Subgroups — #30772861

Order 17152.

492 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	#22568820	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#16863209	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#16863209	C₂	1	1	yes	maximal, characteristic, complemented
4	#22561837	C₂	1	1	yes	maximal, characteristic, complemented
5	#22558320	C₂	1	1	yes	maximal, characteristic, complemented
6	#22558312	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#472×C₆₇	C₂	1	1	yes	maximal, characteristic
8	#16864163	—	2	1	no	—
9	C₂×#14523983	C₂²	1	1	yes	characteristic
10	C₂×#14523983	C₂²	1	1	yes	characteristic
11	#16863445	—	2	2	no	—
12	#16863445	—	2	1	no	—
13	C₂×#14523979	—	2	2	no	—
14	C₂×#14523979	C₂²	1	1	yes	characteristic
15	C₂×#14523979	C₂²	1	1	yes	characteristic
16	#16863209	—	2	2	no	—
17	#16863209	—	2	2	no	—
18	#16863180	—	2	2	no	—
19	C₂×AGL(1,ℤ/8)×C₆₇	—	2	1	no	—
20	C₂²×D₈×C₆₇	C₂²	1	1	yes	characteristic
21	C₂×#131×C₆₇	C₂²	1	1	yes	characteristic
22	#472×C₆₇	—	2	2	no	—
23	#472×C₆₇	C₄	1	4	yes	—
24	C₂×#108×C₆₇	C₂²	1	1	yes	characteristic
25	C₂×#100×C₆₇	—	2	1	no	—
26	#14524050	—	4	2	no	—
27	#14524050	—	4	2	no	—
28	C₂×#11761612	—	2	4	no	—
29	C₂×#11761612	—	2	1	no	—
30	C₂×#11761612	—	2	1	no	—
31	#14523983	—	2	1	no	—
32	#14523983	—	2	1	no	—
33	#14523983	—	2	1	no	—
34	#14523983	—	2	1	no	—
35	C₂×C₆₇⋊C₁₆	—	2	1	no	—
36	C₂×C₆₇⋊C₁₆	—	2	1	no	—
37	#14523979	—	2	4	no	—
38	#14523979	—	2	1	no	—
39	#14523979	—	2	1	no	—
40	#14523979	D₄	1	2	yes	—
41	#14523979	D₄	1	2	yes	—
42	C₂×#41×C₆₇	D₄	1	1	yes	characteristic
43	C₂²×D₄×C₆₇	—	2	1	no	—
44	AGL(1,ℤ/8)×C₆₇	—	2	4	no	—
45	AGL(1,ℤ/8)×C₆₇	—	4	2	no	—
46	C₂×SD₁₆×C₆₇	—	4	1	no	—
47	C₂×D₈×C₆₇	—	2	1	no	—
48	C₂×D₈×C₆₇	—	2	1	no	—
49	C₂×D₈×C₆₇	—	2	1	no	—
50	C₂×D₈×C₆₇	—	2	1	no	—
51	C₂×D₈×C₆₇	—	2	1	no	—
52	C₂×D₈×C₆₇	—	4	1	no	—
53	C₂×D₈×C₆₇	C₂×C₄	1	2	yes	—
54	#131×C₆₇	—	2	1	no	—
55	#131×C₆₇	—	2	1	no	—
56	#131×C₆₇	C₂×C₄	1	2	yes	—
57	#131×C₆₇	D₄	1	2	yes	—
58	C₂×M₁₆×C₆₇	—	2	1	no	—
59	C₂×M₁₆×C₆₇	—	2	1	no	—
60	C₂×M₁₆×C₆₇	D₄	1	1	yes	characteristic
61	C₂²×C₈×C₆₇	—	2	1	no	—
62	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
63	#108×C₆₇	—	2	1	no	—
64	#108×C₆₇	C₂×C₄	1	2	yes	—
65	#100×C₆₇	—	2	4	no	—
66	#11761612	—	2	2	no	—
67	#11761612	—	2	2	no	—
68	#11761612	—	4	4	no	—
69	#11761612	—	4	1	no	—
70	#11761612	—	4	1	no	—
71	C₂×C₆₇⋊C₈	—	4	4	no	—
72	C₂×C₆₇⋊C₈	—	4	1	no	—
73	C₂×C₆₇⋊C₈	—	4	1	no	—
74	C₆₇⋊C₁₆	—	4	1	no	—
75	C₆₇⋊C₁₆	—	4	1	no	—
76	C₂⁴×C₆₇	—	4	1	no	—
77	#41×C₆₇	—	2	1	no	—
78	#41×C₆₇	—	2	1	no	—
79	#41×C₆₇	C₂²⋊C₄	1	2	yes	—
80	#41×C₆₇	D₈	1	2	yes	—
81	C₂×Q₈×C₆₇	SD₁₆	1	1	yes	characteristic
82	C₂×D₄×C₆₇	—	2	1	no	—
83	C₂×D₄×C₆₇	—	2	2	no	—
84	C₂×D₄×C₆₇	—	4	1	no	—
