Subgroups — C₆₇×#21751 #30701002

Order 17152.

1294 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#536×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#2310	C₂	1	2	yes	maximal, complemented
3	C₆₇×#1899	C₂	1	8	yes	maximal, complemented
4	C₆₇×#1820	C₂	1	2	yes	maximal, complemented
5	C₆₇×#1782	C₂	1	2	yes	maximal
6	C₂²×#41×C₆₇	C₂²	1	1	yes	characteristic
7	C₂³×Q₈×C₆₇	C₂²	1	1	yes	characteristic
8	#536×C₆₇	—	2	8	no	—
9	C₂×C₄⋊Q₈×C₆₇	C₄	1	2	yes	complemented
10	C₂×#124×C₆₇	C₄	1	2	yes	complemented
11	C₂×#123×C₆₇	C₂²	1	4	yes	—
12	C₂×C₂²⋊Q₈×C₆₇	C₄	1	4	yes	complemented
13	C₂×#117×C₆₇	C₂²	1	4	yes	—
14	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
15	C₂×#99×C₆₇	C₂²	1	8	yes	—
16	#391×C₆₇	—	2	8	no	—
17	#381×C₆₇	C₂²	1	4	yes	—
18	#381×C₆₇	C₂²	1	4	yes	—
19	#342×C₆₇	C₂²	1	8	yes	—
20	C₂×#41×C₆₇	—	2	4	no	—
21	C₂×#41×C₆₇	—	4	4	no	—
22	C₂²×Q₈×C₆₇	—	4	4	no	—
23	C₂²×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
24	C₂²×Q₈×C₆₇	C₂×C₄	1	2	yes	—
25	C₂²×Q₈×C₆₇	C₂×C₄	1	2	yes	—
26	C₂²×Q₈×C₆₇	D₄	1	8	yes	—
27	C₂²×D₄×C₆₇	C₂×C₄	1	1	yes	characteristic
28	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
29	C₂³×C₄×C₆₇	C₂×C₄	1	2	yes	—
30	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
31	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
32	C₄⋊Q₈×C₆₇	—	2	8	no	—
33	#124×C₆₇	—	2	8	no	—
34	#123×C₆₇	—	2	8	no	—
35	#123×C₆₇	—	2	8	no	—
36	C₂²⋊Q₈×C₆₇	—	2	16	no	—
37	#117×C₆₇	—	2	4	no	—
38	#117×C₆₇	—	4	8	no	—
39	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
40	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
41	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	8	yes	—
42	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	8	yes	—
43	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
44	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
45	C₂×C₄²×C₆₇	—	2	4	no	—
46	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
47	#99×C₆₇	—	4	8	no	—
48	#95×C₆₇	—	2	8	no	—
49	#95×C₆₇	—	2	8	no	—
50	C₂⁴×C₆₇	C₂²×C₄	1	2	yes	—
51	C₂⁴×C₆₇	C₂⁴	1	1	yes	characteristic
52	#41×C₆₇	—	4	16	no	—
53	C₂×Q₈×C₆₇	—	2	2	no	—
54	C₂×Q₈×C₆₇	—	4	4	no	—
55	C₂×Q₈×C₆₇	—	4	4	no	—
56	C₂×Q₈×C₆₇	—	4	8	no	—
57	C₂×Q₈×C₆₇	—	4	8	no	—
58	C₂×Q₈×C₆₇	—	4	2	no	—
59	C₂×Q₈×C₆₇	C₂²⋊C₄	1	8	yes	—
60	C₂×Q₈×C₆₇	C₂²⋊C₄	1	8	yes	—
61	C₂×D₄×C₆₇	—	2	4	no	—
62	C₂×D₄×C₆₇	—	2	8	no	—
63	C₂×D₄×C₆₇	—	2	2	no	—
64	C₂×D₄×C₆₇	—	4	2	no	—
65	C₂²×C₄×C₆₇	—	2	8	no	—
66	C₂²×C₄×C₆₇	—	2	4	no	—
67	C₂²×C₄×C₆₇	—	2	4	no	—
68	C₂²×C₄×C₆₇	—	2	8	no	—
69	C₂²×C₄×C₆₇	—	4	4	no	—
70	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
71	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
72	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
73	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
74	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
75	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
76	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
77	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
78	C₄⋊C₄×C₆₇	—	2	4	no	—
79	C₄⋊C₄×C₆₇	—	2	16	no	—
80	C₄⋊C₄×C₆₇	—	4	8	no	—
81	C₂²⋊C₄×C₆₇	—	2	4	no	—
82	C₂²⋊C₄×C₆₇	—	2	16	no	—
83	C₂²⋊C₄×C₆₇	—	2	16	no	—
84	C₄²×C₆₇	—	2	4	no	—
85	C₄²×C₆₇	—	4	8	no	—
86	C₂³×C₆₇	—	2	2	no	—
87	C₂³×C₆₇	—	2	4	no	—
