Subgroups — #30772760

Order 17152.

470 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22573312	C₂	1	1	yes	maximal, characteristic, complemented
2	#22561830	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#16863178	C₂	1	2	yes	maximal, complemented
4	#22558295	C₂	1	2	yes	maximal, complemented
5	C₆₇×#2930	C₂	1	1	yes	maximal, characteristic
6	#16863813	—	2	4	no	—
7	C₂×#14523983	—	2	2	no	—
8	C₂×#14523983	C₂²	1	2	yes	—
9	C₂×#14523983	C₂²	1	2	yes	—
10	#16863178	—	2	4	no	—
11	#16863178	—	2	2	no	—
12	C₂×#131×C₆₇	C₂²	1	1	yes	characteristic
13	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
14	#525×C₆₇	C₄	1	1	yes	characteristic
15	#436×C₆₇	—	2	2	no	—
16	#435×C₆₇	—	2	2	no	—
17	#424×C₆₇	C₄	1	1	yes	characteristic
18	#408×C₆₇	C₄	1	1	yes	characteristic
19	#407×C₆₇	C₄	1	1	yes	characteristic
20	#405×C₆₇	C₂²	1	1	yes	characteristic
21	#14523983	—	2	4	no	—
22	#14523983	—	2	2	no	—
23	#14523983	—	2	2	no	—
24	#14523983	—	2	2	no	—
25	#14523983	D₄	1	4	yes	complemented
26	C₂×C₆₇⋊C₁₆	—	2	4	no	—
27	C₂×C₆₇⋊C₁₆	—	2	2	no	—
28	C₂×C₆₇⋊C₁₆	—	2	2	no	—
29	C₂×#41×C₆₇	C₂×C₄	1	1	yes	characteristic
30	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
31	#131×C₆₇	—	2	2	no	—
32	#131×C₆₇	—	2	2	no	—
33	C₂×M₁₆×C₆₇	—	2	2	no	—
34	C₂×M₁₆×C₆₇	—	2	1	no	—
35	C₂×M₁₆×C₆₇	—	2	1	no	—
36	C₂×M₁₆×C₆₇	—	2	2	no	—
37	C₂×M₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
38	C₂×M₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
39	C₂×M₁₆×C₆₇	D₄	1	2	yes	—
40	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₈×C₆₇	D₄	1	2	yes	—
42	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
43	#123×C₆₇	C₈	1	2	yes	—
44	C₂²⋊Q₈×C₆₇	—	2	1	no	—
45	#121×C₆₇	—	2	1	no	—
46	#120×C₆₇	C₈	1	2	yes	—
47	C₄×D₄×C₆₇	—	2	2	no	—
48	#117×C₆₇	C₂×C₄	1	1	yes	characteristic
49	C₄⋊C₈×C₆₇	—	2	2	no	—
50	C₂²⋊C₈×C₆₇	—	2	2	no	—
51	C₂²⋊C₈×C₆₇	—	2	2	no	—
52	#97×C₆₇	—	2	1	no	—
53	C₄×C₈×C₆₇	—	2	1	no	—
54	C₆₇⋊C₁₆	—	2	4	no	—
55	C₆₇⋊C₁₆	—	4	2	no	—
56	C₂⁴×C₆₇	C₂×C₈	1	1	yes	characteristic
57	#41×C₆₇	—	2	2	no	—
58	#41×C₆₇	—	2	2	no	—
59	C₂×Q₈×C₆₇	M₁₆	1	1	yes	characteristic
60	C₂×D₄×C₆₇	—	2	2	no	—
61	C₂×D₄×C₆₇	C₂×C₈	1	2	yes	—
62	C₂×D₄×C₆₇	M₁₆	1	1	yes	characteristic
63	C₂²×C₄×C₆₇	—	2	1	no	—
64	C₂²×C₄×C₆₇	—	2	2	no	—
65	C₂²×C₄×C₆₇	—	2	1	no	—
66	C₂²×C₄×C₆₇	—	2	2	no	—
