Subgroups — #30774900

Order 17152.

414 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863801	C₂	1	1	yes	maximal, characteristic
2	C₂×#16863200	C₂	1	2	yes	maximal
3	#22558300	C₂	1	2	yes	maximal
4	#22558287	C₂	1	1	yes	maximal, characteristic
5	C₆₇×#1969	C₂	1	1	yes	maximal, characteristic
6	C₂×#14524043	—	2	1	no	—
7	C₂²×C₆₇⋊C₁₆	C₂²	1	2	yes	—
8	#16863801	—	2	4	no	—
9	C₂×#14523977	—	2	1	no	—
10	C₂×#14523977	C₂²	1	2	yes	—
11	#16863200	—	2	4	no	—
12	C₂²×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic
13	C₂×#107×C₆₇	C₂²	1	1	yes	characteristic
14	C₂×#103×C₆₇	—	2	1	no	—
15	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
16	#391×C₆₇	—	2	1	no	—
17	C₂²×Dic₁₃₄	—	4	1	no	—
18	#14524043	—	2	4	no	—
19	#14524043	—	4	2	no	—
20	C₂×C₄⋊Dic₆₇	—	2	2	no	—
21	C₂×C₄⋊Dic₆₇	—	4	1	no	—
22	C₂×C₆₇⋊C₁₆	—	2	2	no	—
23	C₂×C₆₇⋊C₁₆	—	2	2	no	—
24	C₂×C₆₇⋊C₁₆	—	2	2	no	—
25	#14523977	—	2	4	no	—
26	#14523977	—	2	2	no	—
27	#14523977	D₄	1	4	yes	—
28	C₂²×Q₈×C₆₇	—	2	1	no	—
29	C₂×Q₁₆×C₆₇	—	2	2	no	—
30	C₂×Q₁₆×C₆₇	—	2	2	no	—
31	C₂×Q₁₆×C₆₇	—	2	1	no	—
32	C₂×Q₁₆×C₆₇	Q₈	1	2	yes	complemented
33	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
34	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
35	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
36	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
37	#107×C₆₇	—	2	2	no	—
38	#103×C₆₇	—	4	2	no	—
39	C₄×C₈×C₆₇	—	2	2	no	—
40	#95×C₆₇	—	4	1	no	—
41	#95×C₆₇	—	4	1	no	—
42	C₂×Dic₁₃₄	—	4	4	no	—
43	C₂×Dic₁₃₄	—	8	2	no	—
44	C₂×Dic₁₃₄	—	8	2	no	—
45	C₂²×Dic₆₇	—	4	2	no	—
46	C₄⋊Dic₆₇	—	2	4	no	—
47	C₄⋊Dic₆₇	—	4	4	no	—
48	C₄⋊Dic₆₇	—	4	2	no	—
49	C₆₇⋊C₁₆	—	4	2	no	—
50	C₂×Q₈×C₆₇	—	2	2	no	—
51	C₂×Q₈×C₆₇	—	4	1	no	—
52	C₂×Q₈×C₆₇	—	4	1	no	—
53	C₂×Q₈×C₆₇	—	4	1	no	—
54	C₂×Q₈×C₆₇	—	4	1	no	—
55	C₂×Q₈×C₆₇	—	4	1	no	—
56	C₂²×C₄×C₆₇	—	4	1	no	—
57	C₂²×C₄×C₆₇	—	4	1	no	—
58	C₂²×C₄×C₆₇	D₈	1	1	yes	characteristic
59	C₂²×C₄×C₆₇	D₈	1	1	yes	characteristic
60	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
61	Q₁₆×C₆₇	—	2	4	no	—
62	Q₁₆×C₆₇	—	4	2	no	—
63	C₂×C₈×C₆₇	—	2	1	no	—
64	C₂×C₈×C₆₇	—	2	1	no	—
65	C₂×C₈×C₆₇	SD₁₆	1	2	yes	—
66	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
67	C₂×C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
68	C₂×C₈×C₆₇	#41	1	2	yes	—
69	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
70	C₄⋊C₄×C₆₇	—	4	1	no	—
71	C₄⋊C₄×C₆₇	—	4	1	no	—
72	C₄²×C₆₇	—	2	1	no	—
73	C₄²×C₆₇	—	2	1	no	—
74	C₂³×C₆₇	C₂×D₈	1	1	yes	characteristic
75	Q₈×C₆₇	—	4	2	no	—
76	Q₈×C₆₇	—	4	2	no	—
77	Q₈×C₆₇	—	8	1	no	—
78	Q₈×C₆₇	—	8	1	no	—
79	C₂×C₄×C₆₇	—	2	1	no	—
80	C₂×C₄×C₆₇	—	2	1	no	—
81	C₂×C₄×C₆₇	—	2	1	no	—
82	C₂×C₄×C₆₇	—	2	1	no	—
83	C₂×C₄×C₆₇	—	4	2	no	—
84	C₂×C₄×C₆₇	—	8	1	no	—
85	C₂×C₄×C₆₇	—	8	1	no	—
86	C₂×C₄×C₆₇	—	8	1	no	—
87	C₂×C₄×C₆₇	—	8	1	no	—
88	C₂×C₄×C₆₇	—	8	1	no	—
89	C₂×C₄×C₆₇	SD₃₂	1	2	yes	—
90	C₂×C₄×C₆₇	Q₃₂	1	2	yes	—
