Subgroups — C₆₇×#21750 #30701001

Order 17152.

1606 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#535×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#2308	C₂	1	2	yes	maximal, complemented
3	C₆₇×#1898	C₂	1	8	yes	maximal, complemented
4	C₆₇×#1820	C₂	1	2	yes	maximal, complemented
5	C₆₇×#1782	C₂	1	2	yes	maximal, complemented
6	C₂²×#41×C₆₇	C₂²	1	1	yes	characteristic
7	C₂³×D₄×C₆₇	C₂²	1	1	yes	characteristic
8	#535×C₆₇	—	2	8	no	—
9	C₂×#127×C₆₇	C₄	1	2	yes	complemented
10	C₂×#124×C₆₇	C₄	1	2	yes	complemented
11	C₂×#121×C₆₇	C₄	1	4	yes	complemented
12	C₂×#120×C₆₇	C₂²	1	4	yes	—
13	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
14	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	4	yes	—
15	C₂×#99×C₆₇	C₂²	1	8	yes	—
16	#386×C₆₇	—	2	8	no	—
17	#381×C₆₇	C₂²	1	4	yes	—
18	#379×C₆₇	C₂²	1	4	yes	—
19	#342×C₆₇	C₂²	1	8	yes	—
20	C₂⁵×C₆₇	C₂³	1	2	yes	—
21	C₂×#41×C₆₇	—	2	4	no	—
22	C₂×#41×C₆₇	—	4	4	no	—
23	C₂²×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic, complemented
24	C₂²×D₄×C₆₇	—	4	4	no	—
25	C₂²×D₄×C₆₇	C₂×C₄	1	2	yes	—
26	C₂²×D₄×C₆₇	C₂×C₄	1	2	yes	—
27	C₂²×D₄×C₆₇	C₂×C₄	1	1	yes	characteristic, complemented
28	C₂²×D₄×C₆₇	C₂×C₄	1	8	yes	complemented
29	C₂²×D₄×C₆₇	D₄	1	8	yes	—
30	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
31	C₂³×C₄×C₆₇	—	2	4	no	—
32	C₂³×C₄×C₆₇	C₂×C₄	1	2	yes	complemented
33	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
34	#127×C₆₇	—	2	8	no	—
35	#124×C₆₇	—	2	8	no	—
36	#121×C₆₇	—	2	16	no	—
37	#120×C₆₇	—	2	8	no	—
38	#120×C₆₇	—	2	8	no	—
39	#117×C₆₇	—	2	4	no	—
40	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	complemented
41	C₂×C₂²⋊C₄×C₆₇	—	2	4	no	—
42	C₂×C₂²⋊C₄×C₆₇	—	2	8	no	—
43	C₂×C₂²⋊C₄×C₆₇	—	4	8	no	—
44	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	8	yes	complemented
45	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
46	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
47	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	complemented
48	#99×C₆₇	—	4	8	no	—
49	#95×C₆₇	—	2	8	no	—
50	C₂⁴×C₆₇	—	2	8	no	—
51	C₂⁴×C₆₇	—	8	4	no	—
52	C₂⁴×C₆₇	C₂²×C₄	1	2	yes	—
53	C₂⁴×C₆₇	C₂²×C₄	1	4	yes	—
54	C₂⁴×C₆₇	C₂×D₄	1	4	yes	—
55	C₂⁴×C₆₇	C₂×D₄	1	4	yes	—
56	C₂⁴×C₆₇	C₂⁴	1	1	yes	characteristic
57	#41×C₆₇	—	4	16	no	—
58	C₂×Q₈×C₆₇	—	2	2	no	—
59	C₂×Q₈×C₆₇	—	4	2	no	—
60	C₂×D₄×C₆₇	—	2	4	no	—
61	C₂×D₄×C₆₇	—	2	8	no	—
62	C₂×D₄×C₆₇	—	2	16	no	—
63	C₂×D₄×C₆₇	—	2	2	no	—
64	C₂×D₄×C₆₇	—	4	4	no	—
65	C₂×D₄×C₆₇	—	4	4	no	—
66	C₂×D₄×C₆₇	—	4	8	no	—
67	C₂×D₄×C₆₇	—	4	8	no	—
68	C₂×D₄×C₆₇	—	4	2	no	—
69	C₂×D₄×C₆₇	—	8	8	no	—
70	C₂×D₄×C₆₇	C₂²⋊C₄	1	8	yes	—
71	C₂×D₄×C₆₇	C₂²⋊C₄	1	8	yes	—
72	C₂²×C₄×C₆₇	—	2	4	no	—
73	C₂²×C₄×C₆₇	—	2	4	no	—
74	C₂²×C₄×C₆₇	—	2	8	no	—
75	C₂²×C₄×C₆₇	—	4	4	no	—
