Subgroups — #34030329

Order 18688.

498 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22626348	C₂	1	2	yes	maximal, complemented
2	#22626348	C₂	1	2	yes	maximal, complemented
3	#22626346	C₂	1	2	yes	maximal, complemented
4	#22626345	C₂	1	2	yes	maximal, complemented
5	#22626267	C₂	1	2	yes	maximal
6	#22626261	C₂	1	1	yes	maximal, characteristic
7	#22626141	C₂	1	2	yes	maximal
8	#22626138	C₂	1	1	yes	maximal, characteristic
9	#22618862	C₂	1	1	yes	maximal, characteristic
10	C₂×#14556666	C₂²	1	1	yes	characteristic
11	C₂×#14556666	C₂²	1	1	yes	characteristic
12	C₂×#14556664	C₂²	1	2	yes	—
13	#17989508	C₂²	1	1	yes	characteristic
14	#17989508	C₂²	1	2	yes	—
15	#17989508	C₂²	1	1	yes	characteristic
16	#17989508	C₂²	1	2	yes	—
17	C₄⋊#11924031	C₂²	1	2	yes	—
18	C₄⋊#11924031	C₂²	1	2	yes	—
19	C₄⋊#11924031	C₂²	1	2	yes	—
20	#17989476	C₂²	1	2	yes	—
21	#17989475	C₂²	1	2	yes	—
22	#17989475	C₂²	1	2	yes	—
23	#17989473	C₂²	1	2	yes	—
24	#17989473	C₂²	1	2	yes	—
25	C₄×#11924031	C₂²	1	2	yes	—
26	#17989141	C₄	1	1	yes	characteristic
27	#17988717	C₄	1	1	yes	characteristic
28	#17988715	C₄	1	2	yes	—
29	Q₈×#10596831	C₂²	1	1	yes	characteristic
30	#17988711	C₂²	1	2	yes	—
31	#17988711	C₂²	1	1	yes	characteristic
32	#17988612	C₄	1	2	yes	—
33	#17988609	C₄	1	1	yes	characteristic
34	#17988544	C₂²	1	2	yes	—
35	C₄×C₈×D₇₃	C₂²	1	1	yes	characteristic
36	#446×C₇₃	C₄	1	1	yes	characteristic
37	#14556666	—	2	2	no	—
38	#14556666	Q₈	1	4	yes	complemented
39	#14556664	—	2	4	no	—
40	C₂×#11924031	C₂³	1	2	yes	—
41	C₂×#11924031	C₂³	1	2	yes	—
42	C₂×#11924031	C₂³	1	2	yes	—
43	C₂×#11924031	C₂³	1	2	yes	—
44	C₂×#11924011	C₂×C₄	1	2	yes	—
45	C₂×C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
46	C₂×C₈×D₇₃	C₂³	1	1	yes	characteristic
47	#14556590	C₈	1	1	yes	characteristic
48	#14556589	C₈	1	1	yes	characteristic
49	#14556581	C₈	1	2	yes	—
50	#14556571	C₈	1	1	yes	characteristic
51	#14556569	C₈	1	2	yes	—
52	#14556567	C₈	1	1	yes	characteristic
53	C₄×Dic₁₄₆	C₂×C₄	1	2	yes	—
54	C₄×Dic₁₄₆	C₂×C₄	1	1	yes	characteristic
55	#14556548	C₂×C₄	1	2	yes	—
56	C₄²×D₇₃	C₂×C₄	1	1	yes	characteristic
57	C₄⋊#10596831	C₂×C₄	1	1	yes	characteristic
58	C₄⋊#10596831	C₂×C₄	1	2	yes	—
59	#14556520	C₂×C₄	1	2	yes	—
60	#14556520	C₂×C₄	1	2	yes	—
61	#14556520	C₂×C₄	1	2	yes	—
62	#14556501	C₂×C₄	1	2	yes	—
63	C₄×#10596831	C₂³	1	2	yes	—
64	C₄×#10596831	C₂³	1	1	yes	characteristic
65	C₈×Dic₇₃	C₂×C₄	1	1	yes	characteristic
66	C₈×Dic₇₃	C₂³	1	1	yes	characteristic
67	C₈×Dic₇₃	C₂³	1	2	yes	—
68	C₄×Q₈×C₇₃	C₂×C₄	1	1	yes	characteristic
