Subgroups — #30874832

Order 17152.

642 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22569160	C₂	1	2	yes	maximal
2	C₂×C₈×Dic₁₃₄	C₂	1	1	yes	maximal, characteristic
3	#22560983	C₂	1	4	yes	maximal
4	#22560982	C₂	1	2	yes	maximal
5	#22560345	C₂	1	2	yes	maximal
6	C₄×#14523979	C₂	1	1	yes	maximal, characteristic
7	#22560110	C₂	1	2	yes	maximal
8	C₆₇×#1837	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄×Dic₁₃₄	C₂²	1	1	yes	characteristic
10	#16863525	—	2	4	no	—
11	#16863523	—	2	2	no	—
12	C₈×Dic₁₃₄	C₄	1	2	yes	—
13	C₈×Dic₁₃₄	C₄	1	4	yes	—
14	C₈×Dic₁₃₄	C₄	1	2	yes	—
15	#16863520	—	2	2	no	—
16	C₂×#14523979	C₂²	1	2	yes	—
17	C₂×#14523978	C₂²	1	4	yes	—
18	#16863419	C₂²	1	4	yes	—
19	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
20	C₂×Dic₆₇⋊C₈	C₂²	1	2	yes	—
21	C₄×#11761612	C₂²	1	2	yes	—
22	M₁₆×Dic₆₇	C₂²	1	4	yes	—
23	C₂×C₈×Dic₆₇	C₂²	1	2	yes	—
24	#16863161	C₂²	1	1	yes	characteristic
25	#16863161	C₂²	1	1	yes	characteristic
26	#16863160	C₂²	1	4	yes	—
27	#423×C₆₇	C₂²	1	2	yes	—
28	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
29	#341×C₆₇	C₂²	1	2	yes	—
30	#341×C₆₇	C₂²	1	2	yes	—
31	C₂²×Dic₁₃₄	Q₈	1	1	yes	characteristic
32	C₂×#11761612	C₂³	1	2	yes	—
33	C₂×#11761612	C₂³	1	2	yes	—
34	C₂²×C₆₇⋊C₈	C₂³	1	2	yes	—
35	C₄×Dic₁₃₄	C₂×C₄	1	2	yes	—
36	C₄×Dic₁₃₄	C₂×C₄	1	1	yes	characteristic
37	C₄×Dic₁₃₄	C₂×C₄	1	1	yes	characteristic
38	C₄×Dic₁₃₄	D₄	1	1	yes	characteristic
39	C₄×Dic₁₃₄	D₄	1	1	yes	characteristic
40	C₄×Dic₁₃₄	D₄	1	2	yes	—
41	C₂×C₄⋊Dic₆₇	Q₈	1	1	yes	characteristic
42	C₂×Dic₆₇⋊C₄	Q₈	1	2	yes	—
43	C₂×C₄×Dic₆₇	C₂³	1	2	yes	—
44	#14523979	—	2	4	no	—
45	#14523978	—	2	8	no	—
46	C₄⋊(C₆₇⋊C₈)	C₂×C₄	1	2	yes	—
47	C₄⋊(C₆₇⋊C₈)	C₂×C₄	1	2	yes	—
48	Dic₆₇⋊C₈	—	2	4	no	—
49	Dic₆₇⋊C₈	—	2	4	no	—
50	Dic₆₇⋊C₈	C₂×C₄	1	4	yes	—
51	Dic₆₇⋊C₈	C₂×C₄	1	4	yes	—
52	#14523954	—	2	2	no	—
53	#14523953	—	2	4	no	—
54	C₄×C₆₇⋊C₈	—	2	2	no	—
55	C₈×Dic₆₇	—	2	4	no	—
56	C₈×Dic₆₇	C₂×C₄	1	4	yes	—
57	C₈×Dic₆₇	C₂×C₄	1	4	yes	—
58	C₂×M₁₆×C₆₇	C₂³	1	4	yes	—
59	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
60	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
61	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
62	C₄⋊C₈×C₆₇	—	2	2	no	—
63	C₄⋊C₈×C₆₇	—	2	2	no	—
64	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
65	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
66	C₂×Dic₁₃₄	—	2	4	no	—
67	C₂×Dic₁₃₄	C₄⋊C₄	1	1	yes	characteristic
68	C₂×Dic₁₃₄	C₄⋊C₄	1	2	yes	—
69	C₂×Dic₁₃₄	C₄⋊C₄	1	1	yes	characteristic
70	C₂²×Dic₆₇	C₂×Q₈	1	2	yes	—
71	C₂²×Dic₆₇	C₂×Q₈	1	2	yes	—
72	#11761612	—	2	2	no	—