85	C₂×D₄×C₆₇	—	4	1	no	—
86	C₂×D₄×C₆₇	—	4	1	no	—
87	C₂×D₄×C₆₇	—	4	1	no	—
88	C₂×D₄×C₆₇	—	4	1	no	—
89	C₂×D₄×C₆₇	SD₁₆	1	1	yes	characteristic
90	C₂²×C₄×C₆₇	—	2	1	no	—
91	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
92	SD₁₆×C₆₇	—	4	4	no	—
93	D₈×C₆₇	—	2	2	no	—
94	D₈×C₆₇	—	2	2	no	—
95	D₈×C₆₇	—	4	4	no	—
96	D₈×C₆₇	—	4	1	no	—
97	D₈×C₆₇	—	4	1	no	—
98	M₁₆×C₆₇	—	2	1	no	—
99	M₁₆×C₆₇	—	2	2	no	—
100	M₁₆×C₆₇	—	2	2	no	—
101	M₁₆×C₆₇	—	2	2	no	—
102	M₁₆×C₆₇	—	4	1	no	—
103	M₁₆×C₆₇	C₂²⋊C₄	1	2	yes	—
104	C₂×C₈×C₆₇	—	2	1	no	—
105	C₂×C₈×C₆₇	—	2	2	no	—
106	C₂×C₈×C₆₇	—	2	2	no	—
107	C₂×C₈×C₆₇	—	4	1	no	—
108	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
109	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
110	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
111	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
112	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
113	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
114	C₂³×C₆₇	—	2	1	no	—
115	C₂³×C₆₇	—	4	2	no	—
116	C₂³×C₆₇	—	8	1	no	—
117	C₂³×C₆₇	—	8	1	no	—
118	C₂³×C₆₇	—	16	2	no	—
119	C₂³×C₆₇	#137	1	1	yes	characteristic
120	Q₈×C₆₇	—	2	1	no	—
121	Q₈×C₆₇	#102	1	2	yes	—
122	D₄×C₆₇	—	2	2	no	—
123	D₄×C₆₇	—	2	2	no	—
124	D₄×C₆₇	—	2	1	no	—
125	D₄×C₆₇	—	4	2	no	—
126	D₄×C₆₇	—	4	2	no	—
127	D₄×C₆₇	—	8	1	no	—
128	D₄×C₆₇	—	8	1	no	—
129	D₄×C₆₇	#102	1	2	yes	—
130	C₂×C₄×C₆₇	—	2	1	no	—
131	C₂×C₄×C₆₇	—	2	2	no	—
132	C₂×C₄×C₆₇	—	2	2	no	—
133	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
134	C₂×C₄×C₆₇	#120	1	1	yes	characteristic
135	C₂×C₄×C₆₇	#123	1	1	yes	characteristic
136	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
137	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
138	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
139	C₈×C₆₇	—	2	2	no	—
140	C₈×C₆₇	—	4	2	no	—
141	C₈×C₆₇	C₄×D₄	1	1	yes	characteristic
142	C₈×C₆₇	C₄×D₄	1	1	yes	characteristic
143	C₈×C₆₇	#121	1	1	yes	characteristic
144	C₈×C₆₇	#121	1	1	yes	characteristic
145	C₆₇⋊C₈	—	4	2	no	—
146	C₆₇⋊C₈	—	4	2	no	—
147	C₂²×C₆₇	—	2	2	no	—
148	C₂²×C₆₇	—	4	1	no	—
149	C₂²×C₆₇	—	8	1	no	—
150	C₂²×C₆₇	—	8	2	no	—
151	C₂²×C₆₇	—	16	2	no	—
152	C₂²×C₆₇	—	16	2	no	—
153	C₂²×C₆₇	—	16	2	no	—
154	C₂²×C₆₇	#417	1	1	yes	characteristic
155	C₂²×C₆₇	#448	1	1	yes	characteristic
156	C₂²×C₆₇	#480	1	1	yes	characteristic
157	C₄×C₆₇	—	2	2	no	—
158	C₄×C₆₇	#386	1	1	yes	characteristic
159	C₄×C₆₇	C₂×#102	1	1	yes	characteristic
160	C₄×C₆₇	#447	1	1	yes	characteristic
161	C₄×C₆₇	#481	1	1	yes	characteristic
162	#13007	—	67	1	no	maximal, Sylow
163	C₂×C₆₇	—	2	1	no	—
164	C₂×C₆₇	—	2	1	no	—
165	C₂×C₆₇	—	2	1	no	—
166	C₂×C₆₇	—	4	2	no	—
167	C₂×C₆₇	—	16	2	no	—
168	C₂×C₆₇	#1874	1	1	yes	characteristic
