Subgroups — C₂×C₆₇×#2212 #30712842

Order 17152.

706 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#510×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#509×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₂×#495×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#443×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#2212	C₂	1	8	yes	maximal, direct product factor
6	C₂×#364×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#362×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#361×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#135×C₆₇	C₂²	1	1	yes	characteristic
10	C₂²×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic
11	C₂²×SD₁₆×C₆₇	—	2	1	no	—
12	C₂²×D₈×C₆₇	C₂²	1	1	yes	characteristic
13	C₂×#131×C₆₇	—	2	1	no	—
14	C₂×#124×C₆₇	—	2	1	no	—
15	#510×C₆₇	—	2	2	no	—
16	#510×C₆₇	C₂²	1	4	yes	complemented
17	#509×C₆₇	—	2	2	no	—
18	#509×C₆₇	C₂²	1	4	yes	—
19	C₂×Q₃₂×C₆₇	—	2	1	no	—
20	C₂×SD₃₂×C₆₇	—	2	1	no	—
21	C₂×SD₃₂×C₆₇	—	2	1	no	—
22	C₂×D₁₆×C₆₇	—	2	1	no	—
23	C₂×M₃₂×C₆₇	C₂²	1	1	yes	characteristic
24	C₂×M₃₂×C₆₇	C₂²	1	1	yes	characteristic
25	#495×C₆₇	—	2	2	no	—
26	#495×C₆₇	C₂²	1	4	yes	complemented
27	#443×C₆₇	—	2	2	no	—
28	#443×C₆₇	C₂²	1	4	yes	complemented
29	C₂×#108×C₆₇	C₂²	1	1	yes	characteristic
30	C₂×#104×C₆₇	—	2	1	no	—
31	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
32	#364×C₆₇	C₂²	1	4	yes	complemented
33	#362×C₆₇	C₂²	1	4	yes	complemented
34	#361×C₆₇	C₂²	1	4	yes	complemented
35	C₂×#41×C₆₇	—	2	1	no	—
36	C₂²×Q₈×C₆₇	—	2	1	no	—
37	C₂²×D₄×C₆₇	—	2	1	no	—
38	#135×C₆₇	—	2	1	no	—
39	#135×C₆₇	—	2	1	no	—
40	#135×C₆₇	D₄	1	2	yes	—
41	#135×C₆₇	C₂³	1	2	yes	—
42	C₂×Q₁₆×C₆₇	—	2	1	no	—
43	C₂×Q₁₆×C₆₇	—	2	1	no	—
44	C₂×Q₁₆×C₆₇	—	2	1	no	—
45	C₂×Q₁₆×C₆₇	—	2	1	no	—
46	C₂×Q₁₆×C₆₇	—	2	1	no	—
47	C₂×Q₁₆×C₆₇	D₄	1	1	yes	characteristic
48	C₂×Q₁₆×C₆₇	C₂³	1	2	yes	—
49	C₂×SD₁₆×C₆₇	—	2	4	no	—
50	C₂×SD₁₆×C₆₇	—	2	1	no	—
51	C₂×SD₁₆×C₆₇	—	4	1	no	—
52	C₂×SD₁₆×C₆₇	—	4	1	no	—
53	C₂×SD₁₆×C₆₇	—	4	2	no	—
54	C₂×D₈×C₆₇	—	2	1	no	—
55	C₂×D₈×C₆₇	—	2	1	no	—
56	C₂×D₈×C₆₇	—	2	1	no	—
57	C₂×D₈×C₆₇	—	2	1	no	—
58	C₂×D₈×C₆₇	—	2	1	no	—
59	C₂×D₈×C₆₇	D₄	1	1	yes	characteristic
60	C₂×D₈×C₆₇	C₂³	1	2	yes	—
61	#131×C₆₇	—	2	4	no	—
62	#131×C₆₇	—	4	2	no	—
63	C₂×M₁₆×C₆₇	—	2	1	no	—
64	C₂×M₁₆×C₆₇	—	4	1	no	—
65	C₂²×C₈×C₆₇	—	4	1	no	—
66	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
