Subgroups — #34030328

Order 18688.

498 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	#22626348	C₂	1	2	yes	maximal, complemented
2	#22626348	C₂	1	2	yes	maximal, complemented
3	#22626347	C₂	1	2	yes	maximal, complemented
4	#22626345	C₂	1	2	yes	maximal, complemented
5	#22626268	C₂	1	1	yes	maximal, characteristic
6	#22626266	C₂	1	1	yes	maximal, characteristic
7	C₂×#17989508	C₂	1	1	yes	maximal, characteristic
8	#22626142	C₂	1	1	yes	maximal, characteristic
9	#22626140	C₂	1	1	yes	maximal, characteristic
10	#22626137	C₂	1	1	yes	maximal, characteristic
11	#22618864	C₂	1	1	yes	maximal, characteristic
12	C₂×#14556665	C₂²	1	1	yes	characteristic
13	C₂×#14556665	C₂²	1	1	yes	characteristic
14	C₂²×#11924031	C₂²	1	1	yes	characteristic
15	C₂²×#11924031	C₂²	1	1	yes	characteristic
16	#17989508	C₂²	1	1	yes	characteristic
17	#17989508	C₂²	1	2	yes	—
18	#17989508	C₂²	1	1	yes	characteristic
19	#17989508	C₂²	1	1	yes	characteristic
20	#17989508	C₂²	1	1	yes	characteristic
21	#17989508	C₂²	1	2	yes	—
22	C₄⋊#11924031	C₂²	1	2	yes	—
23	C₄⋊#11924031	C₂²	1	2	yes	—
24	#17989476	C₂²	1	2	yes	—
25	#17989476	C₂²	1	2	yes	—
26	#17989475	C₂²	1	2	yes	—
27	#17989473	C₂²	1	2	yes	—
28	#17989473	C₂²	1	2	yes	—
29	C₄×#11924031	C₂²	1	2	yes	—
30	#17989140	C₄	1	1	yes	characteristic
31	#17988716	C₄	1	2	yes	—
32	#17988716	C₄	1	2	yes	—
33	#17988711	C₂²	1	2	yes	—
34	#17988711	C₂²	1	2	yes	—
35	#17988613	C₄	1	1	yes	characteristic
36	#17988610	C₄	1	1	yes	characteristic
37	#17988545	C₂²	1	1	yes	characteristic
38	#17988543	C₂²	1	1	yes	characteristic
39	C₂×C₈×Dic₇₃	C₂²	1	1	yes	characteristic
40	#424×C₇₃	C₄	1	1	yes	characteristic
41	#14556665	—	2	2	no	—
42	#14556665	Q₈	1	4	yes	complemented
43	C₂×#11924031	—	2	2	no	—
44	C₂×#11924031	—	2	2	no	—
45	C₂×#11924031	C₂³	1	1	yes	characteristic
46	C₂×#11924031	C₂³	1	1	yes	characteristic
47	C₂×#11924031	C₂³	1	1	yes	characteristic
48	C₂×#11924031	C₂³	1	1	yes	characteristic
49	C₂×#11924031	C₂³	1	2	yes	—
50	C₂×#11924031	C₂³	1	2	yes	—
51	C₂×#11924012	C₂×C₄	1	1	yes	characteristic
52	C₂²×#10596831	C₂³	1	1	yes	characteristic
53	#14556590	C₈	1	1	yes	characteristic
