Subgroups — #30838909

Order 17152.

714 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	#22571327	C₂	1	2	yes	maximal
2	C₂×#16863776	C₂	1	1	yes	maximal, characteristic, complemented
3	#22559698	C₂	1	4	yes	maximal, complemented
4	#22559574	C₂	1	4	yes	maximal, complemented
5	#22558957	C₂	1	1	yes	maximal, characteristic, complemented
6	#22558882	C₂	1	2	yes	maximal, complemented
7	C₆₇×#2931	C₂	1	1	yes	maximal, characteristic
8	C₂²×Dic₂₆₈	—	2	1	no	—
9	C₂²×Dic₂₆₈	—	2	1	no	—
10	C₂×#14524043	C₂²	1	2	yes	—
11	#16863799	—	2	2	no	—
12	#16863776	—	2	1	no	—
13	#16863776	—	2	1	no	—
14	#16863776	C₂²	1	2	yes	—
15	#16863776	C₂²	1	2	yes	—
16	#16863742	—	2	2	no	—
17	#16863707	—	2	4	no	—
18	#16863676	C₂²	1	8	yes	—
19	#16863449	C₂²	1	2	yes	—
20	#16863149	C₂²	1	2	yes	—
21	#16863147	C₂²	1	2	yes	—
22	#16863143	C₂²	1	8	yes	—
23	C₂×#131×C₆₇	—	2	1	no	—
24	#532×C₆₇	C₂²	1	1	yes	characteristic
25	#436×C₆₇	—	2	1	no	—
26	#436×C₆₇	C₂²	1	2	yes	—
27	C₈×D₄×C₆₇	—	2	1	no	—
28	C₈×D₄×C₆₇	C₂²	1	2	yes	—
29	#423×C₆₇	C₂²	1	1	yes	characteristic
30	#408×C₆₇	—	2	1	no	—
31	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
32	C₂²×Dic₁₃₄	—	2	2	no	—
33	C₂×Dic₂₆₈	—	2	2	no	—
34	C₂×Dic₂₆₈	—	2	2	no	—
35	C₂×Dic₂₆₈	—	2	2	no	—
36	C₂×Dic₂₆₈	—	2	2	no	—
37	C₂×Dic₂₆₈	—	4	4	no	—
38	C₂×Dic₂₆₈	—	4	4	no	—
39	#14524043	—	2	2	no	—
40	#14524043	—	2	2	no	—
41	#14524043	—	2	2	no	—
42	#14524043	C₂³	1	4	yes	—
43	C₂×C₄⋊Dic₆₇	—	2	2	no	—
44	C₂×C₄⋊Dic₆₇	—	2	2	no	—
45	#14523976	—	2	8	no	—
46	C₄⋊(C₆₇⋊C₈)	C₂³	1	4	yes	—
47	#14523964	—	2	8	no	—
48	C₂²⋊(C₆₇⋊C₈)	—	2	2	no	—
49	C₄×C₆₇⋊C₈	—	2	2	no	—
50	C₂×#41×C₆₇	—	2	1	no	—
51	#131×C₆₇	—	4	4	no	—
52	C₂×M₁₆×C₆₇	—	2	1	no	—
53	C₂×M₁₆×C₆₇	—	2	2	no	—
54	C₂²×C₈×C₆₇	—	2	2	no	—
55	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
56	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
57	C₄⋊Q₈×C₆₇	D₄	1	1	yes	characteristic
58	#127×C₆₇	D₄	1	1	yes	characteristic
59	#124×C₆₇	—	2	1	no	—
60	#121×C₆₇	—	2	1	no	—
61	#121×C₆₇	D₄	1	2	yes	—
62	C₄×D₄×C₆₇	—	2	1	no	—
63	C₄×D₄×C₆₇	C₂³	1	2	yes	—
64	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
65	C₄⋊C₈×C₆₇	—	2	1	no	—
66	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
67	C₂²⋊C₈×C₆₇	—	2	2	no	—
68	C₂²⋊C₈×C₆₇	—	2	2	no	—
69	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
70	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
71	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
72	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
73	C₂×Dic₁₃₄	—	2	4	no	—
74	C₂×Dic₁₃₄	—	4	4	no	—
75	C₂×Dic₁₃₄	—	4	4	no	—
76	C₂×Dic₁₃₄	—	4	2	no	—
77	C₂²×Dic₆₇	—	4	2	no	—
78	Dic₂₆₈	—	4	4	no	—
79	Dic₂₆₈	—	4	4	no	—
80	Dic₂₆₈	—	4	4	no	—
81	Dic₂₆₈	—	4	4	no	—
82	C₂×C₆₇⋊C₈	—	2	4	no	—
83	C₄⋊Dic₆₇	—	2	4	no	—
84	C₄⋊Dic₆₇	—	2	4	no	—
85	C₄⋊Dic₆₇	—	4	2	no	—
86	C₄⋊Dic₆₇	—	4	2	no	—
87	#41×C₆₇	—	4	2	no	—
