Subgroups — #30856200

Order 17152.

608 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22568636	C₂	1	1	yes	maximal, characteristic
2	#22560359	C₂	1	1	yes	maximal, characteristic, complemented
3	#22560356	C₂	1	1	yes	maximal, characteristic, complemented
4	#22560331	C₂	1	2	yes	maximal, complemented
5	#22560311	C₂	1	2	yes	maximal
6	#22560258	C₂	1	1	yes	maximal, characteristic
7	#22560254	C₂	1	1	yes	maximal, characteristic
8	#22560094	C₂	1	2	yes	maximal, complemented
9	#22560010	C₂	1	2	yes	maximal
10	#22559964	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₇×#1747	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14524051	C₂²	1	1	yes	characteristic
13	C₂×#14524049	C₂²	1	1	yes	characteristic
14	#16863864	C₂²	1	1	yes	characteristic
15	C₂×#14523978	C₂²	1	2	yes	—
16	#16863432	C₂²	1	2	yes	—
17	#16863378	C₂²	1	2	yes	—
18	#16863369	C₂²	1	2	yes	—
19	#16863353	C₄	1	2	yes	complemented
20	#16863352	C₂²	1	1	yes	characteristic
21	#16863351	C₂²	1	2	yes	—
22	#16863350	C₄	1	2	yes	complemented
23	M₁₆×Dic₆₇	C₂²	1	1	yes	characteristic
24	C₄×#11761611	C₄	1	2	yes	complemented
25	C₂×#14523953	C₂²	1	1	yes	characteristic
26	#97×D₆₇	C₄	1	2	yes	complemented
27	#16863168	C₂²	1	2	yes	—
28	#16863166	C₂²	1	2	yes	—
29	#16863157	C₂²	1	2	yes	—
30	#16863155	C₂²	1	2	yes	—
31	C₆₇⋊#338	C₂²	1	1	yes	characteristic
32	#16863153	C₂²	1	1	yes	characteristic
33	#16863152	C₂²	1	2	yes	—
34	#405×C₆₇	C₂²	1	2	yes	—
35	#405×C₆₇	C₂²	1	1	yes	characteristic
36	#338×C₆₇	C₂²	1	1	yes	characteristic
37	H(3,ℤ/4)×C₆₇	C₂²	1	2	yes	—
38	H(3,ℤ/4)×C₆₇	C₂²	1	1	yes	characteristic
39	C₂×#11761628	C₂³	1	1	yes	characteristic
40	#14524051	—	2	2	no	—
41	#14524051	—	2	2	no	—
42	#14524049	—	2	2	no	—
43	#14524049	—	2	2	no	—
44	C₂×#11761612	C₂³	1	2	yes	—
45	C₂×#11761612	C₂³	1	1	yes	characteristic
46	C₂×#11761611	C₂×C₄	1	2	yes	—
47	C₂×#11761611	C₂×C₄	1	2	yes	—
48	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
49	C₂×C₈×D₆₇	C₂×C₄	1	2	yes	—
50	C₂×C₈×D₆₇	C₂×C₄	1	2	yes	—
51	#14524008	—	2	1	no	—
52	#14524005	—	2	1	no	—
53	(C₂²⋊D₆₇)⋊C₄	—	2	2	no	—
54	#14524002	C₂³	1	2	yes	—
55	#14524000	C₂×C₄	1	2	yes	—
56	C₂×C₄×Dic₆₇	C₂³	1	1	yes	characteristic
57	C₄²×D₆₇	C₂×C₄	1	2	yes	—
58	#14523978	—	2	2	no	—
59	#14523978	—	2	2	no	—
60	#14523976	—	2	2	no	—
61	#14523974	—	2	2	no	—
62	#14523961	—	2	2	no	—
63	#14523959	—	2	2	no	—