67	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
68	C₂²×C₄×C₆₇	C₂×C₈	1	1	yes	characteristic
69	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
70	M₁₆×C₆₇	—	2	2	no	—
71	M₁₆×C₆₇	—	2	4	no	—
72	M₁₆×C₆₇	—	2	2	no	—
73	M₁₆×C₆₇	—	4	1	no	—
74	M₁₆×C₆₇	—	4	2	no	—
75	M₁₆×C₆₇	—	4	1	no	—
76	C₂×C₈×C₆₇	—	2	2	no	—
77	C₂×C₈×C₆₇	—	2	2	no	—
78	C₂×C₈×C₆₇	—	2	1	no	—
79	C₂×C₈×C₆₇	—	2	1	no	—
80	C₂×C₈×C₆₇	—	2	2	no	—
81	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
82	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
83	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
84	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
85	C₂×C₈×C₆₇	#41	1	2	yes	—
86	C₄⋊C₄×C₆₇	—	2	2	no	—
87	C₄⋊C₄×C₆₇	—	2	1	no	—
88	C₂²⋊C₄×C₆₇	—	2	2	no	—
89	C₂²⋊C₄×C₆₇	—	2	2	no	—
90	C₂²⋊C₄×C₆₇	C₂×C₈	1	2	yes	—
91	C₄²×C₆₇	—	2	1	no	—
92	C₂³×C₆₇	—	2	1	no	—
93	C₂³×C₆₇	—	2	2	no	—
94	C₂³×C₆₇	—	8	1	no	—
95	C₂³×C₆₇	C₂²⋊C₈	1	2	yes	—
96	C₂³×C₆₇	C₂²×C₈	1	1	yes	characteristic
97	Q₈×C₆₇	—	2	2	no	—
98	D₄×C₆₇	—	2	4	no	—
99	D₄×C₆₇	—	2	2	no	—
100	D₄×C₆₇	—	8	2	no	—
101	C₂×C₄×C₆₇	—	2	2	no	—
102	C₂×C₄×C₆₇	—	2	2	no	—
103	C₂×C₄×C₆₇	—	2	2	no	—
104	C₂×C₄×C₆₇	—	2	1	no	—
105	C₂×C₄×C₆₇	—	2	2	no	—
106	C₂×C₄×C₆₇	—	4	1	no	—
107	C₂×C₄×C₆₇	—	4	1	no	—
108	C₂×C₄×C₆₇	—	4	2	no	—
109	C₂×C₄×C₆₇	C₂²⋊C₈	1	2	yes	—
110	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
111	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
112	C₂×C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
113	C₂×C₄×C₆₇	#131	1	2	yes	—
114	C₈×C₆₇	—	2	2	no	—
115	C₈×C₆₇	—	4	1	no	—
116	C₈×C₆₇	—	4	1	no	—
117	C₈×C₆₇	#120	1	1	yes	characteristic
118	C₈×C₆₇	#121	1	2	yes	—
119	C₈×C₆₇	#123	1	1	yes	characteristic
120	C₂²×C₆₇	—	2	2	no	—
121	C₂²×C₆₇	—	4	1	no	—
122	C₂²×C₆₇	—	4	2	no	—
123	C₂²×C₆₇	—	4	1	no	—
124	C₂²×C₆₇	—	4	1	no	—
125	C₂²×C₆₇	—	4	1	no	—
126	C₂²×C₆₇	—	8	2	no	—
127	C₂²×C₆₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
128	C₂²×C₆₇	C₈×D₄	1	2	yes	—
129	C₄×C₆₇	—	2	2	no	—
130	C₄×C₆₇	—	4	2	no	—
131	C₄×C₆₇	—	4	1	no	—
132	C₄×C₆₇	#386	1	1	yes	characteristic
133	C₄×C₆₇	#408	1	1	yes	characteristic
134	C₄×C₆₇	#435	1	2	yes	—
135	#12989	—	67	1	no	maximal, Sylow
136	C₂×C₆₇	—	2	1	no	—
137	C₂×C₆₇	—	2	2	no	—
138	C₂×C₆₇	—	4	2	no	—
139	C₂×C₆₇	—	8	1	no	—
140	C₂×C₆₇	#1869	1	1	yes	characteristic
