Subgroups — C₂²×#449×C₆₇ #30739602

Order 17152.

5454 conjugacy classes of subgroups.

The subgroup lattice has 446 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#41×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂³×SD₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂³×D₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×#121×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#449×C₆₇	C₂	1	24	yes	maximal, direct product factor
6	C₂²×#103×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×#102×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₂²⋊C₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×#41×C₆₇	—	2	8	no	—
10	C₂²×#41×C₆₇	—	2	8	no	—
11	C₂²×#41×C₆₇	—	2	6	no	—
12	C₂²×#41×C₆₇	C₂²	1	12	yes	complemented
13	C₂³×Q₈×C₆₇	C₂²	1	1	yes	characteristic
14	C₂³×D₄×C₆₇	—	2	1	no	—
15	C₂³×D₄×C₆₇	—	2	1	no	—
16	C₂³×D₄×C₆₇	C₂²	1	1	yes	characteristic
17	C₂³×D₄×C₆₇	C₂²	1	1	yes	characteristic
18	C₂⁴×C₄×C₆₇	—	2	1	no	—
19	C₂⁴×C₄×C₆₇	C₂²	1	1	yes	characteristic
20	C₂²×SD₁₆×C₆₇	—	2	8	no	—
21	C₂²×SD₁₆×C₆₇	C₂²	1	12	yes	complemented
22	C₂²×D₈×C₆₇	—	2	8	no	—
23	C₂²×D₈×C₆₇	C₂²	1	12	yes	complemented
24	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
25	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
26	C₂×#121×C₆₇	—	2	6	no	—
27	C₂×#121×C₆₇	C₂²	1	12	yes	complemented
28	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
29	C₂²×C₂²⋊C₄×C₆₇	—	2	1	no	—
30	#449×C₆₇	C₂²	1	64	yes	direct product factor
31	C₂×#103×C₆₇	C₂²	1	12	yes	complemented
32	C₂×#102×C₆₇	C₂²	1	12	yes	complemented
33	C₂×C₂²⋊C₈×C₆₇	C₂²	1	12	yes	complemented
34	C₂⁵×C₆₇	—	2	1	no	—
35	C₂⁵×C₆₇	—	2	1	no	—
36	C₂⁵×C₆₇	D₄	1	1	yes	characteristic
37	C₂×#41×C₆₇	—	2	96	no	—
38	C₂×#41×C₆₇	—	2	48	no	—
39	C₂×#41×C₆₇	—	2	48	no	—
40	C₂×#41×C₆₇	—	2	24	no	—
41	C₂×#41×C₆₇	C₂³	1	16	yes	complemented
42	C₂²×Q₈×C₆₇	—	2	4	no	—
43	C₂²×Q₈×C₆₇	—	2	3	no	—
44	C₂²×Q₈×C₆₇	D₄	1	8	yes	complemented
45	C₂²×Q₈×C₆₇	C₂³	1	6	yes	—
46	C₂²×D₄×C₆₇	—	2	16	no	—
47	C₂²×D₄×C₆₇	—	2	12	no	—
48	C₂²×D₄×C₆₇	—	2	12	no	—