88	C₂³×C₆₇	—	4	2	no	—
89	C₂³×C₆₇	—	4	4	no	—
90	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
91	C₂³×C₆₇	C₂²×D₄	1	2	yes	—
92	C₂³×C₆₇	H(5,𝔽₂)	1	4	yes	—
93	Q₈×C₆₇	—	4	8	no	—
94	Q₈×C₆₇	—	4	8	no	—
95	Q₈×C₆₇	—	4	4	no	—
96	D₄×C₆₇	—	4	4	no	—
97	D₄×C₆₇	—	4	8	no	—
98	C₂×C₄×C₆₇	—	2	2	no	—
99	C₂×C₄×C₆₇	—	2	2	no	—
100	C₂×C₄×C₆₇	—	2	8	no	—
101	C₂×C₄×C₆₇	—	2	4	no	—
102	C₂×C₄×C₆₇	—	2	4	no	—
103	C₂×C₄×C₆₇	—	2	8	no	—
104	C₂×C₄×C₆₇	—	2	8	no	—
105	C₂×C₄×C₆₇	—	2	4	no	—
106	C₂×C₄×C₆₇	—	2	2	no	—
107	C₂×C₄×C₆₇	—	2	4	no	—
108	C₂×C₄×C₆₇	—	4	8	no	—
109	C₂×C₄×C₆₇	—	4	4	no	—
110	C₂×C₄×C₆₇	—	4	16	no	—
111	C₂×C₄×C₆₇	—	4	8	no	—
112	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
113	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	8	yes	—
114	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
115	C₂²×C₆₇	—	2	4	no	—
116	C₂²×C₆₇	—	2	2	no	—
117	C₂²×C₆₇	—	4	4	no	—
118	C₂²×C₆₇	—	4	2	no	—
119	C₂²×C₆₇	—	4	4	no	—
120	C₂²×C₆₇	—	8	4	no	—
121	C₂²×C₆₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
122	C₂²×C₆₇	#518	1	2	yes	—
123	C₂²×C₆₇	#534	1	4	yes	—
124	C₄×C₆₇	—	2	2	no	—
125	C₄×C₆₇	—	2	2	no	—
126	C₄×C₆₇	—	2	8	no	—
127	C₄×C₆₇	—	4	2	no	—
128	C₄×C₆₇	—	4	4	no	—
129	C₄×C₆₇	—	8	8	no	—
130	#21751	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
131	C₂×C₆₇	—	2	4	no	—
132	C₂×C₆₇	—	2	2	no	—
133	C₂×C₆₇	—	8	2	no	—
134	C₂×C₆₇	#2308	1	1	yes	characteristic
135	C₂×C₆₇	#2884	1	2	yes	—
136	C₂×#536	C₂×C₆₇	1	1	yes	characteristic
137	#2310	C₂×C₆₇	1	2	yes	complemented
138	#1899	C₂×C₆₇	1	8	yes	complemented
139	#1820	C₂×C₆₇	1	2	yes	complemented
140	#1782	C₂×C₆₇	1	2	yes	—
141	C₆₇	#21751	1	1	yes	characteristic, direct product factor, Sylow
142	C₂²×#41	C₂²×C₆₇	1	1	yes	characteristic
143	C₂³×Q₈	C₂²×C₆₇	1	1	yes	characteristic
144	#536	—	2	8	no	—
145	C₂×C₄⋊Q₈	C₄×C₆₇	1	2	yes	complemented
146	C₂×#124	C₄×C₆₇	1	2	yes	complemented
147	C₂×#123	C₂²×C₆₇	1	4	yes	—
148	C₂×C₂²⋊Q₈	C₄×C₆₇	1	4	yes	complemented
149	C₂×#117	C₂²×C₆₇	1	4	yes	—
150	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
151	C₂×#99	C₂²×C₆₇	1	8	yes	—
152	#391	—	2	8	no	—
153	#381	C₂²×C₆₇	1	4	yes	—
154	#381	C₂²×C₆₇	1	4	yes	—
155	#342	C₂²×C₆₇	1	8	yes	—
156	C₂×#41	—	2	4	no	—
157	C₂×#41	—	4	4	no	—
158	C₂²×Q₈	—	4	4	no	—
159	C₂²×Q₈	C₂×C₄×C₆₇	1	1	yes	characteristic
160	C₂²×Q₈	C₂×C₄×C₆₇	1	2	yes	—
161	C₂²×Q₈	C₂×C₄×C₆₇	1	2	yes	—
162	C₂²×Q₈	D₄×C₆₇	1	8	yes	—
163	C₂²×D₄	C₂×C₄×C₆₇	1	1	yes	characteristic
164	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
165	C₂³×C₄	C₂×C₄×C₆₇	1	2	yes	—
166	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
167	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
168	C₄⋊Q₈	—	2	8	no	—
169	#124	—	2	8	no	—
170	#123	—	2	8	no	—
171	#123	—	2	8	no	—
172	C₂²⋊Q₈	—	2	16	no	—
173	#117	—	2	4	no	—
174	#117	—	4	8	no	—
175	C₂×C₄⋊C₄	—	2	4	no	—
176	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
177	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	8	yes	—
178	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	8	yes	—
179	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
180	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
181	C₂×C₄²	—	2	4	no	—
182	C₂×C₄²	C₂×C₄×C₆₇	1	2	yes	—
183	#99	—	4	8	no	—
184	#95	—	2	8	no	—
