Subgroups — #31012780

Order 17152.

556 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	#22572640	C₂	1	1	yes	maximal, characteristic, complemented
2	#22562432	C₂	1	1	yes	maximal, characteristic, complemented
3	#22562397	C₂	1	2	yes	maximal, complemented
4	#22562351	C₂	1	1	yes	maximal, characteristic, complemented
5	#22562303	C₂	1	2	yes	maximal, complemented
6	#22562246	C₂	1	1	yes	maximal, characteristic, complemented
7	#22562149	C₂	1	2	yes	maximal, complemented
8	C₄×#14523989	C₂	1	1	yes	maximal, characteristic, complemented
9	D₄×C₆₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
10	#22561916	C₂	1	2	yes	maximal, complemented
11	C₆₇×#3287	C₂	1	1	yes	maximal, characteristic
12	#16864116	—	2	1	no	—
13	C₂×#14523989	C₂²	1	1	yes	characteristic
14	C₂×#14523987	C₂²	1	2	yes	—
15	C₂²×C₆₇⋊C₁₆	C₂²	1	2	yes	—
16	#16863801	C₂²	1	1	yes	characteristic
17	#16863738	—	2	2	no	—
18	#16863682	—	2	1	no	—
19	#16863637	C₂²	1	2	yes	—
20	C₄×Dic₂₆₈	C₂²	1	1	yes	characteristic
21	#16863434	C₂²	1	2	yes	—
22	#16863223	C₂²	1	2	yes	—
23	#16863221	C₂²	1	1	yes	characteristic
24	C₄⋊(C₆₇⋊C₁₆)	C₂²	1	1	yes	characteristic
25	#16863200	C₂²	1	1	yes	characteristic
26	#16863200	C₂²	1	2	yes	—
27	#16863199	C₂²	1	1	yes	characteristic
28	#16863197	C₂²	1	2	yes	—
29	#16863197	C₂²	1	2	yes	—
30	#16863197	C₂²	1	2	yes	—
31	C₂²⋊(C₆₇⋊C₁₆)	C₂²	1	2	yes	—
32	C₄×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
33	C₂×#135×C₆₇	C₂²	1	2	yes	—
34	#550×C₆₇	—	2	1	no	—
35	#493×C₆₇	C₂²	1	1	yes	characteristic
36	#466×C₆₇	C₂²	1	2	yes	—
37	#460×C₆₇	—	2	1	no	—
38	#449×C₆₇	—	2	2	no	—
39	C₄×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic
40	C₈×D₄×C₆₇	C₂²	1	1	yes	characteristic
41	C₂×Dic₂₆₈	C₂³	1	1	yes	characteristic
42	#14524043	—	2	1	no	—
43	#14524036	—	4	2	no	—
44	#14524030	—	4	1	no	—
45	#14524022	—	2	2	no	—
46	C₄×Dic₁₃₄	—	2	1	no	—
47	#14524008	—	4	1	no	—
48	#14524002	—	2	2	no	—
49	#14523989	—	2	2	no	—
50	#14523987	—	4	2	no	—
51	C₂×C₆₇⋊C₁₆	—	4	2	no	—
52	C₂×C₆₇⋊C₁₆	C₂³	1	1	yes	characteristic
53	C₂×C₆₇⋊C₁₆	C₂³	1	2	yes	—
54	C₂×C₆₇⋊C₁₆	C₂³	1	1	yes	characteristic
55	#14523977	—	2	1	no	—
56	#14523977	C₂³	1	1	yes	characteristic
57	#14523977	C₂³	1	2	yes	—
58	#14523975	—	2	2	no	—
59	#14523968	—	2	1	no	—
60	#14523968	—	2	2	no	—
