Subgroups — C₆₇×#21801 #30701052

Order 17152.

1386 conjugacy classes of subgroups.

The subgroup lattice has 292 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×#2921	C₂	1	2	yes	maximal, complemented
3	C₆₇×#1967	C₂	1	4	yes	maximal, complemented
4	C₆₇×#1965	C₂	1	4	yes	maximal, complemented
5	C₆₇×#1785	C₂	1	2	yes	maximal, complemented
6	C₆₇×#1783	C₂	1	2	yes	maximal, complemented
7	C₂²×#41×C₆₇	C₂²	1	1	yes	characteristic
8	C₂³×D₄×C₆₇	C₂²	1	1	yes	characteristic
9	#535×C₆₇	—	2	8	no	—
10	C₂×#127×C₆₇	C₂²	1	2	yes	—
11	C₂×#124×C₆₇	C₂²	1	2	yes	—
12	C₂×#121×C₆₇	C₄	1	4	yes	—
13	C₂×#120×C₆₇	C₄	1	4	yes	—
14	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
15	#433×C₆₇	—	2	8	no	—
16	#423×C₆₇	C₂²	1	4	yes	complemented
17	C₂×#101×C₆₇	C₂²	1	4	yes	complemented
18	C₂×#100×C₆₇	C₂²	1	8	yes	—
19	C₂×#100×C₆₇	C₂²	1	4	yes	complemented
20	#407×C₆₇	C₂²	1	4	yes	—
21	C₄×M₁₆×C₆₇	C₂²	1	4	yes	complemented
22	C₂⁵×C₆₇	C₂×C₄	1	2	yes	—
23	C₂×#41×C₆₇	—	2	4	no	—
24	C₂×#41×C₆₇	—	4	4	no	—
25	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
26	C₂²×D₄×C₆₇	—	2	4	no	—
27	C₂²×D₄×C₆₇	—	4	4	no	—
28	C₂²×D₄×C₆₇	C₂×C₄	1	2	yes	—
29	C₂²×D₄×C₆₇	C₂×C₄	1	2	yes	—
30	C₂²×D₄×C₆₇	C₂×C₄	1	8	yes	—
31	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
32	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
33	C₂³×C₄×C₆₇	C₂×C₄	1	2	yes	—
34	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
35	C₂×M₁₆×C₆₇	C₂³	1	8	yes	—
36	#127×C₆₇	—	2	4	no	—
37	#127×C₆₇	—	2	4	no	—
38	#124×C₆₇	—	2	4	no	—
39	#124×C₆₇	—	2	4	no	—
40	#121×C₆₇	—	2	16	no	—
41	#120×C₆₇	—	2	16	no	—
42	#117×C₆₇	D₄	1	8	yes	—
43	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
44	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	8	yes	—
45	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
46	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
47	C₄⋊C₈×C₆₇	—	2	8	no	—
48	#101×C₆₇	—	4	4	no	—
49	#100×C₆₇	—	4	8	no	—
50	#100×C₆₇	—	4	4	no	—
51	C₂²⋊C₈×C₆₇	—	2	8	no	—
52	#97×C₆₇	—	2	4	no	—
53	C₄×C₈×C₆₇	—	2	4	no	—
54	C₂⁴×C₆₇	—	2	4	no	—
55	C₂⁴×C₆₇	—	4	4	no	—
56	C₂⁴×C₆₇	—	8	4	no	—
57	C₂⁴×C₆₇	C₂²×C₄	1	4	yes	—
58	C₂⁴×C₆₇	C₂²×C₄	1	2	yes	—
59	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
60	#41×C₆₇	—	4	16	no	—
61	C₂×Q₈×C₆₇	—	2	2	no	—
62	C₂×Q₈×C₆₇	—	2	4	no	—
63	C₂×D₄×C₆₇	—	2	4	no	—
64	C₂×D₄×C₆₇	—	2	4	no	—
65	C₂×D₄×C₆₇	—	2	16	no	—
66	C₂×D₄×C₆₇	—	2	2	no	—
67	C₂×D₄×C₆₇	—	2	2	no	—
68	C₂×D₄×C₆₇	—	2	2	no	—
69	C₂×D₄×C₆₇	—	2	8	no	—
70	C₂×D₄×C₆₇	—	2	4	no	—