76	C₂²×C₄×C₆₇	—	4	8	no	—
77	C₂²×C₄×C₆₇	—	8	8	no	—
78	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
79	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
80	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
81	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
82	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
83	C₄⋊C₄×C₆₇	—	2	4	no	—
84	C₂²⋊C₄×C₆₇	—	2	16	no	—
85	C₂²⋊C₄×C₆₇	—	2	4	no	—
86	C₂²⋊C₄×C₆₇	—	2	16	no	—
87	C₂²⋊C₄×C₆₇	—	4	8	no	—
88	C₂²⋊C₄×C₆₇	—	8	8	no	—
89	C₄²×C₆₇	—	2	4	no	—
90	C₂³×C₆₇	—	2	4	no	—
91	C₂³×C₆₇	—	2	4	no	—
92	C₂³×C₆₇	—	2	8	no	—
93	C₂³×C₆₇	—	2	8	no	—
94	C₂³×C₆₇	—	2	2	no	—
95	C₂³×C₆₇	—	2	4	no	—
96	C₂³×C₆₇	—	4	8	no	—
97	C₂³×C₆₇	—	4	2	no	—
98	C₂³×C₆₇	—	4	8	no	—
99	C₂³×C₆₇	—	4	8	no	—
100	C₂³×C₆₇	—	4	4	no	—
101	C₂³×C₆₇	—	8	16	no	—
102	C₂³×C₆₇	C₂×C₂²⋊C₄	1	8	yes	—
103	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
104	C₂³×C₆₇	C₂²×D₄	1	2	yes	—
105	C₂³×C₆₇	H(5,𝔽₂)	1	4	yes	—
106	Q₈×C₆₇	—	4	4	no	—
107	D₄×C₆₇	—	4	8	no	—
108	D₄×C₆₇	—	4	8	no	—
109	D₄×C₆₇	—	4	4	no	—
110	D₄×C₆₇	—	4	8	no	—
111	D₄×C₆₇	—	8	16	no	—
112	C₂×C₄×C₆₇	—	2	2	no	—
113	C₂×C₄×C₆₇	—	2	2	no	—
114	C₂×C₄×C₆₇	—	2	8	no	—
115	C₂×C₄×C₆₇	—	2	4	no	—
116	C₂×C₄×C₆₇	—	2	2	no	—
117	C₂×C₄×C₆₇	—	2	4	no	—
118	C₂×C₄×C₆₇	—	4	8	no	—
119	C₂×C₄×C₆₇	—	4	4	no	—
120	C₂×C₄×C₆₇	—	4	16	no	—
121	C₂×C₄×C₆₇	—	4	8	no	—
122	C₂×C₄×C₆₇	—	8	16	no	—
123	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
124	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
125	C₂²×C₆₇	—	2	4	no	—
126	C₂²×C₆₇	—	2	2	no	—
127	C₂²×C₆₇	—	2	8	no	—
128	C₂²×C₆₇	—	2	16	no	—
129	C₂²×C₆₇	—	4	4	no	—
130	C₂²×C₆₇	—	4	8	no	—
131	C₂²×C₆₇	—	4	8	no	—
132	C₂²×C₆₇	—	4	2	no	—
133	C₂²×C₆₇	—	4	4	no	—
134	C₂²×C₆₇	—	8	8	no	—
135	C₂²×C₆₇	—	8	8	no	—
136	C₂²×C₆₇	—	8	4	no	—
137	C₂²×C₆₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
138	C₂²×C₆₇	#518	1	2	yes	—
139	C₂²×C₆₇	#534	1	4	yes	—
140	C₄×C₆₇	—	2	2	no	—
141	C₄×C₆₇	—	2	2	no	—
142	C₄×C₆₇	—	4	2	no	—
143	C₄×C₆₇	—	4	4	no	—
144	C₄×C₆₇	—	8	8	no	—
145	#21750	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
146	C₂×C₆₇	—	2	4	no	—
147	C₂×C₆₇	—	2	2	no	—
148	C₂×C₆₇	—	4	8	no	—
149	C₂×C₆₇	—	8	2	no	—
150	C₂×C₆₇	#2308	1	1	yes	characteristic
151	C₂×C₆₇	#2883	1	2	yes	—
152	C₂×#535	C₂×C₆₇	1	1	yes	characteristic
153	#2308	C₂×C₆₇	1	2	yes	complemented
154	#1898	C₂×C₆₇	1	8	yes	complemented
155	#1820	C₂×C₆₇	1	2	yes	complemented
156	#1782	C₂×C₆₇	1	2	yes	complemented
157	C₆₇	#21750	1	1	yes	characteristic, direct product factor, Sylow
158	C₂²×#41	C₂²×C₆₇	1	1	yes	characteristic
159	C₂³×D₄	C₂²×C₆₇	1	1	yes	characteristic
160	#535	—	2	8	no	—
161	C₂×#127	C₄×C₆₇	1	2	yes	complemented
162	C₂×#124	C₄×C₆₇	1	2	yes	complemented