61	C₂×#41×C₆₇	—	2	2	no	—
62	#135×C₆₇	—	2	4	no	—
63	#135×C₆₇	—	4	2	no	—
64	C₂×Q₁₆×C₆₇	C₂³	1	1	yes	characteristic
65	C₂×SD₁₆×C₆₇	—	2	2	no	—
66	C₂×D₈×C₆₇	—	2	1	no	—
67	C₂×D₈×C₆₇	C₂³	1	2	yes	—
68	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
69	#127×C₆₇	—	2	1	no	—
70	#127×C₆₇	—	4	1	no	—
71	#121×C₆₇	—	2	2	no	—
72	#121×C₆₇	—	4	2	no	—
73	C₄×Q₈×C₆₇	—	2	1	no	—
74	C₄×D₄×C₆₇	—	4	1	no	—
75	C₄×D₄×C₆₇	D₄	1	1	yes	characteristic
76	#107×C₆₇	C₂³	1	1	yes	characteristic
77	C₄⋊C₈×C₆₇	D₄	1	1	yes	characteristic
78	#103×C₆₇	—	2	1	no	—
79	#102×C₆₇	—	2	2	no	—
80	C₂²⋊C₈×C₆₇	D₄	1	2	yes	—
81	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
82	C₂×Dic₁₃₄	—	2	1	no	—
83	C₂×Dic₁₃₄	—	4	1	no	—
84	Dic₂₆₈	—	2	1	no	—
85	Dic₂₆₈	—	2	1	no	—
86	C₄⋊Dic₆₇	—	2	1	no	—
87	C₄⋊Dic₆₇	—	2	2	no	—
88	C₄⋊Dic₆₇	—	4	1	no	—
89	Dic₆₇⋊C₄	—	4	2	no	—
90	Dic₆₇⋊C₄	—	4	1	no	—
91	C₂²⋊Dic₆₇	—	4	2	no	—
92	C₂²⋊Dic₆₇	—	4	2	no	—
93	C₄×Dic₆₇	—	4	2	no	—
94	C₄×Dic₆₇	—	4	1	no	—
95	C₆₇⋊C₁₆	—	2	1	no	—
96	C₆₇⋊C₁₆	—	2	1	no	—
97	C₆₇⋊C₁₆	—	2	2	no	—
98	#41×C₆₇	—	4	4	no	—
99	#41×C₆₇	—	8	2	no	—
100	C₂×Q₈×C₆₇	—	2	1	no	—
101	C₂×D₄×C₆₇	—	2	2	no	—
102	C₂×D₄×C₆₇	—	4	2	no	—
103	C₂×D₄×C₆₇	—	4	2	no	—
104	C₂×D₄×C₆₇	—	4	1	no	—
105	C₂×D₄×C₆₇	—	4	1	no	—
106	C₂×D₄×C₆₇	D₈	1	1	yes	characteristic
107	C₂²×C₄×C₆₇	—	4	2	no	—
108	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
109	Q₁₆×C₆₇	—	2	1	no	—
110	Q₁₆×C₆₇	#41	1	2	yes	—
111	SD₁₆×C₆₇	—	4	4	no	—
112	D₈×C₆₇	—	2	2	no	—
113	D₈×C₆₇	—	2	2	no	—
114	D₈×C₆₇	—	8	1	no	—
115	C₂×C₈×C₆₇	—	2	2	no	—
116	C₂×C₈×C₆₇	—	2	2	no	—
117	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
118	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
119	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
120	C₄⋊C₄×C₆₇	—	2	1	no	—
121	C₄⋊C₄×C₆₇	—	4	1	no	—
122	C₄⋊C₄×C₆₇	D₈	1	1	yes	characteristic
123	C₂²⋊C₄×C₆₇	—	4	2	no	—
124	C₂²⋊C₄×C₆₇	D₈	1	2	yes	—
125	C₄²×C₆₇	—	4	1	no	—
126	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
127	C₂³×C₆₇	—	4	2	no	—
128	C₂³×C₆₇	C₂×D₈	1	2	yes	—
129	Q₈×C₆₇	—	2	2	no	—
130	Q₈×C₆₇	—	4	1	no	—
131	D₄×C₆₇	—	2	2	no	—
132	D₄×C₆₇	—	4	2	no	—
133	D₄×C₆₇	—	4	2	no	—
134	D₄×C₆₇	—	8	4	no	—
135	D₄×C₆₇	—	8	2	no	—
136	D₄×C₆₇	—	16	1	no	—
137	D₄×C₆₇	—	16	2	no	—
138	C₂×C₄×C₆₇	—	2	2	no	—
139	C₂×C₄×C₆₇	—	2	2	no	—