49	C₂²×D₄×C₆₇	—	2	3	no	—
50	C₂²×D₄×C₆₇	—	2	4	no	—
51	C₂²×D₄×C₆₇	—	2	4	no	—
52	C₂²×D₄×C₆₇	—	2	4	no	—
53	C₂²×D₄×C₆₇	—	2	3	no	—
54	C₂²×D₄×C₆₇	—	4	8	no	—
55	C₂²×D₄×C₆₇	D₄	1	8	yes	complemented
56	C₂²×D₄×C₆₇	C₂³	1	6	yes	—
57	C₂²×D₄×C₆₇	C₂³	1	6	yes	—
58	C₂³×C₄×C₆₇	—	2	16	no	—
59	C₂³×C₄×C₆₇	—	2	1	no	—
60	C₂³×C₄×C₆₇	—	2	12	no	—
61	C₂³×C₄×C₆₇	—	2	4	no	—
62	C₂³×C₄×C₆₇	—	2	4	no	—
63	C₂³×C₄×C₆₇	—	2	1	no	—
64	C₂³×C₄×C₆₇	D₄	1	6	yes	—
65	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
66	C₂³×C₄×C₆₇	C₂³	1	6	yes	—
67	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
68	C₂×SD₁₆×C₆₇	—	2	48	no	—
69	C₂×SD₁₆×C₆₇	C₂³	1	16	yes	complemented
70	C₂×D₈×C₆₇	—	2	48	no	—
71	C₂×D₈×C₆₇	C₂³	1	16	yes	complemented
72	C₂²×C₈×C₆₇	—	2	4	no	—
73	C₂²×C₈×C₆₇	—	2	4	no	—
74	C₂²×C₈×C₆₇	C₂³	1	6	yes	—
75	C₂²×C₈×C₆₇	C₂³	1	6	yes	—
76	#121×C₆₇	—	2	24	no	—
77	#121×C₆₇	C₂³	1	16	yes	complemented
78	C₂×C₄⋊C₄×C₆₇	—	2	3	no	—
79	C₂×C₄⋊C₄×C₆₇	C₂³	1	6	yes	—
80	C₂×C₂²⋊C₄×C₆₇	—	2	12	no	—
81	#103×C₆₇	C₂³	1	16	yes	complemented
82	#102×C₆₇	C₂³	1	16	yes	complemented
83	C₂²⋊C₈×C₆₇	C₂³	1	16	yes	complemented
84	C₂⁴×C₆₇	—	2	4	no	—
85	C₂⁴×C₆₇	—	2	8	no	—
86	C₂⁴×C₆₇	—	2	4	no	—
87	C₂⁴×C₆₇	—	2	4	no	—
88	C₂⁴×C₆₇	—	2	6	no	—
89	C₂⁴×C₆₇	—	2	6	no	—
90	C₂⁴×C₆₇	—	4	4	no	—
91	C₂⁴×C₆₇	—	4	4	no	—
92	C₂⁴×C₆₇	—	4	4	no	—
93	C₂⁴×C₆₇	—	4	4	no	—
94	C₂⁴×C₆₇	—	4	4	no	—
95	C₂⁴×C₆₇	C₂×D₄	1	6	yes	—
96	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
97	#41×C₆₇	—	2	192	no	—
98	#41×C₆₇	—	2	32	no	—
99	#41×C₆₇	—	2	32	no	—
100	C₂×Q₈×C₆₇	—	2	24	no	—
101	C₂×Q₈×C₆₇	—	2	12	no	—
102	C₂×Q₈×C₆₇	—	2	6	no	—
103	C₂×Q₈×C₆₇	C₂×D₄	1	24	yes	complemented
104	C₂×Q₈×C₆₇	C₂⁴	1	4	yes	—
105	C₂×D₄×C₆₇	—	2	96	no	—
106	C₂×D₄×C₆₇	—	2	16	no	—
107	C₂×D₄×C₆₇	—	2	16	no	—
108	C₂×D₄×C₆₇	—	2	24	no	—
109	C₂×D₄×C₆₇	—	2	6	no	—
110	C₂×D₄×C₆₇	—	2	12	no	—
111	C₂×D₄×C₆₇	—	2	12	no	—
112	C₂×D₄×C₆₇	—	2	12	no	—
113	C₂×D₄×C₆₇	—	2	6	no	—
114	C₂×D₄×C₆₇	—	4	48	no	—
115	C₂×D₄×C₆₇	—	4	12	no	—
116	C₂×D₄×C₆₇	C₂×D₄	1	24	yes	complemented
117	C₂×D₄×C₆₇	C₂⁴	1	4	yes	—