88	#41×C₆₇	—	4	2	no	—
89	C₂×Q₈×C₆₇	—	2	1	no	—
90	C₂×D₄×C₆₇	—	2	2	no	—
91	C₂×D₄×C₆₇	—	2	1	no	—
92	C₂×D₄×C₆₇	—	2	1	no	—
93	C₂×D₄×C₆₇	C₂×D₄	1	2	yes	—
94	C₂²×C₄×C₆₇	—	2	2	no	—
95	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
96	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
97	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
98	M₁₆×C₆₇	—	4	2	no	—
99	M₁₆×C₆₇	—	4	4	no	—
100	C₂×C₈×C₆₇	—	2	2	no	—
101	C₂×C₈×C₆₇	—	2	2	no	—
102	C₂×C₈×C₆₇	—	4	4	no	—
103	C₂×C₈×C₆₇	D₈	1	4	yes	—
104	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
105	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
106	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
107	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
108	C₄⋊C₄×C₆₇	—	2	1	no	—
109	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
110	C₂²⋊C₄×C₆₇	—	2	2	no	—
111	C₂²⋊C₄×C₆₇	—	2	2	no	—
112	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
113	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
114	C₄²×C₆₇	#41	1	1	yes	characteristic
115	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
116	C₂³×C₆₇	—	2	2	no	—
117	C₂³×C₆₇	#121	1	1	yes	characteristic
118	Q₈×C₆₇	—	4	2	no	—
119	D₄×C₆₇	—	2	4	no	—
120	D₄×C₆₇	—	4	4	no	—
121	D₄×C₆₇	—	4	2	no	—
122	D₄×C₆₇	—	4	2	no	—
123	C₂×C₄×C₆₇	—	2	2	no	—
124	C₂×C₄×C₆₇	—	2	2	no	—
125	C₂×C₄×C₆₇	—	2	2	no	—
126	C₂×C₄×C₆₇	—	4	2	no	—
127	C₂×C₄×C₆₇	—	4	2	no	—
128	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
129	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
130	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
131	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
132	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
133	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
134	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
135	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
136	C₂×Dic₆₇	—	4	4	no	—
137	C₂×Dic₆₇	—	8	4	no	—
138	Dic₁₃₄	—	4	4	no	—
139	Dic₁₃₄	—	4	4	no	—
140	Dic₁₃₄	—	8	4	no	—
141	C₈×C₆₇	—	2	1	no	—
142	C₈×C₆₇	—	2	1	no	—
143	C₈×C₆₇	—	4	2	no	—
144	C₈×C₆₇	C₂×D₈	1	2	yes	—
145	C₈×C₆₇	C₂×D₈	1	2	yes	—
146	C₆₇⋊C₈	—	4	4	no	—
147	C₂²×C₆₇	—	2	1	no	—
148	C₂²×C₆₇	—	2	1	no	—
149	C₂²×C₆₇	—	4	2	no	—
150	C₂²×C₆₇	—	4	2	no	—
151	C₂²×C₆₇	—	4	2	no	—
152	C₂²×C₆₇	#459	1	2	yes	—
153	C₂²×C₆₇	C₂×#121	1	1	yes	characteristic
154	C₄×C₆₇	—	2	1	no	—
155	C₄×C₆₇	—	2	1	no	—
156	C₄×C₆₇	—	4	2	no	—
157	C₄×C₆₇	#459	1	2	yes	—
158	C₄×C₆₇	#473	1	2	yes	—
159	C₄×C₆₇	C₂²×D₈	1	1	yes	characteristic
160	C₄×C₆₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
161	C₄×C₆₇	#578	1	2	yes	—
162	Dic₆₇	—	8	4	no	—
163	#19188	—	67	1	no	maximal, Sylow
164	C₂×C₆₇	—	2	2	no	—
165	C₂×C₆₇	—	8	2	no	—
166	C₂×C₆₇	C₂×#459	1	1	yes	characteristic
167	C₂×C₆₇	#3059	1	1	yes	characteristic
168	C₂×C₆₇	C₂×#473	1	1	yes	characteristic
