Subgroups — C₂²×#359×C₆₇ #30712634

Order 17152.

1174 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#135×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂³×C₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×#108×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#359×C₆₇	C₂	1	12	yes	maximal, direct product factor
5	C₂²×#41×C₆₇	—	2	1	no	—
6	C₂×#135×C₆₇	—	2	3	no	—
7	C₂×#135×C₆₇	—	2	3	no	—
8	C₂×#135×C₆₇	C₄	1	6	yes	—
9	C₂×#135×C₆₇	C₂²	1	6	yes	—
10	C₂²×Q₁₆×C₆₇	C₄	1	1	yes	characteristic
11	C₂²×SD₁₆×C₆₇	—	2	1	no	—
12	C₂²×D₈×C₆₇	C₄	1	1	yes	characteristic
13	C₂²×M₁₆×C₆₇	—	2	1	no	—
14	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
15	C₂²×C₁₆×C₆₇	—	2	1	no	—
16	C₂²×C₁₆×C₆₇	—	2	3	no	—
17	C₂²×C₁₆×C₆₇	C₂²	1	6	yes	complemented
18	C₂×#108×C₆₇	—	2	3	no	—
19	C₂×#108×C₆₇	C₂²	1	6	yes	complemented
20	#359×C₆₇	C₂²	1	16	yes	direct product factor
21	C₂×#41×C₆₇	—	2	6	no	—
22	C₂×#41×C₆₇	—	2	6	no	—
23	C₂×#41×C₆₇	—	4	3	no	—
24	C₂×#41×C₆₇	—	4	3	no	—
25	C₂²×Q₈×C₆₇	—	2	1	no	—
26	C₂²×D₄×C₆₇	—	2	1	no	—
27	C₂²×D₄×C₆₇	—	4	1	no	—
28	C₂³×C₄×C₆₇	—	4	1	no	—
29	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
30	#135×C₆₇	—	2	12	no	—
31	#135×C₆₇	—	2	6	no	—
32	#135×C₆₇	—	2	6	no	—
33	#135×C₆₇	C₂×C₄	1	12	yes	—
34	#135×C₆₇	C₂³	1	4	yes	—
35	C₂×Q₁₆×C₆₇	—	2	3	no	—
36	C₂×Q₁₆×C₆₇	C₂×C₄	1	6	yes	—
37	C₂×SD₁₆×C₆₇	—	2	6	no	—
38	C₂×SD₁₆×C₆₇	—	2	6	no	—
39	C₂×D₈×C₆₇	—	2	3	no	—
40	C₂×D₈×C₆₇	C₂×C₄	1	6	yes	—
41	C₂×M₁₆×C₆₇	—	2	6	no	—
42	C₂×M₁₆×C₆₇	—	4	3	no	—
43	C₂²×C₈×C₆₇	—	4	1	no	—
44	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
45	C₂²×C₈×C₆₇	C₂×C₄	1	3	yes	—
46	C₂²×C₈×C₆₇	D₄	1	3	yes	—
47	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
48	C₂²×C₈×C₆₇	D₄	1	3	yes	—
49	C₂²×C₈×C₆₇	C₂³	1	3	yes	—
50	C₂×C₁₆×C₆₇	—	2	6	no	—
51	C₂×C₁₆×C₆₇	—	2	6	no	—
52	C₂×C₁₆×C₆₇	C₂³	1	4	yes	complemented
53	#108×C₆₇	—	2	6	no	—
54	#108×C₆₇	C₂³	1	4	yes	complemented
55	C₂⁴×C₆₇	—	4	1	no	—
56	C₂⁴×C₆₇	SD₁₆	1	1	yes	characteristic
57	#41×C₆₇	—	2	12	no	—
58	#41×C₆₇	—	2	4	no	—
59	#41×C₆₇	—	4	12	no	—