67	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
68	#124×C₆₇	—	2	4	no	—
69	#124×C₆₇	—	4	2	no	—
70	C₂×C₂²⋊C₄×C₆₇	—	4	2	no	—
71	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
72	Q₃₂×C₆₇	—	2	4	no	—
73	SD₃₂×C₆₇	—	2	4	no	—
74	SD₃₂×C₆₇	—	2	4	no	—
75	D₁₆×C₆₇	—	2	4	no	—
76	M₃₂×C₆₇	—	2	1	no	—
77	M₃₂×C₆₇	—	2	1	no	—
78	M₃₂×C₆₇	C₂³	1	2	yes	—
79	M₃₂×C₆₇	C₂³	1	2	yes	—
80	C₂×C₁₆×C₆₇	—	2	1	no	—
81	C₂×C₁₆×C₆₇	—	2	1	no	—
82	#108×C₆₇	—	2	1	no	—
83	#108×C₆₇	C₂³	1	2	yes	—
84	#104×C₆₇	—	2	4	no	—
85	C₄×C₈×C₆₇	D₄	1	2	yes	—
86	C₄×C₈×C₆₇	C₂³	1	2	yes	—
87	C₂⁴×C₆₇	—	4	1	no	—
88	#41×C₆₇	—	2	2	no	—
89	#41×C₆₇	—	2	2	no	—
90	#41×C₆₇	—	4	1	no	—
91	#41×C₆₇	—	4	1	no	—
92	C₂×Q₈×C₆₇	—	2	1	no	—
93	C₂×Q₈×C₆₇	—	2	2	no	—
94	C₂×Q₈×C₆₇	—	4	1	no	—
95	C₂×Q₈×C₆₇	—	4	1	no	—
96	C₂×Q₈×C₆₇	—	4	1	no	—
97	C₂×Q₈×C₆₇	—	4	1	no	—
98	C₂×Q₈×C₆₇	—	4	1	no	—
99	C₂×D₄×C₆₇	—	2	1	no	—
100	C₂×D₄×C₆₇	—	2	2	no	—
101	C₂×D₄×C₆₇	—	4	1	no	—
102	C₂×D₄×C₆₇	—	4	1	no	—
103	C₂×D₄×C₆₇	—	4	1	no	—
104	C₂×D₄×C₆₇	—	4	1	no	—
105	C₂×D₄×C₆₇	—	4	1	no	—
106	C₂×D₄×C₆₇	—	4	1	no	—
107	C₂²×C₄×C₆₇	—	2	1	no	—
108	C₂²×C₄×C₆₇	—	4	1	no	—
109	C₂²×C₄×C₆₇	—	4	1	no	—
110	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
111	Q₁₆×C₆₇	—	2	1	no	—
112	Q₁₆×C₆₇	—	2	2	no	—
113	Q₁₆×C₆₇	—	2	2	no	—
114	Q₁₆×C₆₇	—	4	1	no	—
115	Q₁₆×C₆₇	—	4	1	no	—
116	Q₁₆×C₆₇	C₂×D₄	1	2	yes	—
117	SD₁₆×C₆₇	—	2	2	no	—
118	SD₁₆×C₆₇	—	2	2	no	—
119	SD₁₆×C₆₇	—	4	4	no	—
120	SD₁₆×C₆₇	—	4	4	no	—
121	D₈×C₆₇	—	2	1	no	—
122	D₈×C₆₇	—	2	2	no	—
123	D₈×C₆₇	—	2	2	no	—
124	D₈×C₆₇	—	4	1	no	—
125	D₈×C₆₇	—	4	1	no	—
126	D₈×C₆₇	C₂×D₄	1	2	yes	—
127	M₁₆×C₆₇	—	2	2	no	—
128	M₁₆×C₆₇	—	4	1	no	—
129	M₁₆×C₆₇	—	4	4	no	—
130	C₂×C₈×C₆₇	—	4	1	no	—
131	C₂×C₈×C₆₇	—	4	4	no	—
132	C₂×C₈×C₆₇	D₈	1	1	yes	characteristic
133	C₂×C₈×C₆₇	D₈	1	1	yes	characteristic
134	C₂×C₈×C₆₇	D₈	1	2	yes	—
135	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
136	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
137	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
138	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
139	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
140	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
141	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
142	C₂²⋊C₄×C₆₇	—	4	8	no	—
143	C₄²×C₆₇	—	2	1	no	—
144	C₄²×C₆₇	C₂×D₄	1	2	yes	—
145	C₁₆×C₆₇	—	2	2	no	—
146	C₁₆×C₆₇	—	2	2	no	—
147	C₂³×C₆₇	—	4	1	no	—
148	C₂³×C₆₇	—	4	2	no	—