91	C₂×C₄×C₆₇	#120	1	1	yes	characteristic
92	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
93	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
94	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
95	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
96	C₂×C₄×C₆₇	#137	1	1	yes	characteristic
97	C₂×Dic₆₇	—	4	4	no	—
98	C₂×Dic₆₇	—	8	2	no	—
99	C₂×Dic₆₇	—	8	2	no	—
100	Dic₁₃₄	—	8	8	no	—
101	C₈×C₆₇	—	2	2	no	—
102	C₈×C₆₇	C₂²⋊Q₈	1	2	yes	—
103	C₈×C₆₇	#124	1	2	yes	—
104	C₂²×C₆₇	#447	1	1	yes	characteristic
105	C₂²×C₆₇	#474	1	1	yes	characteristic
106	C₂²×C₆₇	#486	1	1	yes	characteristic
107	C₂²×C₆₇	C₂×SD₃₂	1	1	yes	characteristic
108	C₂²×C₆₇	C₂×Q₃₂	1	1	yes	characteristic
109	C₂²×C₆₇	#509	1	1	yes	characteristic
110	C₂²×C₆₇	#510	1	1	yes	characteristic
111	C₄×C₆₇	—	2	2	no	—
112	C₄×C₆₇	—	2	2	no	—
113	C₄×C₆₇	—	8	2	no	—
114	C₄×C₆₇	—	16	1	no	—
115	C₄×C₆₇	#393	1	1	yes	characteristic
116	C₄×C₆₇	#448	1	1	yes	characteristic
117	C₄×C₆₇	#476	1	1	yes	characteristic
118	C₄×C₆₇	#489	1	1	yes	characteristic
119	Dic₆₇	—	8	4	no	—
120	#13364	—	67	1	no	maximal, Sylow
121	C₂×C₆₇	#2023	1	1	yes	characteristic
122	C₂×C₆₇	#2187	1	1	yes	characteristic
123	C₂×C₆₇	#2188	1	1	yes	characteristic
124	C₂×C₆₇	#2225	1	1	yes	characteristic
125	C₂×C₆₇	#2227	1	1	yes	characteristic
126	C₂×C₆₇	#2241	1	1	yes	characteristic
127	C₂×C₆₇	#2243	1	1	yes	characteristic
128	C₂×#500	—	67	1	no	—
129	C₂×#358	—	67	2	no	—
130	#1969	D₆₇	1	1	yes	characteristic
131	#1380	—	67	2	no	—
132	#1374	—	67	1	no	—
133	C₆₇	#13364	1	1	yes	characteristic, complemented, Sylow
134	C₂²×Q₁₆	C₂×D₆₇	1	1	yes	characteristic
135	C₂×C₄⋊Q₈	—	134	1	no	—
136	C₂²×C₁₆	—	67	2	no	—
137	#500	—	134	4	no	—
138	C₂×#107	—	67	2	no	—
139	C₂×#107	—	134	1	no	—
140	C₂×#107	C₂×D₆₇	1	1	yes	characteristic
141	C₂×#103	—	2	1	no	—
142	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
143	#391	—	2	1	no	—
144	#358	—	134	4	no	—
145	C₂²×Q₈	—	2	1	no	—
146	C₂²×Q₈	—	268	1	no	—
147	C₂×Q₁₆	—	2	2	no	—
148	C₂×Q₁₆	—	2	2	no	—
149	C₂×Q₁₆	—	2	1	no	—
150	C₂×Q₁₆	Dic₁₃₄	1	2	yes	complemented
151	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
152	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
153	C₄⋊Q₈	—	268	2	no	—
154	C₄⋊Q₈	—	134	4	no	—
155	C₂×C₄⋊C₄	—	2	1	no	—
156	C₂×C₄⋊C₄	—	268	1	no	—
157	C₂×C₄⋊C₄	—	134	2	no	—
158	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
159	C₂×C₁₆	—	134	2	no	—
160	C₂×C₁₆	—	134	2	no	—
161	C₂×C₁₆	—	134	2	no	—
162	#107	—	2	2	no	—
163	#107	—	67	4	no	—
164	#107	—	134	4	no	—
165	#107	—	134	2	no	—
166	#103	—	4	2	no	—
167	C₄×C₈	—	2	2	no	—
168	#95	—	4	1	no	—
169	#95	—	4	1	no	—
170	C₂×Q₈	—	2	2	no	—
171	C₂×Q₈	—	4	1	no	—
172	C₂×Q₈	—	4	1	no	—
173	C₂×Q₈	—	4	1	no	—
174	C₂×Q₈	—	4	1	no	—
175	C₂×Q₈	—	4	1	no	—
176	C₂×Q₈	—	268	4	no	—
177	C₂×Q₈	—	536	2	no	—
178	C₂×Q₈	—	536	2	no	—
179	C₂²×C₄	—	4	1	no	—
180	C₂²×C₄	—	4	1	no	—