169	C₂×#472	D₆₇	1	1	yes	characteristic
170	#2916	—	67	1	no	—
171	C₂×#361	—	67	1	no	—
172	C₂×#361	—	67	1	no	—
173	#2117	—	67	1	no	—
174	#1388	—	67	1	no	—
175	#1385	—	67	1	no	—
176	C₆₇	#13007	1	1	yes	characteristic, complemented, Sylow
177	C₂×AGL(1,ℤ/8)	—	2	1	no	—
178	C₂²×D₈	C₂×D₆₇	1	1	yes	characteristic
179	#568	—	134	1	no	—
180	C₂×#131	C₂×D₆₇	1	1	yes	characteristic
181	C₂×M₃₂	—	67	1	no	—
182	C₂×M₃₂	—	67	1	no	—
183	#472	—	2	2	no	—
184	#472	Dic₆₇	1	4	yes	—
185	#430	—	134	2	no	—
186	#430	—	134	1	no	—
187	C₂×#108	—	67	1	no	—
188	C₂×#108	—	67	1	no	—
189	C₂×#108	—	134	2	no	—
190	C₂×#108	C₂×D₆₇	1	1	yes	characteristic
191	C₂×#100	—	2	1	no	—
192	#361	—	134	2	no	—
193	#361	—	134	2	no	—
194	#350	—	134	2	no	—
195	C₂×#41	C₂²⋊D₆₇	1	1	yes	characteristic
196	C₂²×D₄	—	2	1	no	—
197	AGL(1,ℤ/8)	—	2	4	no	—
198	AGL(1,ℤ/8)	—	4	2	no	—
199	C₂×SD₁₆	—	4	1	no	—
200	C₂×D₈	—	2	1	no	—
201	C₂×D₈	—	2	1	no	—
202	C₂×D₈	—	2	1	no	—
203	C₂×D₈	—	2	1	no	—
204	C₂×D₈	—	2	1	no	—
205	C₂×D₈	—	4	1	no	—
206	C₂×D₈	C₂×Dic₆₇	1	2	yes	—
207	#131	—	2	1	no	—
208	#131	—	2	1	no	—
209	#131	—	268	2	no	—
210	#131	—	268	2	no	—
211	#131	C₂×Dic₆₇	1	2	yes	—
212	#131	C₂²⋊D₆₇	1	2	yes	—
213	C₂×M₁₆	—	2	1	no	—
214	C₂×M₁₆	—	2	1	no	—
215	C₂×M₁₆	—	134	4	no	—
216	C₂×M₁₆	—	134	1	no	—
217	C₂×M₁₆	—	134	1	no	—
218	C₂×M₁₆	C₂²⋊D₆₇	1	1	yes	characteristic
219	C₂²×C₈	—	2	1	no	—
220	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
221	M₃₂	—	134	1	no	—
222	M₃₂	—	134	1	no	—
223	M₃₂	—	134	1	no	—
224	M₃₂	—	134	1	no	—
225	C₂×C₁₆	—	134	1	no	—
226	C₂×C₁₆	—	134	1	no	—
227	#108	—	2	1	no	—
228	#108	—	67	2	no	—
229	#108	—	67	2	no	—
230	#108	—	134	4	no	—
231	#108	—	134	1	no	—
232	#108	—	134	1	no	—
233	#108	C₂×Dic₆₇	1	2	yes	—
234	#100	—	2	4	no	—
235	C₂⁴	—	4	1	no	—
236	#41	—	2	1	no	—
237	#41	—	2	1	no	—
238	#41	C₂²⋊Dic₆₇	1	2	yes	—
239	#41	#11761613	1	2	yes	—
240	C₂×Q₈	#11761616	1	1	yes	characteristic
241	C₂×D₄	—	2	1	no	—
242	C₂×D₄	—	2	2	no	—
243	C₂×D₄	—	4	1	no	—
244	C₂×D₄	—	4	1	no	—
245	C₂×D₄	—	4	1	no	—
246	C₂×D₄	—	4	1	no	—
247	C₂×D₄	—	4	1	no	—
248	C₂×D₄	#11761616	1	1	yes	characteristic
249	C₂²×C₄	—	2	1	no	—
250	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
251	SD₁₆	—	4	4	no	—
252	D₈	—	2	2	no	—
253	D₈	—	2	2	no	—
254	D₈	—	4	4	no	—
255	D₈	—	4	1	no	—
256	D₈	—	4	1	no	—
257	M₁₆	—	2	1	no	—
258	M₁₆	—	2	2	no	—
259	M₁₆	—	2	2	no	—
260	M₁₆	—	2	2	no	—
261	M₁₆	—	4	1	no	—
262	M₁₆	—	268	4	no	—
263	M₁₆	—	268	1	no	—
264	M₁₆	—	268	1	no	—
265	M₁₆	—	134	2	no	—
266	M₁₆	—	134	2	no	—
267	M₁₆	C₂²⋊Dic₆₇	1	2	yes	—
268	C₂×C₈	—	2	1	no	—
269	C₂×C₈	—	2	2	no	—
270	C₂×C₈	—	2	2	no	—
271	C₂×C₈	—	4	1	no	—
272	C₂×C₈	—	268	4	no	—
273	C₂×C₈	—	268	1	no	—
274	C₂×C₈	—	268	1	no	—