149	C₂³×C₆₇	—	8	4	no	—
150	C₂³×C₆₇	—	8	1	no	—
151	C₂³×C₆₇	—	8	1	no	—
152	C₂³×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
153	Q₈×C₆₇	—	2	2	no	—
154	Q₈×C₆₇	—	4	1	no	—
155	Q₈×C₆₇	—	4	2	no	—
156	Q₈×C₆₇	—	4	2	no	—
157	Q₈×C₆₇	—	8	1	no	—
158	Q₈×C₆₇	—	8	1	no	—
159	D₄×C₆₇	—	2	2	no	—
160	D₄×C₆₇	—	4	2	no	—
161	D₄×C₆₇	—	4	2	no	—
162	D₄×C₆₇	—	4	2	no	—
163	D₄×C₆₇	—	4	2	no	—
164	D₄×C₆₇	—	4	1	no	—
165	D₄×C₆₇	—	8	1	no	—
166	D₄×C₆₇	—	8	1	no	—
167	C₂×C₄×C₆₇	—	2	2	no	—
168	C₂×C₄×C₆₇	—	4	2	no	—
169	C₂×C₄×C₆₇	—	4	1	no	—
170	C₂×C₄×C₆₇	—	4	2	no	—
171	C₂×C₄×C₆₇	—	4	2	no	—
172	C₂×C₄×C₆₇	—	4	2	no	—
173	C₂×C₄×C₆₇	—	8	1	no	—
174	C₂×C₄×C₆₇	—	8	1	no	—
175	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
176	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
177	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
178	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
179	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
180	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
181	C₈×C₆₇	—	4	2	no	—
182	C₈×C₆₇	#121	1	1	yes	characteristic
183	C₈×C₆₇	#121	1	1	yes	characteristic
184	C₈×C₆₇	C₂×D₈	1	2	yes	—
185	C₈×C₆₇	C₂×D₈	1	2	yes	—
186	C₈×C₆₇	C₂²×D₄	1	1	yes	characteristic
187	C₈×C₆₇	C₂×#41	1	1	yes	characteristic
188	C₂²×C₆₇	—	2	2	no	—
189	C₂²×C₆₇	—	4	2	no	—
190	C₂²×C₆₇	—	8	4	no	—
191	C₂²×C₆₇	—	8	2	no	—
192	C₂²×C₆₇	—	8	2	no	—
193	C₂²×C₆₇	—	16	2	no	—
194	C₂²×C₆₇	—	16	2	no	—
195	C₂²×C₆₇	#459	1	1	yes	characteristic
196	C₂²×C₆₇	#459	1	1	yes	characteristic
197	C₂²×C₆₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
198	C₄×C₆₇	—	4	2	no	—
199	C₄×C₆₇	—	4	2	no	—
200	C₄×C₆₇	—	8	2	no	—
201	C₄×C₆₇	#459	1	1	yes	characteristic
202	C₄×C₆₇	#459	1	1	yes	characteristic
203	C₄×C₆₇	C₂×#121	1	1	yes	characteristic
204	C₄×C₆₇	C₂²×D₈	1	1	yes	characteristic
205	C₂×#2212	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
206	C₂×C₆₇	—	2	1	no	—
207	C₂×C₆₇	—	2	1	no	—
208	C₂×C₆₇	—	8	2	no	—
209	C₂×C₆₇	—	16	2	no	—
210	C₂×C₆₇	#2212	1	2	yes	direct product factor
211	C₂×C₆₇	C₂×#459	1	1	yes	characteristic
212	C₂×#510	C₂×C₆₇	1	1	yes	characteristic, complemented
213	C₂×#509	C₂×C₆₇	1	1	yes	characteristic
214	C₂×#495	C₂×C₆₇	1	1	yes	characteristic, complemented
215	C₂×#443	C₂×C₆₇	1	1	yes	characteristic, complemented
216	#2212	C₂×C₆₇	1	8	yes	direct product factor
217	C₂×#364	C₂×C₆₇	1	1	yes	characteristic, complemented
