Subgroups — C₆₇×#33922 #30713173

Order 17152.

488 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#3339	C₂	1	2	yes	maximal, complemented
2	C₆₇×#2253	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#2234	C₂	1	2	yes	maximal, complemented
4	C₆₇×#2232	C₂	1	2	yes	maximal, complemented
5	C₆₇×#2228	C₂	1	2	yes	maximal, complemented
6	C₆₇×#2218	C₂	1	2	yes	maximal, complemented
7	C₄×D₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₇×#2169	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₇×#2159	C₂	1	2	yes	maximal, complemented
10	#553×C₆₇	—	2	2	no	—
11	C₂×D₁₆×C₆₇	C₂²	1	1	yes	characteristic
12	C₂×M₃₂×C₆₇	C₂²	1	2	yes	—
13	#499×C₆₇	C₂²	1	2	yes	—
14	#482×C₆₇	—	2	2	no	—
15	#481×C₆₇	—	2	2	no	—
16	#478×C₆₇	—	2	2	no	—
17	#466×C₆₇	C₂²	1	2	yes	—
18	#466×C₆₇	C₂²	1	2	yes	—
19	C₄×D₈×C₆₇	C₂²	1	2	yes	—
20	C₈×D₄×C₆₇	C₂²	1	1	yes	characteristic
21	#427×C₆₇	C₂²	1	2	yes	—
22	C₂×#107×C₆₇	C₂²	1	2	yes	—
23	#367×C₆₇	C₂²	1	2	yes	—
24	#367×C₆₇	C₂²	1	2	yes	—
25	#366×C₆₇	C₂²	1	1	yes	characteristic
26	#366×C₆₇	C₂²	1	1	yes	characteristic
27	#366×C₆₇	C₂²	1	1	yes	characteristic
28	#366×C₆₇	C₂²	1	2	yes	—
29	C₄⋊C₁₆×C₆₇	C₂²	1	1	yes	characteristic
30	#357×C₆₇	C₂²	1	2	yes	—
31	#357×C₆₇	C₂²	1	2	yes	—
32	#357×C₆₇	C₂²	1	2	yes	—
33	C₂²⋊C₁₆×C₆₇	C₂²	1	2	yes	—
34	C₄×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
35	C₂×D₈×C₆₇	C₂³	1	2	yes	—
36	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
37	#126×C₆₇	—	4	2	no	—
38	#125×C₆₇	—	2	2	no	—
39	#123×C₆₇	—	4	2	no	—
40	#121×C₆₇	—	2	2	no	—
41	#121×C₆₇	—	2	2	no	—
42	#121×C₆₇	—	4	2	no	—
43	C₄×D₄×C₆₇	—	2	2	no	—
44	C₄×D₄×C₆₇	—	4	2	no	—
45	C₄×D₄×C₆₇	D₄	1	1	yes	characteristic
46	#117×C₆₇	—	2	2	no	—
47	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
48	D₁₆×C₆₇	—	2	2	no	—
49	M₃₂×C₆₇	—	2	4	no	—
50	C₂×C₁₆×C₆₇	C₂³	1	2	yes	—
51	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
52	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
53	#107×C₆₇	—	2	2	no	—
54	#107×C₆₇	C₂³	1	2	yes	—
55	#107×C₆₇	C₂³	1	2	yes	—
56	#107×C₆₇	C₂³	1	2	yes	—
57	#106×C₆₇	—	2	2	no	—
58	#106×C₆₇	—	2	2	no	—
59	C₄⋊C₈×C₆₇	D₄	1	1	yes	characteristic
60	#102×C₆₇	—	2	2	no	—
61	#102×C₆₇	—	2	2	no	—
62	#102×C₆₇	—	2	2	no	—
63	C₂²⋊C₈×C₆₇	D₄	1	2	yes	—
64	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
65	C₂×D₄×C₆₇	—	2	2	no	—
66	C₂×D₄×C₆₇	—	4	2	no	—
67	C₂×D₄×C₆₇	—	4	2	no	—
68	C₂×D₄×C₆₇	D₈	1	1	yes	characteristic
69	C₂²×C₄×C₆₇	—	4	2	no	—
70	C₂²×C₄×C₆₇	—	4	2	no	—
