Subgroups — #30748865

Order 17152.

404 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22558968	C₂	1	1	yes	maximal, characteristic, complemented
2	#22558965	C₂	1	1	yes	maximal, characteristic, complemented
3	#22558944	C₂	1	2	yes	maximal
4	C₆₇⋊#1309	C₂	1	2	yes	maximal, complemented
5	C₆₇×#2931	C₂	1	1	yes	maximal, characteristic
6	#16863420	C₂²	1	2	yes	—
7	#16863420	C₂²	1	2	yes	—
8	#16863149	—	2	1	no	—
9	#16863149	—	2	1	no	—
10	#16863147	—	2	1	no	—
11	#16863147	—	2	1	no	—
12	#16863145	—	2	2	no	—
13	#16863140	—	2	2	no	—
14	C₂×#131×C₆₇	C₄	1	2	yes	—
15	#532×C₆₇	C₂²	1	1	yes	characteristic
16	#436×C₆₇	—	2	1	no	—
17	#436×C₆₇	—	2	1	no	—
18	C₈×D₄×C₆₇	—	2	1	no	—
19	C₈×D₄×C₆₇	—	2	1	no	—
20	#423×C₆₇	C₂²	1	1	yes	characteristic
21	#408×C₆₇	C₄	1	2	yes	—
22	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
23	C₂×#11761612	—	2	2	no	—
24	C₂×#11761612	—	2	2	no	—
25	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
26	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
27	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
28	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
29	C₂×#41×C₆₇	C₂×C₄	1	2	yes	—
30	#131×C₆₇	—	4	4	no	—
31	C₂×M₁₆×C₆₇	—	2	2	no	—
32	C₂×M₁₆×C₆₇	C₂×C₄	1	2	yes	—
33	C₂²×C₈×C₆₇	—	2	2	no	—
34	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
36	C₄⋊Q₈×C₆₇	D₄	1	1	yes	characteristic
37	#127×C₆₇	D₄	1	1	yes	characteristic
38	#124×C₆₇	Q₈	1	2	yes	—
39	#121×C₆₇	—	2	1	no	—
40	#121×C₆₇	—	2	1	no	—
41	C₄×D₄×C₆₇	—	2	1	no	—
42	C₄×D₄×C₆₇	—	2	1	no	—
43	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
44	C₄⋊C₈×C₆₇	—	2	1	no	—
45	C₄⋊C₈×C₆₇	—	2	1	no	—
46	C₂²⋊C₈×C₆₇	—	2	2	no	—
47	C₂²⋊C₈×C₆₇	—	2	2	no	—
48	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
49	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
50	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
51	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
52	#11761612	—	8	2	no	—
53	#11761612	—	8	2	no	—
54	C₂×C₆₇⋊C₈	—	4	2	no	—
55	C₂×C₆₇⋊C₈	—	4	2	no	—
56	#41×C₆₇	—	4	2	no	—
57	#41×C₆₇	—	4	2	no	—
58	C₂×Q₈×C₆₇	C₄⋊C₄	1	2	yes	—
59	C₂×D₄×C₆₇	—	2	2	no	—
60	C₂×D₄×C₆₇	—	2	1	no	—
61	C₂×D₄×C₆₇	—	2	1	no	—
62	C₂×D₄×C₆₇	C₄⋊C₄	1	2	yes	—
63	C₂²×C₄×C₆₇	—	2	2	no	—
64	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
66	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
67	M₁₆×C₆₇	—	4	4	no	—
68	M₁₆×C₆₇	—	4	2	no	—
69	C₂×C₈×C₆₇	—	2	2	no	—
70	C₂×C₈×C₆₇	—	2	2	no	—
71	C₂×C₈×C₆₇	—	4	1	no	—
72	C₂×C₈×C₆₇	—	4	4	no	—
73	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
74	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
76	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
77	C₄⋊C₄×C₆₇	—	2	1	no	—
78	C₄⋊C₄×C₆₇	—	2	1	no	—
79	C₂²⋊C₄×C₆₇	—	2	2	no	—
80	C₂²⋊C₄×C₆₇	—	2	2	no	—
81	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
82	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
83	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
84	C₄²×C₆₇	C₂×Q₈	1	1	yes	characteristic
85	C₂³×C₆₇	—	2	2	no	—
86	C₂³×C₆₇	#117	1	1	yes	characteristic
87	Q₈×C₆₇	—	4	2	no	—