73	#11761612	—	2	2	no	—
74	#11761612	—	2	2	no	—
75	#11761612	#41	1	4	yes	—
76	C₂×C₆₇⋊C₈	—	2	2	no	—
77	C₂×C₆₇⋊C₈	—	2	4	no	—
78	C₂×C₆₇⋊C₈	—	2	2	no	—
79	C₂×C₆₇⋊C₈	C₂²×C₄	1	4	yes	—
80	C₄⋊Dic₆₇	C₄⋊C₄	1	2	yes	—
81	C₄⋊Dic₆₇	C₄⋊C₄	1	1	yes	characteristic
82	C₄⋊Dic₆₇	C₄⋊C₄	1	1	yes	characteristic
83	Dic₆₇⋊C₄	C₄⋊C₄	1	2	yes	—
84	Dic₆₇⋊C₄	C₄⋊C₄	1	2	yes	—
85	Dic₆₇⋊C₄	C₄⋊C₄	1	2	yes	—
86	Dic₆₇⋊C₄	C₄⋊C₄	1	2	yes	—
87	C₄×Dic₆₇	C₂²×C₄	1	2	yes	—
88	C₄×Dic₆₇	C₂²×C₄	1	2	yes	—
89	C₄×Dic₆₇	C₂×D₄	1	2	yes	—
90	C₄×Dic₆₇	C₂×D₄	1	2	yes	—
91	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
92	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
93	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
94	M₁₆×C₆₇	—	2	4	no	—
95	M₁₆×C₆₇	—	2	4	no	—
96	C₂×C₈×C₆₇	—	2	2	no	—
97	C₂×C₈×C₆₇	—	2	2	no	—
98	C₂×C₈×C₆₇	—	2	4	no	—
99	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
100	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
101	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
102	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
103	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
104	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
105	C₂³×C₆₇	C₂²×Q₈	1	1	yes	characteristic
106	C₂×C₄×C₆₇	—	2	2	no	—
107	C₂×C₄×C₆₇	—	4	1	no	—
108	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
109	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
110	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
111	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
112	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
113	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
114	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
115	C₂×C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
116	C₂×Dic₆₇	—	2	4	no	—
117	C₂×Dic₆₇	—	4	2	no	—
118	C₂×Dic₆₇	C₂×C₄⋊C₄	1	2	yes	—
119	C₂×Dic₆₇	C₂×C₄⋊C₄	1	2	yes	—
120	C₂×Dic₆₇	C₂×C₄⋊C₄	1	2	yes	—
121	C₂×Dic₆₇	C₂×C₄⋊C₄	1	2	yes	—
122	Dic₁₃₄	—	2	2	no	—
123	Dic₁₃₄	—	2	4	no	—
124	Dic₁₃₄	#108	1	4	yes	complemented
125	C₈×C₆₇	—	2	2	no	—
126	C₈×C₆₇	—	2	2	no	—
127	C₈×C₆₇	—	4	4	no	—
128	C₆₇⋊C₈	—	2	4	no	—
129	C₆₇⋊C₈	—	2	4	no	—
130	C₆₇⋊C₈	—	4	2	no	—
131	C₂²×C₆₇	—	2	1	no	—
132	C₂²×C₆₇	—	2	1	no	—
133	C₂²×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
134	C₂²×C₆₇	#557	1	2	yes	—
135	C₄×C₆₇	—	2	1	no	—
136	C₄×C₆₇	—	2	1	no	—
137	C₄×C₆₇	—	2	1	no	—
138	C₄×C₆₇	C₂×#108	1	2	yes	—
139	C₄×C₆₇	C₄×#41	1	1	yes	characteristic
140	C₄×C₆₇	#519	1	1	yes	characteristic
141	C₄×C₆₇	#548	1	2	yes	—
142	Dic₆₇	—	2	2	no	—