60	#41×C₆₇	—	4	6	no	—
61	#41×C₆₇	—	4	6	no	—
62	C₂×Q₈×C₆₇	—	2	6	no	—
63	C₂×Q₈×C₆₇	—	4	3	no	—
64	C₂×D₄×C₆₇	—	2	6	no	—
65	C₂×D₄×C₆₇	—	4	6	no	—
66	C₂×D₄×C₆₇	—	4	6	no	—
67	C₂×D₄×C₆₇	—	4	3	no	—
68	C₂²×C₄×C₆₇	—	4	1	no	—
69	C₂²×C₄×C₆₇	—	4	6	no	—
70	C₂²×C₄×C₆₇	—	4	6	no	—
71	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
72	C₂²×C₄×C₆₇	C₂²⋊C₄	1	3	yes	—
73	C₂²×C₄×C₆₇	D₈	1	3	yes	—
74	C₂²×C₄×C₆₇	D₈	1	1	yes	characteristic
75	C₂²×C₄×C₆₇	SD₁₆	1	3	yes	—
76	C₂²×C₄×C₆₇	C₂×D₄	1	3	yes	—
77	Q₁₆×C₆₇	—	2	6	no	—
78	Q₁₆×C₆₇	C₂²×C₄	1	4	yes	—
79	SD₁₆×C₆₇	—	2	12	no	—
80	SD₁₆×C₆₇	—	2	4	no	—
81	D₈×C₆₇	—	2	6	no	—
82	D₈×C₆₇	C₂²×C₄	1	4	yes	—
83	M₁₆×C₆₇	—	2	4	no	—
84	M₁₆×C₆₇	—	4	6	no	—
85	C₂×C₈×C₆₇	—	4	6	no	—
86	C₂×C₈×C₆₇	C₂²⋊C₄	1	3	yes	—
87	C₂×C₈×C₆₇	C₂²⋊C₄	1	3	yes	—
88	C₂×C₈×C₆₇	C₂²⋊C₄	1	6	yes	—
89	C₂×C₈×C₆₇	C₂²×C₄	1	3	yes	—
90	C₂×C₈×C₆₇	C₂²×C₄	1	3	yes	—
91	C₂×C₈×C₆₇	C₂×D₄	1	3	yes	—
92	C₂×C₈×C₆₇	C₂×D₄	1	3	yes	—
93	C₂×C₈×C₆₇	C₂×D₄	1	3	yes	—
94	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
95	C₁₆×C₆₇	—	2	4	no	—
96	C₂³×C₆₇	—	2	4	no	—
97	C₂³×C₆₇	—	4	6	no	—
98	C₂³×C₆₇	—	8	4	no	—
99	C₂³×C₆₇	#102	1	3	yes	—
100	C₂³×C₆₇	#102	1	1	yes	characteristic
101	C₂³×C₆₇	C₂×SD₁₆	1	3	yes	—
102	Q₈×C₆₇	—	2	4	no	—
103	Q₈×C₆₇	—	4	6	no	—
104	D₄×C₆₇	—	2	4	no	—
105	D₄×C₆₇	—	4	4	no	—
106	D₄×C₆₇	—	4	12	no	—
107	D₄×C₆₇	—	4	6	no	—
108	C₂×C₄×C₆₇	—	4	6	no	—
109	C₂×C₄×C₆₇	—	4	4	no	—
110	C₂×C₄×C₆₇	—	4	12	no	—
111	C₂×C₄×C₆₇	#102	1	3	yes	—
112	C₂×C₄×C₆₇	#102	1	3	yes	—
113	C₂×C₄×C₆₇	#102	1	6	yes	—
114	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	3	yes	—
115	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	3	yes	—
116	C₂×C₄×C₆₇	C₂×D₈	1	3	yes	—
117	C₂×C₄×C₆₇	C₂×D₈	1	3	yes	—
118	C₂×C₄×C₆₇	C₂×SD₁₆	1	3	yes	—
119	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
120	C₈×C₆₇	—	4	4	no	—
121	C₈×C₆₇	C₂×C₂²⋊C₄	1	3	yes	—
122	C₈×C₆₇	C₂×C₂²⋊C₄	1	3	yes	—
123	C₈×C₆₇	C₂³×C₄	1	1	yes	characteristic
124	C₈×C₆₇	C₂²×D₄	1	1	yes	characteristic
125	C₂²×C₆₇	—	2	6	no	—
126	C₂²×C₆₇	—	2	6	no	—
127	C₂²×C₆₇	—	4	4	no	—