118	C₂×D₄×C₆₇	C₂⁴	1	4	yes	—
119	C₂²×C₄×C₆₇	—	2	96	no	—
120	C₂²×C₄×C₆₇	—	2	4	no	—
121	C₂²×C₄×C₆₇	—	2	8	no	—
122	C₂²×C₄×C₆₇	—	2	6	no	—
123	C₂²×C₄×C₆₇	—	2	24	no	—
124	C₂²×C₄×C₆₇	—	2	16	no	—
125	C₂²×C₄×C₆₇	—	2	12	no	—
126	C₂²×C₄×C₆₇	—	2	12	no	—
127	C₂²×C₄×C₆₇	—	2	6	no	—
128	C₂²×C₄×C₆₇	—	4	4	no	—
129	C₂²×C₄×C₆₇	C₂×D₄	1	6	yes	—
130	C₂²×C₄×C₆₇	C₂×D₄	1	12	yes	—
131	C₂²×C₄×C₆₇	C₂×D₄	1	3	yes	—
132	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
133	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
134	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
135	C₂²×C₄×C₆₇	C₂⁴	1	3	yes	—
136	SD₁₆×C₆₇	—	2	32	no	—
137	D₈×C₆₇	—	2	32	no	—
138	C₂×C₈×C₆₇	—	2	12	no	—
139	C₂×C₈×C₆₇	—	2	12	no	—
140	C₂×C₈×C₆₇	C₂⁴	1	4	yes	—
141	C₂×C₈×C₆₇	C₂⁴	1	4	yes	—
142	C₄⋊C₄×C₆₇	—	2	6	no	—
143	C₄⋊C₄×C₆₇	C₂⁴	1	4	yes	—
144	C₂²⋊C₄×C₆₇	—	2	16	no	—
145	C₂³×C₆₇	—	2	12	no	—
146	C₂³×C₆₇	—	2	24	no	—
147	C₂³×C₆₇	—	2	12	no	—
148	C₂³×C₆₇	—	2	12	no	—
149	C₂³×C₆₇	—	2	4	no	—
150	C₂³×C₆₇	—	2	4	no	—
151	C₂³×C₆₇	—	4	12	no	—
152	C₂³×C₆₇	—	4	12	no	—
153	C₂³×C₆₇	—	4	16	no	—
154	C₂³×C₆₇	—	4	32	no	—
155	C₂³×C₆₇	—	4	12	no	—
156	C₂³×C₆₇	—	4	12	no	—
157	C₂³×C₆₇	—	4	12	no	—
158	C₂³×C₆₇	—	8	16	no	—
159	C₂³×C₆₇	#120	1	4	yes	—
160	C₂³×C₆₇	#135	1	4	yes	—
161	C₂³×C₆₇	AGL(1,ℤ/8)	1	4	yes	—
162	C₂³×C₆₇	C₂²×D₄	1	4	yes	—
163	C₂³×C₆₇	C₂²×D₄	1	3	yes	—
164	Q₈×C₆₇	—	2	24	no	—
165	Q₈×C₆₇	—	2	4	no	—
166	Q₈×C₆₇	C₂²×D₄	1	8	yes	complemented
167	D₄×C₆₇	—	2	64	no	—
168	D₄×C₆₇	—	2	24	no	—
169	D₄×C₆₇	—	2	4	no	—
170	D₄×C₆₇	—	2	4	no	—
171	D₄×C₆₇	—	2	4	no	—
172	D₄×C₆₇	—	4	32	no	—
173	D₄×C₆₇	—	4	12	no	—
174	D₄×C₆₇	C₂²×D₄	1	8	yes	complemented
175	C₂×C₄×C₆₇	—	2	64	no	—
176	C₂×C₄×C₆₇	—	2	12	no	—
177	C₂×C₄×C₆₇	—	2	24	no	—
178	C₂×C₄×C₆₇	—	2	4	no	—
179	C₂×C₄×C₆₇	—	2	24	no	—
180	C₂×C₄×C₆₇	—	2	4	no	—
181	C₂×C₄×C₆₇	—	2	4	no	—
182	C₂×C₄×C₆₇	—	2	4	no	—
183	C₂×C₄×C₆₇	—	4	12	no	—
184	C₂×C₄×C₆₇	#120	1	12	yes	—
185	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
186	C₂×C₄×C₆₇	C₂²×D₄	1	3	yes	—
187	C₂×C₄×C₆₇	C₂²×D₄	1	6	yes	—
188	C₂×C₄×C₆₇	C₂²×D₄	1	6	yes	—