54	#14556590	C₈	1	1	yes	characteristic
55	#14556582	C₈	1	1	yes	characteristic
56	#14556582	C₈	1	1	yes	characteristic
57	#14556568	C₈	1	2	yes	—
58	#14556568	C₈	1	2	yes	—
59	C₄×Dic₁₄₆	C₂×C₄	1	2	yes	—
60	C₄×Dic₁₄₆	C₂×C₄	1	2	yes	—
61	#14556550	C₂×C₄	1	1	yes	characteristic
62	C₂×C₄×Dic₇₃	C₂×C₄	1	1	yes	characteristic
63	#14556520	C₂×C₄	1	2	yes	—
64	#14556520	C₂×C₄	1	2	yes	—
65	#14556520	C₂×C₄	1	2	yes	—
66	#14556520	C₂×C₄	1	2	yes	—
67	#14556502	C₂×C₄	1	2	yes	—
68	#14556501	C₂×C₄	1	2	yes	—
69	C₄×#10596831	C₂³	1	2	yes	—
70	C₈×Dic₇₃	C₂×C₄	1	1	yes	characteristic
71	C₈×Dic₇₃	C₂×C₄	1	1	yes	characteristic
72	C₈×Dic₇₃	C₂³	1	1	yes	characteristic
73	C₈×Dic₇₃	C₂³	1	2	yes	—
74	C₈×Dic₇₃	C₂³	1	1	yes	characteristic
75	C₂×M₁₆×C₇₃	C₂×C₄	1	1	yes	characteristic
76	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
77	#117×C₇₃	C₂×C₄	1	1	yes	characteristic
78	C₄⋊C₈×C₇₃	C₂×C₄	1	2	yes	—
79	C₄⋊C₈×C₇₃	C₂×C₄	1	2	yes	—
80	#11924031	—	2	1	no	—
81	#11924031	—	2	1	no	—
82	#11924031	—	2	1	no	—
83	#11924031	—	2	1	no	—
84	#11924031	—	2	2	no	—
85	#11924031	C₂×Q₈	1	4	yes	—
86	C₂×Dic₁₄₆	C₂×C₈	1	2	yes	—
87	C₂×Dic₁₄₆	C₂×C₈	1	2	yes	—
88	C₂²×Dic₇₃	C₂×C₈	1	1	yes	characteristic
89	C₂²×Dic₇₃	C₂×C₈	1	1	yes	characteristic
90	#11924012	—	2	2	no	—
91	C₂×#10596831	C₂²×C₄	1	2	yes	—
92	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
93	C₂×#10596831	C₂×Q₈	1	2	yes	—
94	C₂×#10596831	#41	1	2	yes	—
95	C₂×#10596831	C₂⁴	1	1	yes	characteristic
96	C₄⋊Dic₇₃	C₂×C₈	1	1	yes	characteristic
97	C₄⋊Dic₇₃	C₂×C₈	1	1	yes	characteristic
98	C₄⋊Dic₇₃	C₂×C₈	1	2	yes	—
99	C₄⋊Dic₇₃	C₂×C₈	1	2	yes	—
100	#11924007	C₂×C₈	1	2	yes	—
101	#11924007	C₂×C₈	1	2	yes	—
102	#11924007	C₂×C₈	1	2	yes	—
103	#11924007	C₂×C₈	1	2	yes	—
104	C₂²⋊Dic₇₃	C₂×C₈	1	2	yes	—
105	C₄×Dic₇₃	C₂×C₈	1	1	yes	characteristic
106	C₄×Dic₇₃	C₂×C₈	1	1	yes	characteristic
107	C₄×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
108	C₄×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
109	C₄×Dic₇₃	C₂²×C₄	1	2	yes	—
110	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
111	M₁₆×C₇₃	—	2	2	no	—
112	C₂×C₈×C₇₃	—	2	2	no	—
113	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
