Subgroups — C₆₇×#19968 #30699219

Order 17152.

780 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#3294	C₂	1	2	yes	maximal, complemented
2	C₆₇×#3283	C₂	1	1	yes	maximal, characteristic, complemented
3	D₄×D₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#1709	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#1681	C₂	1	2	yes	maximal, complemented
6	C₆₇×#1669	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#1667	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₇×#1639	C₂	1	2	yes	maximal, complemented
9	C₆₇×#1625	C₂	1	2	yes	maximal, complemented
10	C₈×D₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×D₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#135×C₆₇	—	2	1	no	—
13	C₂²×D₈×C₆₇	—	2	1	no	—
14	#547×C₆₇	C₂²	1	2	yes	complemented
15	D₄²×C₆₇	C₂²	1	2	yes	—
16	#494×C₆₇	C₂²	1	1	yes	characteristic, complemented
17	#493×C₆₇	C₂²	1	1	yes	characteristic
18	#486×C₆₇	C₂²	1	2	yes	complemented
19	#486×C₆₇	C₂²	1	2	yes	complemented
20	#480×C₆₇	—	2	2	no	—
21	#474×C₆₇	C₂²	1	2	yes	—
22	#467×C₆₇	—	2	1	no	—
23	#466×C₆₇	—	2	1	no	—
24	#466×C₆₇	—	2	2	no	—
25	#461×C₆₇	C₂²	1	2	yes	complemented
26	#459×C₆₇	C₂²	1	2	yes	—
27	#459×C₆₇	C₂²	1	2	yes	complemented
28	#452×C₆₇	—	2	2	no	—
29	#447×C₆₇	—	2	2	no	—
30	#447×C₆₇	—	2	2	no	—
31	C₄×D₈×C₆₇	C₂²	1	1	yes	characteristic
32	C₄×D₈×C₆₇	C₂²	1	1	yes	characteristic
33	C₈×D₄×C₆₇	C₂²	1	1	yes	characteristic
34	C₈×D₄×C₆₇	C₂²	1	1	yes	characteristic, complemented
35	C₈×D₄×C₆₇	C₂²	1	2	yes	complemented
36	C₂×#102×C₆₇	—	2	2	no	—
37	#335×C₆₇	C₂²	1	1	yes	characteristic, complemented
38	#335×C₆₇	C₂²	1	1	yes	characteristic, complemented
39	#332×C₆₇	C₂²	1	2	yes	complemented
40	#331×C₆₇	C₂²	1	2	yes	complemented
41	D₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
42	D₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic, complemented
43	D₄⋊C₈×C₆₇	C₂²	1	2	yes	complemented
44	C₈²×C₆₇	C₂²	1	1	yes	characteristic, complemented
45	C₂×#41×C₆₇	—	2	2	no	—
46	C₂²×D₄×C₆₇	—	2	2	no	—
47	C₂²×D₄×C₆₇	—	2	2	no	—
48	#135×C₆₇	—	4	4	no	—
49	C₂×Q₁₆×C₆₇	—	2	1	no	—
50	C₂×Q₁₆×C₆₇	—	2	1	no	—
51	C₂×SD₁₆×C₆₇	—	2	2	no	—
52	C₂×D₈×C₆₇	—	2	2	no	—
53	C₂×D₈×C₆₇	—	2	2	no	—
54	C₂×D₈×C₆₇	—	2	1	no	—
55	C₂×D₈×C₆₇	—	2	1	no	—
56	C₂×D₈×C₆₇	—	4	4	no	—