140	C₂×C₄×C₆₇	—	4	1	no	—
141	C₂×C₄×C₆₇	—	4	1	no	—
142	C₂×C₄×C₆₇	—	8	4	no	—
143	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
144	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
145	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
146	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
147	C₂×Dic₆₇	—	4	1	no	—
148	C₂×Dic₆₇	—	4	1	no	—
149	C₂×Dic₆₇	—	4	2	no	—
150	Dic₁₃₄	—	4	1	no	—
151	Dic₁₃₄	—	4	1	no	—
152	Dic₁₃₄	—	8	2	no	—
153	C₈×C₆₇	—	2	2	no	—
154	C₈×C₆₇	—	2	1	no	—
155	C₈×C₆₇	C₂×#41	1	1	yes	characteristic
156	C₈×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
157	C₂²×C₆₇	—	2	2	no	—
158	C₂²×C₆₇	—	2	2	no	—
159	C₂²×C₆₇	—	2	2	no	—
160	C₂²×C₆₇	—	8	2	no	—
161	C₂²×C₆₇	—	8	2	no	—
162	C₂²×C₆₇	—	8	2	no	—
163	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
164	C₄×C₆₇	—	2	1	no	—
165	C₄×C₆₇	—	2	2	no	—
166	C₄×C₆₇	—	4	2	no	—
167	C₄×C₆₇	—	8	1	no	—
168	C₄×C₆₇	#508	1	2	yes	—
169	C₄×C₆₇	#546	1	1	yes	characteristic
170	C₄×C₆₇	#577	1	1	yes	characteristic
171	Dic₆₇	—	8	2	no	—
172	Dic₆₇	—	8	1	no	—
173	Dic₆₇	—	8	1	no	—
174	#33871	—	67	1	no	maximal, Sylow
175	C₂×C₆₇	—	4	2	no	—
176	C₂×C₆₇	—	16	2	no	—
177	C₂×C₆₇	#3294	1	1	yes	characteristic
178	C₂×C₆₇	C₂×#508	1	1	yes	characteristic
179	C₂×C₆₇	#3416	1	1	yes	characteristic
180	#3342	—	67	1	no	—
181	#3287	D₆₇	1	1	yes	characteristic
182	#2240	—	67	1	no	—
183	#2234	—	67	2	no	—
184	#2225	—	67	1	no	—
185	#2217	—	67	2	no	—
186	#2207	—	67	1	no	—
187	#2189	—	67	2	no	—
188	C₄×Q₃₂	—	67	1	no	—
189	D₄×C₁₆	—	67	1	no	—
190	#2138	—	67	2	no	—
191	C₆₇	#33871	1	1	yes	characteristic, complemented, Sylow
192	C₂×#135	C₂×D₆₇	1	2	yes	—
193	#556	—	134	1	no	—
194	#550	—	2	1	no	—
195	C₂×Q₃₂	—	67	1	no	—
196	C₂×SD₃₂	—	67	2	no	—
197	C₂²×C₁₆	—	67	2	no	—
198	#500	—	67	1	no	—
199	#493	C₂×D₆₇	1	1	yes	characteristic
200	#485	—	134	2	no	—
201	#475	—	134	1	no	—
202	#467	—	67	2	no	—
203	#466	C₂×D₆₇	1	2	yes	—
204	#460	—	2	1	no	—
205	#449	—	2	2	no	—
206	C₄×Q₁₆	—	67	1	no	—
207	C₄×Q₁₆	C₂×D₆₇	1	1	yes	characteristic
208	C₈×D₄	C₂×D₆₇	1	1	yes	characteristic
209	#427	—	67	2	no	—
210	#367	—	67	2	no	—
211	#366	—	67	1	no	—
212	C₄⋊C₁₆	—	67	1	no	—
213	#358	—	67	1	no	—
214	#358	—	67	2	no	—
215	#358	—	67	1	no	—
216	#357	—	67	2	no	—
217	#357	—	67	2	no	—
218	#357	—	67	2	no	—