218	C₂×#362	C₂×C₆₇	1	1	yes	characteristic, complemented
219	C₂×#361	C₂×C₆₇	1	1	yes	characteristic, complemented
220	C₆₇	C₂×#2212	1	1	yes	characteristic, direct product factor, Sylow
221	C₂×#135	C₂²×C₆₇	1	1	yes	characteristic
222	C₂²×Q₁₆	C₂²×C₆₇	1	1	yes	characteristic
223	C₂²×SD₁₆	—	2	1	no	—
224	C₂²×D₈	C₂²×C₆₇	1	1	yes	characteristic
225	C₂×#131	—	2	1	no	—
226	C₂×#124	—	2	1	no	—
227	#510	—	2	2	no	—
228	#510	C₂²×C₆₇	1	4	yes	complemented
229	#509	—	2	2	no	—
230	#509	C₂²×C₆₇	1	4	yes	—
231	C₂×Q₃₂	—	2	1	no	—
232	C₂×SD₃₂	—	2	1	no	—
233	C₂×SD₃₂	—	2	1	no	—
234	C₂×D₁₆	—	2	1	no	—
235	C₂×M₃₂	C₂²×C₆₇	1	1	yes	characteristic
236	C₂×M₃₂	C₂²×C₆₇	1	1	yes	characteristic
237	#495	—	2	2	no	—
238	#495	C₂²×C₆₇	1	4	yes	complemented
239	#443	—	2	2	no	—
240	#443	C₂²×C₆₇	1	4	yes	complemented
241	C₂×#108	C₂²×C₆₇	1	1	yes	characteristic
242	C₂×#104	—	2	1	no	—
243	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
244	#364	C₂²×C₆₇	1	4	yes	complemented
245	#362	C₂²×C₆₇	1	4	yes	complemented
246	#361	C₂²×C₆₇	1	4	yes	complemented
247	C₂×#41	—	2	1	no	—
248	C₂²×Q₈	—	2	1	no	—
249	C₂²×D₄	—	2	1	no	—
250	#135	—	2	1	no	—
251	#135	—	2	1	no	—
252	#135	D₄×C₆₇	1	2	yes	—
253	#135	C₂³×C₆₇	1	2	yes	—
254	C₂×Q₁₆	—	2	1	no	—
255	C₂×Q₁₆	—	2	1	no	—
256	C₂×Q₁₆	—	2	1	no	—
257	C₂×Q₁₆	—	2	1	no	—
258	C₂×Q₁₆	—	2	1	no	—
259	C₂×Q₁₆	D₄×C₆₇	1	1	yes	characteristic
260	C₂×Q₁₆	C₂³×C₆₇	1	2	yes	—
261	C₂×SD₁₆	—	2	4	no	—
262	C₂×SD₁₆	—	2	1	no	—
263	C₂×SD₁₆	—	4	1	no	—
264	C₂×SD₁₆	—	4	1	no	—
265	C₂×SD₁₆	—	4	2	no	—
266	C₂×D₈	—	2	1	no	—
267	C₂×D₈	—	2	1	no	—
268	C₂×D₈	—	2	1	no	—
269	C₂×D₈	—	2	1	no	—
270	C₂×D₈	—	2	1	no	—
271	C₂×D₈	D₄×C₆₇	1	1	yes	characteristic
272	C₂×D₈	C₂³×C₆₇	1	2	yes	—
273	#131	—	2	4	no	—
274	#131	—	4	2	no	—
275	C₂×M₁₆	—	2	1	no	—
276	C₂×M₁₆	—	4	1	no	—
277	C₂²×C₈	—	4	1	no	—
278	C₂²×C₈	D₄×C₆₇	1	1	yes	characteristic
279	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
280	#124	—	2	4	no	—
281	#124	—	4	2	no	—
282	C₂×C₂²⋊C₄	—	4	2	no	—
283	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
284	Q₃₂	—	2	4	no	—
285	SD₃₂	—	2	4	no	—
286	SD₃₂	—	2	4	no	—
287	D₁₆	—	2	4	no	—
288	M₃₂	—	2	1	no	—
289	M₃₂	—	2	1	no	—
290	M₃₂	C₂³×C₆₇	1	2	yes	—
291	M₃₂	C₂³×C₆₇	1	2	yes	—
292	C₂×C₁₆	—	2	1	no	—
293	C₂×C₁₆	—	2	1	no	—
294	#108	—	2	1	no	—
295	#108	C₂³×C₆₇	1	2	yes	—
296	#104	—	2	4	no	—
297	C₄×C₈	D₄×C₆₇	1	2	yes	—
298	C₄×C₈	C₂³×C₆₇	1	2	yes	—
299	C₂⁴	—	4	1	no	—