275	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
276	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
277	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
278	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
279	C₂×C₈	#11761627	1	1	yes	characteristic
280	C₂×C₈	#11761627	1	1	yes	characteristic
281	C₁₆	—	268	1	no	—
282	C₁₆	—	268	1	no	—
283	C₂³	—	2	1	no	—
284	C₂³	—	4	2	no	—
285	C₂³	—	8	1	no	—
286	C₂³	—	8	1	no	—
287	C₂³	—	16	2	no	—
288	C₂³	#14524073	1	1	yes	characteristic
289	Q₈	—	2	1	no	—
290	Q₈	#14523964	1	2	yes	—
291	D₄	—	2	2	no	—
292	D₄	—	2	2	no	—
293	D₄	—	2	1	no	—
294	D₄	—	4	2	no	—
295	D₄	—	4	2	no	—
296	D₄	—	8	1	no	—
297	D₄	—	8	1	no	—
298	D₄	#14523964	1	2	yes	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	2	no	—
301	C₂×C₄	—	2	2	no	—
302	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
303	C₂×C₄	#14524011	1	1	yes	characteristic
304	C₂×C₄	#14524024	1	1	yes	characteristic
305	C₂×C₄	C₂×#11761613	1	1	yes	characteristic
306	C₂×C₄	C₂×#11761616	1	1	yes	characteristic
307	C₂×C₄	#14524068	1	1	yes	characteristic
308	C₈	—	2	2	no	—
309	C₈	—	4	2	no	—
310	C₈	—	268	2	no	—
311	C₈	—	268	2	no	—
312	C₈	D₄×Dic₆₇	1	1	yes	characteristic
313	C₈	D₄×Dic₆₇	1	1	yes	characteristic
314	C₈	#14524015	1	1	yes	characteristic
315	C₈	#14524015	1	1	yes	characteristic
316	C₂²	—	2	2	no	—
317	C₂²	—	4	1	no	—
318	C₂²	—	8	1	no	—
319	C₂²	—	8	2	no	—
320	C₂²	—	16	2	no	—
321	C₂²	—	16	2	no	—
322	C₂²	—	16	2	no	—
323	C₂²	#16863397	1	1	yes	characteristic
324	C₂²	#16863534	1	1	yes	characteristic
325	C₂²	#16863707	1	1	yes	characteristic
326	C₄	—	2	2	no	—
327	C₄	#16863276	1	1	yes	characteristic
328	C₄	C₂×#14523964	1	1	yes	characteristic
329	C₄	#16863529	1	1	yes	characteristic
330	C₄	#16863712	1	1	yes	characteristic
331	C₂	—	2	1	no	—
332	C₂	—	2	1	no	—
333	C₂	—	2	1	no	—
334	C₂	—	4	2	no	—
335	C₂	—	16	2	no	—
336	C₂	#22560527	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#22560527
commutator	C₂×C₈×C₆₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₆₇
Fitting, prime radical	C₂×#472×C₆₇	C₂
socle	C₂×C₆₇	#1874
cosocle	C₂²×C₈×C₆₇	C₂³
hypercenter	C₂×#472	D₆₇
hypocenter	C₆₇	#13007
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#131 ◁ C₂×#472

Complements

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

N	H ≅ G/N	type
1. #22568820	C₂	semidirect
2. C₂×#16863209	C₂	semidirect
3. C₂×#16863209	C₂	semidirect
4. #22561837	C₂	semidirect
5. #22558320	C₂	semidirect
6. #22558312	C₂	semidirect
176. C₆₇	#13007	semidirect

Maximal subgroups

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

subgroup	index
#22568820	2
C₂×#16863209	2
#22561837	2
#22558320	2
#22558312	2
C₂×#472×C₆₇	2
#13007	67