88	D₄×C₆₇	—	4	2	no	—
89	D₄×C₆₇	—	4	4	no	—
90	D₄×C₆₇	—	4	2	no	—
91	D₄×C₆₇	—	4	1	no	—
92	D₄×C₆₇	—	4	1	no	—
93	C₂×C₄×C₆₇	—	2	1	no	—
94	C₂×C₄×C₆₇	—	2	1	no	—
95	C₂×C₄×C₆₇	—	2	2	no	—
96	C₂×C₄×C₆₇	—	2	1	no	—
97	C₂×C₄×C₆₇	—	2	1	no	—
98	C₂×C₄×C₆₇	—	2	2	no	—
99	C₂×C₄×C₆₇	—	4	2	no	—
100	C₂×C₄×C₆₇	—	4	2	no	—
101	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
102	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
103	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
104	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
105	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
106	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
107	C₈×C₆₇	—	2	1	no	—
108	C₈×C₆₇	—	2	1	no	—
109	C₈×C₆₇	—	2	1	no	—
110	C₈×C₆₇	—	4	2	no	—
111	C₈×C₆₇	#100	1	2	yes	—
112	C₆₇⋊C₈	—	8	2	no	—
113	C₆₇⋊C₈	—	8	2	no	—
114	C₂²×C₆₇	—	2	1	no	—
115	C₂²×C₆₇	—	2	1	no	—
116	C₂²×C₆₇	—	2	1	no	—
117	C₂²×C₆₇	—	4	2	no	—
118	C₂²×C₆₇	—	4	2	no	—
119	C₂²×C₆₇	—	4	2	no	—
120	C₂²×C₆₇	#380	1	1	yes	characteristic
121	C₄×C₆₇	—	2	1	no	—
122	C₄×C₆₇	—	2	1	no	—
123	C₄×C₆₇	—	2	1	no	—
124	C₄×C₆₇	—	4	2	no	—
125	C₄×C₆₇	C₂×#100	1	1	yes	characteristic
126	C₄×C₆₇	#413	1	1	yes	characteristic
127	C₄×C₆₇	#471	1	2	yes	—
128	C₄×C₆₇	#472	1	1	yes	characteristic
129	C₄×C₆₇	#473	1	1	yes	characteristic
130	#8640	—	67	1	no	maximal, Sylow
131	C₂×C₆₇	—	4	1	no	—
132	C₂×C₆₇	—	8	2	no	—
133	C₂×C₆₇	#1800	1	1	yes	characteristic
134	C₂×C₆₇	#1815	1	1	yes	characteristic
135	C₂×C₆₇	#1816	1	1	yes	characteristic
136	#2931	D₆₇	1	1	yes	characteristic
137	#1567	—	67	1	no	—
138	#1566	—	67	1	no	—
139	#1560	—	67	2	no	—
140	#1309	—	67	2	no	—
141	C₆₇	#8640	1	1	yes	characteristic, complemented, Sylow
142	C₂×#131	Dic₆₇	1	2	yes	—
143	#532	C₂×D₆₇	1	1	yes	characteristic
144	#436	—	2	1	no	—
145	#436	—	2	1	no	—
146	C₈×D₄	—	2	1	no	—
147	C₈×D₄	—	2	1	no	—
148	#423	—	67	2	no	—
149	#423	—	67	2	no	—
150	#423	C₂×D₆₇	1	1	yes	characteristic
151	#408	Dic₆₇	1	2	yes	—
152	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
153	#335	—	134	1	no	—
154	#335	—	134	1	no	—
155	#334	—	134	1	no	—
156	#334	—	134	1	no	—
157	#333	—	134	2	no	—
158	#331	—	134	2	no	—
159	C₂×#41	C₂×Dic₆₇	1	2	yes	—
160	#131	—	4	4	no	—
161	C₂×M₁₆	—	2	2	no	—
162	C₂×M₁₆	—	134	2	no	—
163	C₂×M₁₆	—	134	2	no	—
164	C₂×M₁₆	C₂×Dic₆₇	1	2	yes	—
165	C₂²×C₈	—	2	2	no	—
166	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
167	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
168	C₄⋊Q₈	D₂₆₈	1	1	yes	characteristic
169	#127	D₂₆₈	1	1	yes	characteristic
170	#124	Dic₁₃₄	1	2	yes	—
171	#121	—	2	1	no	—
172	#121	—	2	1	no	—
173	C₄×D₄	—	2	1	no	—
174	C₄×D₄	—	2	1	no	—
175	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
176	C₄⋊C₈	—	2	1	no	—
177	C₄⋊C₈	—	2	1	no	—
178	C₄⋊C₈	—	134	2	no	—
179	C₄⋊C₈	—	134	2	no	—
180	C₄⋊C₈	—	134	2	no	—
181	C₄⋊C₈	—	134	2	no	—
182	C₂²⋊C₈	—	2	2	no	—
183	C₂²⋊C₈	—	2	2	no	—
184	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
185	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
186	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