69	C₄⋊C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
70	C₄⋊C₈×C₇₃	C₂×C₄	1	2	yes	—
71	#97×C₇₃	C₂×C₄	1	2	yes	—
72	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
73	#11924031	—	2	2	no	—
74	#11924031	—	2	2	no	—
75	#11924031	—	2	2	no	—
76	#11924031	C₂×Q₈	1	4	yes	—
77	C₂×Dic₁₄₆	C₂×C₈	1	2	yes	—
78	C₂×Dic₁₄₆	C₂×C₈	1	1	yes	characteristic
79	C₂×C₄×D₇₃	C₂×C₈	1	1	yes	characteristic
80	C₂×C₄×D₇₃	C₂×C₈	1	1	yes	characteristic
81	C₂×C₄×D₇₃	C₂²×C₄	1	1	yes	characteristic
82	#11924011	—	2	4	no	—
83	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
84	C₂×#10596831	C₂²×C₄	1	2	yes	—
85	C₂×#10596831	C₂⁴	1	1	yes	characteristic
86	C₈×D₇₃	—	2	2	no	—
87	C₈×D₇₃	C₂×Q₈	1	2	yes	—
88	C₈×D₇₃	#41	1	2	yes	—
89	C₄⋊Dic₇₃	C₂×C₈	1	1	yes	characteristic
90	C₄⋊Dic₇₃	C₂×C₈	1	2	yes	—
91	#11924007	C₂×C₈	1	2	yes	—
92	#11924007	C₂×C₈	1	2	yes	—
93	#11924007	C₂×C₈	1	2	yes	—
94	#11924007	C₂×C₈	1	2	yes	—
95	#11924007	C₂×C₈	1	2	yes	—
96	#11924005	C₂×C₈	1	2	yes	—
97	#11924005	C₂×C₈	1	2	yes	—
98	C₄×Dic₇₃	C₂×C₈	1	1	yes	characteristic
99	C₄×Dic₇₃	C₂×C₈	1	1	yes	characteristic
100	C₄×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
101	C₄×Dic₇₃	C₂²×C₄	1	2	yes	—
102	C₂×Q₈×C₇₃	C₂×C₈	1	1	yes	characteristic
103	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
104	C₂×C₈×C₇₃	C₂²×C₄	1	2	yes	—
105	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
106	C₄⋊C₄×C₇₃	C₂×C₈	1	2	yes	—
107	C₄⋊C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
108	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
109	C₄²×C₇₃	C₂²×C₄	1	2	yes	—
110	C₂²×D₇₃	C₂²×C₈	1	1	yes	characteristic
111	Q₈×C₇₃	—	4	1	no	—
112	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
113	C₂×C₄×C₇₃	C₂²×C₈	1	2	yes	—
114	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
115	C₂×C₄×C₇₃	C₂²×C₈	1	2	yes	—
116	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
117	C₂×Dic₇₃	C₂²×C₈	1	1	yes	characteristic
118	C₂×Dic₇₃	C₂²×C₈	1	2	yes	—
119	C₂×Dic₇₃	C₂²×C₈	1	2	yes	—
120	C₂×Dic₇₃	C₂²×C₈	1	1	yes	characteristic
121	C₂×Dic₇₃	C₂²×C₈	1	1	yes	characteristic
122	C₄×D₇₃	—	2	2	no	—
123	C₄×D₇₃	—	4	1	no	—
124	C₄×D₇₃	C₄×Q₈	1	2	yes	—
125	C₄×D₇₃	#131	1	2	yes	—
126	Dic₁₄₆	—	2	4	no	—
127	Dic₁₄₆	—	4	1	no	—
128	C₈×C₇₃	—	2	1	no	—
129	C₈×C₇₃	—	2	2	no	—
130	C₈×C₇₃	C₄×Q₈	1	2	yes	—
131	#10596831	—	2	1	no	—
132	#10596831	—	2	2	no	—
133	#10596831	C₂²×Q₈	1	1	yes	characteristic
134	#10596831	C₂×#41	1	1	yes	characteristic
135	C₂²×C₇₃	C₂³×C₈	1	1	yes	characteristic
136	C₂×D₇₃	—	2	1	no	—
137	C₂×D₇₃	—	2	1	no	—
138	C₂×D₇₃	C₈×Q₈	1	2	yes	—
139	C₄×C₇₃	—	2	2	no	—
