Subgroups — C₆₇×#21883 #30701134

Order 17152.

954 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#541×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#2921	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#2905	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#1998	C₂	1	2	yes	maximal, complemented
5	C₆₇×#1966	C₂	1	2	yes	maximal, complemented
6	C₆₇×#1901	C₂	1	4	yes	maximal
7	C₆₇×#1801	C₂	1	4	yes	maximal, complemented
8	C₂²×M₁₆×C₆₇	C₂²	1	2	yes	—
9	#541×C₆₇	—	2	4	no	—
10	#541×C₆₇	—	2	4	no	—
11	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
12	C₂×#126×C₆₇	C₄	1	4	yes	—
13	C₂×#125×C₆₇	C₄	1	4	yes	—
14	C₂×#125×C₆₇	C₂²	1	1	yes	characteristic
15	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	2	yes	—
16	C₂×C₄×Q₈×C₆₇	C₂²	1	2	yes	—
17	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
18	#433×C₆₇	—	2	4	no	—
19	#424×C₆₇	—	2	4	no	—
20	#423×C₆₇	C₂²	1	2	yes	—
21	#423×C₆₇	C₂²	1	2	yes	—
22	#423×C₆₇	C₂²	1	2	yes	—
23	C₂×#101×C₆₇	C₂²	1	8	yes	—
24	#407×C₆₇	C₂²	1	2	yes	—
25	C₄×M₁₆×C₆₇	C₂²	1	2	yes	—
26	#343×C₆₇	C₂²	1	8	yes	—
27	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
28	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
29	C₂×M₁₆×C₆₇	—	2	2	no	—
30	C₂×M₁₆×C₆₇	—	4	4	no	—
31	C₂×M₁₆×C₆₇	C₂³	1	4	yes	—
32	C₂×M₁₆×C₆₇	C₂³	1	4	yes	—
33	C₂²×C₈×C₆₇	—	2	2	no	—
34	C₄⋊Q₈×C₆₇	—	2	2	no	—
35	C₄⋊Q₈×C₆₇	—	2	2	no	—
36	#126×C₆₇	—	2	16	no	—
37	#125×C₆₇	—	2	16	no	—
38	#125×C₆₇	—	2	2	no	—
39	#125×C₆₇	—	2	2	no	—
40	C₂²⋊Q₈×C₆₇	—	2	4	no	—
41	C₂²⋊Q₈×C₆₇	—	2	4	no	—
42	C₄×Q₈×C₆₇	—	2	4	no	—
43	C₄×Q₈×C₆₇	—	2	4	no	—
44	#117×C₆₇	D₄	1	4	yes	—
45	#117×C₆₇	D₄	1	4	yes	—
46	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	4	yes	—
47	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	4	yes	—
48	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	4	yes	—
49	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	4	yes	—
50	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
51	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
52	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	4	yes	—
53	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
54	C₂×C₄²×C₆₇	C₂×C₄	1	4	yes	—
55	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
56	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
57	C₄⋊C₈×C₆₇	—	2	4	no	—
58	C₄⋊C₈×C₆₇	—	2	4	no	—
59	C₄⋊C₈×C₆₇	—	2	4	no	—
60	#101×C₆₇	—	4	8	no	—
61	C₂²⋊C₈×C₆₇	—	2	4	no	—
62	#97×C₆₇	—	2	2	no	—
63	C₄×C₈×C₆₇	—	2	2	no	—
64	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
65	C₂×Q₈×C₆₇	—	2	2	no	—
66	C₂×Q₈×C₆₇	—	2	2	no	—
67	C₂×Q₈×C₆₇	—	2	4	no	—
68	C₂×Q₈×C₆₇	—	2	4	no	—
69	C₂²×C₄×C₆₇	—	2	2	no	—
70	C₂²×C₄×C₆₇	—	2	2	no	—
71	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
72	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
73	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
74	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
75	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
76	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
77	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
78	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
79	M₁₆×C₆₇	—	2	4	no	—
80	M₁₆×C₆₇	—	2	8	no	—
81	M₁₆×C₆₇	—	2	4	no	—
82	M₁₆×C₆₇	—	4	4	no	—
83	C₂×C₈×C₆₇	—	2	4	no	—
84	C₂×C₈×C₆₇	—	2	4	no	—
85	C₂×C₈×C₆₇	—	4	4	no	—
86	C₄⋊C₄×C₆₇	—	2	8	no	—
87	C₄⋊C₄×C₆₇	—	2	8	no	—
88	C₄⋊C₄×C₆₇	—	2	8	no	—
89	C₄⋊C₄×C₆₇	—	2	8	no	—
90	C₄⋊C₄×C₆₇	—	2	2	no	—
91	C₄⋊C₄×C₆₇	—	2	2	no	—
92	C₄⋊C₄×C₆₇	C₂²⋊C₄	1	8	yes	—
93	C₂²⋊C₄×C₆₇	—	2	8	no	—
94	C₂²⋊C₄×C₆₇	C₂²⋊C₄	1	8	yes	—
95	C₄²×C₆₇	—	2	8	no	—