143	Dic₆₇	—	2	2	no	—
144	Dic₆₇	—	2	2	no	—
145	Dic₆₇	C₂×#108	1	4	yes	—
146	#22746	—	67	1	no	maximal, Sylow
147	C₂×C₆₇	—	2	2	no	—
148	C₂×C₆₇	#2351	1	1	yes	characteristic
149	C₂×C₆₇	C₂²×#108	1	1	yes	characteristic
150	C₂×C₆₇	#2916	1	1	yes	characteristic
151	Q₈×M₁₆	—	67	2	no	—
152	C₂×C₈×Q₈	—	67	1	no	—
153	#1950	—	67	2	no	—
154	#1950	—	67	4	no	—
155	#1837	—	67	2	no	—
156	#1837	D₆₇	1	1	yes	characteristic
157	C₄×#108	—	67	1	no	—
158	C₄×#108	—	67	2	no	—
159	C₆₇	#22746	1	1	yes	characteristic, complemented, Sylow
160	C₂×C₄×Q₈	—	67	1	no	—
161	#446	—	134	2	no	—
162	#446	—	134	4	no	—
163	C₈×Q₈	—	67	2	no	—
164	C₈×Q₈	—	67	4	no	—
165	C₈×Q₈	—	67	2	no	—
166	C₈×Q₈	—	134	2	no	—
167	C₂×#108	—	67	4	no	—
168	C₂×#108	—	67	2	no	—
169	#423	—	67	4	no	—
170	#423	C₂×D₆₇	1	2	yes	—
171	C₂×C₄⋊C₈	—	67	2	no	—
172	C₂×C₄⋊C₈	—	67	1	no	—
173	C₄×M₁₆	—	67	4	no	—
174	C₄×M₁₆	—	67	2	no	—
175	C₂×C₄×C₈	—	67	2	no	—
176	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
177	#341	—	67	4	no	—
178	#341	—	67	1	no	—
179	#341	—	67	1	no	—
180	#341	C₂×D₆₇	1	2	yes	—
181	#341	C₂×D₆₇	1	2	yes	—
182	C₂²×Q₈	—	67	1	no	—
183	C₂×M₁₆	—	67	2	no	—
184	C₂×M₁₆	—	67	2	no	—
185	C₂×M₁₆	C₂²×D₆₇	1	4	yes	—
186	C₂²×C₈	—	67	2	no	—
187	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
188	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
189	C₄×Q₈	—	67	1	no	—
190	C₄×Q₈	—	67	2	no	—
191	C₄×Q₈	—	67	1	no	—
192	C₄×Q₈	—	67	1	no	—
193	C₄×Q₈	—	67	2	no	—
194	C₄×Q₈	—	67	1	no	—
195	C₂×C₄⋊C₄	—	67	2	no	—
196	C₂×C₄⋊C₄	—	67	1	no	—
197	C₂×C₄²	—	67	2	no	—
198	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
199	#108	—	134	4	no	—
200	#108	—	134	8	no	—
201	C₄⋊C₈	—	2	2	no	—
202	C₄⋊C₈	—	2	2	no	—
203	C₄⋊C₈	—	67	4	no	—
204	C₄⋊C₈	—	67	4	no	—
205	C₄⋊C₈	—	67	2	no	—
206	C₄⋊C₈	—	67	2	no	—
207	C₄⋊C₈	—	134	4	no	—
208	C₄⋊C₈	—	134	4	no	—
209	#97	—	134	4	no	—
210	#97	—	134	2	no	—
211	C₄×C₈	—	67	4	no	—
212	C₄×C₈	—	67	4	no	—
213	C₄×C₈	—	134	4	no	—
214	C₄×C₈	—	134	2	no	—
215	C₄×C₈	C₄×D₆₇	1	2	yes	—
216	C₄×C₈	C₄×D₆₇	1	2	yes	—
217	C₂×Q₈	—	67	1	no	—
218	C₂×Q₈	—	67	2	no	—
219	C₂×Q₈	—	67	1	no	—
220	C₂×Q₈	—	134	4	no	—
221	C₂²×C₄	—	67	2	no	—
222	C₂²×C₄	—	67	2	no	—
223	C₂²×C₄	Q₈×D₆₇	1	1	yes	characteristic
224	C₂²×C₄	Q₈×D₆₇	1	1	yes	characteristic
225	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
226	M₁₆	—	2	4	no	—
227	M₁₆	—	2	4	no	—
228	M₁₆	—	67	4	no	—
229	M₁₆	—	134	2	no	—
230	M₁₆	—	134	2	no	—
231	M₁₆	—	134	2	no	—
232	C₂×C₈	—	2	2	no	—
233	C₂×C₈	—	2	2	no	—
234	C₂×C₈	—	2	4	no	—
235	C₂×C₈	—	67	4	no	—
236	C₂×C₈	—	134	2	no	—
237	C₂×C₈	—	134	2	no	—
