Subgroups — #33102756

Order 18688.

970 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17989316	C₂	1	1	yes	maximal, characteristic
2	Q₈×#11924012	C₂	1	1	yes	maximal, characteristic
3	#22610775	C₂	1	3	yes	maximal
4	#22610774	C₂	1	3	yes	maximal
5	#22609924	C₂	1	3	yes	maximal
6	C₄×#14556509	C₂	1	3	yes	maximal
7	Q₈×M₁₆×C₇₃	C₂	1	1	yes	maximal, characteristic
8	#17989316	—	2	1	no	—
9	#17989316	—	2	3	no	—
10	#17989316	—	2	1	no	—
11	#17989316	—	2	3	no	—
12	C₂×#14556591	C₂²	1	3	yes	—
13	C₂×#14556582	C₄	1	6	yes	—
14	C₂×Q₈×Dic₇₃	C₄	1	2	yes	—
15	C₂×C₄×Dic₁₄₆	C₂²	1	3	yes	—
16	#17988718	—	2	3	no	—
17	Q₈×#10596831	—	2	1	no	—
18	#17988611	C₂²	1	3	yes	—
19	C₂×#14556509	C₂²	1	3	yes	—
20	C₂×#14556509	C₂²	1	3	yes	—
21	C₂×#14556509	C₂²	1	6	yes	—
22	C₄×#11924012	C₂²	1	3	yes	—
23	#17988355	C₂²	1	3	yes	—
24	#17988355	C₂²	1	1	yes	characteristic
25	C₂×C₄×Q₈×C₇₃	C₂²	1	1	yes	characteristic
26	#446×C₇₃	—	2	3	no	—
27	C₈×Q₈×C₇₃	—	2	1	no	—
28	#423×C₇₃	C₂²	1	3	yes	—
29	C₄×M₁₆×C₇₃	C₂²	1	3	yes	—
30	C₂²×Dic₁₄₆	C₂³	1	3	yes	—
31	C₂×#11924012	C₂³	1	3	yes	—
32	C₂×#11924012	C₂³	1	1	yes	characteristic
33	#14556591	—	2	3	no	—
34	#14556591	—	2	3	no	—
35	#14556591	—	2	3	no	—
36	#14556591	—	2	3	no	—
37	#14556582	—	2	12	no	—
38	#14556582	—	2	6	no	—
39	#14556582	—	2	6	no	—
40	Q₈×Dic₇₃	—	2	6	no	—
41	Q₈×Dic₇₃	—	2	2	no	—
42	C₄×Dic₁₄₆	—	2	3	no	—
43	C₄×Dic₁₄₆	—	2	3	no	—
44	C₄×Dic₁₄₆	—	2	3	no	—
45	C₄×Dic₁₄₆	—	2	3	no	—
46	C₂×C₄⋊Dic₇₃	C₂×C₄	1	6	yes	—
47	C₂×C₄⋊Dic₇₃	C₂×C₄	1	6	yes	—
48	C₂×C₄⋊Dic₇₃	C₂³	1	1	yes	characteristic
49	C₂×#11924007	C₂×C₄	1	6	yes	—
50	C₂×C₄×Dic₇₃	C₂×C₄	1	6	yes	—
51	C₄⋊#10596831	—	2	3	no	—
52	C₄⋊#10596831	—	2	3	no	—
53	#14556509	—	2	6	no	—
54	#14556509	—	2	6	no	—
55	#14556509	—	4	6	no	—
56	#14556502	—	2	3	no	—
57	C₄×#10596831	—	2	3	no	—
58	C₂²×Q₈×C₇₃	C₂×C₄	1	1	yes	characteristic
59	C₂×M₁₆×C₇₃	C₂³	1	3	yes	—
60	C₂×M₁₆×C₇₃	C₂³	1	1	yes	characteristic
61	C₄×Q₈×C₇₃	D₄	1	1	yes	characteristic
62	C₄×Q₈×C₇₃	D₄	1	3	yes	—
63	C₄×Q₈×C₇₃	D₄	1	1	yes	characteristic
64	C₄×Q₈×C₇₃	D₄	1	3	yes	—