71	C₂×D₄×C₆₇	—	4	4	no	—
72	C₂×D₄×C₆₇	—	4	4	no	—
73	C₂×D₄×C₆₇	—	4	16	no	—
74	C₂×D₄×C₆₇	—	8	8	no	—
75	C₂²×C₄×C₆₇	—	2	4	no	—
76	C₂²×C₄×C₆₇	—	2	4	no	—
77	C₂²×C₄×C₆₇	—	4	4	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	4	yes	—
81	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
82	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
83	M₁₆×C₆₇	—	2	8	no	—
84	M₁₆×C₆₇	—	2	8	no	—
85	C₂×C₈×C₆₇	—	2	8	no	—
86	C₄⋊C₄×C₆₇	C₂²⋊C₄	1	8	yes	—
87	C₂²⋊C₄×C₆₇	—	2	16	no	—
88	C₂²⋊C₄×C₆₇	C₂²⋊C₄	1	8	yes	—
89	C₄²×C₆₇	C₂×D₄	1	4	yes	—
90	C₄²×C₆₇	C₂×D₄	1	4	yes	—
91	C₂³×C₆₇	—	2	2	no	—
92	C₂³×C₆₇	—	2	4	no	—
93	C₂³×C₆₇	—	2	2	no	—
94	C₂³×C₆₇	—	2	4	no	—
95	C₂³×C₆₇	—	2	8	no	—
96	C₂³×C₆₇	—	4	2	no	—
97	C₂³×C₆₇	—	4	8	no	—
98	C₂³×C₆₇	—	4	4	no	—
99	C₂³×C₆₇	—	4	8	no	—
100	C₂³×C₆₇	—	8	8	no	—
101	C₂³×C₆₇	—	8	16	no	—
102	C₂³×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
103	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
104	Q₈×C₆₇	—	4	4	no	—
105	D₄×C₆₇	—	4	8	no	—
106	D₄×C₆₇	—	4	8	no	—
107	D₄×C₆₇	—	4	4	no	—
108	D₄×C₆₇	—	4	4	no	—
109	D₄×C₆₇	—	4	4	no	—
110	D₄×C₆₇	—	8	16	no	—
111	C₂×C₄×C₆₇	—	2	2	no	—
112	C₂×C₄×C₆₇	—	2	8	no	—
113	C₂×C₄×C₆₇	—	2	2	no	—
114	C₂×C₄×C₆₇	—	2	4	no	—
115	C₂×C₄×C₆₇	—	4	4	no	—
116	C₂×C₄×C₆₇	—	4	8	no	—
117	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	8	yes	—
118	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
119	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
120	C₂×C₄×C₆₇	H(5,𝔽₂)	1	4	yes	—
121	C₈×C₆₇	—	4	8	no	—
122	C₂²×C₆₇	—	2	2	no	—
123	C₂²×C₆₇	—	2	2	no	—
124	C₂²×C₆₇	—	2	4	no	—
125	C₂²×C₆₇	—	4	4	no	—
126	C₂²×C₆₇	—	4	2	no	—
127	C₂²×C₆₇	—	4	4	no	—
128	C₂²×C₆₇	—	4	4	no	—
129	C₂²×C₆₇	—	4	8	no	—
130	C₂²×C₆₇	—	8	4	no	—
131	C₂²×C₆₇	—	8	8	no	—
132	C₂²×C₆₇	—	8	16	no	—
133	C₂²×C₆₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
134	C₂²×C₆₇	#518	1	2	yes	—
135	C₄×C₆₇	—	2	2	no	—
136	C₄×C₆₇	—	2	8	no	—
137	C₄×C₆₇	—	4	2	no	—
138	C₄×C₆₇	#535	1	4	yes	—
139	#21801	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
140	C₂×C₆₇	—	2	2	no	—
141	C₂×C₆₇	—	4	2	no	—
142	C₂×C₆₇	—	8	2	no	—
143	C₂×C₆₇	—	8	4	no	—
144	C₂×C₆₇	#2309	1	1	yes	characteristic
145	C₂×C₆₇	#2888	1	2	yes	—
146	C₂×#535	C₂×C₆₇	1	1	yes	characteristic
147	#2921	C₂×C₆₇	1	2	yes	complemented
148	#1967	C₂×C₆₇	1	4	yes	complemented
149	#1965	C₂×C₆₇	1	4	yes	complemented
150	#1785	C₂×C₆₇	1	2	yes	complemented
151	#1783	C₂×C₆₇	1	2	yes	complemented
152	C₆₇	#21801	1	1	yes	characteristic, direct product factor, Sylow