141	#3407	—	67	1	no	—
142	#2930	D₆₇	1	1	yes	characteristic
143	#2115	—	67	1	no	—
144	C₂×#349	—	67	2	no	—
145	#1378	—	67	2	no	—
146	C₆₇	#12989	1	1	yes	characteristic, complemented, Sylow
147	C₂×#131	C₂×D₆₇	1	1	yes	characteristic
148	C₂²×M₁₆	C₂×D₆₇	1	1	yes	characteristic
149	#525	Dic₆₇	1	1	yes	characteristic
150	#504	—	134	4	no	—
151	C₂×M₃₂	—	67	2	no	—
152	C₂×M₃₂	—	67	2	no	—
153	C₂×M₃₂	—	134	2	no	—
154	#436	—	2	2	no	—
155	#435	—	2	2	no	—
156	#424	Dic₆₇	1	1	yes	characteristic
157	#408	Dic₆₇	1	1	yes	characteristic
158	#407	Dic₆₇	1	1	yes	characteristic
159	#405	C₂×D₆₇	1	1	yes	characteristic
160	#349	—	134	4	no	—
161	#349	—	134	2	no	—
162	C₂×#41	C₂×Dic₆₇	1	1	yes	characteristic
163	C₂³×C₄	C₂×Dic₆₇	1	1	yes	characteristic
164	#131	—	2	2	no	—
165	#131	—	2	2	no	—
166	C₂×M₁₆	—	2	2	no	—
167	C₂×M₁₆	—	2	1	no	—
168	C₂×M₁₆	—	2	1	no	—
169	C₂×M₁₆	—	2	2	no	—
170	C₂×M₁₆	C₂×Dic₆₇	1	1	yes	characteristic
171	C₂×M₁₆	C₂×Dic₆₇	1	1	yes	characteristic
172	C₂×M₁₆	C₂²⋊D₆₇	1	2	yes	—
173	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
174	C₂²×C₈	C₂²⋊D₆₇	1	2	yes	—
175	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
176	#123	C₆₇⋊C₈	1	2	yes	—
177	C₂²⋊Q₈	—	2	1	no	—
178	#121	—	2	1	no	—
179	#120	C₆₇⋊C₈	1	2	yes	—
180	C₄×D₄	—	2	2	no	—
181	#117	C₂×Dic₆₇	1	1	yes	characteristic
182	M₃₂	—	67	4	no	—
183	M₃₂	—	134	2	no	—
184	M₃₂	—	134	2	no	—
185	M₃₂	—	134	4	no	—
186	M₃₂	—	134	2	no	—
187	C₂×C₁₆	—	134	2	no	—
188	C₂×C₁₆	—	134	4	no	—
189	C₂×C₁₆	—	134	2	no	—
190	C₄⋊C₈	—	2	2	no	—
191	C₂²⋊C₈	—	2	2	no	—
192	C₂²⋊C₈	—	2	2	no	—
193	#97	—	2	1	no	—
194	C₄×C₈	—	2	1	no	—
195	C₂⁴	C₂×C₆₇⋊C₈	1	1	yes	characteristic
196	#41	—	2	2	no	—
197	#41	—	2	2	no	—
198	C₂×Q₈	#11761612	1	1	yes	characteristic
199	C₂×D₄	—	2	2	no	—
200	C₂×D₄	C₂×C₆₇⋊C₈	1	2	yes	—
201	C₂×D₄	#11761612	1	1	yes	characteristic
202	C₂²×C₄	—	2	1	no	—
203	C₂²×C₄	—	2	2	no	—
204	C₂²×C₄	—	2	1	no	—
205	C₂²×C₄	—	2	2	no	—
206	C₂²×C₄	C₂²⋊Dic₆₇	1	2	yes	—
207	C₂²×C₄	C₂×C₆₇⋊C₈	1	1	yes	characteristic
208	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
209	M₁₆	—	2	2	no	—
210	M₁₆	—	2	4	no	—
211	M₁₆	—	2	2	no	—
212	M₁₆	—	4	1	no	—
213	M₁₆	—	4	2	no	—
214	M₁₆	—	4	1	no	—
215	C₂×C₈	—	2	2	no	—
216	C₂×C₈	—	2	2	no	—
217	C₂×C₈	—	2	1	no	—
218	C₂×C₈	—	2	1	no	—
219	C₂×C₈	—	2	2	no	—
220	C₂×C₈	C₂²⋊Dic₆₇	1	2	yes	—