96	C₄²×C₆₇	C₂×D₄	1	2	yes	—
97	C₄²×C₆₇	C₂×D₄	1	2	yes	—
98	C₄²×C₆₇	C₂×D₄	1	2	yes	—
99	C₄²×C₆₇	C₂×D₄	1	2	yes	—
100	C₂³×C₆₇	—	2	2	no	—
101	C₂³×C₆₇	—	4	2	no	—
102	C₂³×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
103	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
104	Q₈×C₆₇	—	4	2	no	—
105	Q₈×C₆₇	—	4	2	no	—
106	Q₈×C₆₇	—	4	4	no	—
107	C₂×C₄×C₆₇	—	2	2	no	—
108	C₂×C₄×C₆₇	—	2	2	no	—
109	C₂×C₄×C₆₇	—	2	2	no	—
110	C₂×C₄×C₆₇	—	2	4	no	—
111	C₂×C₄×C₆₇	—	2	4	no	—
112	C₂×C₄×C₆₇	—	2	4	no	—
113	C₂×C₄×C₆₇	—	2	8	no	—
114	C₂×C₄×C₆₇	—	4	2	no	—
115	C₂×C₄×C₆₇	—	4	2	no	—
116	C₂×C₄×C₆₇	—	4	4	no	—
117	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	4	yes	—
118	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	4	yes	—
119	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
120	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
121	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
122	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
123	C₂×C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
124	C₈×C₆₇	—	4	4	no	—
125	C₈×C₆₇	—	4	4	no	—
126	C₂²×C₆₇	—	2	2	no	—
127	C₂²×C₆₇	—	2	2	no	—
128	C₂²×C₆₇	—	2	4	no	—
129	C₂²×C₆₇	—	8	2	no	—
130	C₂²×C₆₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
131	C₂²×C₆₇	#520	1	2	yes	—
132	C₄×C₆₇	—	2	2	no	—
133	C₄×C₆₇	—	2	4	no	—
134	C₄×C₆₇	—	2	4	no	—
135	C₄×C₆₇	—	4	4	no	—
136	C₄×C₆₇	—	4	4	no	—
137	C₄×C₆₇	#535	1	1	yes	characteristic
138	C₄×C₆₇	#536	1	1	yes	characteristic
139	C₄×C₆₇	#537	1	2	yes	—
140	#21883	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
141	C₂×C₆₇	—	2	2	no	—
142	C₂×C₆₇	—	4	2	no	—
143	C₂×C₆₇	#2311	1	1	yes	characteristic
144	C₂×C₆₇	#2889	1	2	yes	—
145	C₂×#541	C₂×C₆₇	1	1	yes	characteristic
146	#2921	C₂×C₆₇	1	1	yes	characteristic
147	#2905	C₂×C₆₇	1	1	yes	characteristic
148	#1998	C₂×C₆₇	1	2	yes	complemented
149	#1966	C₂×C₆₇	1	2	yes	complemented
150	#1901	C₂×C₆₇	1	4	yes	—
151	#1801	C₂×C₆₇	1	4	yes	complemented
152	C₆₇	#21883	1	1	yes	characteristic, direct product factor, Sylow
153	C₂²×M₁₆	C₂²×C₆₇	1	2	yes	—
154	#541	—	2	4	no	—
155	#541	—	2	4	no	—
156	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
157	C₂×#126	C₄×C₆₇	1	4	yes	—
158	C₂×#125	C₄×C₆₇	1	4	yes	—
159	C₂×#125	C₂²×C₆₇	1	1	yes	characteristic
160	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	2	yes	—
161	C₂×C₄×Q₈	C₂²×C₆₇	1	2	yes	—
162	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
163	#433	—	2	4	no	—
164	#424	—	2	4	no	—
165	#423	C₂²×C₆₇	1	2	yes	—
166	#423	C₂²×C₆₇	1	2	yes	—
167	#423	C₂²×C₆₇	1	2	yes	—
168	C₂×#101	C₂²×C₆₇	1	8	yes	—
169	#407	C₂²×C₆₇	1	2	yes	—
170	C₄×M₁₆	C₂²×C₆₇	1	2	yes	—
171	#343	C₂²×C₆₇	1	8	yes	—
172	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
173	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
174	C₂×M₁₆	—	2	2	no	—
175	C₂×M₁₆	—	4	4	no	—
176	C₂×M₁₆	C₂³×C₆₇	1	4	yes	—
177	C₂×M₁₆	C₂³×C₆₇	1	4	yes	—
178	C₂²×C₈	—	2	2	no	—
179	C₄⋊Q₈	—	2	2	no	—
180	C₄⋊Q₈	—	2	2	no	—
181	#126	—	2	16	no	—
182	#125	—	2	16	no	—
183	#125	—	2	2	no	—
184	#125	—	2	2	no	—
185	C₂²⋊Q₈	—	2	4	no	—
186	C₂²⋊Q₈	—	2	4	no	—
187	C₄×Q₈	—	2	4	no	—
188	C₄×Q₈	—	2	4	no	—
189	#117	D₄×C₆₇	1	4	yes	—
190	#117	D₄×C₆₇	1	4	yes	—
191	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	4	yes	—
192	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	4	yes	—
193	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	4	yes	—
194	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	4	yes	—
195	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
196	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
