#431	1	1	yes	characteristic
162	C₂×C₇₃	C₂×#504	1	1	yes	characteristic
163	C₂×C₇₃	#3407	1	1	yes	characteristic
164	C₄×#131	—	73	1	no	—
165	#2164	—	73	4	no	—
166	#2163	—	73	2	no	—
167	#2162	—	73	2	no	—
168	#2150	—	73	1	no	—
169	C₂×C₄⋊C₁₆	—	73	1	no	—
170	#2113	—	73	2	no	—
171	#2108	—	73	2	no	—
172	C₇₃	#33423	1	1	yes	characteristic, complemented, Sylow
173	C₂×#131	—	73	1	no	—
174	C₂×#131	—	73	1	no	—
175	C₄×#41	#64408	1	1	yes	characteristic
176	C₂×M₃₂	—	73	2	no	—
177	C₂²×C₁₆	—	73	2	no	—
178	#405	—	73	1	no	—
179	#405	—	73	2	no	—
180	#405	—	73	2	no	—
181	#405	—	73	1	no	—
182	C₄×M₁₆	—	73	1	no	—
183	C₄×M₁₆	—	73	2	no	—
184	C₂×C₄×C₈	—	73	1	no	—
185	C₂×C₄×C₈	—	73	2	no	—
186	C₄⋊C₁₆	—	73	1	no	—
187	C₄⋊C₁₆	—	73	1	no	—
188	C₄⋊C₁₆	—	73	2	no	—
189	C₄⋊C₁₆	—	73	2	no	—
190	C₄⋊C₁₆	—	73	2	no	—
191	C₂²⋊C₁₆	—	73	4	no	—
192	C₂²⋊C₁₆	—	73	4	no	—
193	#346	—	73	4	no	—
194	C₄×C₁₆	—	73	4	no	—
195	C₂×#41	C₂×#64408	1	1	yes	characteristic
196	#131	—	146	2	no	—
197	#131	—	146	2	no	—
198	#131	—	146	4	no	—
199	C₂×M₁₆	—	73	1	no	—
200	C₂×M₁₆	—	73	2	no	—
201	C₂×M₁₆	—	73	2	no	—
202	C₂×M₁₆	—	73	1	no	—
203	C₂²×C₈	—	73	1	no	—
204	C₂²×C₈	—	73	2	no	—
205	C₂²×C₈	—	73	2	no	—
206	C₂²×C₈	—	73	1	no	—
207	C₄×Q₈	#10596833	1	1	yes	characteristic
208	C₄×Q₈	#10596833	1	1	yes	characteristic
209	C₄×D₄	#10596833	1	2	yes	—
210	C₄×D₄	#10596833	1	1	yes	characteristic
211	C₄×D₄	#10596833	1	2	yes	—
212	C₄×D₄	#10596833	1	1	yes	characteristic
213	#117	C₂×#64408	1	2	yes	—
214	#117	C₂×#64408	1	1	yes	characteristic
215	C₂×C₄²	C₂×#64408	1	2	yes	—
216	C₂×C₄²	C₂×#64408	1	1	yes	characteristic
217	M₃₂	—	146	4	no	—
218	C₂×C₁₆	—	73	2	no	—
219	C₂×C₁₆	—	73	4	no	—
220	C₂×C₁₆	—	73	2	no	—
221	C₂×C₁₆	—	146	4	no	—
222	#97	—	73	2	no	—
223	#97	—	73	2	no	—
224	#97	—	73	2	no	—
225	C₄×C₈	—	73	1	no	—
226	C₄×C₈	—	73	2	no	—
227	C₄×C₈	—	73	1	no	—
228	C₄×C₈	—	73	2	no	—
229	C₄×C₈	—	73	2	no	—
230	C₄×C₈	—	73	1	no	—
231	C₄×C₈	—	73	1	no	—
232	#41	—	2	2	no	—
233	#41	—	2	2	no	—
234	C₂×Q₈	C₂×#10596833	1	1	yes	characteristic
235	C₂×D₄	C₂×#10596833	1	2	yes	—
236	C₂×D₄	C₂×#10596833	1	1	yes	characteristic
237	C₂²×C₄	C₂×#10596833	1	2	yes	—
238	C₂²×C₄	C₂×#10596833	1	1	yes	characteristic
239	C₂²×C₄	C₂×#10596833	1	2	yes	—
240	C₂²×C₄	C₂×#10596833	1	1	yes	characteristic
241	C₂²×C₄	C₂²×#64408	1	2	yes	—
242	C₂²×C₄	C₂²×#64408	1	1	yes	characteristic
243	M₁₆	—	73	4	no	—
244	M₁₆	—	146	4	no	—
245	M₁₆	—	146	4	no	—
246	M₁₆	—	146	2	no	—
247	C₂×C₈	—	73	1	no	—
248	C₂×C₈	—	73	2	no	—
249	C₂×C₈	—	73	1	no	—
250	C₂×C₈	—	73	2	no	—
251	C₂×C₈	—	73	2	no	—
252	C₂×C₈	—	73	1	no	—
253	C₂×C₈	—	73	2	no	—
254	C₂×C₈	—	73	1	no	—
255	C₂×C₈	—	146	2	no	—
256	C₂×C₈	—	146	2	no	—
257	C₂×C₈	—	146	4	no	—
258	C₂×C₈	—	146	2	no	—
259	C₄⋊C₄	C₂×#10596833	1	2	yes	—
260	C₄⋊C₄	C₂×#10596833	1	2	yes	—
