Subgroups — C₂²⋊C₄×M₁₆×C₆₇ #30699639

Order 17152.

1774 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×M₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#407×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×M₁₆×C₆₇	C₂	1	2	yes	maximal, complemented
4	C₂×C₄×C₂²⋊C₄×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₆₇×#1753	C₂	1	4	yes	maximal, complemented
6	C₆₇×#1752	C₂	1	2	yes	maximal, complemented
7	C₈×C₂²⋊C₄×C₆₇	C₂	1	2	yes	maximal, complemented
8	C₆₇×#1728	C₂	1	2	yes	maximal, complemented
9	C₂⁴×C₄×C₆₇	C₂²	1	1	yes	characteristic
10	C₂²×M₁₆×C₆₇	—	2	8	no	—
11	C₂²×M₁₆×C₆₇	C₄	1	8	yes	complemented
12	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
13	C₂²×M₁₆×C₆₇	C₂²	1	2	yes	—
14	C₂²×M₁₆×C₆₇	C₂²	1	2	yes	—
15	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
16	C₂³×C₈×C₆₇	C₂²	1	2	yes	—
17	C₂²×C₂²⋊C₄×C₆₇	C₄	1	2	yes	—
18	C₂²×C₄²×C₆₇	C₂²	1	2	yes	—
19	#407×C₆₇	C₄	1	8	yes	complemented
20	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
21	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
22	C₄×M₁₆×C₆₇	—	2	8	no	—
23	C₂×#97×C₆₇	C₂²	1	4	yes	complemented
24	C₂×C₄×C₈×C₆₇	C₂²	1	4	yes	complemented
25	C₄×C₂²⋊C₄×C₆₇	C₄	1	2	yes	—
26	C₄×C₂²⋊C₄×C₆₇	C₄	1	2	yes	—
27	C₄×C₂²⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
28	C₄×C₂²⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
29	C₄×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
30	C₂×#95×C₆₇	C₄	1	2	yes	—
31	#336×C₆₇	C₂²	1	4	yes	complemented
32	#336×C₆₇	C₂²	1	4	yes	complemented
33	C₂⁵×C₆₇	C₂×C₄	1	1	yes	characteristic
34	C₂³×C₄×C₆₇	—	2	4	no	—
35	C₂³×C₄×C₆₇	—	2	4	no	—
36	C₂³×C₄×C₆₇	C₂×C₄	1	4	yes	—
37	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
38	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
39	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
40	C₂³×C₄×C₆₇	C₂×C₄	1	4	yes	—
41	C₂³×C₄×C₆₇	C₂×C₄	1	4	yes	—
42	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
43	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
44	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
45	C₂×M₁₆×C₆₇	—	2	4	no	—
46	C₂×M₁₆×C₆₇	—	2	4	no	—
47	C₂×M₁₆×C₆₇	—	2	4	no	—
48	C₂×M₁₆×C₆₇	—	2	32	no	—
49	C₂×M₁₆×C₆₇	—	2	8	no	—
50	C₂×M₁₆×C₆₇	C₂×C₄	1	4	yes	—
51	C₂×M₁₆×C₆₇	C₂×C₄	1	8	yes	—
52	C₂×M₁₆×C₆₇	C₂×C₄	1	4	yes	—
53	C₂×M₁₆×C₆₇	C₂×C₄	1	8	yes	complemented
54	C₂×M₁₆×C₆₇	D₄	1	8	yes	—
55	C₂²×C₈×C₆₇	—	2	8	no	—
56	C₂²×C₈×C₆₇	C₂×C₄	1	8	yes	complemented