189	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
190	C₈×C₆₇	—	2	4	no	—
191	C₈×C₆₇	—	2	4	no	—
192	C₂²×C₆₇	—	2	4	no	—
193	C₂²×C₆₇	—	2	8	no	—
194	C₂²×C₆₇	—	2	4	no	—
195	C₂²×C₆₇	—	2	4	no	—
196	C₂²×C₆₇	—	4	24	no	—
197	C₂²×C₆₇	—	4	48	no	—
198	C₂²×C₆₇	—	4	4	no	—
199	C₂²×C₆₇	—	4	4	no	—
200	C₂²×C₆₇	—	4	4	no	—
201	C₂²×C₆₇	—	4	4	no	—
202	C₂²×C₆₇	—	4	4	no	—
203	C₂²×C₆₇	—	8	24	no	—
204	C₂²×C₆₇	#449	1	16	yes	direct product factor
205	C₂²×C₆₇	C₂×#120	1	6	yes	—
206	C₂²×C₆₇	C₂×#135	1	6	yes	—
207	C₂²×C₆₇	C₂×AGL(1,ℤ/8)	1	6	yes	—
208	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
209	C₄×C₆₇	—	2	4	no	—
210	C₄×C₆₇	—	2	8	no	—
211	C₄×C₆₇	—	4	4	no	—
212	C₄×C₆₇	C₂×#120	1	6	yes	—
213	C₄×C₆₇	C₂³×D₄	1	1	yes	characteristic
214	C₄×C₆₇	C₂³×D₄	1	1	yes	characteristic
215	C₂²×#449	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
216	C₂×C₆₇	—	4	4	no	—
217	C₂×C₆₇	—	4	8	no	—
218	C₂×C₆₇	—	8	4	no	—
219	C₂×C₆₇	C₂×#449	1	12	yes	direct product factor
220	C₂×C₆₇	C₂²×#120	1	1	yes	characteristic
221	C₂×C₆₇	C₂²×#135	1	1	yes	characteristic
222	C₂×C₆₇	C₂²×AGL(1,ℤ/8)	1	1	yes	characteristic
223	C₂³×#41	C₂×C₆₇	1	1	yes	characteristic, complemented
224	C₂³×SD₁₆	C₂×C₆₇	1	1	yes	characteristic, complemented
225	C₂³×D₈	C₂×C₆₇	1	1	yes	characteristic, complemented
226	C₂²×#121	C₂×C₆₇	1	1	yes	characteristic, complemented
227	C₂×#449	C₂×C₆₇	1	24	yes	direct product factor
228	C₂²×#103	C₂×C₆₇	1	1	yes	characteristic, complemented
229	C₂²×#102	C₂×C₆₇	1	1	yes	characteristic, complemented
230	C₂²×C₂²⋊C₈	C₂×C₆₇	1	1	yes	characteristic, complemented
231	C₆₇	C₂²×#449	1	1	yes	characteristic, direct product factor, Sylow
232	C₂²×#41	—	2	8	no	—
233	C₂²×#41	—	2	8	no	—
234	C₂²×#41	—	2	6	no	—
235	C₂²×#41	C₂²×C₆₇	1	12	yes	complemented
236	C₂³×Q₈	C₂²×C₆₇	1	1	yes	characteristic
237	C₂³×D₄	—	2	1	no	—
238	C₂³×D₄	—	2	1	no	—
239	C₂³×D₄	C₂²×C₆₇	1	1	yes	characteristic
240	C₂³×D₄	C₂²×C₆₇	1	1	yes	characteristic
241	C₂⁴×C₄	—	2	1	no	—
242	C₂⁴×C₄	C₂²×C₆₇	1	1	yes	characteristic
243	C₂²×SD₁₆	—	2	8	no	—