140	C₄×C₇₃	—	2	1	no	—
141	C₄×C₇₃	—	2	1	no	—
142	C₄×C₇₃	—	2	2	no	—
143	C₄×C₇₃	C₂×C₄×Q₈	1	1	yes	characteristic
144	C₄×C₇₃	C₂×#131	1	1	yes	characteristic
145	Dic₇₃	—	2	2	no	—
146	Dic₇₃	—	2	2	no	—
147	Dic₇₃	—	2	1	no	—
148	Dic₇₃	C₈×Q₈	1	2	yes	—
149	Dic₇₃	#504	1	2	yes	—
150	#33476	—	73	1	no	maximal, Sylow
151	C₂×C₇₃	C₂×C₈×Q₈	1	1	yes	characteristic
152	C₂×C₇₃	C₂×#504	1	1	yes	characteristic
153	C₂×C₇₃	#3407	1	1	yes	characteristic
154	D₇₃	—	2	2	no	—
155	#2960	—	73	1	no	—
156	#2183	—	73	2	no	—
157	#2183	—	73	2	no	—
158	#2183	—	73	2	no	—
159	Q₈×C₁₆	—	73	2	no	—
160	#2151	—	73	2	no	—
161	C₂×C₄⋊C₁₆	—	73	1	no	—
162	#2108	—	73	2	no	—
163	C₄×M₃₂	—	73	1	no	—
164	C₇₃	#33476	1	1	yes	characteristic, complemented, Sylow
165	#533	—	73	1	no	—
166	C₂×M₃₂	—	73	1	no	—
167	C₂×M₃₂	—	73	1	no	—
168	C₂²×C₁₆	—	73	2	no	—
169	#446	—	73	2	no	—
170	#446	—	73	1	no	—
171	#446	#64408	1	1	yes	characteristic
172	C₈×Q₈	—	73	1	no	—
173	C₈×Q₈	—	73	2	no	—
174	C₈×Q₈	—	73	1	no	—
175	#424	—	73	2	no	—
176	#423	—	73	1	no	—
177	#405	—	73	2	no	—
178	C₂×C₄×C₈	—	73	1	no	—
179	C₄⋊C₁₆	—	73	2	no	—
180	C₄⋊C₁₆	—	73	1	no	—
181	C₄⋊C₁₆	—	73	1	no	—
182	C₄⋊C₁₆	—	73	2	no	—
183	C₄⋊C₁₆	—	73	2	no	—
184	C₄⋊C₁₆	—	73	2	no	—
185	C₄⋊C₁₆	—	73	2	no	—
186	#346	—	73	2	no	—
187	#346	—	73	2	no	—
188	#346	—	73	2	no	—
189	C₄×C₁₆	—	73	2	no	—
190	C₄×C₁₆	—	73	2	no	—
191	C₄×C₁₆	—	73	2	no	—
192	C₂×M₁₆	—	73	2	no	—
193	C₂²×C₈	—	73	1	no	—
194	C₂²×C₈	—	73	1	no	—
195	C₄⋊Q₈	—	73	1	no	—
196	C₄⋊Q₈	—	73	1	no	—
197	#125	—	73	2	no	—
198	C₂²⋊Q₈	—	73	1	no	—
199	C₂²⋊Q₈	—	73	2	no	—
200	C₂²⋊Q₈	—	73	1	no	—
201	C₄×Q₈	—	73	2	no	—
202	C₄×Q₈	—	73	1	no	—
203	C₄×Q₈	C₂×#64408	1	1	yes	characteristic
204	#117	—	73	2	no	—
205	C₂×C₄²	—	73	1	no	—
206	M₃₂	—	73	4	no	—
207	M₃₂	—	146	2	no	—
208	C₂×C₁₆	—	73	2	no	—
209	C₂×C₁₆	—	73	2	no	—
210	C₂×C₁₆	—	73	2	no	—
211	C₂×C₁₆	—	73	2	no	—
212	C₂×C₁₆	—	146	4	no	—
213	C₄⋊C₈	—	73	2	no	—
214	C₄⋊C₈	—	73	2	no	—
215	C₄⋊C₈	—	73	2	no	—
216	C₄⋊C₈	—	73	1	no	—
217	C₄⋊C₈	—	73	2	no	—
218	C₄⋊C₈	C₂×#64408	1	1	yes	characteristic
219	C₄⋊C₈	C₂×#64408	1	2	yes	—
220	#97	—	73	2	no	—
221	#97	C₂×#64408	1	2	yes	—
222	C₄×C₈	—	73	1	no	—
223	C₄×C₈	—	73	1	no	—
224	C₄×C₈	—	73	2	no	—
225	C₄×C₈	—	73	2	no	—
226	C₄×C₈	—	73	1	no	—
227	C₄×C₈	C₂×#64408	1	1	yes	characteristic
228	C₂×Q₈	—	73	1	no	—
229	C₂×Q₈	—	73	2	no	—
230	C₂×Q₈	#11924036	1	1	yes	characteristic
231	C₂²×C₄	—	73	1	no	—
232	C₂²×C₄	—	73	1	no	—
233	C₂²×C₄	—	73	1	no	—
234	M₁₆	—	146	4	no	—
235	C₂×C₈	—	73	1	no	—
236	C₂×C₈	—	73	2	no	—
237	C₂×C₈	—	73	2	no	—
238	C₂×C₈	—	73	1	no	—
239	C₂×C₈	—	73	2	no	—