65	C₂×C₄⋊C₄×C₇₃	C₂×C₄	1	3	yes	—
66	C₂×C₄²×C₇₃	C₂³	1	3	yes	—
67	C₄⋊C₈×C₇₃	—	2	3	no	—
68	C₄⋊C₈×C₇₃	—	2	3	no	—
69	#97×C₇₃	—	2	3	no	—
70	C₄×C₈×C₇₃	—	2	3	no	—
71	C₂×Dic₁₄₆	—	2	3	no	—
72	C₂×Dic₁₄₆	—	2	3	no	—
73	C₂×Dic₁₄₆	—	2	3	no	—
74	C₂×Dic₁₄₆	—	2	3	no	—
75	C₂×Dic₁₄₆	—	2	3	no	—
76	C₂×Dic₁₄₆	—	2	3	no	—
77	C₂²×Dic₇₃	C₂²×C₄	1	6	yes	—
78	C₂²×Dic₇₃	C₂²×C₄	1	2	yes	—
79	#11924012	—	2	1	no	—
80	#11924012	—	2	3	no	—
81	#11924012	—	2	3	no	—
82	#11924012	#41	1	2	yes	—
83	C₂×#10596831	—	2	1	no	—
84	C₂×#10596831	—	2	3	no	—
85	C₄⋊Dic₇₃	—	2	6	no	—
86	C₄⋊Dic₇₃	—	2	6	no	—
87	C₄⋊Dic₇₃	—	2	6	no	—
88	C₄⋊Dic₇₃	—	2	6	no	—
89	C₄⋊Dic₇₃	—	2	1	no	—
90	C₄⋊Dic₇₃	—	2	1	no	—
91	#11924007	—	2	6	no	—
92	#11924007	—	2	6	no	—
93	C₄×Dic₇₃	—	2	6	no	—
94	C₄×Dic₇₃	—	2	6	no	—
95	C₂×Q₈×C₇₃	—	2	4	no	—
96	C₂×Q₈×C₇₃	C₂²⋊C₄	1	3	yes	—
97	C₂×Q₈×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
98	C₂²×C₄×C₇₃	C₂²×C₄	1	3	yes	—
99	C₂²×C₄×C₇₃	C₂²×C₄	1	3	yes	—
100	C₂²×C₄×C₇₃	C₂⁴	1	1	yes	characteristic
101	M₁₆×C₇₃	—	2	1	no	—
102	M₁₆×C₇₃	—	2	3	no	—
103	M₁₆×C₇₃	—	2	3	no	—
104	M₁₆×C₇₃	#41	1	2	yes	—
105	C₂×C₈×C₇₃	—	2	1	no	—
106	C₂×C₈×C₇₃	—	2	3	no	—
107	C₄⋊C₄×C₇₃	C₂²⋊C₄	1	6	yes	—
108	C₄⋊C₄×C₇₃	C₂²⋊C₄	1	3	yes	—
109	C₄⋊C₄×C₇₃	C₂²⋊C₄	1	3	yes	—
110	C₄²×C₇₃	C₂×D₄	1	3	yes	—
111	C₄²×C₇₃	C₂×D₄	1	3	yes	—
112	C₄²×C₇₃	C₂×D₄	1	3	yes	—
113	C₄²×C₇₃	C₂×D₄	1	3	yes	—
114	C₂³×C₇₃	C₂³×C₄	1	1	yes	characteristic
115	Q₈×C₇₃	—	2	6	no	—
116	Q₈×C₇₃	#101	1	4	yes	complemented
117	C₂×C₄×C₇₃	—	2	6	no	—
118	C₂×C₄×C₇₃	—	4	3	no	—
119	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	3	yes	—
120	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	3	yes	—
121	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	3	yes	—
122	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	3	yes	—
123	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
124	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
125	C₂×C₄×C₇₃	C₂×#41	1	1	yes	characteristic
126	C₂×C₄×C₇₃	C₂×#41	1	1	yes	characteristic
127	C₂×C₄×C₇₃	H(5,𝔽₂)	1	1	yes	characteristic