219	C₂²⋊C₁₆	—	67	2	no	—
220	C₄×C₁₆	—	67	1	no	—
221	C₂×#41	—	2	2	no	—
222	#135	—	2	4	no	—
223	#135	—	4	2	no	—
224	C₂×Q₁₆	—	67	1	no	—
225	C₂×Q₁₆	C₂²×D₆₇	1	1	yes	characteristic
226	C₂×SD₁₆	—	2	2	no	—
227	C₂×D₈	—	2	1	no	—
228	C₂×D₈	C₂²×D₆₇	1	2	yes	—
229	C₂²×C₈	C₂²×D₆₇	1	2	yes	—
230	C₄⋊Q₈	—	134	1	no	—
231	#127	—	2	1	no	—
232	#127	—	4	1	no	—
233	#126	—	268	2	no	—
234	#124	—	268	1	no	—
235	C₂²⋊Q₈	—	134	2	no	—
236	#121	—	2	2	no	—
237	#121	—	4	2	no	—
238	C₄×Q₈	—	2	1	no	—
239	C₄×Q₈	—	268	1	no	—
240	C₄×Q₈	—	134	1	no	—
241	C₄×D₄	—	4	1	no	—
242	C₄×D₄	C₂²⋊D₆₇	1	1	yes	characteristic
243	#117	—	134	2	no	—
244	Q₃₂	—	134	2	no	—
245	SD₃₂	—	268	2	no	—
246	C₂×C₁₆	—	268	2	no	—
247	C₂×C₁₆	—	67	1	no	—
248	C₂×C₁₆	—	67	1	no	—
249	C₂×C₁₆	—	67	2	no	—
250	#107	—	67	1	no	—
251	#107	—	67	2	no	—
252	#107	—	134	1	no	—
253	#107	C₂²×D₆₇	1	1	yes	characteristic
254	#106	—	134	2	no	—
255	C₄⋊C₈	C₂²⋊D₆₇	1	1	yes	characteristic
256	#103	—	2	1	no	—
257	#103	—	134	1	no	—
258	#103	—	134	2	no	—
259	#102	—	2	2	no	—
260	C₂²⋊C₈	C₂²⋊D₆₇	1	2	yes	—
261	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
262	#41	—	4	4	no	—
263	#41	—	8	2	no	—
264	C₂×Q₈	—	2	1	no	—
265	C₂×Q₈	—	268	1	no	—
266	C₂×Q₈	—	134	1	no	—
267	C₂×D₄	—	2	2	no	—
268	C₂×D₄	—	4	2	no	—
269	C₂×D₄	—	4	2	no	—
270	C₂×D₄	—	4	1	no	—
271	C₂×D₄	—	4	1	no	—
272	C₂×D₄	#11761613	1	1	yes	characteristic
273	C₂²×C₄	—	4	2	no	—
274	C₂²×C₄	C₂×C₂²⋊D₆₇	1	2	yes	—
275	Q₁₆	—	2	1	no	—
276	Q₁₆	—	134	1	no	—
277	Q₁₆	—	134	1	no	—
278	Q₁₆	#11761627	1	2	yes	—
279	SD₁₆	—	4	4	no	—
280	D₈	—	2	2	no	—
281	D₈	—	2	2	no	—
282	D₈	—	8	1	no	—
283	C₂×C₈	—	2	2	no	—
284	C₂×C₈	—	2	2	no	—
285	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
286	C₂×C₈	C₂×C₂²⋊D₆₇	1	2	yes	—
287	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
288	C₄⋊C₄	—	2	1	no	—
289	C₄⋊C₄	—	4	1	no	—
290	C₄⋊C₄	—	268	2	no	—
291	C₄⋊C₄	—	268	1	no	—
292	C₄⋊C₄	—	268	1	no	—
293	C₄⋊C₄	—	134	1	no	—
294	C₄⋊C₄	—	134	2	no	—
295	C₄⋊C₄	#11761613	1	1	yes	characteristic
296	C₂²⋊C₄	—	4	2	no	—
297	C₂²⋊C₄	—	268	2	no	—
298	C₂²⋊C₄	—	268	2	no	—
299	C₂²⋊C₄	#11761613	1	2	yes	—
300	C₄²	—	4	1	no	—
301	C₄²	—	268	2	no	—
302	C₄²	—	268	1	no	—
303	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
304	C₁₆	—	134	1	no	—
305	C₁₆	—	134	1	no	—
306	C₁₆	—	134	2	no	—