128	C₂²×C₆₇	—	8	12	no	—
129	C₂²×C₆₇	#359	1	4	yes	direct product factor
130	C₂²×C₆₇	C₂×#102	1	3	yes	—
131	C₂²×C₆₇	C₂×#102	1	3	yes	—
132	C₂²×C₆₇	C₂²×SD₁₆	1	1	yes	characteristic
133	C₄×C₆₇	—	4	4	no	—
134	C₄×C₆₇	C₂×#102	1	3	yes	—
135	C₄×C₆₇	C₂×#102	1	3	yes	—
136	C₄×C₆₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
137	C₄×C₆₇	C₂²×D₈	1	1	yes	characteristic
138	C₂²×#359	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
139	C₂×C₆₇	—	2	1	no	—
140	C₂×C₆₇	—	2	3	no	—
141	C₂×C₆₇	—	8	4	no	—
142	C₂×C₆₇	C₂×#359	1	6	yes	direct product factor
143	C₂×C₆₇	C₂²×#102	1	1	yes	characteristic
144	C₂²×#135	C₂×C₆₇	1	1	yes	characteristic
145	C₂³×C₁₆	C₂×C₆₇	1	1	yes	characteristic, complemented
146	C₂²×#108	C₂×C₆₇	1	1	yes	characteristic, complemented
147	C₂×#359	C₂×C₆₇	1	12	yes	direct product factor
148	C₆₇	C₂²×#359	1	1	yes	characteristic, direct product factor, Sylow
149	C₂²×#41	—	2	1	no	—
150	C₂×#135	—	2	3	no	—
151	C₂×#135	—	2	3	no	—
152	C₂×#135	C₄×C₆₇	1	6	yes	—
153	C₂×#135	C₂²×C₆₇	1	6	yes	—
154	C₂²×Q₁₆	C₄×C₆₇	1	1	yes	characteristic
155	C₂²×SD₁₆	—	2	1	no	—
156	C₂²×D₈	C₄×C₆₇	1	1	yes	characteristic
157	C₂²×M₁₆	—	2	1	no	—
158	C₂³×C₈	C₂²×C₆₇	1	1	yes	characteristic
159	C₂²×C₁₆	—	2	1	no	—
160	C₂²×C₁₆	—	2	3	no	—
161	C₂²×C₁₆	C₂²×C₆₇	1	6	yes	complemented
162	C₂×#108	—	2	3	no	—
163	C₂×#108	C₂²×C₆₇	1	6	yes	complemented
164	#359	C₂²×C₆₇	1	16	yes	direct product factor
165	C₂×#41	—	2	6	no	—
166	C₂×#41	—	2	6	no	—
167	C₂×#41	—	4	3	no	—
168	C₂×#41	—	4	3	no	—
169	C₂²×Q₈	—	2	1	no	—
170	C₂²×D₄	—	2	1	no	—
171	C₂²×D₄	—	4	1	no	—
172	C₂³×C₄	—	4	1	no	—
173	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
174	#135	—	2	12	no	—
175	#135	—	2	6	no	—
176	#135	—	2	6	no	—
177	#135	C₂×C₄×C₆₇	1	12	yes	—
178	#135	C₂³×C₆₇	1	4	yes	—
179	C₂×Q₁₆	—	2	3	no	—
180	C₂×Q₁₆	C₂×C₄×C₆₇	1	6	yes	—
181	C₂×SD₁₆	—	2	6	no	—
182	C₂×SD₁₆	—	2	6	no	—
183	C₂×D₈	—	2	3	no	—
184	C₂×D₈	C₂×C₄×C₆₇	1	6	yes	—
185	C₂×M₁₆	—	2	6	no	—
186	C₂×M₁₆	—	4	3	no	—
187	C₂²×C₈	—	4	1	no	—
188	C₂²×C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