128	C₂×C₄×C₇₃	2₋¹⁺⁴	1	1	yes	characteristic
129	C₂×Dic₇₃	—	2	2	no	—
130	C₂×Dic₇₃	—	2	6	no	—
131	C₂×Dic₇₃	—	2	6	no	—
132	C₂×Dic₇₃	—	2	6	no	—
133	C₂×Dic₇₃	—	4	2	no	—
134	Dic₁₄₆	—	4	3	no	—
135	Dic₁₄₆	—	4	3	no	—
136	Dic₁₄₆	—	4	3	no	—
137	Dic₁₄₆	—	4	3	no	—
138	C₈×C₇₃	—	2	2	no	—
139	C₈×C₇₃	—	4	3	no	—
140	#10596831	—	2	2	no	—
141	#10596831	—	4	3	no	—
142	C₂²×C₇₃	—	2	1	no	—
143	C₂²×C₇₃	—	2	1	no	—
144	C₂²×C₇₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
145	C₂²×C₇₃	C₄×#41	1	1	yes	characteristic
146	C₂²×C₇₃	#520	1	1	yes	characteristic
147	C₄×C₇₃	—	2	3	no	—
148	C₄×C₇₃	—	2	3	no	—
149	C₄×C₇₃	—	2	3	no	—
150	C₄×C₇₃	C₂×#101	1	6	yes	—
151	C₄×C₇₃	#548	1	1	yes	characteristic
152	C₄×C₇₃	#548	1	1	yes	characteristic
153	C₄×C₇₃	#550	1	1	yes	characteristic
154	C₄×C₇₃	#550	1	1	yes	characteristic
155	Dic₇₃	—	4	6	no	—
156	Dic₇₃	—	4	2	no	—
157	#22538	—	73	1	no	maximal, Sylow
158	C₂×C₇₃	—	2	1	no	—
159	C₂×C₇₃	—	2	1	no	—
160	C₂×C₇₃	#2341	1	1	yes	characteristic
161	C₂×C₇₃	C₂²×#101	1	1	yes	characteristic
162	C₂×C₇₃	#2889	1	1	yes	characteristic
163	C₂×#557	—	73	1	no	—
164	Q₈×M₁₆	—	73	1	no	—
165	Q₈×M₁₆	D₇₃	1	1	yes	characteristic
166	#1966	—	73	3	no	—
167	#1966	—	73	3	no	—
168	#1801	—	73	3	no	—
169	C₄×#101	—	73	3	no	—
170	C₇₃	#22538	1	1	yes	characteristic, complemented, Sylow
171	#557	—	146	1	no	—
172	#557	—	146	3	no	—
173	#557	—	146	1	no	—
174	#557	—	146	3	no	—
175	C₂×C₄⋊Q₈	—	73	3	no	—
176	C₂×#125	—	73	6	no	—
177	C₂×C₄×Q₈	—	73	2	no	—
178	C₂×C₄×Q₈	—	73	3	no	—
179	C₂×C₄×Q₈	C₂×D₇₃	1	1	yes	characteristic
180	#446	—	2	3	no	—
181	#446	—	146	3	no	—
182	C₈×Q₈	—	2	1	no	—
183	C₈×Q₈	—	146	1	no	—
184	#423	—	73	3	no	—
185	#423	C₂×D₇₃	1	3	yes	—
186	C₂×#101	—	73	6	no	—
187	C₂×#101	—	73	3	no	—
188	C₂×#101	—	73	3	no	—
189	C₄×M₁₆	—	73	3	no	—
190	C₄×M₁₆	C₂×D₇₃	1	3	yes	—
191	#343	—	73	3	no	—
192	#343	—	73	1	no	—
193	C₂²×Q₈	—	73	3	no	—
194	C₂²×Q₈	C₄×D₇₃	1	1	yes	characteristic
195	C₂×M₁₆	—	73	1	no	—
196	C₂×M₁₆	—	73	3	no	—
197	C₂×M₁₆	C₂²×D₇₃	1	3	yes	—
198	C₂×M₁₆	C₂²×D₇₃	1	1	yes	characteristic