244	C₂²×SD₁₆	C₂²×C₆₇	1	12	yes	complemented
245	C₂²×D₈	—	2	8	no	—
246	C₂²×D₈	C₂²×C₆₇	1	12	yes	complemented
247	C₂³×C₈	C₂²×C₆₇	1	1	yes	characteristic
248	C₂³×C₈	C₂²×C₆₇	1	1	yes	characteristic
249	C₂×#121	—	2	6	no	—
250	C₂×#121	C₂²×C₆₇	1	12	yes	complemented
251	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
252	C₂²×C₂²⋊C₄	—	2	1	no	—
253	#449	C₂²×C₆₇	1	64	yes	direct product factor
254	C₂×#103	C₂²×C₆₇	1	12	yes	complemented
255	C₂×#102	C₂²×C₆₇	1	12	yes	complemented
256	C₂×C₂²⋊C₈	C₂²×C₆₇	1	12	yes	complemented
257	C₂⁵	—	2	1	no	—
258	C₂⁵	—	2	1	no	—
259	C₂⁵	D₄×C₆₇	1	1	yes	characteristic
260	C₂×#41	—	2	96	no	—
261	C₂×#41	—	2	48	no	—
262	C₂×#41	—	2	48	no	—
263	C₂×#41	—	2	24	no	—
264	C₂×#41	C₂³×C₆₇	1	16	yes	complemented
265	C₂²×Q₈	—	2	4	no	—
266	C₂²×Q₈	—	2	3	no	—
267	C₂²×Q₈	D₄×C₆₇	1	8	yes	complemented
268	C₂²×Q₈	C₂³×C₆₇	1	6	yes	—
269	C₂²×D₄	—	2	16	no	—
270	C₂²×D₄	—	2	12	no	—
271	C₂²×D₄	—	2	12	no	—
272	C₂²×D₄	—	2	3	no	—
273	C₂²×D₄	—	2	4	no	—
274	C₂²×D₄	—	2	4	no	—
275	C₂²×D₄	—	2	4	no	—
276	C₂²×D₄	—	2	3	no	—
277	C₂²×D₄	—	4	8	no	—
278	C₂²×D₄	D₄×C₆₇	1	8	yes	complemented
279	C₂²×D₄	C₂³×C₆₇	1	6	yes	—
280	C₂²×D₄	C₂³×C₆₇	1	6	yes	—
281	C₂³×C₄	—	2	16	no	—
282	C₂³×C₄	—	2	1	no	—
283	C₂³×C₄	—	2	12	no	—
284	C₂³×C₄	—	2	4	no	—
285	C₂³×C₄	—	2	4	no	—
286	C₂³×C₄	—	2	1	no	—
287	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
288	C₂³×C₄	D₄×C₆₇	1	6	yes	—
289	C₂³×C₄	C₂³×C₆₇	1	6	yes	—
290	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
291	C₂×SD₁₆	—	2	48	no	—
292	C₂×SD₁₆	C₂³×C₆₇	1	16	yes	complemented
293	C₂×D₈	—	2	48	no	—
294	C₂×D₈	C₂³×C₆₇	1	16	yes	complemented
295	C₂²×C₈	—	2	4	no	—
296	C₂²×C₈	—	2	4	no	—
297	C₂²×C₈	C₂³×C₆₇	1	6	yes	—
298	C₂²×C₈	C₂³×C₆₇	1	6	yes	—
299	#121	—	2	24	no	—
300	#121	C₂³×C₆₇	1	16	yes	complemented
301	C₂×C₄⋊C₄	—	2	3	no	—
302	C₂×C₄⋊C₄	C₂³×C₆₇	1	6	yes	—
303	C₂×C₂²⋊C₄	—	2	12	no	—
304	#103	C₂³×C₆₇	1	16	yes	complemented
305	#102	C₂³×C₆₇	1	16	yes	complemented
306	C₂²⋊C₈	C₂³×C₆₇	1	16	yes	complemented
307	C₂⁴	—	2	4	no	—
308	C₂⁴	—	2	8	no	—
309	C₂⁴	—	2	4	no	—