57	C₂×D₈×C₆₇	D₄	1	1	yes	characteristic
58	C₂×D₈×C₆₇	D₄	1	1	yes	characteristic
59	C₂²×C₈×C₆₇	—	2	1	no	—
60	C₂²×C₈×C₆₇	—	2	1	no	—
61	C₂²×C₈×C₆₇	—	2	2	no	—
62	C₄⋊Q₈×C₆₇	C₂³	1	2	yes	—
63	#127×C₆₇	C₂³	1	2	yes	—
64	#123×C₆₇	—	4	2	no	—
65	C₂²⋊Q₈×C₆₇	—	2	2	no	—
66	#121×C₆₇	—	2	2	no	—
67	#121×C₆₇	—	2	2	no	—
68	#121×C₆₇	—	2	2	no	—
69	#120×C₆₇	—	4	2	no	—
70	C₄×D₄×C₆₇	C₂³	1	2	yes	—
71	C₄×D₄×C₆₇	C₂³	1	1	yes	characteristic
72	C₄×D₄×C₆₇	C₂³	1	1	yes	characteristic
73	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
74	#107×C₆₇	—	2	2	no	—
75	#107×C₆₇	D₄	1	1	yes	characteristic
76	#107×C₆₇	D₄	1	1	yes	characteristic
77	C₄⋊C₈×C₆₇	C₂³	1	1	yes	characteristic
78	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
79	C₄⋊C₈×C₆₇	C₂³	1	1	yes	characteristic
80	#103×C₆₇	—	2	2	no	—
81	#102×C₆₇	—	2	2	no	—
82	#102×C₆₇	—	2	2	no	—
83	#102×C₆₇	—	2	2	no	—
84	#102×C₆₇	—	2	2	no	—
85	#102×C₆₇	—	2	2	no	—
86	#102×C₆₇	—	4	2	no	—
87	#102×C₆₇	D₄	1	2	yes	—
88	#102×C₆₇	D₄	1	1	yes	characteristic
89	#102×C₆₇	D₄	1	1	yes	characteristic
90	C₂²⋊C₈×C₆₇	—	2	1	no	—
91	C₂²⋊C₈×C₆₇	—	2	2	no	—
92	C₂²⋊C₈×C₆₇	—	2	1	no	—
93	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
94	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
95	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
96	C₂⁴×C₆₇	—	4	2	no	—
97	#41×C₆₇	—	4	4	no	—
98	#41×C₆₇	—	8	2	no	—
99	C₂×Q₈×C₆₇	—	2	2	no	—
100	C₂×D₄×C₆₇	—	2	2	no	—
101	C₂×D₄×C₆₇	—	2	2	no	—
102	C₂×D₄×C₆₇	—	2	2	no	—
103	C₂×D₄×C₆₇	—	4	2	no	—
104	C₂×D₄×C₆₇	—	4	2	no	—
105	C₂×D₄×C₆₇	—	4	4	no	—
106	C₂×D₄×C₆₇	—	4	4	no	—
107	C₂×D₄×C₆₇	—	4	2	no	—
108	C₂×D₄×C₆₇	—	8	2	no	—
109	C₂×D₄×C₆₇	—	8	2	no	—
110	C₂×D₄×C₆₇	C₂×D₄	1	1	yes	characteristic
111	C₂×D₄×C₆₇	C₂×D₄	1	2	yes	—
112	C₂×D₄×C₆₇	C₂×D₄	1	1	yes	characteristic
113	C₂²×C₄×C₆₇	—	2	2	no	—
114	C₂²×C₄×C₆₇	—	2	1	no	—
115	C₂²×C₄×C₆₇	—	2	1	no	—
116	C₂²×C₄×C₆₇	—	4	2	no	—
117	Q₁₆×C₆₇	—	4	2	no	—
118	Q₁₆×C₆₇	—	8	1	no	—
119	SD₁₆×C₆₇	—	4	4	no	—
120	D₈×C₆₇	—	2	2	no	—
121	D₈×C₆₇	—	4	4	no	—
122	D₈×C₆₇	—	4	2	no	—
123	D₈×C₆₇	—	8	1	no	—
124	D₈×C₆₇	—	8	2	no	—
125	D₈×C₆₇	D₈	1	4	yes	complemented
126	C₂×C₈×C₆₇	—	2	2	no	—
127	C₂×C₈×C₆₇	—	2	1	no	—
128	C₂×C₈×C₆₇	—	2	1	no	—
129	C₂×C₈×C₆₇	—	2	1	no	—
130	C₂×C₈×C₆₇	—	4	2	no	—
131	C₂×C₈×C₆₇	—	4	2	no	—
132	C₂×C₈×C₆₇	—	8	2	no	—