71	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
72	D₈×C₆₇	—	2	2	no	—
73	D₈×C₆₇	—	2	2	no	—
74	C₂×C₈×C₆₇	—	2	2	no	—
75	C₂×C₈×C₆₇	—	2	2	no	—
76	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
77	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
78	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
79	C₄⋊C₄×C₆₇	—	2	2	no	—
80	C₄⋊C₄×C₆₇	—	2	2	no	—
81	C₄⋊C₄×C₆₇	—	2	2	no	—
82	C₄⋊C₄×C₆₇	—	4	2	no	—
83	C₄⋊C₄×C₆₇	—	4	2	no	—
84	C₄⋊C₄×C₆₇	—	4	2	no	—
85	C₄⋊C₄×C₆₇	D₈	1	1	yes	characteristic
86	C₂²⋊C₄×C₆₇	—	4	2	no	—
87	C₂²⋊C₄×C₆₇	—	4	2	no	—
88	C₂²⋊C₄×C₆₇	—	4	2	no	—
89	C₂²⋊C₄×C₆₇	—	4	2	no	—
90	C₂²⋊C₄×C₆₇	D₈	1	2	yes	—
91	C₄²×C₆₇	—	4	2	no	—
92	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
93	C₁₆×C₆₇	—	2	2	no	—
94	C₁₆×C₆₇	—	2	1	no	—
95	C₁₆×C₆₇	#41	1	2	yes	—
96	C₂³×C₆₇	—	4	2	no	—
97	C₂³×C₆₇	C₂×D₈	1	2	yes	—
98	D₄×C₆₇	—	4	1	no	—
99	D₄×C₆₇	—	4	2	no	—
100	D₄×C₆₇	—	4	2	no	—
101	D₄×C₆₇	—	8	4	no	—
102	D₄×C₆₇	—	16	2	no	—
103	C₂×C₄×C₆₇	—	2	2	no	—
104	C₂×C₄×C₆₇	—	2	2	no	—
105	C₂×C₄×C₆₇	—	4	2	no	—
106	C₂×C₄×C₆₇	—	4	2	no	—
107	C₂×C₄×C₆₇	—	4	2	no	—
108	C₂×C₄×C₆₇	—	16	2	no	—
109	C₂×C₄×C₆₇	—	16	2	no	—
110	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
111	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
112	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
113	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
114	C₈×C₆₇	—	2	2	no	—
115	C₈×C₆₇	—	2	1	no	—
116	C₈×C₆₇	C₂×#41	1	1	yes	characteristic
117	C₈×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
118	C₂²×C₆₇	—	2	2	no	—
119	C₂²×C₆₇	—	2	2	no	—
120	C₂²×C₆₇	—	2	2	no	—
121	C₂²×C₆₇	—	8	2	no	—
122	C₂²×C₆₇	—	8	2	no	—
123	C₂²×C₆₇	—	8	2	no	—
124	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
125	C₄×C₆₇	—	2	2	no	—
126	C₄×C₆₇	—	2	1	no	—
127	C₄×C₆₇	—	2	1	no	—
128	C₄×C₆₇	—	8	2	no	—
129	C₄×C₆₇	—	8	2	no	—
130	C₄×C₆₇	—	8	2	no	—
131	C₄×C₆₇	#547	1	1	yes	characteristic
132	C₄×C₆₇	#578	1	1	yes	characteristic
133	#33922	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
134	C₂×C₆₇	—	4	2	no	—
135	C₂×C₆₇	—	16	2	no	—
136	C₂×C₆₇	#3334	1	1	yes	characteristic
137	C₂×C₆₇	#3414	1	1	yes	characteristic
138	C₂×C₆₇	#3415	1	1	yes	characteristic
139	#3339	C₂×C₆₇	1	2	yes	complemented
140	#2253	C₂×C₆₇	1	1	yes	characteristic, complemented
141	#2234	C₂×C₆₇	1	2	yes	complemented
142	#2232	C₂×C₆₇	1	2	yes	complemented
143	#2228	C₂×C₆₇	1	2	yes	complemented
144	#2218	C₂×C₆₇	1	2	yes	complemented
145	C₄×D₁₆	C₂×C₆₇	1	1	yes	characteristic, complemented