57	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
58	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
59	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
60	C₂×C₂²⋊C₄×C₆₇	—	2	8	no	—
61	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
62	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	4	yes	—
63	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
64	C₂×C₄²×C₆₇	—	2	4	no	—
65	C₂×C₄²×C₆₇	—	2	4	no	—
66	C₂×C₄²×C₆₇	C₂×C₄	1	4	yes	—
67	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
68	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
69	C₂²⋊C₈×C₆₇	C₂×C₄	1	8	yes	complemented
70	C₂²⋊C₈×C₆₇	C₂×C₄	1	8	yes	complemented
71	#97×C₆₇	—	2	8	no	—
72	C₄×C₈×C₆₇	—	2	8	no	—
73	#95×C₆₇	C₂×C₄	1	2	yes	—
74	#95×C₆₇	C₂×C₄	1	4	yes	—
75	#95×C₆₇	C₂×C₄	1	2	yes	—
76	C₂⁴×C₆₇	—	2	4	no	—
77	C₂⁴×C₆₇	—	2	4	no	—
78	C₂⁴×C₆₇	—	4	2	no	—
79	C₂⁴×C₆₇	C₄²	1	4	yes	—
80	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
81	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
82	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
83	C₂²×C₄×C₆₇	—	2	4	no	—
84	C₂²×C₄×C₆₇	—	2	8	no	—
85	C₂²×C₄×C₆₇	—	2	8	no	—
86	C₂²×C₄×C₆₇	—	2	4	no	—
87	C₂²×C₄×C₆₇	—	2	4	no	—
88	C₂²×C₄×C₆₇	—	2	16	no	—
89	C₂²×C₄×C₆₇	—	2	2	no	—
90	C₂²×C₄×C₆₇	—	2	2	no	—
91	C₂²×C₄×C₆₇	—	2	2	no	—
92	C₂²×C₄×C₆₇	—	2	2	no	—
93	C₂²×C₄×C₆₇	—	2	8	no	—
94	C₂²×C₄×C₆₇	—	2	8	no	—
95	C₂²×C₄×C₆₇	—	4	4	no	—
96	C₂²×C₄×C₆₇	C₄²	1	4	yes	—
97	C₂²×C₄×C₆₇	C₄²	1	4	yes	—
98	C₂²×C₄×C₆₇	C₄²	1	4	yes	—
99	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
100	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
101	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
102	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
103	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
104	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
105	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
106	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
107	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
108	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
109	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
110	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
111	C₂²×C₄×C₆₇	#41	1	4	yes	—
112	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
113	M₁₆×C₆₇	—	2	8	no	—
114	M₁₆×C₆₇	—	2	16	no	—
115	M₁₆×C₆₇	—	2	16	no	—
116	M₁₆×C₆₇	C₂²⋊C₄	1	16	yes	direct product factor
117	C₂×C₈×C₆₇	—	2	4	no	—
118	C₂×C₈×C₆₇	—	2	4	no	—