199	C₄⋊Q₈	—	146	3	no	—
200	C₄⋊Q₈	—	146	3	no	—
201	C₄⋊Q₈	—	146	3	no	—
202	C₄⋊Q₈	—	146	3	no	—
203	#125	—	146	12	no	—
204	#125	—	146	6	no	—
205	#125	—	146	6	no	—
206	C₄×Q₈	—	146	6	no	—
207	C₄×Q₈	—	146	2	no	—
208	C₄×Q₈	—	146	3	no	—
209	C₄×Q₈	—	146	3	no	—
210	C₄×Q₈	—	146	3	no	—
211	C₄×Q₈	—	146	3	no	—
212	C₄×Q₈	D₂₉₂	1	1	yes	characteristic
213	C₄×Q₈	D₂₉₂	1	3	yes	—
214	C₄×Q₈	C₂²⋊D₇₃	1	1	yes	characteristic
215	C₄×Q₈	C₂²⋊D₇₃	1	3	yes	—
216	C₂×C₄⋊C₄	—	73	6	no	—
217	C₂×C₄⋊C₄	—	73	6	no	—
218	C₂×C₄⋊C₄	—	73	6	no	—
219	C₂×C₄⋊C₄	—	73	1	no	—
220	C₂×C₄⋊C₄	C₄×D₇₃	1	3	yes	—
221	C₂×C₄²	—	73	6	no	—
222	C₂×C₄²	C₂²×D₇₃	1	3	yes	—
223	C₄⋊C₈	—	2	3	no	—
224	C₄⋊C₈	—	2	3	no	—
225	C₄⋊C₈	—	146	3	no	—
226	C₄⋊C₈	—	146	3	no	—
227	#101	—	292	6	no	—
228	#101	—	146	6	no	—
229	#101	—	146	6	no	—
230	#97	—	2	3	no	—
231	#97	—	146	3	no	—
232	C₄×C₈	—	2	3	no	—
233	C₄×C₈	—	146	3	no	—
234	C₂×Q₈	—	2	4	no	—
235	C₂×Q₈	—	146	3	no	—
236	C₂×Q₈	—	146	3	no	—
237	C₂×Q₈	—	146	3	no	—
238	C₂×Q₈	—	146	3	no	—
239	C₂×Q₈	—	146	3	no	—
240	C₂×Q₈	—	146	3	no	—
241	C₂×Q₈	#11924005	1	3	yes	—
242	C₂×Q₈	#11924005	1	1	yes	characteristic
243	C₂²×C₄	—	73	2	no	—
244	C₂²×C₄	—	73	6	no	—
245	C₂²×C₄	C₂×C₄×D₇₃	1	3	yes	—
246	C₂²×C₄	C₂×C₄×D₇₃	1	3	yes	—
247	C₂²×C₄	C₂³×D₇₃	1	1	yes	characteristic
248	M₁₆	—	2	1	no	—
249	M₁₆	—	2	3	no	—
250	M₁₆	—	2	3	no	—
251	M₁₆	—	73	2	no	—
252	M₁₆	—	146	3	no	—
253	M₁₆	—	146	3	no	—
254	M₁₆	—	146	1	no	—
255	M₁₆	#11924029	1	2	yes	—
256	C₂×C₈	—	2	1	no	—
257	C₂×C₈	—	2	3	no	—
258	C₂×C₈	—	146	3	no	—
259	C₂×C₈	—	146	1	no	—
260	C₄⋊C₄	—	146	6	no	—
261	C₄⋊C₄	—	146	6	no	—
262	C₄⋊C₄	—	146	6	no	—
263	C₄⋊C₄	—	146	6	no	—
264	C₄⋊C₄	—	146	6	no	—
265	C₄⋊C₄	—	146	6	no	—
266	C₄⋊C₄	—	146	1	no	—
267	C₄⋊C₄	—	146	1	no	—
268	C₄⋊C₄	#11924005	1	6	yes	—
269	C₄⋊C₄	#11924005	1	3	yes	—
270	C₄⋊C₄	#11924005	1	3	yes	—
271	C₄²	—	146	6	no	—
272	C₄²	—	146	6	no	—
273	C₄²	C₂×C₂²⋊D₇₃	1	3	yes	—
274	C₄²	C₂×C₂²⋊D₇₃	1	3	yes	—
275	C₄²	C₂×D₂₉₂	1	3	yes	—
276	C₄²	C₂×D₂₉₂	1	3	yes	—
277	C₂³	C₂²×C₄×D₇₃	1	1	yes	characteristic
278	Q₈	—	2	6	no	—