133	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
134	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
135	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
136	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
137	C₄⋊C₄×C₆₇	—	2	2	no	—
138	C₄⋊C₄×C₆₇	—	2	2	no	—
139	C₄⋊C₄×C₆₇	—	4	2	no	—
140	C₄⋊C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
141	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
142	C₄⋊C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
143	C₂²⋊C₄×C₆₇	—	2	1	no	—
144	C₂²⋊C₄×C₆₇	—	2	2	no	—
145	C₂²⋊C₄×C₆₇	—	2	1	no	—
146	C₂²⋊C₄×C₆₇	—	4	2	no	—
147	C₂²⋊C₄×C₆₇	—	4	2	no	—
148	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
149	C₂³×C₆₇	—	2	2	no	—
150	C₂³×C₆₇	—	2	1	no	—
151	C₂³×C₆₇	—	2	1	no	—
152	C₂³×C₆₇	—	4	2	no	—
153	C₂³×C₆₇	—	8	4	no	—
154	C₂³×C₆₇	—	8	4	no	—
155	C₂³×C₆₇	—	16	2	no	—
156	Q₈×C₆₇	—	4	2	no	—
157	Q₈×C₆₇	—	4	2	no	—
158	D₄×C₆₇	—	2	1	no	—
159	D₄×C₆₇	—	2	1	no	—
160	D₄×C₆₇	—	2	2	no	—
161	D₄×C₆₇	—	2	1	no	—
162	D₄×C₆₇	—	4	2	no	—
163	D₄×C₆₇	—	4	2	no	—
164	D₄×C₆₇	—	4	2	no	—
165	D₄×C₆₇	—	4	2	no	—
166	D₄×C₆₇	—	8	2	no	—
167	D₄×C₆₇	—	8	4	no	—
168	D₄×C₆₇	—	8	4	no	—
169	D₄×C₆₇	—	8	4	no	—
170	D₄×C₆₇	C₂×D₈	1	4	yes	—
171	D₄×C₆₇	#135	1	2	yes	complemented
172	C₂×C₄×C₆₇	—	2	2	no	—
173	C₂×C₄×C₆₇	—	2	1	no	—
174	C₂×C₄×C₆₇	—	2	1	no	—
175	C₂×C₄×C₆₇	—	4	1	no	—
176	C₂×C₄×C₆₇	—	4	1	no	—
177	C₂×C₄×C₆₇	—	4	2	no	—
178	C₂×C₄×C₆₇	—	4	2	no	—
179	C₂×C₄×C₆₇	—	4	1	no	—
180	C₂×C₄×C₆₇	—	4	1	no	—
181	C₂×C₄×C₆₇	—	4	2	no	—
182	C₂×C₄×C₆₇	—	8	4	no	—
183	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
184	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
185	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
186	C₈×C₆₇	—	2	1	no	—
187	C₈×C₆₇	—	2	1	no	—
188	C₈×C₆₇	—	2	1	no	—
189	C₈×C₆₇	—	4	1	no	—
190	C₈×C₆₇	—	4	1	no	—
191	C₈×C₆₇	—	4	1	no	—
192	C₈×C₆₇	—	4	2	no	—
193	C₈×C₆₇	C₂×D₈	1	2	yes	—
194	C₂²×C₆₇	—	2	4	no	—
195	C₂²×C₆₇	—	2	2	no	—
196	C₂²×C₆₇	—	4	1	no	—
197	C₂²×C₆₇	—	4	2	no	—
198	C₂²×C₆₇	—	4	2	no	—
199	C₂²×C₆₇	—	4	1	no	—
200	C₂²×C₆₇	—	4	1	no	—
201	C₂²×C₆₇	—	4	1	no	—
202	C₂²×C₆₇	—	4	1	no	—
203	C₂²×C₆₇	—	8	2	no	—
204	C₂²×C₆₇	—	8	2	no	—
205	C₂²×C₆₇	—	8	2	no	—
206	C₂²×C₆₇	—	16	8	no	—
207	C₂²×C₆₇	D₄²	1	1	yes	characteristic
208	C₄×C₆₇	—	2	1	no	—