146	#2169	C₂×C₆₇	1	1	yes	characteristic, complemented
147	#2159	C₂×C₆₇	1	2	yes	complemented
148	C₆₇	#33922	1	1	yes	characteristic, direct product factor, Sylow
149	#553	—	2	2	no	—
150	C₂×D₁₆	C₂²×C₆₇	1	1	yes	characteristic
151	C₂×M₃₂	C₂²×C₆₇	1	2	yes	—
152	#499	C₂²×C₆₇	1	2	yes	—
153	#482	—	2	2	no	—
154	#481	—	2	2	no	—
155	#478	—	2	2	no	—
156	#466	C₂²×C₆₇	1	2	yes	—
157	#466	C₂²×C₆₇	1	2	yes	—
158	C₄×D₈	C₂²×C₆₇	1	2	yes	—
159	C₈×D₄	C₂²×C₆₇	1	1	yes	characteristic
160	#427	C₂²×C₆₇	1	2	yes	—
161	C₂×#107	C₂²×C₆₇	1	2	yes	—
162	#367	C₂²×C₆₇	1	2	yes	—
163	#367	C₂²×C₆₇	1	2	yes	—
164	#366	C₂²×C₆₇	1	1	yes	characteristic
165	#366	C₂²×C₆₇	1	2	yes	—
166	#366	C₂²×C₆₇	1	1	yes	characteristic
167	#366	C₂²×C₆₇	1	1	yes	characteristic
168	C₄⋊C₁₆	C₂²×C₆₇	1	1	yes	characteristic
169	#357	C₂²×C₆₇	1	2	yes	—
170	#357	C₂²×C₆₇	1	2	yes	—
171	#357	C₂²×C₆₇	1	2	yes	—
172	C₂²⋊C₁₆	C₂²×C₆₇	1	2	yes	—
173	C₄×C₁₆	C₂²×C₆₇	1	1	yes	characteristic
174	C₂×D₈	C₂³×C₆₇	1	2	yes	—
175	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
176	#126	—	4	2	no	—
177	#125	—	2	2	no	—
178	#123	—	4	2	no	—
179	#121	—	2	2	no	—
180	#121	—	2	2	no	—
181	#121	—	4	2	no	—
182	C₄×D₄	—	2	2	no	—
183	C₄×D₄	—	4	2	no	—
184	C₄×D₄	D₄×C₆₇	1	1	yes	characteristic
185	#117	—	2	2	no	—
186	C₂×C₄⋊C₄	—	2	2	no	—
187	D₁₆	—	2	2	no	—
188	M₃₂	—	2	4	no	—
189	C₂×C₁₆	C₂³×C₆₇	1	1	yes	characteristic
190	C₂×C₁₆	C₂³×C₆₇	1	2	yes	—
191	C₂×C₁₆	C₂³×C₆₇	1	1	yes	characteristic
192	#107	—	2	2	no	—
193	#107	C₂³×C₆₇	1	2	yes	—
194	#107	C₂³×C₆₇	1	2	yes	—
195	#107	C₂³×C₆₇	1	2	yes	—
196	#106	—	2	2	no	—
197	#106	—	2	2	no	—
198	C₄⋊C₈	D₄×C₆₇	1	1	yes	characteristic
199	#102	—	2	2	no	—
200	#102	—	2	2	no	—
201	#102	—	2	2	no	—
202	C₂²⋊C₈	D₄×C₆₇	1	2	yes	—
203	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
204	C₂×D₄	—	2	2	no	—
205	C₂×D₄	—	4	2	no	—
206	C₂×D₄	—	4	2	no	—
207	C₂×D₄	D₈×C₆₇	1	1	yes	characteristic
208	C₂²×C₄	—	4	2	no	—
209	C₂²×C₄	—	4	2	no	—
210	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
211	D₈	—	2	2	no	—
212	D₈	—	2	2	no	—
213	C₂×C₈	—	2	2	no	—
214	C₂×C₈	—	2	2	no	—
215	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
216	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
217	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
218	C₄⋊C₄	—	2	2	no	—
219	C₄⋊C₄	—	2	2	no	—
220	C₄⋊C₄	—	2	2	no	—
221	C₄⋊C₄	—	4	2	no	—
222	C₄⋊C₄	—	4	2	no	—
223	C₄⋊C₄	—	4	2	no	—
224	C₄⋊C₄	D₈×C₆₇	1	1	yes	characteristic
225	C₂²⋊C₄	—	4	2	no	—
226	C₂²⋊C₄	—	4	2	no	—