169	#3226	—	67	2	no	—
170	C₂×#494	—	67	1	no	—
171	#2931	D₆₇	1	1	yes	characteristic
172	#1689	—	67	4	no	—
173	#1669	—	67	4	no	—
174	#1564	—	67	1	no	—
175	#1546	—	67	2	no	—
176	C₆₇	#19188	1	1	yes	characteristic, complemented, Sylow
177	C₂²×Q₁₆	—	134	1	no	—
178	C₂²×Q₁₆	—	134	1	no	—
179	C₂×#131	—	2	1	no	—
180	#532	C₂×D₆₇	1	1	yes	characteristic
181	C₂×C₄⋊Q₈	—	67	2	no	—
182	#500	—	134	2	no	—
183	#494	—	67	2	no	—
184	#494	—	67	2	no	—
185	#494	—	134	1	no	—
186	#494	—	134	1	no	—
187	#486	—	134	2	no	—
188	#480	—	134	4	no	—
189	#474	—	67	8	no	—
190	#436	—	2	1	no	—
191	#436	C₂×D₆₇	1	2	yes	—
192	C₈×D₄	—	2	1	no	—
193	C₈×D₄	C₂×D₆₇	1	2	yes	—
194	#431	—	67	2	no	—
195	#423	C₂×D₆₇	1	1	yes	characteristic
196	#408	—	2	1	no	—
197	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
198	#335	—	67	2	no	—
199	#334	—	67	2	no	—
200	#332	—	67	8	no	—
201	C₂×#41	—	2	1	no	—
202	C₂²×Q₈	—	134	2	no	—
203	C₂×Q₁₆	—	268	4	no	—
204	C₂×Q₁₆	—	268	4	no	—
205	C₂×Q₁₆	—	134	2	no	—
206	C₂×Q₁₆	—	134	2	no	—
207	C₂×Q₁₆	—	134	2	no	—
208	C₂×Q₁₆	—	134	2	no	—
209	#131	—	4	4	no	—
210	C₂×M₁₆	—	2	1	no	—
211	C₂×M₁₆	—	2	2	no	—
212	C₂²×C₈	—	2	2	no	—
213	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
214	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
215	C₄⋊Q₈	—	67	4	no	—
216	C₄⋊Q₈	—	134	2	no	—
217	C₄⋊Q₈	—	134	2	no	—
218	C₄⋊Q₈	—	134	2	no	—
219	C₄⋊Q₈	D₂₆₈	1	1	yes	characteristic
220	#127	D₂₆₈	1	1	yes	characteristic
221	#124	—	2	1	no	—
222	#121	—	2	1	no	—
223	#121	D₂₆₈	1	2	yes	—
224	C₄×D₄	—	2	1	no	—
225	C₄×D₄	C₂²×D₆₇	1	2	yes	—
226	C₂×C₄⋊C₄	—	134	2	no	—
227	C₂×C₄⋊C₄	—	134	2	no	—
228	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
229	#107	—	134	8	no	—
230	C₄⋊C₈	—	2	1	no	—
231	C₄⋊C₈	—	67	4	no	—
232	C₄⋊C₈	C₂²×D₆₇	1	2	yes	—
233	#102	—	134	8	no	—
234	C₂²⋊C₈	—	2	2	no	—
235	C₂²⋊C₈	—	2	2	no	—
236	C₂²⋊C₈	—	134	2	no	—
237	C₄×C₈	—	134	2	no	—
238	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
239	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
240	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
241	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
242	#41	—	4	2	no	—
243	#41	—	4	2	no	—
244	C₂×Q₈	—	2	1	no	—
245	C₂×Q₈	—	268	2	no	—
246	C₂×Q₈	—	268	4	no	—
247	C₂×Q₈	—	268	4	no	—
248	C₂×Q₈	—	134	4	no	—
249	C₂×D₄	—	2	2	no	—
250	C₂×D₄	—	2	1	no	—
251	C₂×D₄	—	2	1	no	—
252	C₂×D₄	C₂×D₂₆₈	1	2	yes	—
253	C₂²×C₄	—	2	2	no	—
254	C₂²×C₄	—	268	2	no	—
255	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
256	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
257	C₂²×C₄	#11761628	1	1	yes	characteristic
258	Q₁₆	—	268	4	no	—
259	Q₁₆	—	268	4	no	—
260	Q₁₆	—	268	4	no	—
261	Q₁₆	—	268	4	no	—
262	M₁₆	—	4	2	no	—
263	M₁₆	—	4	4	no	—
264	C₂×C₈	—	2	2	no	—
265	C₂×C₈	—	2	2	no	—
266	C₂×C₈	—	4	4	no	—
267	C₂×C₈	—	134	4	no	—