64	#14523957	—	2	4	no	—
65	#14523954	C₂×C₄	1	2	yes	—
66	#14523953	—	2	2	no	—
67	#14523953	C₂×C₄	1	1	yes	characteristic
68	#14523953	C₂×C₄	1	2	yes	—
69	#14523953	C₂×C₄	1	1	yes	characteristic
70	#14523953	C₂×C₄	1	1	yes	characteristic
71	#14523953	C₂×C₄	1	1	yes	characteristic
72	C₄×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
73	C₈×Dic₆₇	—	2	2	no	—
74	C₈×Dic₆₇	C₂×C₄	1	2	yes	—
75	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
76	C₂×M₁₆×C₆₇	C₂³	1	1	yes	characteristic
77	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
78	#117×C₆₇	C₂³	1	2	yes	—
79	#117×C₆₇	C₂³	1	1	yes	characteristic
80	#97×C₆₇	—	2	2	no	—
81	#97×C₆₇	C₂×C₄	1	2	yes	—
82	C₄×C₈×C₆₇	—	2	2	no	—
83	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
84	#11761628	—	2	2	no	—
85	#11761628	—	2	2	no	—
86	C₂×Dic₁₃₄	#41	1	1	yes	characteristic
87	C₂×D₂₆₈	#41	1	1	yes	characteristic
88	C₂×C₂²⋊D₆₇	#41	1	2	yes	—
89	C₂²×Dic₆₇	—	2	1	no	—
90	C₂×C₄×D₆₇	—	2	2	no	—
91	C₂×C₄×D₆₇	C₂²×C₄	1	2	yes	—
92	#11761612	—	2	2	no	—
93	#11761612	—	2	2	no	—
94	#11761612	—	2	2	no	—
95	#11761611	—	2	2	no	—
96	#11761611	—	2	2	no	—
97	#11761611	—	2	4	no	—
98	C₂×C₆₇⋊C₈	—	2	2	no	—
99	C₂×C₆₇⋊C₈	C₂²×C₄	1	2	yes	—
100	C₂×C₆₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
101	C₂×C₆₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
102	C₂×C₆₇⋊C₈	#41	1	2	yes	—
103	C₂×C₆₇⋊C₈	#41	1	2	yes	—
104	C₈×D₆₇	—	2	4	no	—
105	C₈×D₆₇	—	2	2	no	—
106	C₈×D₆₇	—	2	2	no	—
107	C₄⋊Dic₆₇	—	2	2	no	—
108	Dic₆₇⋊C₄	—	2	2	no	—
109	C₂²⋊Dic₆₇	—	2	2	no	—
110	(C₂×D₆₇)⋊C₄	—	2	2	no	—
111	C₄×Dic₆₇	—	2	1	no	—
112	C₄×Dic₆₇	—	2	2	no	—
113	C₄×Dic₆₇	—	2	2	no	—
114	C₄×Dic₆₇	C₂²×C₄	1	1	yes	characteristic
115	C₄×Dic₆₇	C₂²×C₄	1	1	yes	characteristic
116	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
117	M₁₆×C₆₇	—	2	2	no	—
118	M₁₆×C₆₇	—	2	2	no	—
119	M₁₆×C₆₇	—	2	2	no	—
120	C₂×C₈×C₆₇	—	2	2	no	—
121	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
122	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
123	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
124	C₂×C₈×C₆₇	#41	1	2	yes	—
125	C₂×C₈×C₆₇	#41	1	2	yes	—
126	C₄⋊C₄×C₆₇	—	2	1	no	—
127	C₄⋊C₄×C₆₇	—	2	2	no	—
128	C₂²⋊C₄×C₆₇	—	2	1	no	—
129	C₂²⋊C₄×C₆₇	—	2	2	no	—
130	C₄²×C₆₇	—	2	2	no	—
131	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
132	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