153	C₂²×#41	C₂²×C₆₇	1	1	yes	characteristic
154	C₂³×D₄	C₂²×C₆₇	1	1	yes	characteristic
155	#535	—	2	8	no	—
156	C₂×#127	C₂²×C₆₇	1	2	yes	—
157	C₂×#124	C₂²×C₆₇	1	2	yes	—
158	C₂×#121	C₄×C₆₇	1	4	yes	—
159	C₂×#120	C₄×C₆₇	1	4	yes	—
160	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
161	#433	—	2	8	no	—
162	#423	C₂²×C₆₇	1	4	yes	complemented
163	C₂×#101	C₂²×C₆₇	1	4	yes	complemented
164	C₂×#100	C₂²×C₆₇	1	8	yes	—
165	C₂×#100	C₂²×C₆₇	1	4	yes	complemented
166	#407	C₂²×C₆₇	1	4	yes	—
167	C₄×M₁₆	C₂²×C₆₇	1	4	yes	complemented
168	C₂⁵	C₂×C₄×C₆₇	1	2	yes	—
169	C₂×#41	—	2	4	no	—
170	C₂×#41	—	4	4	no	—
171	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
172	C₂²×D₄	—	2	4	no	—
173	C₂²×D₄	—	4	4	no	—
174	C₂²×D₄	C₂×C₄×C₆₇	1	2	yes	—
175	C₂²×D₄	C₂×C₄×C₆₇	1	2	yes	—
176	C₂²×D₄	C₂×C₄×C₆₇	1	8	yes	—
177	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
178	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
179	C₂³×C₄	C₂×C₄×C₆₇	1	2	yes	—
180	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
181	C₂×M₁₆	C₂³×C₆₇	1	8	yes	—
182	#127	—	2	4	no	—
183	#127	—	2	4	no	—
184	#124	—	2	4	no	—
185	#124	—	2	4	no	—
186	#121	—	2	16	no	—
187	#120	—	2	16	no	—
188	#117	D₄×C₆₇	1	8	yes	—
189	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
190	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	8	yes	—
191	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
192	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
193	C₄⋊C₈	—	2	8	no	—
194	#101	—	4	4	no	—
195	#100	—	4	8	no	—
196	#100	—	4	4	no	—
197	C₂²⋊C₈	—	2	8	no	—
198	#97	—	2	4	no	—
199	C₄×C₈	—	2	4	no	—
200	C₂⁴	—	2	4	no	—
201	C₂⁴	—	4	4	no	—
202	C₂⁴	—	8	4	no	—
203	C₂⁴	C₂²×C₄×C₆₇	1	4	yes	—
204	C₂⁴	C₂²×C₄×C₆₇	1	2	yes	—
205	C₂⁴	C₂²×C₄×C₆₇	1	1	yes	characteristic
206	#41	—	4	16	no	—
207	C₂×Q₈	—	2	2	no	—
208	C₂×Q₈	—	2	4	no	—
209	C₂×D₄	—	2	4	no	—
210	C₂×D₄	—	2	4	no	—
211	C₂×D₄	—	2	16	no	—
212	C₂×D₄	—	2	2	no	—
213	C₂×D₄	—	2	2	no	—
214	C₂×D₄	—	2	2	no	—
215	C₂×D₄	—	2	8	no	—
216	C₂×D₄	—	2	4	no	—
217	C₂×D₄	—	4	4	no	—
218	C₂×D₄	—	4	4	no	—
219	C₂×D₄	—	4	16	no	—
220	C₂×D₄	—	8	8	no	—
221	C₂²×C₄	—	2	4	no	—
222	C₂²×C₄	—	2	4	no	—
223	C₂²×C₄	—	4	4	no	—
224	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
225	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
226	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
227	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
228	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
229	M₁₆	—	2	8	no	—
230	M₁₆	—	2	8	no	—