227	C₂²⋊C₄	—	4	2	no	—
228	C₂²⋊C₄	—	4	2	no	—
229	C₂²⋊C₄	D₈×C₆₇	1	2	yes	—
230	C₄²	—	4	2	no	—
231	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
232	C₁₆	—	2	2	no	—
233	C₁₆	—	2	1	no	—
234	C₁₆	#41×C₆₇	1	2	yes	—
235	C₂³	—	4	2	no	—
236	C₂³	C₂×D₈×C₆₇	1	2	yes	—
237	D₄	—	4	1	no	—
238	D₄	—	4	2	no	—
239	D₄	—	4	2	no	—
240	D₄	—	8	4	no	—
241	D₄	—	16	2	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	4	2	no	—
245	C₂×C₄	—	4	2	no	—
246	C₂×C₄	—	4	2	no	—
247	C₂×C₄	—	16	2	no	—
248	C₂×C₄	—	16	2	no	—
249	C₂×C₄	C₂×D₈×C₆₇	1	2	yes	—
250	C₂×C₄	C₂×D₈×C₆₇	1	1	yes	characteristic
251	C₂×C₄	C₂×D₈×C₆₇	1	1	yes	characteristic
252	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
253	C₈	—	2	2	no	—
254	C₈	—	2	1	no	—
255	C₈	C₂×#41×C₆₇	1	1	yes	characteristic
256	C₈	2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
257	C₂²	—	2	2	no	—
258	C₂²	—	2	2	no	—
259	C₂²	—	2	2	no	—
260	C₂²	—	8	2	no	—
261	C₂²	—	8	2	no	—
262	C₂²	—	8	2	no	—
263	C₂²	C₂²×D₈×C₆₇	1	1	yes	characteristic
264	C₄	—	2	2	no	—
265	C₄	—	2	1	no	—
266	C₄	—	2	1	no	—
267	C₄	—	8	2	no	—
268	C₄	—	8	2	no	—
269	C₄	—	8	2	no	—
270	C₄	#547×C₆₇	1	1	yes	characteristic
271	C₄	#578×C₆₇	1	1	yes	characteristic
272	C₂	—	4	2	no	—
273	C₂	—	16	2	no	—
274	C₂	C₆₇×#3334	1	1	yes	characteristic
275	C₂	C₆₇×#3414	1	1	yes	characteristic
276	C₂	C₆₇×#3415	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₇	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₆₇
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ C₄×D₄×C₆₇ ◁ C₈×D₄×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₆₇×#3339	C₂	semidirect
2. C₆₇×#2253	C₂	semidirect
3. C₆₇×#2234	C₂	semidirect
4. C₆₇×#2232	C₂	semidirect
5. C₆₇×#2228	C₂	semidirect
6. C₆₇×#2218	C₂	semidirect
7. C₄×D₁₆×C₆₇	C₂	semidirect
8. C₆₇×#2169	C₂	semidirect
9. C₆₇×#2159	C₂	semidirect
133. #33922	C₆₇	direct (N × H)
139. #3339	C₂×C₆₇	semidirect
140. #2253	C₂×C₆₇	semidirect
141. #2234	C₂×C₆₇	semidirect
142. #2232	C₂×C₆₇	semidirect
143. #2228	C₂×C₆₇	semidirect
144. #2218	C₂×C₆₇	semidirect
145. C₄×D₁₆	C₂×C₆₇	semidirect
146. #2169	C₂×C₆₇	semidirect
147. #2159	C₂×C₆₇	semidirect
148. C₆₇	#33922	direct (N × H)

Maximal subgroups

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

subgroup	index
C₆₇×#3339	2
C₆₇×#2253	2
C₆₇×#2234	2
C₆₇×#2232	2
C₆₇×#2228	2
C₆₇×#2218	2
C₄×D₁₆×C₆₇	2
C₆₇×#2169	2
C₆₇×#2159	2
#33922	67

Subgroups — C67×#33922 #30713173

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33922 #30713173