185	#95	—	2	8	no	—
186	C₂⁴	C₂²×C₄×C₆₇	1	2	yes	—
187	C₂⁴	C₂⁴×C₆₇	1	1	yes	characteristic
188	#41	—	4	16	no	—
189	C₂×Q₈	—	2	2	no	—
190	C₂×Q₈	—	4	4	no	—
191	C₂×Q₈	—	4	4	no	—
192	C₂×Q₈	—	4	8	no	—
193	C₂×Q₈	—	4	8	no	—
194	C₂×Q₈	—	4	2	no	—
195	C₂×Q₈	C₂²⋊C₄×C₆₇	1	8	yes	—
196	C₂×Q₈	C₂²⋊C₄×C₆₇	1	8	yes	—
197	C₂×D₄	—	2	4	no	—
198	C₂×D₄	—	2	8	no	—
199	C₂×D₄	—	2	2	no	—
200	C₂×D₄	—	4	2	no	—
201	C₂²×C₄	—	2	8	no	—
202	C₂²×C₄	—	2	4	no	—
203	C₂²×C₄	—	2	4	no	—
204	C₂²×C₄	—	2	8	no	—
205	C₂²×C₄	—	4	4	no	—
206	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
207	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
208	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
209	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
210	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
211	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
212	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
213	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
214	C₄⋊C₄	—	2	4	no	—
215	C₄⋊C₄	—	2	16	no	—
216	C₄⋊C₄	—	4	8	no	—
217	C₂²⋊C₄	—	2	4	no	—
218	C₂²⋊C₄	—	2	16	no	—
219	C₂²⋊C₄	—	2	16	no	—
220	C₄²	—	2	4	no	—
221	C₄²	—	4	8	no	—
222	C₂³	—	2	2	no	—
223	C₂³	—	2	4	no	—
224	C₂³	—	4	2	no	—
225	C₂³	—	4	4	no	—
226	C₂³	C₂³×C₄×C₆₇	1	1	yes	characteristic
227	C₂³	C₂²×D₄×C₆₇	1	2	yes	—
228	C₂³	H(5,𝔽₂)×C₆₇	1	4	yes	—
229	Q₈	—	4	8	no	—
230	Q₈	—	4	8	no	—
231	Q₈	—	4	4	no	—
232	D₄	—	4	4	no	—
233	D₄	—	4	8	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	8	no	—
237	C₂×C₄	—	2	4	no	—
238	C₂×C₄	—	2	4	no	—
239	C₂×C₄	—	2	8	no	—
240	C₂×C₄	—	2	8	no	—
241	C₂×C₄	—	2	4	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	2	4	no	—
244	C₂×C₄	—	4	8	no	—
245	C₂×C₄	—	4	4	no	—
246	C₂×C₄	—	4	16	no	—
247	C₂×C₄	—	4	8	no	—
248	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
249	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	8	yes	—
250	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
251	C₂²	—	2	4	no	—
252	C₂²	—	2	2	no	—
253	C₂²	—	4	4	no	—
254	C₂²	—	4	2	no	—
255	C₂²	—	4	4	no	—
256	C₂²	—	8	4	no	—
257	C₂²	C₂²×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
258	C₂²	#518×C₆₇	1	2	yes	—
259	C₂²	#534×C₆₇	1	4	yes	—
260	C₄	—	2	2	no	—
261	C₄	—	2	2	no	—
262	C₄	—	2	8	no	—
263	C₄	—	4	2	no	—
264	C₄	—	4	4	no	—
265	C₄	—	8	8	no	—
266	C₂	—	2	4	no	—
267	C₂	—	2	2	no	—
268	C₂	—	8	2	no	—
269	C₂	C₆₇×#2308	1	1	yes	characteristic
270	C₂	C₆₇×#2884	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	C₂²×#41×C₆₇	C₂²
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ C₂³×Q₈×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₆₇×#2310	C₂	semidirect
3. C₆₇×#1899	C₂	semidirect
4. C₆₇×#1820	C₂	semidirect
9. C₂×C₄⋊Q₈×C₆₇	C₄	semidirect
10. C₂×#124×C₆₇	C₄	semidirect
12. C₂×C₂²⋊Q₈×C₆₇	C₄	semidirect
130. #21751	C₆₇	direct (N × H)
137. #2310	C₂×C₆₇	semidirect
138. #1899	C₂×C₆₇	semidirect
139. #1820	C₂×C₆₇	semidirect
141. C₆₇	#21751	direct (N × H)
145. C₂×C₄⋊Q₈	C₄×C₆₇	semidirect
146. C₂×#124	C₄×C₆₇	semidirect
148. C₂×C₂²⋊Q₈	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×#536×C₆₇	2
C₆₇×#2310	2
C₆₇×#1899	2
C₆₇×#1820	2
C₆₇×#1782	2
#21751	67

Subgroups — C67×#21751 #30701002

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#21751 #30701002