307	C₂³	—	4	2	no	—
308	C₂³	C₂×#11761613	1	2	yes	—
309	Q₈	—	2	2	no	—
310	Q₈	—	4	1	no	—
311	Q₈	—	268	1	no	—
312	Q₈	—	268	1	no	—
313	Q₈	—	536	2	no	—
314	D₄	—	2	2	no	—
315	D₄	—	4	2	no	—
316	D₄	—	4	2	no	—
317	D₄	—	8	4	no	—
318	D₄	—	8	2	no	—
319	D₄	—	16	1	no	—
320	D₄	—	16	2	no	—
321	C₂×C₄	—	2	2	no	—
322	C₂×C₄	—	2	2	no	—
323	C₂×C₄	—	4	1	no	—
324	C₂×C₄	—	4	1	no	—
325	C₂×C₄	—	8	4	no	—
326	C₂×C₄	—	268	1	no	—
327	C₂×C₄	—	268	1	no	—
328	C₂×C₄	—	268	2	no	—
329	C₂×C₄	C₂×#11761613	1	2	yes	—
330	C₂×C₄	C₂×#11761613	1	1	yes	characteristic
331	C₂×C₄	C₂×#11761613	1	1	yes	characteristic
332	C₂×C₄	C₂²×C₂²⋊D₆₇	1	1	yes	characteristic
333	C₈	—	2	2	no	—
334	C₈	—	2	1	no	—
335	C₈	C₂×#11761627	1	1	yes	characteristic
336	C₈	#14524088	1	1	yes	characteristic
337	C₂²	—	2	2	no	—
338	C₂²	—	2	2	no	—
339	C₂²	—	2	2	no	—
340	C₂²	—	8	2	no	—
341	C₂²	—	8	2	no	—
342	C₂²	—	8	2	no	—
343	C₂²	C₂²×#11761613	1	1	yes	characteristic
344	C₄	—	2	1	no	—
345	C₄	—	2	2	no	—
346	C₄	—	4	2	no	—
347	C₄	—	8	1	no	—
348	C₄	—	536	2	no	—
349	C₄	—	536	1	no	—
350	C₄	—	536	1	no	—
351	C₄	#16863826	1	2	yes	—
352	C₄	#16864040	1	1	yes	characteristic
353	C₄	#16864207	1	1	yes	characteristic
354	C₂	—	4	2	no	—
355	C₂	—	16	2	no	—
356	C₂	#22572022	1	1	yes	characteristic
357	C₂	C₂×#16863826	1	1	yes	characteristic
358	C₂	#22573356	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×#11761613
commutator, cosocle	C₂×C₈×C₆₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₆₇
Fitting, prime radical	C₆₇×#3287	C₂
socle	C₂²×C₆₇	C₂²×D₈
hypercenter	#3287	D₆₇
hypocenter	C₆₇	#33871
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ C₈×D₄ ◁ #3287

Complements

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

N	H ≅ G/N	type
1. #22572640	C₂	semidirect
2. #22562432	C₂	semidirect
3. #22562397	C₂	semidirect
4. #22562351	C₂	semidirect
5. #22562303	C₂	semidirect
6. #22562246	C₂	semidirect
7. #22562149	C₂	semidirect
8. C₄×#14523989	C₂	semidirect
9. D₄×C₆₇⋊C₁₆	C₂	semidirect
10. #22561916	C₂	semidirect
191. C₆₇	#33871	semidirect

Maximal subgroups

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

subgroup	index
#22572640	2
#22562432	2
#22562397	2
#22562351	2
#22562303	2
#22562246	2
#22562149	2
C₄×#14523989	2
D₄×C₆₇⋊C₁₆	2
#22561916	2
C₆₇×#3287	2
#33871	67