163	C₂×#121	C₄×C₆₇	1	4	yes	complemented
164	C₂×#120	C₂²×C₆₇	1	4	yes	—
165	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
166	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	4	yes	—
167	C₂×#99	C₂²×C₆₇	1	8	yes	—
168	#386	—	2	8	no	—
169	#381	C₂²×C₆₇	1	4	yes	—
170	#379	C₂²×C₆₇	1	4	yes	—
171	#342	C₂²×C₆₇	1	8	yes	—
172	C₂⁵	C₂³×C₆₇	1	2	yes	—
173	C₂×#41	—	2	4	no	—
174	C₂×#41	—	4	4	no	—
175	C₂²×Q₈	C₂×C₄×C₆₇	1	1	yes	characteristic, complemented
176	C₂²×D₄	—	4	4	no	—
177	C₂²×D₄	C₂×C₄×C₆₇	1	2	yes	—
178	C₂²×D₄	C₂×C₄×C₆₇	1	2	yes	—
179	C₂²×D₄	C₂×C₄×C₆₇	1	1	yes	characteristic, complemented
180	C₂²×D₄	C₂×C₄×C₆₇	1	8	yes	complemented
181	C₂²×D₄	D₄×C₆₇	1	8	yes	—
182	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
183	C₂³×C₄	—	2	4	no	—
184	C₂³×C₄	C₂×C₄×C₆₇	1	2	yes	complemented
185	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
186	#127	—	2	8	no	—
187	#124	—	2	8	no	—
188	#121	—	2	16	no	—
189	#120	—	2	8	no	—
190	#120	—	2	8	no	—
191	#117	—	2	4	no	—
192	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	complemented
193	C₂×C₂²⋊C₄	—	2	4	no	—
194	C₂×C₂²⋊C₄	—	2	8	no	—
195	C₂×C₂²⋊C₄	—	4	8	no	—
196	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	8	yes	complemented
197	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
198	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
199	C₂×C₄²	C₂×C₄×C₆₇	1	2	yes	complemented
200	#99	—	4	8	no	—
201	#95	—	2	8	no	—
202	C₂⁴	—	2	8	no	—
203	C₂⁴	—	8	4	no	—
204	C₂⁴	C₂²×C₄×C₆₇	1	2	yes	—
205	C₂⁴	C₂²×C₄×C₆₇	1	4	yes	—
206	C₂⁴	C₂×D₄×C₆₇	1	4	yes	—
207	C₂⁴	C₂×D₄×C₆₇	1	4	yes	—
208	C₂⁴	C₂⁴×C₆₇	1	1	yes	characteristic
209	#41	—	4	16	no	—
210	C₂×Q₈	—	2	2	no	—
211	C₂×Q₈	—	4	2	no	—
212	C₂×D₄	—	2	4	no	—
213	C₂×D₄	—	2	8	no	—
214	C₂×D₄	—	2	16	no	—
215	C₂×D₄	—	2	2	no	—
216	C₂×D₄	—	4	4	no	—
217	C₂×D₄	—	4	4	no	—
218	C₂×D₄	—	4	8	no	—
219	C₂×D₄	—	4	8	no	—
220	C₂×D₄	—	4	2	no	—
221	C₂×D₄	—	8	8	no	—
222	C₂×D₄	C₂²⋊C₄×C₆₇	1	8	yes	—
223	C₂×D₄	C₂²⋊C₄×C₆₇	1	8	yes	—
224	C₂²×C₄	—	2	4	no	—
225	C₂²×C₄	—	2	4	no	—
226	C₂²×C₄	—	2	8	no	—
227	C₂²×C₄	—	4	4	no	—
228	C₂²×C₄	—	4	8	no	—
229	C₂²×C₄	—	8	8	no	—
230	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
231	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
232	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
233	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
234	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
235	C₄⋊C₄	—	2	4	no	—
236	C₂²⋊C₄	—	2	16	no	—
237	C₂²⋊C₄	—	2	4	no	—
238	C₂²⋊C₄	—	2	16	no	—
239	C₂²⋊C₄	—	4	8	no	—
240	C₂²⋊C₄	—	8	8	no	—
241	C₄²	—	2	4	no	—
242	C₂³	—	2	4	no	—
243	C₂³	—	2	4	no	—
244	C₂³	—	2	8	no	—
245	C₂³	—	2	8	no	—
246	C₂³	—	2	2	no	—
247	C₂³	—	2	4	no	—
248	C₂³	—	4	8	no	—
249	C₂³	—	4	2	no	—
250	C₂³	—	4	8	no	—