119	C₂×C₈×C₆₇	—	2	4	no	—
120	C₂×C₈×C₆₇	—	2	16	no	—
121	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
122	C₂×C₈×C₆₇	C₂²×C₄	1	8	yes	—
123	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
124	C₂×C₈×C₆₇	C₂×D₄	1	8	yes	—
125	C₂²⋊C₄×C₆₇	—	2	4	no	—
126	C₂²⋊C₄×C₆₇	—	2	8	no	—
127	C₂²⋊C₄×C₆₇	M₁₆	1	8	yes	direct product factor
128	C₄²×C₆₇	—	2	8	no	—
129	C₄²×C₆₇	—	2	2	no	—
130	C₄²×C₆₇	—	2	2	no	—
131	C₄²×C₆₇	—	2	2	no	—
132	C₄²×C₆₇	—	2	2	no	—
133	C₂³×C₆₇	—	2	2	no	—
134	C₂³×C₆₇	—	2	2	no	—
135	C₂³×C₆₇	—	2	2	no	—
136	C₂³×C₆₇	—	2	2	no	—
137	C₂³×C₆₇	—	2	2	no	—
138	C₂³×C₆₇	—	2	2	no	—
139	C₂³×C₆₇	—	2	2	no	—
140	C₂³×C₆₇	—	2	8	no	—
141	C₂³×C₆₇	—	2	8	no	—
142	C₂³×C₆₇	—	4	4	no	—
143	C₂³×C₆₇	—	4	16	no	—
144	C₂³×C₆₇	C₂×C₄²	1	2	yes	—
145	C₂³×C₆₇	C₂×C₄²	1	2	yes	—
146	C₂³×C₆₇	C₂×C₄²	1	2	yes	—
147	C₂³×C₆₇	C₄×D₄	1	4	yes	—
148	C₂³×C₆₇	C₂×M₁₆	1	4	yes	—
149	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
150	C₂×C₄×C₆₇	—	2	4	no	—
151	C₂×C₄×C₆₇	—	2	2	no	—
152	C₂×C₄×C₆₇	—	2	2	no	—
153	C₂×C₄×C₆₇	—	2	4	no	—
154	C₂×C₄×C₆₇	—	2	8	no	—
155	C₂×C₄×C₆₇	—	2	2	no	—
156	C₂×C₄×C₆₇	—	2	2	no	—
157	C₂×C₄×C₆₇	—	2	2	no	—
158	C₂×C₄×C₆₇	—	2	2	no	—
159	C₂×C₄×C₆₇	—	2	4	no	—
160	C₂×C₄×C₆₇	—	2	4	no	—
161	C₂×C₄×C₆₇	—	2	8	no	—
162	C₂×C₄×C₆₇	—	2	8	no	—
163	C₂×C₄×C₆₇	—	2	4	no	—
164	C₂×C₄×C₆₇	—	2	4	no	—
165	C₂×C₄×C₆₇	—	2	4	no	—
166	C₂×C₄×C₆₇	—	4	16	no	—
167	C₂×C₄×C₆₇	C₂×C₄²	1	2	yes	—
168	C₂×C₄×C₆₇	C₂×C₄²	1	2	yes	—
169	C₂×C₄×C₆₇	C₂×C₄²	1	2	yes	—
170	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	4	yes	—
171	C₂×C₄×C₆₇	#117	1	4	yes	—
172	C₂×C₄×C₆₇	C₄×D₄	1	8	yes	—
173	C₂×C₄×C₆₇	C₄×D₄	1	4	yes	—
174	C₂×C₄×C₆₇	C₂×M₁₆	1	8	yes	complemented
175	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
176	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
177	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
178	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
179	C₈×C₆₇	—	2	4	no	—
180	C₈×C₆₇	—	2	8	no	—
181	C₈×C₆₇	C₂×C₂²⋊C₄	1	8	yes	complemented
182	C₂²×C₆₇	—	2	2	no	—
183	C₂²×C₆₇	—	2	2	no	—
184	C₂²×C₆₇	—	2	2	no	—
185	C₂²×C₆₇	—	2	8	no	—
186	C₂²×C₆₇	—	2	8	no	—
187	C₂²×C₆₇	—	2	2	no	—
188	C₂²×C₆₇	—	2	2	no	—
189	C₂²×C₆₇	—	2	2	no	—
190	C₂²×C₆₇	—	2	2	no	—
191	C₂²×C₆₇	—	4	16	no	—
192	C₂²×C₆₇	—	4	4	no	—
193	C₂²×C₆₇	C₄×C₂²⋊C₄	1	4	yes	—
194	C₂²×C₆₇	C₄×M₁₆	1	4	yes	complemented
195	C₂²×C₆₇	C₂²×C₄²	1	1	yes	characteristic
196	C₂²×C₆₇	C₂×C₄×D₄	1	2	yes	—
197	C₂²×C₆₇	C₂²×M₁₆	1	2	yes	—
198	C₂²×C₆₇	#568	1	2	yes	—