221	C₂×C₈	D₄×D₆₇	1	2	yes	—
222	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
223	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
224	C₂×C₈	#11761627	1	2	yes	—
225	C₄⋊C₄	—	2	2	no	—
226	C₄⋊C₄	—	2	1	no	—
227	C₂²⋊C₄	—	2	2	no	—
228	C₂²⋊C₄	—	2	2	no	—
229	C₂²⋊C₄	C₂×C₆₇⋊C₈	1	2	yes	—
230	C₄²	—	2	1	no	—
231	C₁₆	—	268	2	no	—
232	C₁₆	—	134	4	no	—
233	C₂³	—	2	1	no	—
234	C₂³	—	2	2	no	—
235	C₂³	—	8	1	no	—
236	C₂³	C₂²⋊(C₆₇⋊C₈)	1	2	yes	—
237	C₂³	C₂²×C₆₇⋊C₈	1	1	yes	characteristic
238	Q₈	—	2	2	no	—
239	D₄	—	2	4	no	—
240	D₄	—	2	2	no	—
241	D₄	—	8	2	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	4	1	no	—
248	C₂×C₄	—	4	1	no	—
249	C₂×C₄	—	4	2	no	—
250	C₂×C₄	C₂²⋊(C₆₇⋊C₈)	1	2	yes	—
251	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
252	C₂×C₄	D₄×Dic₆₇	1	2	yes	—
253	C₂×C₄	C₂×#11761612	1	1	yes	characteristic
254	C₂×C₄	#14524050	1	2	yes	—
255	C₈	—	2	2	no	—
256	C₈	—	4	1	no	—
257	C₈	—	4	1	no	—
258	C₈	#14524011	1	1	yes	characteristic
259	C₈	#14524015	1	2	yes	—
260	C₈	#14524024	1	1	yes	characteristic
261	C₂²	—	2	2	no	—
262	C₂²	—	4	1	no	—
263	C₂²	—	4	2	no	—
264	C₂²	—	4	1	no	—
265	C₂²	—	4	1	no	—
266	C₂²	—	4	1	no	—
267	C₂²	—	8	2	no	—
268	C₂²	C₂×C₂²⋊(C₆₇⋊C₈)	1	1	yes	characteristic
269	C₂²	D₄×C₆₇⋊C₈	1	2	yes	—
270	C₄	—	2	2	no	—
271	C₄	—	4	2	no	—
272	C₄	—	4	1	no	—
273	C₄	#16863276	1	1	yes	characteristic
274	C₄	#16863363	1	1	yes	characteristic
275	C₄	#16863468	1	2	yes	—
276	C₂	—	2	1	no	—
277	C₂	—	2	2	no	—
278	C₂	—	4	2	no	—
279	C₂	—	8	1	no	—
280	C₂	#22560502	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#16863276
commutator	C₂³×C₆₇	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₆₇
Fitting	C₆₇×#2930	C₂
socle	C₂×C₆₇	#1869
prime radical	#525×C₆₇	C₄
cosocle	C₂²×C₈×C₆₇	C₂³
hypercenter	#2930	D₆₇
hypocenter	C₆₇	#12989
LCS	G ▷ C₂³×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₈ ◁ #2930

Complements

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

N	H ≅ G/N	type
1. #22573312	C₂	semidirect
2. #22561830	C₂	semidirect
3. C₂×#16863178	C₂	semidirect
4. #22558295	C₂	semidirect
25. #14523983	D₄	semidirect
146. C₆₇	#12989	semidirect

Maximal subgroups

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

subgroup	index
#22573312	2
#22561830	2
C₂×#16863178	2
#22558295	2
C₆₇×#2930	2
#12989	67