133	C₂²×D₆₇	#117	1	2	yes	—
134	C₂×C₄×C₆₇	—	2	2	no	—
135	C₂×C₄×C₆₇	—	4	2	no	—
136	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
137	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
138	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
139	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
140	C₂²⋊D₆₇	—	2	4	no	—
141	C₂×Dic₆₇	—	2	1	no	—
142	C₂×Dic₆₇	—	2	1	no	—
143	C₂×Dic₆₇	—	4	2	no	—
144	C₂×Dic₆₇	—	8	1	no	—
145	C₂×Dic₆₇	#117	1	2	yes	—
146	D₂₆₈	—	2	2	no	—
147	C₄×D₆₇	—	2	2	no	—
148	C₄×D₆₇	—	2	2	no	—
149	C₄×D₆₇	—	8	2	no	—
150	Dic₁₃₄	—	2	2	no	—
151	C₈×C₆₇	—	2	2	no	—
152	C₈×C₆₇	—	4	2	no	—
153	C₈×C₆₇	#117	1	2	yes	—
154	C₈×C₆₇	#117	1	1	yes	characteristic
155	C₈×C₆₇	#117	1	1	yes	characteristic
156	C₆₇⋊C₈	—	2	2	no	—
157	C₆₇⋊C₈	—	4	2	no	—
158	C₆₇⋊C₈	#117	1	1	yes	characteristic
159	C₆₇⋊C₈	#117	1	1	yes	characteristic
160	C₆₇⋊C₈	#117	1	2	yes	—
161	C₂²×C₆₇	—	4	1	no	—
162	C₂²×C₆₇	C₂×#117	1	1	yes	characteristic
163	C₂²×C₆₇	#529	1	2	yes	—
164	C₂×D₆₇	—	2	2	no	—
165	C₂×D₆₇	—	4	2	no	—
166	C₄×C₆₇	—	4	2	no	—
167	C₄×C₆₇	—	8	2	no	—
168	C₄×C₆₇	C₂×#117	1	1	yes	characteristic
169	C₄×C₆₇	C₂×#117	1	1	yes	characteristic
170	C₄×C₆₇	#529	1	2	yes	—
171	Dic₆₇	—	2	2	no	—
172	Dic₆₇	—	4	1	no	—
173	Dic₆₇	—	4	1	no	—
174	Dic₆₇	—	8	2	no	—
175	#20788	—	67	1	no	maximal, Sylow
176	C₂×C₆₇	—	2	1	no	—
177	C₂×C₆₇	—	2	2	no	—
178	C₂×C₆₇	#2283	1	1	yes	characteristic
179	D₆₇	—	4	2	no	—
180	#2875	—	67	1	no	—
181	#1841	—	67	1	no	—
182	#1841	—	67	1	no	—
183	#1833	—	67	2	no	—
184	#1827	—	67	2	no	—
185	#1813	—	67	1	no	—
186	#1812	—	67	1	no	—
187	#1773	—	67	2	no	—
188	#1757	—	67	2	no	—
189	#1747	—	67	1	no	—
190	#1747	D₆₇	1	1	yes	characteristic, complemented
191	C₆₇	#20788	1	1	yes	characteristic, complemented, Sylow
192	C₂×#131	—	67	1	no	—
193	C₂×#131	—	67	1	no	—
194	C₄×#41	—	67	1	no	—
195	C₂×#108	—	67	2	no	—
196	#427	—	67	2	no	—
197	#413	—	67	2	no	—
198	#410	—	67	2	no	—
199	#405	—	67	1	no	—
200	#405	—	67	2	no	—
201	#405	—	67	2	no	—
202	#405	—	67	2	no	—
203	#405	C₂×D₆₇	1	2	yes	—
204	#405	C₂×D₆₇	1	1	yes	characteristic
205	C₄×M₁₆	—	67	1	no	—
206	C₄×M₁₆	—	67	2	no	—
207	C₂×#97	—	67	1	no	—
208	C₂×#97	—	67	2	no	—
209	#344	—	67	2	no	—
210	#344	—	67	2	no	—
211	#339	—	67	2	no	—
212	#339	—	67	2	no	—
213	#338	—	67	1	no	—
214	#338	C₂×D₆₇	1	1	yes	characteristic
