Subgroups — #34019413

Order 18688.

638 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22626733	C₂	1	2	yes	maximal
2	#22625571	C₂	1	2	yes	maximal, complemented
3	#22625545	C₂	1	4	yes	maximal, complemented
4	#22625519	C₂	1	2	yes	maximal, complemented
5	#22625489	C₂	1	1	yes	maximal, characteristic
6	#22625260	C₂	1	2	yes	maximal
7	C₈×#11924038	C₂	1	1	yes	maximal, characteristic
8	C₄×Q₁₆×D₇₃	C₂	1	1	yes	maximal, characteristic
9	#17989639	—	2	4	no	—
10	#17989638	C₂²	1	4	yes	—
11	#17989636	C₂²	1	4	yes	—
12	#17989606	C₂²	1	1	yes	characteristic
13	#17989592	—	2	2	no	—
14	#17989591	C₂²	1	2	yes	—
15	C₄×#11924038	—	2	2	no	—
16	C₄×#11924038	C₂²	1	1	yes	characteristic
17	#17989450	C₂²	1	2	yes	—
18	#17989450	C₂²	1	2	yes	—
19	#17989437	C₂²	1	4	yes	—
20	Q₈⋊#10596833	C₂²	1	4	yes	—
21	#17989429	C₂²	1	2	yes	—
22	C₈×#10596833	C₂²	1	2	yes	—
23	C₂×Q₁₆×D₇₃	C₄	1	1	yes	characteristic
24	C₄×Q₈×D₇₃	C₂²	1	2	yes	—
25	C₄×Dic₂₉₂	C₄	1	1	yes	characteristic
26	Q₁₆×Dic₇₃	C₂²	1	1	yes	characteristic
27	#17988681	C₂²	1	1	yes	characteristic
28	C₄×#11924018	C₄	1	2	yes	—
29	#17988679	C₂²	1	2	yes	—
30	#107×D₇₃	C₄	1	1	yes	characteristic
31	#103×D₇₃	C₄	1	2	yes	—
32	C₄×C₈×D₇₃	C₂²	1	1	yes	characteristic
33	C₄×Q₁₆×C₇₃	C₄	1	1	yes	characteristic
34	C₂×#11924038	—	2	2	no	—
35	C₂×#11924038	D₄	1	1	yes	characteristic
36	C₂×#11924038	D₄	1	1	yes	characteristic
37	#14556658	—	2	4	no	—
38	#14556658	—	2	4	no	—
39	C₄⋊#10596833	—	2	2	no	—
40	C₄⋊#10596833	—	2	1	no	—
41	C₄⋊#10596833	C₂³	1	4	yes	—
42	#14556648	—	2	1	no	—
43	#14556646	—	2	4	no	—
44	#14556646	D₄	1	2	yes	—
45	C₄×#10596833	—	2	4	no	—
46	C₄×#10596833	C₂³	1	2	yes	—
47	C₄×#10596833	C₂³	1	2	yes	—
48	C₂×Q₈×D₇₃	C₂×C₄	1	2	yes	—
49	C₂×Dic₂₉₂	C₂×C₄	1	1	yes	characteristic
50	C₂×#11924018	C₂×C₄	1	2	yes	—
51	Q₁₆×D₇₃	—	2	4	no	—
52	C₂×C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
53	C₂×C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
54	Q₈×Dic₇₃	C₂³	1	2	yes	—
55	C₄×Dic₁₄₆	—	2	2	no	—
56	C₄×Dic₁₄₆	C₂×C₄	1	2	yes	—
57	#14556556	—	2	2	no	—
58	#14556556	C₂³	1	2	yes	—
59	C₄⋊C₄×D₇₃	—	2	2	no	—
60	C₄⋊C₄×D₇₃	C₂×C₄	1	2	yes	—
61	C₄²×D₇₃	—	2	2	no	—