209	C₄×C₆₇	—	4	1	no	—
210	C₄×C₆₇	—	4	1	no	—
211	C₄×C₆₇	—	4	2	no	—
212	C₄×C₆₇	—	8	2	no	—
213	C₄×C₆₇	C₂²×D₈	1	1	yes	characteristic
214	C₄×C₆₇	C₂×#135	1	1	yes	characteristic
215	C₄×C₆₇	#576	1	1	yes	characteristic
216	C₄×C₆₇	#576	1	1	yes	characteristic
217	#19968	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
218	C₂×C₆₇	—	4	4	no	—
219	C₂×C₆₇	—	4	2	no	—
220	C₂×C₆₇	—	8	1	no	—
221	C₂×C₆₇	—	16	2	no	—
222	C₂×C₆₇	D₄×D₈	1	1	yes	characteristic
223	C₂×C₆₇	#3280	1	1	yes	characteristic
224	C₂×C₆₇	#3290	1	1	yes	characteristic
225	#3294	C₂×C₆₇	1	2	yes	complemented
226	#3283	C₂×C₆₇	1	1	yes	characteristic, complemented
227	D₄×D₈	C₂×C₆₇	1	1	yes	characteristic, complemented
228	#1709	C₂×C₆₇	1	1	yes	characteristic, complemented
229	#1681	C₂×C₆₇	1	2	yes	complemented
230	#1669	C₂×C₆₇	1	1	yes	characteristic, complemented
231	#1667	C₂×C₆₇	1	1	yes	characteristic, complemented
232	#1639	C₂×C₆₇	1	2	yes	complemented
233	#1625	C₂×C₆₇	1	2	yes	complemented
234	C₈×D₈	C₂×C₆₇	1	1	yes	characteristic, complemented
235	C₈×D₈	C₂×C₆₇	1	1	yes	characteristic, complemented
236	C₆₇	#19968	1	1	yes	characteristic, direct product factor, Sylow
237	C₂×#135	—	2	1	no	—
238	C₂²×D₈	—	2	1	no	—
239	#547	C₂²×C₆₇	1	2	yes	complemented
240	D₄²	C₂²×C₆₇	1	2	yes	—
241	#494	C₂²×C₆₇	1	1	yes	characteristic, complemented
242	#493	C₂²×C₆₇	1	1	yes	characteristic
243	#486	C₂²×C₆₇	1	2	yes	complemented
244	#486	C₂²×C₆₇	1	2	yes	complemented
245	#480	—	2	2	no	—
246	#474	C₂²×C₆₇	1	2	yes	—
247	#467	—	2	1	no	—
248	#466	—	2	1	no	—
249	#466	—	2	2	no	—
250	#461	C₂²×C₆₇	1	2	yes	complemented
251	#459	C₂²×C₆₇	1	2	yes	—
252	#459	C₂²×C₆₇	1	2	yes	complemented
253	#452	—	2	2	no	—
254	#447	—	2	2	no	—
255	#447	—	2	2	no	—
256	C₄×D₈	C₂²×C₆₇	1	1	yes	characteristic
257	C₄×D₈	C₂²×C₆₇	1	1	yes	characteristic
258	C₈×D₄	C₂²×C₆₇	1	1	yes	characteristic
259	C₈×D₄	C₂²×C₆₇	1	1	yes	characteristic, complemented
260	C₈×D₄	C₂²×C₆₇	1	2	yes	complemented
261	C₂×#102	—	2	2	no	—
262	#335	C₂²×C₆₇	1	1	yes	characteristic, complemented
263	#335	C₂²×C₆₇	1	1	yes	characteristic, complemented
264	#332	C₂²×C₆₇	1	2	yes	complemented
265	#331	C₂²×C₆₇	1	2	yes	complemented
266	D₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
267	D₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic, complemented