189	C₂²×C₈	C₂×C₄×C₆₇	1	3	yes	—
190	C₂²×C₈	D₄×C₆₇	1	3	yes	—
191	C₂²×C₈	D₄×C₆₇	1	1	yes	characteristic
192	C₂²×C₈	D₄×C₆₇	1	3	yes	—
193	C₂²×C₈	C₂³×C₆₇	1	3	yes	—
194	C₂×C₁₆	—	2	6	no	—
195	C₂×C₁₆	—	2	6	no	—
196	C₂×C₁₆	C₂³×C₆₇	1	4	yes	complemented
197	#108	—	2	6	no	—
198	#108	C₂³×C₆₇	1	4	yes	complemented
199	C₂⁴	—	4	1	no	—
200	C₂⁴	SD₁₆×C₆₇	1	1	yes	characteristic
201	#41	—	2	12	no	—
202	#41	—	2	4	no	—
203	#41	—	4	12	no	—
204	#41	—	4	6	no	—
205	#41	—	4	6	no	—
206	C₂×Q₈	—	2	6	no	—
207	C₂×Q₈	—	4	3	no	—
208	C₂×D₄	—	2	6	no	—
209	C₂×D₄	—	4	6	no	—
210	C₂×D₄	—	4	6	no	—
211	C₂×D₄	—	4	3	no	—
212	C₂²×C₄	—	4	1	no	—
213	C₂²×C₄	—	4	6	no	—
214	C₂²×C₄	—	4	6	no	—
215	C₂²×C₄	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
216	C₂²×C₄	C₂²⋊C₄×C₆₇	1	3	yes	—
217	C₂²×C₄	D₈×C₆₇	1	1	yes	characteristic
218	C₂²×C₄	D₈×C₆₇	1	3	yes	—
219	C₂²×C₄	SD₁₆×C₆₇	1	3	yes	—
220	C₂²×C₄	C₂×D₄×C₆₇	1	3	yes	—
221	Q₁₆	—	2	6	no	—
222	Q₁₆	C₂²×C₄×C₆₇	1	4	yes	—
223	SD₁₆	—	2	12	no	—
224	SD₁₆	—	2	4	no	—
225	D₈	—	2	6	no	—
226	D₈	C₂²×C₄×C₆₇	1	4	yes	—
227	M₁₆	—	2	4	no	—
228	M₁₆	—	4	6	no	—
229	C₂×C₈	—	4	6	no	—
230	C₂×C₈	C₂²⋊C₄×C₆₇	1	3	yes	—
231	C₂×C₈	C₂²⋊C₄×C₆₇	1	3	yes	—
232	C₂×C₈	C₂²⋊C₄×C₆₇	1	6	yes	—
233	C₂×C₈	C₂²×C₄×C₆₇	1	3	yes	—
234	C₂×C₈	C₂²×C₄×C₆₇	1	3	yes	—
235	C₂×C₈	C₂×D₄×C₆₇	1	3	yes	—
236	C₂×C₈	C₂×D₄×C₆₇	1	3	yes	—
237	C₂×C₈	C₂×D₄×C₆₇	1	3	yes	—
238	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
239	C₁₆	—	2	4	no	—
240	C₂³	—	2	4	no	—
241	C₂³	—	4	6	no	—
242	C₂³	—	8	4	no	—
243	C₂³	#102×C₆₇	1	3	yes	—
244	C₂³	#102×C₆₇	1	1	yes	characteristic
245	C₂³	C₂×SD₁₆×C₆₇	1	3	yes	—
246	Q₈	—	2	4	no	—
247	Q₈	—	4	6	no	—
248	D₄	—	2	4	no	—
249	D₄	—	4	4	no	—
250	D₄	—	4	12	no	—
251	D₄	—	4	6	no	—
252	C₂×C₄	—	4	6	no	—
253	C₂×C₄	—	4	4	no	—
254	C₂×C₄	—	4	12	no	—
255	C₂×C₄	#102×C₆₇	1	3	yes	—
256	C₂×C₄	#102×C₆₇	1	3	yes	—
257	C₂×C₄	#102×C₆₇	1	6	yes	—
258	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	3	yes	—
259	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	3	yes	—
260	C₂×C₄	C₂×D₈×C₆₇	1	3	yes	—
261	C₂×C₄	C₂×D₈×C₆₇	1	3	yes	—