300	#41	—	2	2	no	—
301	#41	—	2	2	no	—
302	#41	—	4	1	no	—
303	#41	—	4	1	no	—
304	C₂×Q₈	—	2	1	no	—
305	C₂×Q₈	—	2	2	no	—
306	C₂×Q₈	—	4	1	no	—
307	C₂×Q₈	—	4	1	no	—
308	C₂×Q₈	—	4	1	no	—
309	C₂×Q₈	—	4	1	no	—
310	C₂×Q₈	—	4	1	no	—
311	C₂×D₄	—	2	1	no	—
312	C₂×D₄	—	2	2	no	—
313	C₂×D₄	—	4	1	no	—
314	C₂×D₄	—	4	1	no	—
315	C₂×D₄	—	4	1	no	—
316	C₂×D₄	—	4	1	no	—
317	C₂×D₄	—	4	1	no	—
318	C₂×D₄	—	4	1	no	—
319	C₂²×C₄	—	2	1	no	—
320	C₂²×C₄	—	4	1	no	—
321	C₂²×C₄	—	4	1	no	—
322	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
323	Q₁₆	—	2	1	no	—
324	Q₁₆	—	2	2	no	—
325	Q₁₆	—	2	2	no	—
326	Q₁₆	—	4	1	no	—
327	Q₁₆	—	4	1	no	—
328	Q₁₆	C₂×D₄×C₆₇	1	2	yes	—
329	SD₁₆	—	2	2	no	—
330	SD₁₆	—	2	2	no	—
331	SD₁₆	—	4	4	no	—
332	SD₁₆	—	4	4	no	—
333	D₈	—	2	1	no	—
334	D₈	—	2	2	no	—
335	D₈	—	2	2	no	—
336	D₈	—	4	1	no	—
337	D₈	—	4	1	no	—
338	D₈	C₂×D₄×C₆₇	1	2	yes	—
339	M₁₆	—	2	2	no	—
340	M₁₆	—	4	1	no	—
341	M₁₆	—	4	4	no	—
342	C₂×C₈	—	4	1	no	—
343	C₂×C₈	—	4	4	no	—
344	C₂×C₈	D₈×C₆₇	1	1	yes	characteristic
345	C₂×C₈	D₈×C₆₇	1	1	yes	characteristic
346	C₂×C₈	D₈×C₆₇	1	2	yes	—
347	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
348	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
349	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
350	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
351	C₂×C₈	#41×C₆₇	1	1	yes	characteristic
352	C₂×C₈	#41×C₆₇	1	1	yes	characteristic
353	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
354	C₂²⋊C₄	—	4	8	no	—
355	C₄²	—	2	1	no	—
356	C₄²	C₂×D₄×C₆₇	1	2	yes	—
357	C₁₆	—	2	2	no	—
358	C₁₆	—	2	2	no	—
359	C₂³	—	4	1	no	—
360	C₂³	—	4	2	no	—
361	C₂³	—	8	4	no	—
362	C₂³	—	8	1	no	—
363	C₂³	—	8	1	no	—
364	C₂³	AGL(1,ℤ/8)×C₆₇	1	1	yes	characteristic
365	Q₈	—	2	2	no	—
366	Q₈	—	4	1	no	—
367	Q₈	—	4	2	no	—
368	Q₈	—	4	2	no	—
369	Q₈	—	8	1	no	—
370	Q₈	—	8	1	no	—
371	D₄	—	2	2	no	—
372	D₄	—	4	2	no	—
373	D₄	—	4	2	no	—
374	D₄	—	4	1	no	—
375	D₄	—	4	2	no	—
376	D₄	—	4	2	no	—
377	D₄	—	8	1	no	—
378	D₄	—	8	1	no	—
379	C₂×C₄	—	2	2	no	—
380	C₂×C₄	—	4	2	no	—
381	C₂×C₄	—	4	1	no	—
382	C₂×C₄	—	4	2	no	—
383	C₂×C₄	—	4	2	no	—
384	C₂×C₄	—	4	2	no	—
385	C₂×C₄	—	8	1	no	—
386	C₂×C₄	—	8	1	no	—
387	C₂×C₄	#121×C₆₇	1	1	yes	characteristic
388	C₂×C₄	#121×C₆₇	1	1	yes	characteristic
389	C₂×C₄	C₂×D₈×C₆₇	1	1	yes	characteristic
390	C₂×C₄	C₂×D₈×C₆₇	1	1	yes	characteristic
391	C₂×C₄	AGL(1,ℤ/8)×C₆₇	1	1	yes	characteristic