114	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
115	C₂×C₈×C₇₃	C₂²×C₄	1	2	yes	—
116	C₄⋊C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
117	C₄⋊C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
118	C₂²⋊C₄×C₇₃	C₂×C₈	1	2	yes	—
119	C₄²×C₇₃	C₂²×C₄	1	2	yes	—
120	C₂³×C₇₃	C₂²×C₈	1	1	yes	characteristic
121	C₂×C₄×C₇₃	C₄×Q₈	1	2	yes	—
122	C₂×C₄×C₇₃	C₂²×C₈	1	2	yes	—
123	C₂×C₄×C₇₃	C₂²×C₈	1	2	yes	—
124	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
125	C₂×C₄×C₇₃	#131	1	2	yes	—
126	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
127	C₂×Dic₇₃	—	2	2	no	—
128	C₂×Dic₇₃	—	4	1	no	—
129	C₂×Dic₇₃	C₂²×C₈	1	1	yes	characteristic
130	C₂×Dic₇₃	C₂²×C₈	1	1	yes	characteristic
131	C₂×Dic₇₃	C₂²×C₈	1	1	yes	characteristic
132	C₂×Dic₇₃	C₂²×C₈	1	2	yes	—
133	C₂×Dic₇₃	C₂²×C₈	1	2	yes	—
134	C₂×Dic₇₃	C₂²×C₈	1	1	yes	characteristic
135	Dic₁₄₆	—	2	4	no	—
136	Dic₁₄₆	—	4	2	no	—
137	C₈×C₇₃	—	2	1	no	—
138	C₈×C₇₃	—	2	1	no	—
139	C₈×C₇₃	—	2	2	no	—
140	#10596831	—	2	2	no	—
141	#10596831	C₄×Q₈	1	2	yes	—
142	#10596831	C₂²×Q₈	1	1	yes	characteristic
143	#10596831	C₂×#41	1	1	yes	characteristic
144	C₂²×C₇₃	—	2	1	no	—
145	C₂²×C₇₃	—	2	1	no	—
146	C₂²×C₇₃	C₈×Q₈	1	2	yes	—
147	C₂²×C₇₃	C₂³×C₈	1	1	yes	characteristic
148	C₄×C₇₃	—	2	2	no	—
149	C₄×C₇₃	—	2	2	no	—
150	C₄×C₇₃	C₈×Q₈	1	2	yes	—
151	C₄×C₇₃	C₂×C₄×Q₈	1	1	yes	characteristic
152	C₄×C₇₃	C₂×#131	1	1	yes	characteristic
153	Dic₇₃	—	2	2	no	—
154	Dic₇₃	—	2	1	no	—
155	Dic₇₃	—	2	1	no	—
156	Dic₇₃	—	2	2	no	—
157	Dic₇₃	—	2	1	no	—
158	Dic₇₃	#504	1	2	yes	—
159	#33476	—	73	1	no	maximal, Sylow
160	C₂×C₇₃	—	2	2	no	—
161	C₂×C₇₃	C₂×C₈×Q₈	1	1	yes	characteristic
162	C₂×C₇₃	C₂×#504	1	1	yes	characteristic
163	C₂×C₇₃	#3407	1	1	yes	characteristic
164	#2960	—	73	1	no	—
165	#2183	—	73	2	no	—
166	#2183	—	73	2	no	—
167	#2183	—	73	2	no	—
168	Q₈×C₁₆	—	73	2	no	—
169	#2151	—	73	1	no	—
170	#2151	—	73	1	no	—
171	C₂×C₄⋊C₁₆	—	73	1	no	—
172	#2108	—	73	1	no	—
173	#2108	—	73	1	no	—
174	C₄×M₃₂	—	73	1	no	—
175	C₇₃	#33476	1	1	yes	characteristic, complemented, Sylow
176	#533	—	73	1	no	—
177	C₂×M₃₂	—	73	1	no	—
178	C₂×M₃₂	—	73	1	no	—
179	C₂²×C₁₆	—	73	1	no	—
180	C₂²×C₁₆	—	73	1	no	—