199	C₄×C₆₇	—	2	8	no	—
200	C₄×C₆₇	—	2	2	no	—
201	C₄×C₆₇	—	2	2	no	—
202	C₄×C₆₇	—	4	4	no	—
203	C₄×C₆₇	C₄×C₂²⋊C₄	1	4	yes	—
204	C₄×C₆₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
205	C₄×C₆₇	C₂×#117	1	1	yes	characteristic
206	C₄×C₆₇	C₂×C₄×D₄	1	2	yes	—
207	C₂²⋊C₄×M₁₆	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
208	C₂×C₆₇	—	2	4	no	—
209	C₂×C₆₇	—	2	4	no	—
210	C₂×C₆₇	—	4	2	no	—
211	C₂×C₆₇	C₂×C₄×C₂²⋊C₄	1	1	yes	characteristic
212	C₂×C₆₇	C₂×C₄×M₁₆	1	1	yes	characteristic
213	C₂×C₆₇	#2873	1	1	yes	characteristic
214	C₂×C₆₇	D₄×M₁₆	1	4	yes	—
215	C₂³×M₁₆	C₂×C₆₇	1	1	yes	characteristic
216	C₂×#407	C₂×C₆₇	1	1	yes	characteristic
217	C₂×C₄×M₁₆	C₂×C₆₇	1	2	yes	complemented
218	C₂×C₄×C₂²⋊C₄	C₂×C₆₇	1	1	yes	characteristic
219	#1753	C₂×C₆₇	1	4	yes	complemented
220	#1752	C₂×C₆₇	1	2	yes	complemented
221	C₈×C₂²⋊C₄	C₂×C₆₇	1	2	yes	complemented
222	#1728	C₂×C₆₇	1	2	yes	complemented
223	C₆₇	C₂²⋊C₄×M₁₆	1	1	yes	characteristic, direct product factor, Sylow
224	C₂⁴×C₄	C₂²×C₆₇	1	1	yes	characteristic
225	C₂²×M₁₆	—	2	8	no	—
226	C₂²×M₁₆	C₄×C₆₇	1	8	yes	complemented
227	C₂²×M₁₆	C₂²×C₆₇	1	2	yes	—
228	C₂²×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
229	C₂²×M₁₆	C₂²×C₆₇	1	2	yes	—
230	C₂²×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
231	C₂³×C₈	C₂²×C₆₇	1	2	yes	—
232	C₂²×C₂²⋊C₄	C₄×C₆₇	1	2	yes	—
233	C₂²×C₄²	C₂²×C₆₇	1	2	yes	—
234	#407	C₄×C₆₇	1	8	yes	complemented
235	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
236	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
237	C₄×M₁₆	—	2	8	no	—
238	C₂×#97	C₂²×C₆₇	1	4	yes	complemented
239	C₂×C₄×C₈	C₂²×C₆₇	1	4	yes	complemented
240	C₄×C₂²⋊C₄	C₄×C₆₇	1	2	yes	—
241	C₄×C₂²⋊C₄	C₄×C₆₇	1	2	yes	—
242	C₄×C₂²⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
243	C₄×C₂²⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
244	C₄×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
245	C₂×#95	C₄×C₆₇	1	2	yes	—
246	#336	C₂²×C₆₇	1	4	yes	complemented
247	#336	C₂²×C₆₇	1	4	yes	complemented
248	C₂⁵	C₂×C₄×C₆₇	1	1	yes	characteristic
249	C₂³×C₄	—	2	4	no	—
250	C₂³×C₄	—	2	4	no	—
251	C₂³×C₄	C₂×C₄×C₆₇	1	4	yes	—
252	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
253	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
254	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
255	C₂³×C₄	C₂×C₄×C₆₇	1	4	yes	—
256	C₂³×C₄	C₂×C₄×C₆₇	1	4	yes	—
257	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