238	C₂×C₈	—	134	4	no	—
239	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
240	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
241	C₄⋊C₄	—	67	2	no	—
242	C₄⋊C₄	—	67	2	no	—
243	C₄⋊C₄	—	67	2	no	—
244	C₄⋊C₄	—	67	2	no	—
245	C₄⋊C₄	—	67	2	no	—
246	C₄⋊C₄	—	67	1	no	—
247	C₄⋊C₄	—	67	1	no	—
248	C₄²	—	67	2	no	—
249	C₄²	—	67	2	no	—
250	C₄²	—	67	2	no	—
251	C₄²	—	67	2	no	—
252	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
253	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
254	C₄²	D₄×D₆₇	1	1	yes	characteristic
255	C₄²	D₄×D₆₇	1	1	yes	characteristic
256	C₂³	C₂×Q₈×D₆₇	1	1	yes	characteristic
257	Q₈	—	67	4	no	—
258	Q₈	—	134	4	no	—
259	Q₈	—	134	2	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	4	1	no	—
262	C₂×C₄	—	268	2	no	—
263	C₂×C₄	—	67	2	no	—
264	C₂×C₄	—	67	2	no	—
265	C₂×C₄	—	67	2	no	—
266	C₂×C₄	—	67	2	no	—
267	C₂×C₄	—	134	4	no	—
268	C₂×C₄	C₄⋊C₄×D₆₇	1	1	yes	characteristic
269	C₂×C₄	C₄⋊C₄×D₆₇	1	1	yes	characteristic
270	C₂×C₄	C₄⋊C₄×D₆₇	1	1	yes	characteristic
271	C₂×C₄	C₄⋊C₄×D₆₇	1	1	yes	characteristic
272	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
273	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
274	C₂×C₄	#41×D₆₇	1	2	yes	—
275	C₂×C₄	#14524090	1	2	yes	—
276	C₈	—	2	2	no	—
277	C₈	—	2	2	no	—
278	C₈	—	4	4	no	—
279	C₈	—	268	2	no	—
280	C₈	—	134	4	no	—
281	C₈	—	134	4	no	—
282	C₂²	—	2	1	no	—
283	C₂²	—	2	1	no	—
284	C₂²	C₂×C₄⋊C₄×D₆₇	1	1	yes	characteristic
285	C₂²	#16864123	1	2	yes	—
286	C₄	—	2	1	no	—
287	C₄	—	2	1	no	—
288	C₄	—	2	1	no	—
289	C₄	—	67	4	no	—
290	C₄	—	134	2	no	—
291	C₄	—	134	2	no	—
292	C₄	—	134	2	no	—
293	C₄	#108×D₆₇	1	2	yes	—
294	C₄	#16863864	1	1	yes	characteristic
295	C₄	#16863873	1	1	yes	characteristic
296	C₄	#16864054	1	2	yes	—
297	C₂	—	2	2	no	—
298	C₂	#22563126	1	1	yes	characteristic
299	C₂	C₂×#108×D₆₇	1	1	yes	characteristic
300	C₂	#22568817	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₆₇
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₆₇×#1837	C₂
socle	C₂²×C₆₇	C₂²×C₄⋊C₄
prime radical	C₂³×C₆₇	C₂²×Q₈
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	#1837	D₆₇
hypocenter	C₆₇	#22746
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1837

Complements

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

N	H ≅ G/N	type
124. Dic₁₃₄	#108	semidirect
159. C₆₇	#22746	semidirect

Maximal subgroups

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

subgroup	index
#22569160	2
C₂×C₈×Dic₁₃₄	2
#22560983	2
#22560982	2
#22560345	2
C₄×#14523979	2
#22560110	2
C₆₇×#1837	2
#22746	67