181	#446	—	73	2	no	—
182	#446	—	73	2	no	—
183	C₈×Q₈	—	73	2	no	—
184	C₈×Q₈	—	73	2	no	—
185	#424	—	73	1	no	—
186	#424	#64408	1	1	yes	characteristic
187	#423	—	73	1	no	—
188	#405	—	73	1	no	—
189	#405	—	73	1	no	—
190	C₂×C₄×C₈	—	73	1	no	—
191	C₄⋊C₁₆	—	73	1	no	—
192	C₄⋊C₁₆	—	73	1	no	—
193	C₄⋊C₁₆	—	73	1	no	—
194	C₄⋊C₁₆	—	73	1	no	—
195	C₄⋊C₁₆	—	73	2	no	—
196	C₄⋊C₁₆	—	73	2	no	—
197	C₄⋊C₁₆	—	73	2	no	—
198	C₄⋊C₁₆	—	73	2	no	—
199	#346	—	73	2	no	—
200	#346	—	73	2	no	—
201	#346	—	73	2	no	—
202	C₄×C₁₆	—	73	2	no	—
203	C₄×C₁₆	—	73	2	no	—
204	C₄×C₁₆	—	73	2	no	—
205	C₂×M₁₆	—	73	1	no	—
206	C₂×M₁₆	C₂×#64408	1	1	yes	characteristic
207	C₂²×C₈	—	73	1	no	—
208	C₂²×C₈	C₂×#64408	1	1	yes	characteristic
209	C₄⋊Q₈	—	73	1	no	—
210	C₄⋊Q₈	—	73	1	no	—
211	#125	—	73	1	no	—
212	#125	—	73	1	no	—
213	C₂²⋊Q₈	—	73	2	no	—
214	C₂²⋊Q₈	—	73	2	no	—
215	C₄×Q₈	—	73	2	no	—
216	C₄×Q₈	—	73	2	no	—
217	#117	—	73	1	no	—
218	#117	C₂×#64408	1	1	yes	characteristic
219	C₂×C₄²	—	73	1	no	—
220	M₃₂	—	73	4	no	—
221	M₃₂	—	146	2	no	—
222	C₂×C₁₆	—	73	2	no	—
223	C₂×C₁₆	—	73	1	no	—
224	C₂×C₁₆	—	73	1	no	—
225	C₂×C₁₆	—	73	2	no	—
226	C₂×C₁₆	—	73	1	no	—
227	C₂×C₁₆	—	73	1	no	—
228	C₂×C₁₆	—	146	2	no	—
229	C₂×C₁₆	—	146	2	no	—
230	C₄⋊C₈	—	73	2	no	—
231	C₄⋊C₈	—	73	2	no	—
232	C₄⋊C₈	—	73	2	no	—
233	C₄⋊C₈	—	73	2	no	—
234	C₄⋊C₈	C₂×#64408	1	2	yes	—
235	C₄⋊C₈	C₂×#64408	1	2	yes	—
236	#97	—	73	2	no	—
237	#97	—	73	2	no	—
238	C₄×C₈	—	73	1	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	—	73	2	no	—
244	C₂×Q₈	—	73	2	no	—
245	C₂×Q₈	—	73	2	no	—
246	C₂²×C₄	—	73	1	no	—
247	C₂²×C₄	—	73	1	no	—
248	C₂²×C₄	C₂²×#64408	1	1	yes	characteristic
249	M₁₆	—	2	2	no	—
250	M₁₆	—	146	2	no	—
251	C₂×C₈	—	2	2	no	—
252	C₂×C₈	—	73	2	no	—
253	C₂×C₈	—	73	2	no	—
254	C₂×C₈	—	73	1	no	—
255	C₂×C₈	—	73	1	no	—
256	C₂×C₈	—	73	2	no	—
257	C₂×C₈	C₂²×#64408	1	1	yes	characteristic
258	C₂×C₈	C₂²×#64408	1	1	yes	characteristic
259	C₂×C₈	C₂²×#64408	1	2	yes	—
260	C₄⋊C₄	—	73	2	no	—
261	C₄⋊C₄	—	73	2	no	—
262	C₄⋊C₄	—	73	2	no	—
263	C₄⋊C₄	—	73	2	no	—
264	C₄⋊C₄	—	73	1	no	—
265	C₄⋊C₄	—	73	1	no	—
266	C₄⋊C₄	—	73	2	no	—
267	C₄⋊C₄	—	73	2	no	—
268	C₄⋊C₄	#11924036	1	1	yes	characteristic