62	C₄²×D₇₃	C₂³	1	1	yes	characteristic
63	#14556525	C₂×C₄	1	1	yes	characteristic
64	#14556524	C₂×C₄	1	2	yes	—
65	#14556516	C₂×C₄	1	2	yes	—
66	#14556515	C₂×C₄	1	2	yes	—
67	#14556514	C₂×C₄	1	2	yes	—
68	C₄×#10596831	C₂×C₄	1	1	yes	characteristic
69	C₈×Dic₇₃	C₂³	1	1	yes	characteristic
70	C₈×Dic₇₃	C₂³	1	1	yes	characteristic
71	C₂×Q₁₆×C₇₃	C₂×C₄	1	1	yes	characteristic
72	C₄×Q₈×C₇₃	C₂×C₄	1	2	yes	—
73	#107×C₇₃	C₂×C₄	1	1	yes	characteristic
74	#103×C₇₃	C₂×C₄	1	2	yes	—
75	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
76	#11924038	—	2	2	no	—
77	#11924038	—	4	4	no	—
78	#11924038	Q₁₆	1	4	yes	complemented
79	C₂×#10596833	—	2	4	no	—
80	C₂×#10596833	—	2	2	no	—
81	C₂×#10596833	C₂×D₄	1	2	yes	—
82	C₂×#10596833	C₂×D₄	1	2	yes	—
83	C₂×Dic₁₄₆	—	2	2	no	—
84	C₂×Dic₁₄₆	C₂²×C₄	1	2	yes	—
85	Q₈×D₇₃	—	2	4	no	—
86	Q₈×D₇₃	—	4	2	no	—
87	C₂×C₄×D₇₃	—	2	2	no	—
88	C₂×C₄×D₇₃	—	2	2	no	—
89	C₂×C₄×D₇₃	C₂²×C₄	1	1	yes	characteristic
90	C₂×C₄×D₇₃	C₂×D₄	1	1	yes	characteristic
91	C₂×C₄×D₇₃	#41	1	1	yes	characteristic
92	Dic₂₉₂	M₁₆	1	4	yes	complemented
93	#11924018	—	2	4	no	—
94	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
95	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
96	C₈×D₇₃	—	2	2	no	—
97	C₈×D₇₃	—	4	1	no	—
98	C₄⋊Dic₇₃	—	2	2	no	—
99	C₄⋊Dic₇₃	C₂²×C₄	1	2	yes	—
100	#11924007	—	2	2	no	—
101	#11924007	—	2	2	no	—
102	#11924007	—	2	2	no	—
103	C₄×Dic₇₃	—	2	2	no	—
104	C₄×Dic₇₃	—	2	2	no	—
105	C₄×Dic₇₃	—	2	2	no	—
106	C₄×Dic₇₃	C₂×D₄	1	1	yes	characteristic
107	C₄×Dic₇₃	#41	1	1	yes	characteristic
108	C₄×Dic₇₃	C₂⁴	1	1	yes	characteristic
109	C₂×Q₈×C₇₃	C₂²×C₄	1	2	yes	—
110	Q₁₆×C₇₃	—	2	2	no	—
111	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
112	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
113	C₄⋊C₄×C₇₃	—	2	2	no	—
114	C₄⋊C₄×C₇₃	C₂²×C₄	1	2	yes	—
115	C₄²×C₇₃	—	2	2	no	—
116	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
117	C₂²×D₇₃	C₄×D₄	1	1	yes	characteristic
118	Q₈×C₇₃	—	2	2	no	—
119	Q₈×C₇₃	—	2	2	no	—
120	C₂×C₄×C₇₃	—	2	2	no	—
121	C₂×C₄×C₇₃	—	2	2	no	—
122	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
123	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