258	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
259	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
260	C₂×M₁₆	—	2	4	no	—
261	C₂×M₁₆	—	2	4	no	—
262	C₂×M₁₆	—	2	4	no	—
263	C₂×M₁₆	—	2	32	no	—
264	C₂×M₁₆	—	2	8	no	—
265	C₂×M₁₆	C₂×C₄×C₆₇	1	4	yes	—
266	C₂×M₁₆	C₂×C₄×C₆₇	1	8	yes	—
267	C₂×M₁₆	C₂×C₄×C₆₇	1	4	yes	—
268	C₂×M₁₆	C₂×C₄×C₆₇	1	8	yes	complemented
269	C₂×M₁₆	D₄×C₆₇	1	8	yes	—
270	C₂²×C₈	—	2	8	no	—
271	C₂²×C₈	C₂×C₄×C₆₇	1	8	yes	complemented
272	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
273	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
274	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
275	C₂×C₂²⋊C₄	—	2	8	no	—
276	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
277	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	4	yes	—
278	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
279	C₂×C₄²	—	2	4	no	—
280	C₂×C₄²	—	2	4	no	—
281	C₂×C₄²	C₂×C₄×C₆₇	1	4	yes	—
282	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
283	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
284	C₂²⋊C₈	C₂×C₄×C₆₇	1	8	yes	complemented
285	C₂²⋊C₈	C₂×C₄×C₆₇	1	8	yes	complemented
286	#97	—	2	8	no	—
287	C₄×C₈	—	2	8	no	—
288	#95	C₂×C₄×C₆₇	1	2	yes	—
289	#95	C₂×C₄×C₆₇	1	4	yes	—
290	#95	C₂×C₄×C₆₇	1	2	yes	—
291	C₂⁴	—	2	4	no	—
292	C₂⁴	—	2	4	no	—
293	C₂⁴	—	4	2	no	—
294	C₂⁴	C₄²×C₆₇	1	4	yes	—
295	C₂⁴	C₂²×C₄×C₆₇	1	1	yes	characteristic
296	C₂⁴	C₂²×C₄×C₆₇	1	1	yes	characteristic
297	C₂⁴	C₂²×C₄×C₆₇	1	1	yes	characteristic
298	C₂²×C₄	—	2	4	no	—
299	C₂²×C₄	—	2	8	no	—
300	C₂²×C₄	—	2	8	no	—
301	C₂²×C₄	—	2	4	no	—
302	C₂²×C₄	—	2	4	no	—
303	C₂²×C₄	—	2	16	no	—
304	C₂²×C₄	—	2	2	no	—
305	C₂²×C₄	—	2	2	no	—
306	C₂²×C₄	—	2	2	no	—
307	C₂²×C₄	—	2	2	no	—
308	C₂²×C₄	—	2	8	no	—
309	C₂²×C₄	—	2	8	no	—
310	C₂²×C₄	—	4	4	no	—
311	C₂²×C₄	C₄²×C₆₇	1	4	yes	—
312	C₂²×C₄	C₄²×C₆₇	1	4	yes	—
313	C₂²×C₄	C₄²×C₆₇	1	4	yes	—
314	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
315	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
316	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
317	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
318	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
319	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
320	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
321	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
322	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
323	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