181	C₂²×C₄	—	268	2	no	—
182	C₂²×C₄	#11761613	1	1	yes	characteristic
183	C₂²×C₄	#11761613	1	1	yes	characteristic
184	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
185	Q₁₆	—	2	4	no	—
186	Q₁₆	—	4	2	no	—
187	C₂×C₈	—	2	1	no	—
188	C₂×C₈	—	2	1	no	—
189	C₂×C₈	#11761616	1	2	yes	—
190	C₂×C₈	D₄×D₆₇	1	2	yes	—
191	C₂×C₈	C₂×Dic₁₃₄	1	1	yes	characteristic
192	C₂×C₈	#11761627	1	2	yes	—
193	C₂×C₈	#11761628	1	1	yes	characteristic
194	C₄⋊C₄	—	4	1	no	—
195	C₄⋊C₄	—	4	1	no	—
196	C₄⋊C₄	—	268	4	no	—
197	C₄⋊C₄	—	268	2	no	—
198	C₄⋊C₄	—	134	4	no	—
199	C₄²	—	2	1	no	—
200	C₄²	—	2	1	no	—
201	C₁₆	—	268	2	no	—
202	C₂³	C₂×#11761613	1	1	yes	characteristic
203	Q₈	—	4	2	no	—
204	Q₈	—	4	2	no	—
205	Q₈	—	8	1	no	—
206	Q₈	—	8	1	no	—
207	Q₈	—	536	8	no	—
208	C₂×C₄	—	2	1	no	—
209	C₂×C₄	—	2	1	no	—
210	C₂×C₄	—	2	1	no	—
211	C₂×C₄	—	2	1	no	—
212	C₂×C₄	—	4	2	no	—
213	C₂×C₄	—	8	1	no	—
214	C₂×C₄	—	8	1	no	—
215	C₂×C₄	—	8	1	no	—
216	C₂×C₄	—	8	1	no	—
217	C₂×C₄	—	8	1	no	—
218	C₂×C₄	—	268	4	no	—
219	C₂×C₄	—	536	2	no	—
220	C₂×C₄	—	536	2	no	—
221	C₂×C₄	#14523986	1	2	yes	—
222	C₂×C₄	#14523989	1	2	yes	—
223	C₂×C₄	#14524011	1	1	yes	characteristic
224	C₂×C₄	#14524022	1	1	yes	characteristic
225	C₂×C₄	#14524030	1	1	yes	characteristic
226	C₂×C₄	C₂×#11761616	1	1	yes	characteristic
227	C₂×C₄	#14524068	1	1	yes	characteristic
228	C₂×C₄	#14524073	1	1	yes	characteristic
229	C₈	—	2	2	no	—
230	C₈	#14524020	1	2	yes	—
231	C₈	#14524029	1	2	yes	—
232	C₂²	#16863529	1	1	yes	characteristic
233	C₂²	#16863676	1	1	yes	characteristic
234	C₂²	#16863742	1	1	yes	characteristic
235	C₂²	C₂×#14523986	1	1	yes	characteristic
236	C₂²	C₂×#14523989	1	1	yes	characteristic
237	C₂²	#16863831	1	1	yes	characteristic
238	C₂²	#16863836	1	1	yes	characteristic
239	C₄	—	2	2	no	—
240	C₄	—	2	2	no	—
241	C₄	—	8	2	no	—
242	C₄	—	16	1	no	—
243	C₄	—	536	4	no	—
244	C₄	#16863308	1	1	yes	characteristic
245	C₄	#16863534	1	1	yes	characteristic
246	C₄	#16863686	1	1	yes	characteristic
247	C₄	#16863755	1	1	yes	characteristic
248	C₂	#22561345	1	1	yes	characteristic
249	C₂	#22562137	1	1	yes	characteristic
250	C₂	#22562142	1	1	yes	characteristic
251	C₂	#22562351	1	1	yes	characteristic
252	C₂	#22562361	1	1	yes	characteristic
253	C₂	#22562436	1	1	yes	characteristic
254	C₂	#22562445	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#11761613
commutator, cosocle	C₂²×C₈×C₆₇	C₂³
Frattini	C₂²×C₈	C₂²×D₆₇
Fitting	C₆₇×#1969	C₂
socle, prime radical	C₂³×C₆₇	C₂×D₈
hypercenter	#1969	D₆₇
hypocenter	C₆₇	#13364
LCS	G ▷ C₂²×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1969

Complements

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

N	H ≅ G/N	type
32. C₂×Q₁₆×C₆₇	Q₈	semidirect
133. C₆₇	#13364	semidirect
150. C₂×Q₁₆	Dic₁₃₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16863801	2
C₂×#16863200	2
#22558300	2
#22558287	2
C₆₇×#1969	2
#13364	67