215	H(3,ℤ/4)	—	67	1	no	—
216	H(3,ℤ/4)	—	67	2	no	—
217	H(3,ℤ/4)	C₂×D₆₇	1	1	yes	characteristic
218	H(3,ℤ/4)	C₂×D₆₇	1	2	yes	—
219	C₂×#41	—	67	1	no	—
220	#131	—	134	2	no	—
221	#131	—	134	2	no	—
222	#131	—	134	2	no	—
223	#131	—	134	2	no	—
224	C₂×M₁₆	—	67	1	no	—
225	C₂×M₁₆	—	67	2	no	—
226	C₂×M₁₆	—	67	2	no	—
227	C₂×M₁₆	—	67	2	no	—
228	C₂×M₁₆	C₂²×D₆₇	1	2	yes	—
229	C₂×M₁₆	C₂²×D₆₇	1	1	yes	characteristic
230	C₂²×C₈	—	67	1	no	—
231	C₂²×C₈	—	67	2	no	—
232	C₂²×C₈	—	67	2	no	—
233	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
234	C₄×Q₈	—	134	1	no	—
235	C₄×D₄	—	134	2	no	—
236	C₄×D₄	—	134	1	no	—
237	#117	—	67	2	no	—
238	#117	—	67	2	no	—
239	#117	C₂²×D₆₇	1	2	yes	—
240	#117	C₂²×D₆₇	1	1	yes	characteristic
241	C₂×C₄²	—	67	1	no	—
242	C₂×C₄²	—	67	2	no	—
243	#108	—	134	2	no	—
244	#108	—	134	2	no	—
245	#107	—	134	2	no	—
246	#106	—	134	2	no	—
247	#101	—	134	2	no	—
248	#100	—	134	2	no	—
249	#99	—	134	4	no	—
250	#97	—	2	2	no	—
251	#97	—	67	1	no	—
252	#97	—	67	1	no	—
253	#97	—	67	2	no	—
254	#97	—	67	1	no	—
255	#97	—	67	1	no	—
256	#97	—	67	2	no	—
257	#97	—	134	2	no	—
258	#97	C₄×D₆₇	1	2	yes	—
259	C₄×C₈	—	2	2	no	—
260	C₄×C₈	—	67	2	no	—
261	C₄×C₈	—	67	2	no	—
262	C₄×C₈	—	134	2	no	—
263	C₄×C₈	C₄×D₆₇	1	2	yes	—
264	#41	—	134	2	no	—
265	#41	—	134	2	no	—
266	C₂×Q₈	—	67	1	no	—
267	C₂×D₄	—	67	2	no	—
268	C₂×D₄	—	67	1	no	—
269	C₂²×C₄	—	67	2	no	—
270	C₂²×C₄	—	134	2	no	—
271	C₂²×C₄	—	134	1	no	—
272	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
273	M₁₆	—	2	2	no	—
274	M₁₆	—	2	2	no	—
275	M₁₆	—	2	2	no	—
276	M₁₆	—	134	4	no	—
277	M₁₆	—	134	2	no	—
278	M₁₆	—	134	2	no	—
279	M₁₆	—	134	2	no	—
280	M₁₆	—	134	2	no	—
281	M₁₆	—	134	2	no	—
282	C₂×C₈	—	2	2	no	—
283	C₂×C₈	—	67	1	no	—
284	C₂×C₈	—	67	1	no	—
285	C₂×C₈	—	67	2	no	—
286	C₂×C₈	—	67	2	no	—
287	C₂×C₈	—	67	2	no	—
288	C₂×C₈	—	134	2	no	—
289	C₂×C₈	—	134	2	no	—
290	C₂×C₈	—	134	2	no	—
291	C₂×C₈	—	134	4	no	—
292	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
293	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
294	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
295	C₂×C₈	#11761628	1	2	yes	—
296	C₂×C₈	#11761628	1	2	yes	—
297	C₄⋊C₄	—	2	1	no	—
298	C₄⋊C₄	—	2	2	no	—
299	C₄⋊C₄	—	134	2	no	—
300	C₄⋊C₄	—	134	2	no	—
301	C₂²⋊C₄	—	2	1	no	—
302	C₂²⋊C₄	—	2	2	no	—
303	C₂²⋊C₄	—	134	2	no	—
304	C₂²⋊C₄	—	134	2	no	—
305	C₄²	—	2	2	no	—