310	C₂⁴	—	2	4	no	—
311	C₂⁴	—	2	6	no	—
312	C₂⁴	—	2	6	no	—
313	C₂⁴	—	4	4	no	—
314	C₂⁴	—	4	4	no	—
315	C₂⁴	—	4	4	no	—
316	C₂⁴	—	4	4	no	—
317	C₂⁴	—	4	4	no	—
318	C₂⁴	C₂×D₄×C₆₇	1	6	yes	—
319	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
320	#41	—	2	192	no	—
321	#41	—	2	32	no	—
322	#41	—	2	32	no	—
323	C₂×Q₈	—	2	24	no	—
324	C₂×Q₈	—	2	12	no	—
325	C₂×Q₈	—	2	6	no	—
326	C₂×Q₈	C₂×D₄×C₆₇	1	24	yes	complemented
327	C₂×Q₈	C₂⁴×C₆₇	1	4	yes	—
328	C₂×D₄	—	2	96	no	—
329	C₂×D₄	—	2	16	no	—
330	C₂×D₄	—	2	16	no	—
331	C₂×D₄	—	2	24	no	—
332	C₂×D₄	—	2	6	no	—
333	C₂×D₄	—	2	12	no	—
334	C₂×D₄	—	2	12	no	—
335	C₂×D₄	—	2	12	no	—
336	C₂×D₄	—	2	6	no	—
337	C₂×D₄	—	4	48	no	—
338	C₂×D₄	—	4	12	no	—
339	C₂×D₄	C₂×D₄×C₆₇	1	24	yes	complemented
340	C₂×D₄	C₂⁴×C₆₇	1	4	yes	—
341	C₂×D₄	C₂⁴×C₆₇	1	4	yes	—
342	C₂²×C₄	—	2	96	no	—
343	C₂²×C₄	—	2	4	no	—
344	C₂²×C₄	—	2	8	no	—
345	C₂²×C₄	—	2	6	no	—
346	C₂²×C₄	—	2	24	no	—
347	C₂²×C₄	—	2	16	no	—
348	C₂²×C₄	—	2	12	no	—
349	C₂²×C₄	—	2	12	no	—
350	C₂²×C₄	—	2	6	no	—
351	C₂²×C₄	—	4	4	no	—
352	C₂²×C₄	C₂×D₄×C₆₇	1	6	yes	—
353	C₂²×C₄	C₂×D₄×C₆₇	1	12	yes	—
354	C₂²×C₄	C₂×D₄×C₆₇	1	3	yes	—
355	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
356	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
357	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
358	C₂²×C₄	C₂⁴×C₆₇	1	3	yes	—
359	SD₁₆	—	2	32	no	—
360	D₈	—	2	32	no	—
361	C₂×C₈	—	2	12	no	—
362	C₂×C₈	—	2	12	no	—
363	C₂×C₈	C₂⁴×C₆₇	1	4	yes	—
364	C₂×C₈	C₂⁴×C₆₇	1	4	yes	—
365	C₄⋊C₄	—	2	6	no	—
366	C₄⋊C₄	C₂⁴×C₆₇	1	4	yes	—
367	C₂²⋊C₄	—	2	16	no	—
368	C₂³	—	2	12	no	—
369	C₂³	—	2	24	no	—
370	C₂³	—	2	12	no	—
371	C₂³	—	2	12	no	—
372	C₂³	—	2	4	no	—
373	C₂³	—	2	4	no	—
374	C₂³	—	4	16	no	—
375	C₂³	—	4	32	no	—
376	C₂³	—	4	12	no	—
377	C₂³	—	4	12	no	—
378	C₂³	—	4	12	no	—
379	C₂³	—	4	12	no	—
380	C₂³	—	4	12	no	—
381	C₂³	—	8	16	no	—
382	C₂³	#120×C₆₇	1	4	yes	—
383	C₂³	#135×C₆₇	1	4	yes	—
384	C₂³	AGL(1,ℤ/8)×C₆₇	1	4	yes	—
385	C₂³	C₂²×D₄×C₆₇	1	4	yes	—
386	C₂³	C₂²×D₄×C₆₇	1	3	yes	—
387	Q₈	—	2	24	no	—
388	Q₈	—	2	4	no	—