262	C₂×C₄	C₂×SD₁₆×C₆₇	1	3	yes	—
263	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
264	C₈	—	4	4	no	—
265	C₈	C₂×C₂²⋊C₄×C₆₇	1	3	yes	—
266	C₈	C₂×C₂²⋊C₄×C₆₇	1	3	yes	—
267	C₈	C₂³×C₄×C₆₇	1	1	yes	characteristic
268	C₈	C₂²×D₄×C₆₇	1	1	yes	characteristic
269	C₂²	—	2	6	no	—
270	C₂²	—	2	6	no	—
271	C₂²	—	4	4	no	—
272	C₂²	—	8	12	no	—
273	C₂²	#359×C₆₇	1	4	yes	direct product factor
274	C₂²	C₂×#102×C₆₇	1	3	yes	—
275	C₂²	C₂×#102×C₆₇	1	3	yes	—
276	C₂²	C₂²×SD₁₆×C₆₇	1	1	yes	characteristic
277	C₄	—	4	4	no	—
278	C₄	C₂×#102×C₆₇	1	3	yes	—
279	C₄	C₂×#102×C₆₇	1	3	yes	—
280	C₄	C₂²×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
281	C₄	C₂²×D₈×C₆₇	1	1	yes	characteristic
282	C₂	—	2	1	no	—
283	C₂	—	2	3	no	—
284	C₂	—	8	4	no	—
285	C₂	C₂×#359×C₆₇	1	6	yes	direct product factor
286	C₂	C₂²×#102×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₆₇	D₈
commutator	C₈	C₂³×C₄×C₆₇
Frattini, cosocle	C₂×C₈	C₂⁴×C₆₇
socle	C₂³×C₆₇	#102
prime radical	C₂²×#135×C₆₇	C₂
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₆₇ ◁ C₂³×C₄×C₆₇ ◁ C₂³×C₈×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂³×C₁₆×C₆₇	C₂	semidirect
3. C₂²×#108×C₆₇	C₂	semidirect
4. C₂×#359×C₆₇	C₂	direct (N × H)
17. C₂²×C₁₆×C₆₇	C₂²	semidirect
19. C₂×#108×C₆₇	C₂²	semidirect
20. #359×C₆₇	C₂²	direct (N × H)
52. C₂×C₁₆×C₆₇	C₂³	semidirect
54. #108×C₆₇	C₂³	semidirect
129. C₂²×C₆₇	#359	direct (N × H)
138. C₂²×#359	C₆₇	direct (N × H)
142. C₂×C₆₇	C₂×#359	direct (N × H)
145. C₂³×C₁₆	C₂×C₆₇	semidirect
146. C₂²×#108	C₂×C₆₇	semidirect
147. C₂×#359	C₂×C₆₇	direct (N × H)
148. C₆₇	C₂²×#359	direct (N × H)
161. C₂²×C₁₆	C₂²×C₆₇	semidirect
163. C₂×#108	C₂²×C₆₇	semidirect
164. #359	C₂²×C₆₇	direct (N × H)
196. C₂×C₁₆	C₂³×C₆₇	semidirect
198. #108	C₂³×C₆₇	semidirect
273. C₂²	#359×C₆₇	direct (N × H)
285. C₂	C₂×#359×C₆₇	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#135×C₆₇	2
C₂³×C₁₆×C₆₇	2
C₂²×#108×C₆₇	2
C₂×#359×C₆₇	2
C₂²×#359	67

Subgroups — C22×#359×C67 #30712634

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#359×C₆₇ #30712634