392	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
393	C₈	—	4	2	no	—
394	C₈	#121×C₆₇	1	1	yes	characteristic
395	C₈	#121×C₆₇	1	1	yes	characteristic
396	C₈	C₂×D₈×C₆₇	1	2	yes	—
397	C₈	C₂×D₈×C₆₇	1	2	yes	—
398	C₈	C₂²×D₄×C₆₇	1	1	yes	characteristic
399	C₈	C₂×#41×C₆₇	1	1	yes	characteristic
400	C₂²	—	2	2	no	—
401	C₂²	—	4	2	no	—
402	C₂²	—	8	4	no	—
403	C₂²	—	8	2	no	—
404	C₂²	—	8	2	no	—
405	C₂²	—	16	2	no	—
406	C₂²	—	16	2	no	—
407	C₂²	#459×C₆₇	1	1	yes	characteristic
408	C₂²	#459×C₆₇	1	1	yes	characteristic
409	C₂²	C₂×AGL(1,ℤ/8)×C₆₇	1	1	yes	characteristic
410	C₄	—	4	2	no	—
411	C₄	—	4	2	no	—
412	C₄	—	8	2	no	—
413	C₄	#459×C₆₇	1	1	yes	characteristic
414	C₄	#459×C₆₇	1	1	yes	characteristic
415	C₄	C₂×#121×C₆₇	1	1	yes	characteristic
416	C₄	C₂²×D₈×C₆₇	1	1	yes	characteristic
417	C₂	—	2	1	no	—
418	C₂	—	2	1	no	—
419	C₂	—	8	2	no	—
420	C₂	—	16	2	no	—
421	C₂	C₆₇×#2212	1	2	yes	direct product factor
422	C₂	C₂×#459×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₇	#459
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₆₇
prime radical	C₂×#509×C₆₇	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ C₂²×C₄×C₆₇ ◁ C₂×C₄×C₈×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#510×C₆₇	C₂	semidirect
3. C₂×#495×C₆₇	C₂	semidirect
4. C₂×#443×C₆₇	C₂	semidirect
5. C₆₇×#2212	C₂	direct (N × H)
6. C₂×#364×C₆₇	C₂	semidirect
7. C₂×#362×C₆₇	C₂	semidirect
8. C₂×#361×C₆₇	C₂	semidirect
16. #510×C₆₇	C₂²	semidirect
26. #495×C₆₇	C₂²	semidirect
28. #443×C₆₇	C₂²	semidirect
32. #364×C₆₇	C₂²	semidirect
33. #362×C₆₇	C₂²	semidirect
34. #361×C₆₇	C₂²	semidirect
205. C₂×#2212	C₆₇	direct (N × H)
210. C₂×C₆₇	#2212	direct (N × H)
212. C₂×#510	C₂×C₆₇	semidirect
214. C₂×#495	C₂×C₆₇	semidirect
215. C₂×#443	C₂×C₆₇	semidirect
216. #2212	C₂×C₆₇	direct (N × H)
217. C₂×#364	C₂×C₆₇	semidirect
218. C₂×#362	C₂×C₆₇	semidirect
219. C₂×#361	C₂×C₆₇	semidirect
220. C₆₇	C₂×#2212	direct (N × H)
228. #510	C₂²×C₆₇	semidirect
238. #495	C₂²×C₆₇	semidirect
240. #443	C₂²×C₆₇	semidirect
244. #364	C₂²×C₆₇	semidirect
245. #362	C₂²×C₆₇	semidirect
246. #361	C₂²×C₆₇	semidirect
421. C₂	C₆₇×#2212	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#510×C₆₇	2
C₂×#509×C₆₇	2
C₂×#495×C₆₇	2
C₂×#443×C₆₇	2
C₆₇×#2212	2
C₂×#364×C₆₇	2
C₂×#362×C₆₇	2
C₂×#361×C₆₇	2
C₂×#2212	67

Subgroups — C2×C67×#2212 #30712842

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₇×#2212 #30712842