124	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
125	C₂×Dic₇₃	—	2	2	no	—
126	C₂×Dic₇₃	—	2	2	no	—
127	C₂×Dic₇₃	C₄×D₄	1	1	yes	characteristic
128	C₂×Dic₇₃	C₂²×D₄	1	1	yes	characteristic
129	C₂×Dic₇₃	C₂×#41	1	1	yes	characteristic
130	C₄×D₇₃	—	2	1	no	—
131	C₄×D₇₃	—	2	1	no	—
132	C₄×D₇₃	—	4	1	no	—
133	C₄×D₇₃	—	4	4	no	—
134	C₄×D₇₃	—	8	2	no	—
135	C₄×D₇₃	C₂×Q₁₆	1	2	yes	—
136	C₄×D₇₃	#135	1	2	yes	—
137	Dic₁₄₆	—	2	2	no	—
138	Dic₁₄₆	—	4	4	no	—
139	Dic₁₄₆	C₂×M₁₆	1	4	yes	—
140	#10596833	—	2	2	no	—
141	#10596833	—	4	2	no	—
142	#10596833	—	4	4	no	—
143	#10596833	C₂×Q₁₆	1	4	yes	complemented
144	C₈×C₇₃	—	2	1	no	—
145	C₈×C₇₃	C₂×M₁₆	1	2	yes	—
146	#10596831	—	2	1	no	—
147	#10596831	—	2	1	no	—
148	C₂²×C₇₃	C₂×C₄×D₄	1	1	yes	characteristic
149	C₂×D₇₃	—	2	1	no	—
150	C₂×D₇₃	—	2	1	no	—
151	C₂×D₇₃	C₄×Q₁₆	1	2	yes	—
152	C₄×C₇₃	—	2	1	no	—
153	C₄×C₇₃	—	4	2	no	—
154	C₄×C₇₃	—	4	2	no	—
155	C₄×C₇₃	C₄×Q₁₆	1	2	yes	—
156	C₄×C₇₃	C₂²×M₁₆	1	1	yes	characteristic
157	C₄×C₇₃	#568	1	1	yes	characteristic
158	Dic₇₃	—	2	4	no	—
159	Dic₇₃	—	2	1	no	—
160	Dic₇₃	—	2	1	no	—
161	Dic₇₃	—	4	2	no	—
162	Dic₇₃	C₂²×Q₁₆	1	1	yes	characteristic
163	Dic₇₃	C₂×#135	1	1	yes	characteristic
164	M₁₆×Q₁₆	—	73	1	no	maximal, Sylow
165	C₂×C₇₃	D₄×M₁₆	1	1	yes	characteristic
166	C₂×C₇₃	C₂×C₄×Q₁₆	1	1	yes	characteristic
167	C₂×C₇₃	#2957	1	1	yes	characteristic
168	D₇₃	—	2	2	no	—
169	Q₈×M₁₆	—	73	2	no	—
170	C₂×C₄×Q₁₆	—	73	1	no	—
171	#1591	—	73	2	no	—
172	#1584	—	73	4	no	—
173	C₈×Q₁₆	—	73	2	no	—
174	#1567	—	73	1	no	—
175	#1487	—	73	2	no	—
176	C₈×M₁₆	—	73	1	no	—
177	C₇₃	M₁₆×Q₁₆	1	1	yes	characteristic, complemented, Sylow
178	C₂²×Q₁₆	—	73	1	no	—
179	C₂×C₄×Q₈	—	73	2	no	—
180	#446	—	73	4	no	—
181	#446	—	146	4	no	—
182	C₈×Q₈	—	73	4	no	—
183	C₄×Q₁₆	—	73	1	no	—
184	C₄×Q₁₆	—	73	2	no	—
185	C₄×Q₁₆	—	73	1	no	—
186	C₄×Q₁₆	—	73	2	no	—
187	C₄×Q₁₆	—	73	1	no	—
188	C₄×Q₁₆	#64408	1	1	yes	characteristic
189	#431	—	73	1	no	—
190	C₂×#107	—	73	1	no	—
191	#423	—	73	2	no	—
192	#423	—	146	2	no	—
193	C₂×#103	—	73	2	no	—
194	C₄×M₁₆	—	73	1	no	—
195	C₄×M₁₆	—	146	2	no	—
196	C₂×C₄×C₈	—	73	1	no	—
197	#335	—	73	2	no	—
198	#335	—	73	2	no	—