389	Q₈	C₂²×D₄×C₆₇	1	8	yes	complemented
390	D₄	—	2	64	no	—
391	D₄	—	2	24	no	—
392	D₄	—	2	4	no	—
393	D₄	—	2	4	no	—
394	D₄	—	2	4	no	—
395	D₄	—	4	32	no	—
396	D₄	—	4	12	no	—
397	D₄	C₂²×D₄×C₆₇	1	8	yes	complemented
398	C₂×C₄	—	2	64	no	—
399	C₂×C₄	—	2	12	no	—
400	C₂×C₄	—	2	24	no	—
401	C₂×C₄	—	2	4	no	—
402	C₂×C₄	—	2	24	no	—
403	C₂×C₄	—	2	4	no	—
404	C₂×C₄	—	2	4	no	—
405	C₂×C₄	—	2	4	no	—
406	C₂×C₄	—	4	12	no	—
407	C₂×C₄	#120×C₆₇	1	12	yes	—
408	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
409	C₂×C₄	C₂²×D₄×C₆₇	1	3	yes	—
410	C₂×C₄	C₂²×D₄×C₆₇	1	6	yes	—
411	C₂×C₄	C₂²×D₄×C₆₇	1	6	yes	—
412	C₂×C₄	C₂⁵×C₆₇	1	1	yes	characteristic
413	C₈	—	2	4	no	—
414	C₈	—	2	4	no	—
415	C₂²	—	2	4	no	—
416	C₂²	—	2	4	no	—
417	C₂²	—	2	4	no	—
418	C₂²	—	2	8	no	—
419	C₂²	—	4	4	no	—
420	C₂²	—	4	4	no	—
421	C₂²	—	4	24	no	—
422	C₂²	—	4	48	no	—
423	C₂²	—	4	4	no	—
424	C₂²	—	4	4	no	—
425	C₂²	—	4	4	no	—
426	C₂²	—	8	24	no	—
427	C₂²	#449×C₆₇	1	16	yes	direct product factor
428	C₂²	C₂×#120×C₆₇	1	6	yes	—
429	C₂²	C₂×#135×C₆₇	1	6	yes	—
430	C₂²	C₂×AGL(1,ℤ/8)×C₆₇	1	6	yes	—
431	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
432	C₄	—	2	4	no	—
433	C₄	—	2	8	no	—
434	C₄	—	4	4	no	—
435	C₄	C₂×#120×C₆₇	1	6	yes	—
436	C₄	C₂³×D₄×C₆₇	1	1	yes	characteristic
437	C₄	C₂³×D₄×C₆₇	1	1	yes	characteristic
438	C₂	—	4	4	no	—
439	C₂	—	4	8	no	—
440	C₂	—	8	4	no	—
441	C₂	C₂×#449×C₆₇	1	12	yes	direct product factor
442	C₂	C₂²×#120×C₆₇	1	1	yes	characteristic
443	C₂	C₂²×#135×C₆₇	1	1	yes	characteristic
444	C₂	C₂²×AGL(1,ℤ/8)×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂⁴×C₆₇	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₆₇
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₆₇ ◁ C₂⁴×C₄×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂³×#41×C₆₇	C₂	semidirect
2. C₂³×SD₁₆×C₆₇	C₂	semidirect
3. C₂³×D₈×C₆₇	C₂	semidirect
4. C₂²×#121×C₆₇	C₂	semidirect
5. C₂×#449×C₆₇	C₂	direct (N × H)
6. C₂²×#103×C₆₇	C₂	semidirect
7. C₂²×#102×C₆₇	C₂	semidirect
8. C₂²×C₂²⋊C₈×C₆₇	C₂	semidirect
12. C₂²×#41×C₆₇	C₂²	semidirect