231	C₂×C₈	—	2	8	no	—
232	C₄⋊C₄	C₂²⋊C₄×C₆₇	1	8	yes	—
233	C₂²⋊C₄	—	2	16	no	—
234	C₂²⋊C₄	C₂²⋊C₄×C₆₇	1	8	yes	—
235	C₄²	C₂×D₄×C₆₇	1	4	yes	—
236	C₄²	C₂×D₄×C₆₇	1	4	yes	—
237	C₂³	—	2	2	no	—
238	C₂³	—	2	4	no	—
239	C₂³	—	2	2	no	—
240	C₂³	—	2	4	no	—
241	C₂³	—	2	8	no	—
242	C₂³	—	4	2	no	—
243	C₂³	—	4	8	no	—
244	C₂³	—	4	4	no	—
245	C₂³	—	4	8	no	—
246	C₂³	—	8	8	no	—
247	C₂³	—	8	16	no	—
248	C₂³	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
249	C₂³	C₂³×C₄×C₆₇	1	1	yes	characteristic
250	Q₈	—	4	4	no	—
251	D₄	—	4	8	no	—
252	D₄	—	4	8	no	—
253	D₄	—	4	4	no	—
254	D₄	—	4	4	no	—
255	D₄	—	4	4	no	—
256	D₄	—	8	16	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	2	8	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	4	no	—
261	C₂×C₄	—	4	4	no	—
262	C₂×C₄	—	4	8	no	—
263	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	8	yes	—
264	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
265	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
266	C₂×C₄	H(5,𝔽₂)×C₆₇	1	4	yes	—
267	C₈	—	4	8	no	—
268	C₂²	—	2	2	no	—
269	C₂²	—	2	2	no	—
270	C₂²	—	2	4	no	—
271	C₂²	—	4	4	no	—
272	C₂²	—	4	2	no	—
273	C₂²	—	4	4	no	—
274	C₂²	—	4	4	no	—
275	C₂²	—	4	8	no	—
276	C₂²	—	8	8	no	—
277	C₂²	—	8	4	no	—
278	C₂²	—	8	16	no	—
279	C₂²	C₂²×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
280	C₂²	#518×C₆₇	1	2	yes	—
281	C₄	—	2	2	no	—
282	C₄	—	2	8	no	—
283	C₄	—	4	2	no	—
284	C₄	#535×C₆₇	1	4	yes	—
285	C₂	—	2	2	no	—
286	C₂	—	4	2	no	—
287	C₂	—	8	4	no	—
288	C₂	—	8	2	no	—
289	C₂	C₆₇×#2309	1	1	yes	characteristic
290	C₂	C₆₇×#2888	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	C₂×#535×C₆₇	C₂
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
2. C₆₇×#2921	C₂	semidirect
3. C₆₇×#1967	C₂	semidirect
4. C₆₇×#1965	C₂	semidirect
5. C₆₇×#1785	C₂	semidirect
6. C₆₇×#1783	C₂	semidirect
16. #423×C₆₇	C₂²	semidirect
17. C₂×#101×C₆₇	C₂²	semidirect
19. C₂×#100×C₆₇	C₂²	semidirect
21. C₄×M₁₆×C₆₇	C₂²	semidirect
139. #21801	C₆₇	direct (N × H)
147. #2921	C₂×C₆₇	semidirect
148. #1967	C₂×C₆₇	semidirect
149. #1965	C₂×C₆₇	semidirect
150. #1785	C₂×C₆₇	semidirect
151. #1783	C₂×C₆₇	semidirect
152. C₆₇	#21801	direct (N × H)
162. #423	C₂²×C₆₇	semidirect
163. C₂×#101	C₂²×C₆₇	semidirect
165. C₂×#100	C₂²×C₆₇	semidirect
167. C₄×M₁₆	C₂²×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×#2921	2
C₆₇×#1967	2
C₆₇×#1965	2
C₆₇×#1785	2
C₆₇×#1783	2
#21801	67

Subgroups — C67×#21801 #30701052

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#21801 #30701052