251	C₂³	—	4	8	no	—
252	C₂³	—	4	4	no	—
253	C₂³	—	8	16	no	—
254	C₂³	C₂×C₂²⋊C₄×C₆₇	1	8	yes	—
255	C₂³	C₂³×C₄×C₆₇	1	1	yes	characteristic
256	C₂³	C₂²×D₄×C₆₇	1	2	yes	—
257	C₂³	H(5,𝔽₂)×C₆₇	1	4	yes	—
258	Q₈	—	4	4	no	—
259	D₄	—	4	8	no	—
260	D₄	—	4	8	no	—
261	D₄	—	4	4	no	—
262	D₄	—	4	8	no	—
263	D₄	—	8	16	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	8	no	—
267	C₂×C₄	—	2	4	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	2	4	no	—
270	C₂×C₄	—	4	8	no	—
271	C₂×C₄	—	4	4	no	—
272	C₂×C₄	—	4	16	no	—
273	C₂×C₄	—	4	8	no	—
274	C₂×C₄	—	8	16	no	—
275	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
276	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
277	C₂²	—	2	4	no	—
278	C₂²	—	2	2	no	—
279	C₂²	—	2	8	no	—
280	C₂²	—	2	16	no	—
281	C₂²	—	4	4	no	—
282	C₂²	—	4	8	no	—
283	C₂²	—	4	8	no	—
284	C₂²	—	4	2	no	—
285	C₂²	—	4	4	no	—
286	C₂²	—	8	8	no	—
287	C₂²	—	8	8	no	—
288	C₂²	—	8	4	no	—
289	C₂²	C₂²×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
290	C₂²	#518×C₆₇	1	2	yes	—
291	C₂²	#534×C₆₇	1	4	yes	—
292	C₄	—	2	2	no	—
293	C₄	—	2	2	no	—
294	C₄	—	4	2	no	—
295	C₄	—	4	4	no	—
296	C₄	—	8	8	no	—
297	C₂	—	2	4	no	—
298	C₂	—	2	2	no	—
299	C₂	—	4	8	no	—
300	C₂	—	8	2	no	—
301	C₂	C₆₇×#2308	1	1	yes	characteristic
302	C₂	C₆₇×#2883	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₆₇	C₂²×C₂²⋊C₄
commutator	C₂³	C₂³×C₄×C₆₇
Frattini, cosocle	C₂⁴	C₂⁴×C₆₇
prime radical	C₂×#535×C₆₇	C₂
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ C₂³×D₄×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₆₇×#2308	C₂	semidirect
3. C₆₇×#1898	C₂	semidirect
4. C₆₇×#1820	C₂	semidirect
5. C₆₇×#1782	C₂	semidirect
9. C₂×#127×C₆₇	C₄	semidirect
10. C₂×#124×C₆₇	C₄	semidirect
11. C₂×#121×C₆₇	C₄	semidirect
23. C₂²×Q₈×C₆₇	C₂×C₄	semidirect
27. C₂²×D₄×C₆₇	C₂×C₄	semidirect
28. C₂²×D₄×C₆₇	C₂×C₄	semidirect
32. C₂³×C₄×C₆₇	C₂×C₄	semidirect
40. C₂×C₄⋊C₄×C₆₇	C₂×C₄	semidirect
44. C₂×C₂²⋊C₄×C₆₇	C₂×C₄	semidirect
47. C₂×C₄²×C₆₇	C₂×C₄	semidirect
145. #21750	C₆₇	direct (N × H)
153. #2308	C₂×C₆₇	semidirect
154. #1898	C₂×C₆₇	semidirect
155. #1820	C₂×C₆₇	semidirect
156. #1782	C₂×C₆₇	semidirect
157. C₆₇	#21750	direct (N × H)
161. C₂×#127	C₄×C₆₇	semidirect
162. C₂×#124	C₄×C₆₇	semidirect
163. C₂×#121	C₄×C₆₇	semidirect
175. C₂²×Q₈	C₂×C₄×C₆₇	semidirect
179. C₂²×D₄	C₂×C₄×C₆₇	semidirect
180. C₂²×D₄	C₂×C₄×C₆₇	semidirect
184. C₂³×C₄	C₂×C₄×C₆₇	semidirect
192. C₂×C₄⋊C₄	C₂×C₄×C₆₇	semidirect
196. C₂×C₂²⋊C₄	C₂×C₄×C₆₇	semidirect
199. C₂×C₄²	C₂×C₄×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#535×C₆₇	2
C₆₇×#2308	2
C₆₇×#1898	2
C₆₇×#1820	2
C₆₇×#1782	2
#21750	67

Subgroups — C67×#21750 #30701001

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#21750 #30701001