269	C₄⋊C₄	#11924036	1	1	yes	characteristic
270	C₂²⋊C₄	—	73	2	no	—
271	C₂²⋊C₄	#11924036	1	2	yes	—
272	C₄²	—	73	1	no	—
273	C₄²	—	73	1	no	—
274	C₄²	—	73	1	no	—
275	C₄²	—	73	1	no	—
276	C₄²	—	73	2	no	—
277	C₄²	C₂²×#64408	1	2	yes	—
278	C₁₆	—	73	4	no	—
279	C₁₆	—	146	1	no	—
280	C₁₆	—	146	1	no	—
281	C₁₆	—	146	1	no	—
282	C₁₆	—	146	1	no	—
283	C₁₆	—	146	2	no	—
284	C₂³	C₂×#11924036	1	1	yes	characteristic
285	Q₈	—	292	2	no	—
286	Q₈	—	146	4	no	—
287	C₂×C₄	—	292	1	no	—
288	C₂×C₄	—	73	1	no	—
289	C₂×C₄	—	73	1	no	—
290	C₂×C₄	—	73	1	no	—
291	C₂×C₄	—	73	1	no	—
292	C₂×C₄	—	73	2	no	—
293	C₂×C₄	—	73	2	no	—
294	C₂×C₄	—	146	2	no	—
295	C₂×C₄	Q₈×#64408	1	2	yes	—
296	C₂×C₄	C₂×#11924036	1	2	yes	—
297	C₂×C₄	C₂×#11924036	1	2	yes	—
298	C₂×C₄	C₂×#11924036	1	1	yes	characteristic
299	C₂×C₄	#14556678	1	2	yes	—
300	C₂×C₄	C₂³×#64408	1	1	yes	characteristic
301	C₈	—	2	1	no	—
302	C₈	—	2	1	no	—
303	C₈	—	2	2	no	—
304	C₈	—	73	1	no	—
305	C₈	—	73	1	no	—
306	C₈	—	73	2	no	—
307	C₈	—	146	2	no	—
308	C₂²	—	2	1	no	—
309	C₂²	—	2	1	no	—
310	C₂²	#17989636	1	2	yes	—
311	C₂²	C₂²×#11924036	1	1	yes	characteristic
312	C₄	—	2	2	no	—
313	C₄	—	2	2	no	—
314	C₄	—	73	2	no	—
315	C₄	—	146	1	no	—
316	C₄	—	146	2	no	—
317	C₄	—	146	1	no	—
318	C₄	—	146	1	no	—
319	C₄	—	146	2	no	—
320	C₄	#17989636	1	2	yes	—
321	C₄	C₂×Q₈×#64408	1	1	yes	characteristic
322	C₄	C₂×#14556678	1	1	yes	characteristic
323	C₂	—	2	2	no	—
324	C₂	C₂×#17989636	1	1	yes	characteristic
325	C₂	#22626896	1	1	yes	characteristic
326	C₂	#22626899	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×#64408
commutator, socle	C₂²×C₇₃	C₂³×C₈
Fitting	#424×C₇₃	C₄
prime radical	C₂³×C₇₃	C₂²×C₈
cosocle	C₂×#10596831	C₂⁴
hypercenter	#424	#64408
hypocenter	C₇₃	#33476
LCS	G ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ #424

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. #22626347	C₂	semidirect
4. #22626345	C₂	semidirect
42. #14556665	Q₈	semidirect
175. 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
#22626347	2
#22626345	2
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C₂×#17989508	2
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#22626137	2
#22618864	2
#33476	73