21. C₂²×SD₁₆×C₆₇	C₂²	semidirect
23. C₂²×D₈×C₆₇	C₂²	semidirect
27. C₂×#121×C₆₇	C₂²	semidirect
30. #449×C₆₇	C₂²	direct (N × H)
31. C₂×#103×C₆₇	C₂²	semidirect
32. C₂×#102×C₆₇	C₂²	semidirect
33. C₂×C₂²⋊C₈×C₆₇	C₂²	semidirect
41. C₂×#41×C₆₇	C₂³	semidirect
44. C₂²×Q₈×C₆₇	D₄	semidirect
55. C₂²×D₄×C₆₇	D₄	semidirect
69. C₂×SD₁₆×C₆₇	C₂³	semidirect
71. C₂×D₈×C₆₇	C₂³	semidirect
77. #121×C₆₇	C₂³	semidirect
81. #103×C₆₇	C₂³	semidirect
82. #102×C₆₇	C₂³	semidirect
83. C₂²⋊C₈×C₆₇	C₂³	semidirect
103. C₂×Q₈×C₆₇	C₂×D₄	semidirect
116. C₂×D₄×C₆₇	C₂×D₄	semidirect
166. Q₈×C₆₇	C₂²×D₄	semidirect
174. D₄×C₆₇	C₂²×D₄	semidirect
204. C₂²×C₆₇	#449	direct (N × H)
215. C₂²×#449	C₆₇	direct (N × H)
219. C₂×C₆₇	C₂×#449	direct (N × H)
223. C₂³×#41	C₂×C₆₇	semidirect
224. C₂³×SD₁₆	C₂×C₆₇	semidirect
225. C₂³×D₈	C₂×C₆₇	semidirect
226. C₂²×#121	C₂×C₆₇	semidirect
227. C₂×#449	C₂×C₆₇	direct (N × H)
228. C₂²×#103	C₂×C₆₇	semidirect
229. C₂²×#102	C₂×C₆₇	semidirect
230. C₂²×C₂²⋊C₈	C₂×C₆₇	semidirect
231. C₆₇	C₂²×#449	direct (N × H)
235. C₂²×#41	C₂²×C₆₇	semidirect
244. C₂²×SD₁₆	C₂²×C₆₇	semidirect
246. C₂²×D₈	C₂²×C₆₇	semidirect
250. C₂×#121	C₂²×C₆₇	semidirect
253. #449	C₂²×C₆₇	direct (N × H)
254. C₂×#103	C₂²×C₆₇	semidirect
255. C₂×#102	C₂²×C₆₇	semidirect
256. C₂×C₂²⋊C₈	C₂²×C₆₇	semidirect
264. C₂×#41	C₂³×C₆₇	semidirect
267. C₂²×Q₈	D₄×C₆₇	semidirect
278. C₂²×D₄	D₄×C₆₇	semidirect
292. C₂×SD₁₆	C₂³×C₆₇	semidirect
294. C₂×D₈	C₂³×C₆₇	semidirect
300. #121	C₂³×C₆₇	semidirect
304. #103	C₂³×C₆₇	semidirect
305. #102	C₂³×C₆₇	semidirect
306. C₂²⋊C₈	C₂³×C₆₇	semidirect
326. C₂×Q₈	C₂×D₄×C₆₇	semidirect
339. C₂×D₄	C₂×D₄×C₆₇	semidirect
389. Q₈	C₂²×D₄×C₆₇	semidirect
397. D₄	C₂²×D₄×C₆₇	semidirect
427. C₂²	#449×C₆₇	direct (N × H)
441. C₂	C₂×#449×C₆₇	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×#41×C₆₇	2
C₂³×SD₁₆×C₆₇	2
C₂³×D₈×C₆₇	2
C₂²×#121×C₆₇	2
C₂×#449×C₆₇	2
C₂²×#103×C₆₇	2
C₂²×#102×C₆₇	2
C₂²×C₂²⋊C₈×C₆₇	2
C₂²×#449	67

Subgroups — C22×#449×C67 #30739602

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#449×C₆₇ #30739602