268	C₂×C₈	#11761613	1	4	yes	—
269	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
270	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
271	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
272	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
273	C₄⋊C₄	—	2	1	no	—
274	C₄⋊C₄	—	268	2	no	—
275	C₄⋊C₄	—	268	2	no	—
276	C₄⋊C₄	—	134	4	no	—
277	C₄⋊C₄	—	134	4	no	—
278	C₄⋊C₄	C₂×D₂₆₈	1	2	yes	—
279	C₂²⋊C₄	—	2	2	no	—
280	C₂²⋊C₄	—	2	2	no	—
281	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
282	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
283	C₄²	#11761628	1	1	yes	characteristic
284	C₄²	C₂³×D₆₇	1	1	yes	characteristic
285	C₂³	—	2	2	no	—
286	C₂³	#14524017	1	1	yes	characteristic
287	Q₈	—	4	2	no	—
288	Q₈	—	268	4	no	—
289	Q₈	—	268	4	no	—
290	Q₈	—	536	4	no	—
291	D₄	—	2	4	no	—
292	D₄	—	4	4	no	—
293	D₄	—	4	2	no	—
294	D₄	—	4	2	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	4	2	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	—	268	4	no	—
301	C₂×C₄	—	536	4	no	—
302	C₂×C₄	#14524017	1	1	yes	characteristic
303	C₂×C₄	#14524017	1	1	yes	characteristic
304	C₂×C₄	#14524017	1	1	yes	characteristic
305	C₂×C₄	C₂×#11761613	1	2	yes	—
306	C₂×C₄	#14524068	1	2	yes	—
307	C₂×C₄	C₂²×C₂²⋊D₆₇	1	1	yes	characteristic
308	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
309	C₂×C₄	C₂×#11761628	1	1	yes	characteristic
310	C₈	—	2	1	no	—
311	C₈	—	2	1	no	—
312	C₈	—	4	2	no	—
313	C₈	—	268	4	no	—
314	C₈	C₂×#11761613	1	2	yes	—
315	C₈	C₂×#11761613	1	2	yes	—
316	C₂²	—	2	1	no	—
317	C₂²	—	2	1	no	—
318	C₂²	—	4	2	no	—
319	C₂²	—	4	2	no	—
320	C₂²	—	4	2	no	—
321	C₂²	#16863591	1	2	yes	—
322	C₂²	C₂×#14524017	1	1	yes	characteristic
323	C₄	—	2	1	no	—
324	C₄	—	2	1	no	—
325	C₄	—	4	2	no	—
326	C₄	—	536	4	no	—
327	C₄	#16863591	1	2	yes	—
328	C₄	#16863671	1	2	yes	—
329	C₄	C₂²×#11761613	1	1	yes	characteristic
330	C₄	C₂×#14524068	1	1	yes	characteristic
331	C₄	#16864213	1	2	yes	—
332	C₂	—	2	2	no	—
333	C₂	—	8	2	no	—
334	C₂	C₂×#16863591	1	1	yes	characteristic
335	C₂	#22569864	1	1	yes	characteristic
336	C₂	C₂×#16863671	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#14524017
commutator, cosocle	C₄²×C₆₇	C₂⁴
Frattini	C₄²	C₂³×D₆₇
Fitting	C₆₇×#2931	C₂
socle	C₂²×C₆₇	C₂×#121
prime radical	#532×C₆₇	C₂²
hypercenter	#2931	D₆₇
hypocenter	C₆₇	#19188
LCS	G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931

Complements

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

N	H ≅ G/N	type
2. C₂×#16863776	C₂	semidirect
3. #22559698	C₂	semidirect
4. #22559574	C₂	semidirect
5. #22558957	C₂	semidirect
6. #22558882	C₂	semidirect
176. C₆₇	#19188	semidirect

Maximal subgroups

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

subgroup	index
#22571327	2
C₂×#16863776	2
#22559698	2
#22559574	2
#22558957	2
#22558882	2
C₆₇×#2931	2
#19188	67