279	Q₈	—	292	3	no	—
280	Q₈	—	292	3	no	—
281	Q₈	—	292	3	no	—
282	Q₈	—	292	3	no	—
283	Q₈	#14556509	1	4	yes	complemented
284	C₂×C₄	—	2	6	no	—
285	C₂×C₄	—	4	3	no	—
286	C₂×C₄	—	292	2	no	—
287	C₂×C₄	—	146	2	no	—
288	C₂×C₄	—	146	6	no	—
289	C₂×C₄	—	146	6	no	—
290	C₂×C₄	—	146	6	no	—
291	C₂×C₄	C₂×#11924005	1	3	yes	—
292	C₂×C₄	C₂×#11924005	1	3	yes	—
293	C₂×C₄	C₂×#11924005	1	3	yes	—
294	C₂×C₄	C₂×#11924005	1	3	yes	—
295	C₂×C₄	C₂²×C₂²⋊D₇₃	1	1	yes	characteristic
296	C₂×C₄	C₂²×D₂₉₂	1	1	yes	characteristic
297	C₂×C₄	#41×D₇₃	1	1	yes	characteristic
298	C₂×C₄	C₂×#11924029	1	1	yes	characteristic
299	C₂×C₄	#14556637	1	1	yes	characteristic
300	C₂×C₄	#14556639	1	1	yes	characteristic
301	C₈	—	2	2	no	—
302	C₈	—	4	3	no	—
303	C₈	—	292	3	no	—
304	C₈	—	146	2	no	—
305	C₂²	—	2	1	no	—
306	C₂²	—	2	1	no	—
307	C₂²	C₂²×#11924005	1	1	yes	characteristic
308	C₂²	C₄×#11924029	1	1	yes	characteristic
309	C₂²	#17989069	1	1	yes	characteristic
310	C₄	—	2	3	no	—
311	C₄	—	2	3	no	—
312	C₄	—	2	3	no	—
313	C₄	—	292	6	no	—
314	C₄	—	292	2	no	—
315	C₄	C₂×#14556509	1	6	yes	—
316	C₄	#17989248	1	1	yes	characteristic
317	C₄	#17989249	1	1	yes	characteristic
318	C₄	#17989256	1	1	yes	characteristic
319	C₄	#17989259	1	1	yes	characteristic
320	C₂	—	2	1	no	—
321	C₂	—	2	1	no	—
322	C₂	#22612769	1	1	yes	characteristic
323	C₂	C₂²×#14556509	1	1	yes	characteristic
324	C₂	#22618413	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×#11924005
commutator, prime radical	C₂³×C₇₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₇₃
Fitting	Q₈×M₁₆×C₇₃	C₂
socle	C₂²×C₇₃	C₂²×C₂²⋊C₄
cosocle	C₂²×C₄×C₇₃	C₂⁴
hypercenter	Q₈×M₁₆	D₇₃
hypocenter	C₇₃	#22538
LCS	G ▷ C₂³×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄×Q₈ ◁ Q₈×M₁₆

Complements

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

N	H ≅ G/N	type
116. Q₈×C₇₃	#101	semidirect
170. C₇₃	#22538	semidirect
283. Q₈	#14556509	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17989316	2
Q₈×#11924012	2
#22610775	2
#22610774	2
#22609924	2
C₄×#14556509	2
Q₈×M₁₆×C₇₃	2
#22538	73