324	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
325	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
326	C₂²×C₄	#41×C₆₇	1	4	yes	—
327	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
328	M₁₆	—	2	8	no	—
329	M₁₆	—	2	16	no	—
330	M₁₆	—	2	16	no	—
331	M₁₆	C₂²⋊C₄×C₆₇	1	16	yes	direct product factor
332	C₂×C₈	—	2	4	no	—
333	C₂×C₈	—	2	4	no	—
334	C₂×C₈	—	2	4	no	—
335	C₂×C₈	—	2	16	no	—
336	C₂×C₈	C₂²×C₄×C₆₇	1	8	yes	—
337	C₂×C₈	C₂²×C₄×C₆₇	1	4	yes	—
338	C₂×C₈	C₂²×C₄×C₆₇	1	4	yes	—
339	C₂×C₈	C₂×D₄×C₆₇	1	8	yes	—
340	C₂²⋊C₄	—	2	4	no	—
341	C₂²⋊C₄	—	2	8	no	—
342	C₂²⋊C₄	M₁₆×C₆₇	1	8	yes	direct product factor
343	C₄²	—	2	8	no	—
344	C₄²	—	2	2	no	—
345	C₄²	—	2	2	no	—
346	C₄²	—	2	2	no	—
347	C₄²	—	2	2	no	—
348	C₂³	—	2	2	no	—
349	C₂³	—	2	2	no	—
350	C₂³	—	2	2	no	—
351	C₂³	—	2	2	no	—
352	C₂³	—	2	2	no	—
353	C₂³	—	2	2	no	—
354	C₂³	—	2	2	no	—
355	C₂³	—	2	8	no	—
356	C₂³	—	2	8	no	—
357	C₂³	—	4	4	no	—
358	C₂³	—	4	16	no	—
359	C₂³	C₂×C₄²×C₆₇	1	2	yes	—
360	C₂³	C₂×C₄²×C₆₇	1	2	yes	—
361	C₂³	C₂×C₄²×C₆₇	1	2	yes	—
362	C₂³	C₄×D₄×C₆₇	1	4	yes	—
363	C₂³	C₂×M₁₆×C₆₇	1	4	yes	—
364	C₂³	C₂³×C₄×C₆₇	1	1	yes	characteristic
365	C₂×C₄	—	2	4	no	—
366	C₂×C₄	—	2	2	no	—
367	C₂×C₄	—	2	2	no	—
368	C₂×C₄	—	2	4	no	—
369	C₂×C₄	—	2	8	no	—
370	C₂×C₄	—	2	2	no	—
371	C₂×C₄	—	2	2	no	—
372	C₂×C₄	—	2	2	no	—
373	C₂×C₄	—	2	2	no	—
374	C₂×C₄	—	2	4	no	—
375	C₂×C₄	—	2	4	no	—
376	C₂×C₄	—	2	8	no	—
377	C₂×C₄	—	2	8	no	—
378	C₂×C₄	—	2	4	no	—
379	C₂×C₄	—	2	4	no	—
380	C₂×C₄	—	2	4	no	—
381	C₂×C₄	—	4	16	no	—
382	C₂×C₄	C₂×C₄²×C₆₇	1	2	yes	—
383	C₂×C₄	C₂×C₄²×C₆₇	1	2	yes	—
384	C₂×C₄	C₂×C₄²×C₆₇	1	2	yes	—
385	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	4	yes	—
386	C₂×C₄	#117×C₆₇	1	4	yes	—
387	C₂×C₄	C₄×D₄×C₆₇	1	8	yes	—
388	C₂×C₄	C₄×D₄×C₆₇	1	4	yes	—
389	C₂×C₄	C₂×M₁₆×C₆₇	1	8	yes	complemented
390	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
391	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
392	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
393	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
394	C₈	—	2	4	no	—
395	C₈	—	2	8	no	—
396	C₈	C₂×C₂²⋊C₄×C₆₇	1	8	yes	complemented
397	C₂²	—	2	2	no	—
398	C₂²	—	2	2	no	—
399	C₂²	—	2	2	no	—
400	C₂²	—	2	2	no	—
401	C₂²	—	2	8	no	—
402	C₂²	—	2	8	no	—
403	C₂²	—	2	2	no	—
404	C₂²	—	2	2	no	—
405	C₂²	—	2	2	no	—
406	C₂²	—	4	16	no	—
407	C₂²	—	4	4	no	—
408	C₂²	C₄×C₂²⋊C₄×C₆₇	1	4	yes	—
409	C₂²	C₄×M₁₆×C₆₇	1	4	yes	complemented
410	C₂²	C₂²×C₄²×C₆₇	1	1	yes	characteristic