187	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
188	#41	—	4	2	no	—
189	#41	—	4	2	no	—
190	C₂×Q₈	C₄⋊Dic₆₇	1	2	yes	—
191	C₂×D₄	—	2	2	no	—
192	C₂×D₄	—	2	1	no	—
193	C₂×D₄	—	2	1	no	—
194	C₂×D₄	C₄⋊Dic₆₇	1	2	yes	—
195	C₂²×C₄	—	2	2	no	—
196	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
197	C₂²×C₄	#11761628	1	1	yes	characteristic
198	C₂²×C₄	#11761628	1	1	yes	characteristic
199	M₁₆	—	4	4	no	—
200	M₁₆	—	4	2	no	—
201	M₁₆	—	536	2	no	—
202	M₁₆	—	536	2	no	—
203	C₂×C₈	—	2	2	no	—
204	C₂×C₈	—	2	2	no	—
205	C₂×C₈	—	4	1	no	—
206	C₂×C₈	—	4	4	no	—
207	C₂×C₈	—	268	2	no	—
208	C₂×C₈	—	268	2	no	—
209	C₂×C₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
210	C₂×C₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
211	C₂×C₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
212	C₂×C₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
213	C₄⋊C₄	—	2	1	no	—
214	C₄⋊C₄	—	2	1	no	—
215	C₂²⋊C₄	—	2	2	no	—
216	C₂²⋊C₄	—	2	2	no	—
217	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
218	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
219	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
220	C₄²	C₂×Dic₁₃₄	1	1	yes	characteristic
221	C₂³	—	2	2	no	—
222	C₂³	#14524002	1	1	yes	characteristic
223	Q₈	—	4	2	no	—
224	D₄	—	4	2	no	—
225	D₄	—	4	4	no	—
226	D₄	—	4	2	no	—
227	D₄	—	4	1	no	—
228	D₄	—	4	1	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	4	2	no	—
236	C₂×C₄	—	4	2	no	—
237	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
238	C₂×C₄	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
239	C₂×C₄	#14524017	1	1	yes	characteristic
240	C₂×C₄	#14524017	1	1	yes	characteristic
241	C₂×C₄	#14524022	1	1	yes	characteristic
242	C₂×C₄	#14524022	1	1	yes	characteristic
243	C₈	—	2	1	no	—
244	C₈	—	2	1	no	—
245	C₈	—	2	1	no	—
246	C₈	—	4	2	no	—
247	C₈	—	536	2	no	—
248	C₈	—	536	2	no	—
249	C₈	#14523960	1	2	yes	—
250	C₂²	—	2	1	no	—
251	C₂²	—	2	1	no	—
252	C₂²	—	2	1	no	—
253	C₂²	—	4	2	no	—
254	C₂²	—	4	2	no	—
255	C₂²	—	4	2	no	—
256	C₂²	#16863256	1	1	yes	characteristic
257	C₄	—	2	1	no	—
258	C₄	—	2	1	no	—
259	C₄	—	2	1	no	—
260	C₄	—	4	2	no	—
261	C₄	C₂×#14523960	1	1	yes	characteristic
262	C₄	#16863380	1	1	yes	characteristic
263	C₄	#16863659	1	2	yes	—
264	C₄	#16863666	1	1	yes	characteristic
265	C₄	#16863671	1	1	yes	characteristic
266	C₂	—	4	1	no	—
267	C₂	—	8	2	no	—
268	C₂	#22560201	1	1	yes	characteristic
269	C₂	#22560268	1	1	yes	characteristic
270	C₂	#22560273	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#16863256
commutator	C₂²×C₄×C₆₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₆₇×#2931	C₂
socle	C₂²×C₆₇	#380
prime radical	#532×C₆₇	C₂²
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	#2931	D₆₇
hypocenter	C₆₇	#8640
LCS	G ▷ C₂²×C₄×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
1. #22558968	C₂	semidirect
2. #22558965	C₂	semidirect
4. C₆₇⋊#1309	C₂	semidirect
141. C₆₇	#8640	semidirect

Maximal subgroups

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

subgroup	index
#22558968	2
#22558965	2
#22558944	2
C₆₇⋊#1309	2
C₆₇×#2931	2
#8640	67