199	Q₈⋊C₈	—	73	4	no	—
200	Q₈⋊C₈	—	73	4	no	—
201	#322	—	73	2	no	—
202	C₈²	—	73	2	no	—
203	C₂²×Q₈	—	73	2	no	—
204	C₂×Q₁₆	—	73	2	no	—
205	C₂×Q₁₆	—	73	1	no	—
206	C₂×Q₁₆	—	146	4	no	—
207	C₂×Q₁₆	C₂×#64408	1	1	yes	characteristic
208	C₂×M₁₆	—	73	1	no	—
209	C₂×M₁₆	—	73	1	no	—
210	C₂×M₁₆	—	146	2	no	—
211	C₂²×C₈	—	73	1	no	—
212	C₂²×C₈	—	73	1	no	—
213	C₄×Q₈	—	73	2	no	—
214	C₄×Q₈	—	73	2	no	—
215	C₄×Q₈	—	73	2	no	—
216	C₄×Q₈	—	146	2	no	—
217	C₄×Q₈	—	146	2	no	—
218	C₄×Q₈	C₂×#64408	1	2	yes	—
219	C₂×C₄⋊C₄	—	73	2	no	—
220	C₂×C₄⋊C₄	—	146	2	no	—
221	C₂×C₄²	—	73	1	no	—
222	C₂×C₄²	—	146	2	no	—
223	#107	—	73	1	no	—
224	#107	—	73	2	no	—
225	#107	C₂×#64408	1	1	yes	characteristic
226	C₄⋊C₈	—	73	4	no	—
227	C₄⋊C₈	—	146	2	no	—
228	C₄⋊C₈	—	146	1	no	—
229	C₄⋊C₈	—	146	4	no	—
230	C₄⋊C₈	—	146	4	no	—
231	#103	—	73	2	no	—
232	#103	—	73	2	no	—
233	#103	—	73	2	no	—
234	#103	C₂×#64408	1	2	yes	—
235	C₂²⋊C₈	—	146	1	no	—
236	#97	—	73	2	no	—
237	#97	—	146	4	no	—
238	C₄×C₈	—	73	2	no	—
239	C₄×C₈	—	73	1	no	—
240	C₄×C₈	—	73	1	no	—
241	C₄×C₈	—	73	2	no	—
242	C₄×C₈	—	73	1	no	—
243	C₄×C₈	—	146	4	no	—
244	C₄×C₈	C₂×#64408	1	1	yes	characteristic
245	C₂×Q₈	—	292	2	no	—
246	C₂×Q₈	—	73	2	no	—
247	C₂×Q₈	—	146	2	no	—
248	C₂×Q₈	—	146	4	no	—
249	C₂×Q₈	C₂²×#64408	1	2	yes	—
250	C₂²×C₄	—	73	1	no	—
251	C₂²×C₄	—	73	1	no	—
252	C₂²×C₄	—	73	1	no	—
253	C₂²×C₄	—	146	2	no	—
254	C₂²×C₄	—	146	2	no	—
255	Q₁₆	—	2	2	no	—
256	Q₁₆	—	73	4	no	—
257	Q₁₆	—	146	4	no	—
258	M₁₆	—	292	4	no	—
259	M₁₆	—	73	4	no	—
260	M₁₆	—	146	2	no	—
261	C₂×C₈	—	292	1	no	—
262	C₂×C₈	—	73	2	no	—
263	C₂×C₈	—	73	2	no	—
264	C₂×C₈	—	73	1	no	—
265	C₂×C₈	—	73	1	no	—
266	C₂×C₈	—	146	2	no	—
267	C₂×C₈	—	146	4	no	—
268	C₂×C₈	—	146	2	no	—
269	C₂×C₈	C₂²×#64408	1	1	yes	characteristic
270	C₂×C₈	C₂²×#64408	1	1	yes	characteristic
271	C₄⋊C₄	—	2	2	no	—
272	C₄⋊C₄	—	73	2	no	—
273	C₄⋊C₄	—	146	2	no	—
274	C₄⋊C₄	—	146	2	no	—
275	C₄⋊C₄	—	146	2	no	—
276	C₄⋊C₄	—	146	2	no	—
277	C₄⋊C₄	C₂²×#64408	1	2	yes	—
278	C₄²	—	2	2	no	—
279	C₄²	—	73	1	no	—
280	C₄²	—	73	1	no	—
281	C₄²	—	73	1	no	—
282	C₄²	—	146	2	no	—
283	C₄²	—	146	2	no	—
284	C₄²	—	146	2	no	—