306	C₄²	—	67	1	no	—
307	C₄²	—	67	1	no	—
308	C₄²	—	134	2	no	—
309	C₄²	—	134	2	no	—
310	C₄²	—	134	1	no	—
311	C₄²	C₂×C₄×D₆₇	1	2	yes	—
312	C₂³	—	67	2	no	—
313	C₂³	#41×D₆₇	1	1	yes	characteristic
314	Q₈	—	134	2	no	—
315	D₄	—	134	4	no	—
316	D₄	—	134	2	no	—
317	C₂×C₄	—	2	2	no	—
318	C₂×C₄	—	4	2	no	—
319	C₂×C₄	—	268	2	no	—
320	C₂×C₄	—	536	2	no	—
321	C₂×C₄	—	536	1	no	—
322	C₂×C₄	—	67	2	no	—
323	C₂×C₄	—	134	1	no	—
324	C₂×C₄	—	134	1	no	—
325	C₂×C₄	—	134	2	no	—
326	C₂×C₄	—	134	2	no	—
327	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
328	C₂×C₄	#41×D₆₇	1	2	yes	—
329	C₂×C₄	#41×D₆₇	1	1	yes	characteristic
330	C₂×C₄	C₂×#11761628	1	2	yes	—
331	C₈	—	2	2	no	—
332	C₈	—	4	2	no	—
333	C₈	—	268	2	no	—
334	C₈	—	67	1	no	—
335	C₈	—	67	1	no	—
336	C₈	—	67	2	no	—
337	C₈	—	134	2	no	—
338	C₈	#14524000	1	2	yes	—
339	C₈	#14524000	1	1	yes	characteristic
340	C₈	#14524000	1	1	yes	characteristic
341	C₂²	—	4	1	no	—
342	C₂²	—	268	2	no	—
343	C₂²	—	134	2	no	—
344	C₂²	#117×D₆₇	1	1	yes	characteristic
345	C₂²	#16863924	1	2	yes	—
346	C₄	—	4	2	no	—
347	C₄	—	8	2	no	—
348	C₄	—	268	1	no	—
349	C₄	—	268	1	no	—
350	C₄	—	536	2	no	—
351	C₄	—	134	2	no	—
352	C₄	#117×D₆₇	1	1	yes	characteristic
353	C₄	C₂×#14524000	1	1	yes	characteristic
354	C₄	#16863924	1	2	yes	—
355	C₂	—	2	1	no	—
356	C₂	—	2	2	no	—
357	C₂	—	268	2	no	—
358	C₂	#22562601	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×#14524000
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₆₇×#1747	C₂
socle	C₂×C₆₇	#2283
prime radical	C₂×#11761628	C₂³
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	#1747	D₆₇
hypocenter	C₆₇	#20788
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₄ ◁ #405 ◁ #1747

Complements

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

N	H ≅ G/N	type
2. #22560359	C₂	semidirect
3. #22560356	C₂	semidirect
4. #22560331	C₂	semidirect
8. #22560094	C₂	semidirect
10. #22559964	C₂	semidirect
11. C₆₇×#1747	C₂	semidirect
19. #16863353	C₄	semidirect
22. #16863350	C₄	semidirect
24. C₄×#11761611	C₄	semidirect
26. #97×D₆₇	C₄	semidirect
190. #1747	D₆₇	semidirect
191. C₆₇	#20788	semidirect

Maximal subgroups

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

subgroup	index
#22568636	2
#22560359	2
#22560356	2
#22560331	2
#22560311	2
#22560258	2
#22560254	2
#22560094	2
#22560010	2
#22559964	2
C₆₇×#1747	2
#20788	67