261	C₄⋊C₄	C₂×#10596833	1	1	yes	characteristic
262	C₄⋊C₄	C₂×#10596833	1	1	yes	characteristic
263	C₂²⋊C₄	C₂×#10596833	1	2	yes	—
264	C₂²⋊C₄	C₂×#10596833	1	2	yes	—
265	C₂²⋊C₄	C₂×#10596833	1	2	yes	—
266	C₄²	C₂×#10596833	1	2	yes	—
267	C₄²	C₂×#10596833	1	1	yes	characteristic
268	C₄²	C₂×#10596833	1	2	yes	—
269	C₄²	C₂×#10596833	1	1	yes	characteristic
270	C₄²	C₂²×#64408	1	1	yes	characteristic
271	C₄²	C₂²×#64408	1	2	yes	—
272	C₄²	C₂²×#64408	1	1	yes	characteristic
273	C₁₆	—	146	2	no	—
274	C₁₆	—	146	2	no	—
275	C₁₆	—	146	4	no	—
276	C₂³	C₂²×#10596833	1	2	yes	—
277	C₂³	C₂²×#10596833	1	1	yes	characteristic
278	Q₈	—	2	2	no	—
279	D₄	—	2	4	no	—
280	D₄	—	2	2	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	4	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	—	4	1	no	—
286	C₂×C₄	—	4	2	no	—
287	C₂×C₄	#14556670	1	2	yes	—
288	C₂×C₄	C₂²×#10596833	1	1	yes	characteristic
289	C₂×C₄	C₂²×#10596833	1	2	yes	—
290	C₂×C₄	C₂²×#10596833	1	2	yes	—
291	C₂×C₄	C₂²×#10596833	1	1	yes	characteristic
292	C₂×C₄	C₂²×#10596833	1	2	yes	—
293	C₂×C₄	C₂²×#10596833	1	1	yes	characteristic
294	C₂×C₄	C₂²×#10596833	1	1	yes	characteristic
295	C₂×C₄	C₂²×#10596833	1	1	yes	characteristic
296	C₂×C₄	D₇₃⋊M₁₆	1	2	yes	—
297	C₂×C₄	C₂³×#64408	1	1	yes	characteristic
298	C₈	—	73	1	no	—
299	C₈	—	73	1	no	—
300	C₈	—	73	2	no	—
301	C₈	—	73	4	no	—
302	C₈	—	146	2	no	—
303	C₈	—	146	1	no	—
304	C₈	—	146	1	no	—
305	C₂²	—	2	1	no	—
306	C₂²	—	2	1	no	—
307	C₂²	—	2	2	no	—
308	C₂²	—	2	2	no	—
309	C₂²	—	2	2	no	—
310	C₂²	#17989603	1	2	yes	—
311	C₂²	C₂³×#10596833	1	1	yes	characteristic
312	C₄	—	2	2	no	—
313	C₄	—	2	1	no	—
314	C₄	—	2	2	no	—
315	C₄	—	2	1	no	—
316	C₄	—	2	1	no	—
317	C₄	—	2	2	no	—
318	C₄	#17989603	1	2	yes	—
319	C₄	#17989645	1	2	yes	—
320	C₄	C₂×#14556670	1	1	yes	characteristic
321	C₄	C₂×D₇₃⋊M₁₆	1	1	yes	characteristic
322	C₂	—	2	2	no	—
323	C₂	—	4	2	no	—
324	C₂	C₂×#17989603	1	1	yes	characteristic
325	C₂	C₂×#17989645	1	1	yes	characteristic
326	C₂	#22626898	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×#64408
commutator, socle	C₂²×C₇₃	C₂³×C₈
Fitting	C₄×#41×C₇₃	C₄
prime radical	C₂×#41×C₇₃	C₂×C₄
cosocle	C₂×#10596831	C₂⁴
hypercenter	C₄×#41	#64408
hypocenter	C₇₃	#33423
LCS	G ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×#41

Complements

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

N	H ≅ G/N	type
1. #22626296	C₂	semidirect
2. #22626292	C₂	semidirect
3. #22626288	C₂	semidirect
4. #22626263	C₂	semidirect
5. C₂×C₄⋊#11924031	C₂	semidirect
7. #22626140	C₂	semidirect
172. C₇₃	#33423	semidirect

Maximal subgroups

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

subgroup	index
#22626296	2
#22626292	2
#22626288	2
#22626263	2
C₂×C₄⋊#11924031	2
#22626161	2
#22626140	2
C₄×#14556598	2
#33423	73