197	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	4	yes	—
198	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
199	C₂×C₄²	C₂×C₄×C₆₇	1	4	yes	—
200	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
201	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
202	C₄⋊C₈	—	2	4	no	—
203	C₄⋊C₈	—	2	4	no	—
204	C₄⋊C₈	—	2	4	no	—
205	#101	—	4	8	no	—
206	C₂²⋊C₈	—	2	4	no	—
207	#97	—	2	2	no	—
208	C₄×C₈	—	2	2	no	—
209	C₂⁴	C₂²×C₄×C₆₇	1	1	yes	characteristic
210	C₂×Q₈	—	2	2	no	—
211	C₂×Q₈	—	2	2	no	—
212	C₂×Q₈	—	2	4	no	—
213	C₂×Q₈	—	2	4	no	—
214	C₂²×C₄	—	2	2	no	—
215	C₂²×C₄	—	2	2	no	—
216	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
217	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
218	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
219	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
220	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
221	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
222	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
223	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
224	M₁₆	—	2	4	no	—
225	M₁₆	—	2	8	no	—
226	M₁₆	—	2	4	no	—
227	M₁₆	—	4	4	no	—
228	C₂×C₈	—	2	4	no	—
229	C₂×C₈	—	2	4	no	—
230	C₂×C₈	—	4	4	no	—
231	C₄⋊C₄	—	2	8	no	—
232	C₄⋊C₄	—	2	8	no	—
233	C₄⋊C₄	—	2	8	no	—
234	C₄⋊C₄	—	2	8	no	—
235	C₄⋊C₄	—	2	2	no	—
236	C₄⋊C₄	—	2	2	no	—
237	C₄⋊C₄	C₂²⋊C₄×C₆₇	1	8	yes	—
238	C₂²⋊C₄	—	2	8	no	—
239	C₂²⋊C₄	C₂²⋊C₄×C₆₇	1	8	yes	—
240	C₄²	—	2	8	no	—
241	C₄²	C₂×D₄×C₆₇	1	2	yes	—
242	C₄²	C₂×D₄×C₆₇	1	2	yes	—
243	C₄²	C₂×D₄×C₆₇	1	2	yes	—
244	C₄²	C₂×D₄×C₆₇	1	2	yes	—
245	C₂³	—	2	2	no	—
246	C₂³	—	4	2	no	—
247	C₂³	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
248	C₂³	C₂³×C₄×C₆₇	1	1	yes	characteristic
249	Q₈	—	4	2	no	—
250	Q₈	—	4	2	no	—
251	Q₈	—	4	4	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	2	4	no	—
256	C₂×C₄	—	2	4	no	—
257	C₂×C₄	—	2	4	no	—
258	C₂×C₄	—	2	8	no	—
259	C₂×C₄	—	4	2	no	—
260	C₂×C₄	—	4	2	no	—
261	C₂×C₄	—	4	4	no	—
262	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	4	yes	—
263	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	4	yes	—
264	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
265	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
266	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
267	C₂×C₄	H(5,𝔽₂)×C₆₇	1	2	yes	—
268	C₂×C₄	2₋¹⁺⁴×C₆₇	1	2	yes	—
269	C₈	—	4	4	no	—
270	C₈	—	4	4	no	—
271	C₂²	—	2	2	no	—
272	C₂²	—	2	2	no	—
273	C₂²	—	2	4	no	—
274	C₂²	—	8	2	no	—
275	C₂²	C₂²×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
276	C₂²	#520×C₆₇	1	2	yes	—
277	C₄	—	2	2	no	—
278	C₄	—	2	4	no	—
279	C₄	—	2	4	no	—
280	C₄	—	4	4	no	—
281	C₄	—	4	4	no	—
282	C₄	#535×C₆₇	1	1	yes	characteristic
283	C₄	#536×C₆₇	1	1	yes	characteristic
284	C₄	#537×C₆₇	1	2	yes	—
285	C₂	—	2	2	no	—
286	C₂	—	4	2	no	—
287	C₂	C₆₇×#2311	1	1	yes	characteristic
288	C₂	C₆₇×#2889	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₆₇	C₂²×C₂²⋊C₄
commutator	C₂³	C₂³×C₄×C₆₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₂⁴×C₆₇	C₂²×C₄
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ C₂×#117×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₆₇×#1998	C₂	semidirect
5. C₆₇×#1966	C₂	semidirect
7. C₆₇×#1801	C₂	semidirect
140. #21883	C₆₇	direct (N × H)
148. #1998	C₂×C₆₇	semidirect
149. #1966	C₂×C₆₇	semidirect
151. #1801	C₂×C₆₇	semidirect
152. C₆₇	#21883	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#541×C₆₇	2
C₆₇×#2921	2
C₆₇×#2905	2
C₆₇×#1998	2
C₆₇×#1966	2
C₆₇×#1901	2
C₆₇×#1801	2
#21883	67

Subgroups — C67×#21883 #30701134

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#21883 #30701134