411	C₂²	C₂×C₄×D₄×C₆₇	1	2	yes	—
412	C₂²	C₂²×M₁₆×C₆₇	1	2	yes	—
413	C₂²	#568×C₆₇	1	2	yes	—
414	C₄	—	2	2	no	—
415	C₄	—	2	2	no	—
416	C₄	—	2	8	no	—
417	C₄	—	4	4	no	—
418	C₄	C₄×C₂²⋊C₄×C₆₇	1	4	yes	—
419	C₄	C₂²×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
420	C₄	C₂×#117×C₆₇	1	1	yes	characteristic
421	C₄	C₂×C₄×D₄×C₆₇	1	2	yes	—
422	C₂	—	2	4	no	—
423	C₂	—	2	4	no	—
424	C₂	—	4	2	no	—
425	C₂	C₂×C₄×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
426	C₂	C₂×C₄×M₁₆×C₆₇	1	1	yes	characteristic
427	C₂	C₆₇×#2873	1	1	yes	characteristic
428	C₂	D₄×M₁₆×C₆₇	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄×C₆₇	C₂⁴
commutator	C₂²	C₂²×C₄²×C₆₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
socle	C₂³×C₆₇	C₂³×C₄
prime radical	C₂⁵×C₆₇	C₂×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂×C₄×M₁₆×C₆₇	C₂	semidirect
5. C₆₇×#1753	C₂	semidirect
6. C₆₇×#1752	C₂	semidirect
7. C₈×C₂²⋊C₄×C₆₇	C₂	semidirect
8. C₆₇×#1728	C₂	semidirect
11. C₂²×M₁₆×C₆₇	C₄	semidirect
19. #407×C₆₇	C₄	semidirect
23. C₂×#97×C₆₇	C₂²	semidirect
24. C₂×C₄×C₈×C₆₇	C₂²	semidirect
31. #336×C₆₇	C₂²	semidirect
32. #336×C₆₇	C₂²	semidirect
53. C₂×M₁₆×C₆₇	C₂×C₄	semidirect
56. C₂²×C₈×C₆₇	C₂×C₄	semidirect
69. C₂²⋊C₈×C₆₇	C₂×C₄	semidirect
70. C₂²⋊C₈×C₆₇	C₂×C₄	semidirect
116. M₁₆×C₆₇	C₂²⋊C₄	direct (N × H)
127. C₂²⋊C₄×C₆₇	M₁₆	direct (N × H)
174. C₂×C₄×C₆₇	C₂×M₁₆	semidirect
181. C₈×C₆₇	C₂×C₂²⋊C₄	semidirect
194. C₂²×C₆₇	C₄×M₁₆	semidirect
207. C₂²⋊C₄×M₁₆	C₆₇	direct (N × H)
217. C₂×C₄×M₁₆	C₂×C₆₇	semidirect
219. #1753	C₂×C₆₇	semidirect
220. #1752	C₂×C₆₇	semidirect
221. C₈×C₂²⋊C₄	C₂×C₆₇	semidirect
222. #1728	C₂×C₆₇	semidirect
223. C₆₇	C₂²⋊C₄×M₁₆	direct (N × H)
226. C₂²×M₁₆	C₄×C₆₇	semidirect
234. #407	C₄×C₆₇	semidirect
238. C₂×#97	C₂²×C₆₇	semidirect
239. C₂×C₄×C₈	C₂²×C₆₇	semidirect
246. #336	C₂²×C₆₇	semidirect
247. #336	C₂²×C₆₇	semidirect
268. C₂×M₁₆	C₂×C₄×C₆₇	semidirect
271. C₂²×C₈	C₂×C₄×C₆₇	semidirect
284. C₂²⋊C₈	C₂×C₄×C₆₇	semidirect
285. C₂²⋊C₈	C₂×C₄×C₆₇	semidirect
331. M₁₆	C₂²⋊C₄×C₆₇	direct (N × H)
342. C₂²⋊C₄	M₁₆×C₆₇	direct (N × H)
389. C₂×C₄	C₂×M₁₆×C₆₇	semidirect
396. C₈	C₂×C₂²⋊C₄×C₆₇	semidirect
409. C₂²	C₄×M₁₆×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂³×M₁₆×C₆₇	2
C₂×#407×C₆₇	2
C₂×C₄×M₁₆×C₆₇	2
C₂×C₄×C₂²⋊C₄×C₆₇	2
C₆₇×#1753	2
C₆₇×#1752	2
C₈×C₂²⋊C₄×C₆₇	2
C₆₇×#1728	2
C₂²⋊C₄×M₁₆	67

Subgroups — C22⋊C4×M16×C67 #30699639

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²⋊C₄×M₁₆×C₆₇ #30699639