240	C₂×C₈	—	146	2	no	—
241	C₂×C₈	C₂²×#64408	1	1	yes	characteristic
242	C₂×C₈	C₂²×#64408	1	2	yes	—
243	C₂×C₈	C₂²×#64408	1	1	yes	characteristic
244	C₄⋊C₄	—	73	2	no	—
245	C₄⋊C₄	—	73	2	no	—
246	C₄⋊C₄	—	73	2	no	—
247	C₄⋊C₄	—	73	2	no	—
248	C₄⋊C₄	—	73	2	no	—
249	C₄⋊C₄	—	73	2	no	—
250	C₄⋊C₄	—	73	1	no	—
251	C₄⋊C₄	#11924036	1	2	yes	—
252	C₄⋊C₄	#11924036	1	1	yes	characteristic
253	C₂²⋊C₄	—	73	2	no	—
254	C₂²⋊C₄	—	73	2	no	—
255	C₄²	—	73	1	no	—
256	C₄²	—	73	1	no	—
257	C₄²	—	73	1	no	—
258	C₄²	—	73	2	no	—
259	C₄²	C₂²×#64408	1	1	yes	characteristic
260	C₄²	C₂²×#64408	1	2	yes	—
261	C₁₆	—	73	4	no	—
262	C₁₆	—	146	2	no	—
263	C₁₆	—	146	2	no	—
264	C₁₆	—	146	2	no	—
265	C₂³	—	73	1	no	—
266	Q₈	—	4	1	no	—
267	Q₈	—	292	1	no	—
268	Q₈	—	146	4	no	—
269	C₂×C₄	—	292	1	no	—
270	C₂×C₄	—	73	1	no	—
271	C₂×C₄	—	73	1	no	—
272	C₂×C₄	—	73	2	no	—
273	C₂×C₄	—	73	2	no	—
274	C₂×C₄	—	73	1	no	—
275	C₂×C₄	—	73	2	no	—
276	C₂×C₄	—	73	2	no	—
277	C₂×C₄	—	146	2	no	—
278	C₂×C₄	C₂×#11924036	1	1	yes	characteristic
279	C₂×C₄	C₂×#11924036	1	1	yes	characteristic
280	C₂×C₄	C₂×#11924036	1	2	yes	—
281	C₂×C₄	C₂×#11924036	1	2	yes	—
282	C₂×C₄	C₂³×#64408	1	1	yes	characteristic
283	C₈	—	2	1	no	—
284	C₈	—	2	2	no	—
285	C₈	—	73	1	no	—
286	C₈	—	73	1	no	—
287	C₈	—	146	1	no	—
288	C₈	—	146	2	no	—
289	C₈	Q₈×#64408	1	2	yes	—
290	C₂²	—	73	2	no	—
291	C₂²	—	146	1	no	—
292	C₂²	—	146	1	no	—
293	C₂²	C₂²×#11924036	1	1	yes	characteristic
294	C₄	—	2	1	no	—
295	C₄	—	2	2	no	—
296	C₄	—	2	1	no	—
297	C₄	—	2	2	no	—
298	C₄	—	73	2	no	—
299	C₄	—	73	2	no	—
300	C₄	—	146	2	no	—
301	C₄	—	146	1	no	—
302	C₄	—	146	2	no	—
303	C₄	C₂×Q₈×#64408	1	1	yes	characteristic
304	C₄	D₇₃⋊#131	1	1	yes	characteristic
305	C₂	—	146	2	no	—
306	C₂	Q₈×#11924036	1	1	yes	characteristic
307	C₂	#22626896	1	1	yes	characteristic
308	C₂	#22626900	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×#64408
commutator, socle	C₂²×C₇₃	C₂³×C₈
Fitting	#446×C₇₃	C₄
prime radical	C₂²×D₇₃	C₂²×C₈
cosocle	C₂×#10596831	C₂⁴
hypercenter	#446	#64408
hypocenter	C₇₃	#33476
LCS	G ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ #446

Complements

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

N	H ≅ G/N	type
1. #22626348	C₂	semidirect
2. #22626348	C₂	semidirect
3. #22626346	C₂	semidirect
4. #22626345	C₂	semidirect
38. #14556666	Q₈	semidirect
164. C₇₃	#33476	semidirect

Maximal subgroups

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

subgroup	index
#22626348	2
#22626346	2
#22626345	2
#22626267	2
#22626261	2
#22626141	2
#22626138	2
#22618862	2
#33476	73