285	C₄²	C₂²×#64408	1	1	yes	characteristic
286	C₂³	—	73	1	no	—
287	Q₈	—	2	2	no	—
288	Q₈	—	2	2	no	—
289	Q₈	—	292	4	no	—
290	Q₈	—	73	4	no	—
291	Q₈	—	146	2	no	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	292	1	no	—
295	C₂×C₄	—	292	4	no	—
296	C₂×C₄	—	584	2	no	—
297	C₂×C₄	—	73	1	no	—
298	C₂×C₄	—	73	2	no	—
299	C₂×C₄	—	73	2	no	—
300	C₂×C₄	—	73	1	no	—
301	C₂×C₄	—	73	1	no	—
302	C₂×C₄	—	146	2	no	—
303	C₂×C₄	—	146	2	no	—
304	C₂×C₄	—	146	1	no	—
305	C₂×C₄	—	146	1	no	—
306	C₂×C₄	D₄×#64408	1	1	yes	characteristic
307	C₂×C₄	D₄×#64408	1	1	yes	characteristic
308	C₂×C₄	C₂³×#64408	1	1	yes	characteristic
309	C₈	—	2	1	no	—
310	C₈	—	292	4	no	—
311	C₈	—	292	2	no	—
312	C₈	—	73	4	no	—
313	C₈	—	146	2	no	—
314	C₈	—	146	1	no	—
315	C₈	—	146	1	no	—
316	C₈	D₇₃⋊M₁₆	1	2	yes	—
317	C₂²	—	73	2	no	—
318	C₂²	—	146	1	no	—
319	C₂²	—	146	1	no	—
320	C₂²	C₂×D₄×#64408	1	1	yes	characteristic
321	C₄	—	2	1	no	—
322	C₄	—	4	2	no	—
323	C₄	—	4	2	no	—
324	C₄	—	292	2	no	—
325	C₄	—	73	1	no	—
326	C₄	—	73	1	no	—
327	C₄	—	146	4	no	—
328	C₄	—	146	1	no	—
329	C₄	—	146	1	no	—
330	C₄	Q₁₆×#64408	1	2	yes	—
331	C₄	C₂×D₇₃⋊M₁₆	1	1	yes	characteristic
332	C₄	#17989666	1	1	yes	characteristic
333	C₂	—	146	2	no	—
334	C₂	#22626672	1	1	yes	characteristic
335	C₂	C₂×Q₁₆×#64408	1	1	yes	characteristic
336	C₂	#22626708	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×#64408
commutator	C₂×C₄×C₇₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×#64408
Fitting	C₄×Q₁₆×C₇₃	C₄
socle	C₂²×C₇₃	C₂×C₄×D₄
prime radical	C₂²×D₇₃	C₄×D₄
cosocle	C₄×Dic₇₃	C₂⁴
hypercenter	C₄×Q₁₆	#64408
hypocenter	C₇₃	M₁₆×Q₁₆
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×Q₁₆

Complements

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

N	H ≅ G/N	type
2. #22625571	C₂	semidirect
3. #22625545	C₂	semidirect
4. #22625519	C₂	semidirect
78. #11924038	Q₁₆	semidirect
92. Dic₂₉₂	M₁₆	semidirect
143. #10596833	C₂×Q₁₆	semidirect
177. C₇₃	M₁₆×Q₁₆	semidirect

Maximal subgroups

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

subgroup	index
#22626733	2
#22625571	2
#22625545	2
#22625519	2
#22625489	2
#22625260	2
C₈×#11924038	2
C₄×Q₁₆×D₇₃	2
M₁₆×Q₁₆	73