268	D₄⋊C₈	C₂²×C₆₇	1	2	yes	complemented
269	C₈²	C₂²×C₆₇	1	1	yes	characteristic, complemented
270	C₂×#41	—	2	2	no	—
271	C₂²×D₄	—	2	2	no	—
272	C₂²×D₄	—	2	2	no	—
273	#135	—	4	4	no	—
274	C₂×Q₁₆	—	2	1	no	—
275	C₂×Q₁₆	—	2	1	no	—
276	C₂×SD₁₆	—	2	2	no	—
277	C₂×D₈	—	2	2	no	—
278	C₂×D₈	—	2	2	no	—
279	C₂×D₈	—	2	1	no	—
280	C₂×D₈	—	2	1	no	—
281	C₂×D₈	—	4	4	no	—
282	C₂×D₈	D₄×C₆₇	1	1	yes	characteristic
283	C₂×D₈	D₄×C₆₇	1	1	yes	characteristic
284	C₂²×C₈	—	2	1	no	—
285	C₂²×C₈	—	2	1	no	—
286	C₂²×C₈	—	2	2	no	—
287	C₄⋊Q₈	C₂³×C₆₇	1	2	yes	—
288	#127	C₂³×C₆₇	1	2	yes	—
289	#123	—	4	2	no	—
290	C₂²⋊Q₈	—	2	2	no	—
291	#121	—	2	2	no	—
292	#121	—	2	2	no	—
293	#121	—	2	2	no	—
294	#120	—	4	2	no	—
295	C₄×D₄	C₂³×C₆₇	1	1	yes	characteristic
296	C₄×D₄	C₂³×C₆₇	1	2	yes	—
297	C₄×D₄	C₂³×C₆₇	1	1	yes	characteristic
298	C₂×C₄⋊C₄	—	2	2	no	—
299	#107	—	2	2	no	—
300	#107	D₄×C₆₇	1	1	yes	characteristic
301	#107	D₄×C₆₇	1	1	yes	characteristic
302	C₄⋊C₈	C₂³×C₆₇	1	1	yes	characteristic
303	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
304	C₄⋊C₈	C₂³×C₆₇	1	1	yes	characteristic
305	#103	—	2	2	no	—
306	#102	—	2	2	no	—
307	#102	—	2	2	no	—
308	#102	—	2	2	no	—
309	#102	—	2	2	no	—
310	#102	—	2	2	no	—
311	#102	—	4	2	no	—
312	#102	D₄×C₆₇	1	1	yes	characteristic
313	#102	D₄×C₆₇	1	2	yes	—
314	#102	D₄×C₆₇	1	1	yes	characteristic
315	C₂²⋊C₈	—	2	1	no	—
316	C₂²⋊C₈	—	2	2	no	—
317	C₂²⋊C₈	—	2	1	no	—
318	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
319	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
320	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
321	C₂⁴	—	4	2	no	—
322	#41	—	4	4	no	—
323	#41	—	8	2	no	—
324	C₂×Q₈	—	2	2	no	—
325	C₂×D₄	—	2	2	no	—
326	C₂×D₄	—	2	2	no	—
327	C₂×D₄	—	2	2	no	—
328	C₂×D₄	—	4	2	no	—
329	C₂×D₄	—	4	2	no	—
330	C₂×D₄	—	4	4	no	—
331	C₂×D₄	—	4	4	no	—
332	C₂×D₄	—	4	2	no	—
333	C₂×D₄	—	8	2	no	—
334	C₂×D₄	—	8	2	no	—
335	C₂×D₄	C₂×D₄×C₆₇	1	1	yes	characteristic
336	C₂×D₄	C₂×D₄×C₆₇	1	2	yes	—
337	C₂×D₄	C₂×D₄×C₆₇	1	1	yes	characteristic
338	C₂²×C₄	—	2	2	no	—
339	C₂²×C₄	—	2	1	no	—
340	C₂²×C₄	—	2	1	no	—
341	C₂²×C₄	—	4	2	no	—
342	Q₁₆	—	4	2	no	—
343	Q₁₆	—	8	1	no	—
344	SD₁₆	—	4	4	no	—
345	D₈	—	2	2	no	—
346	D₈	—	4	4	no	—
347	D₈	—	4	2	no	—
348	D₈	—	8	1	no	—
349	D₈	—	8	2	no	—
350	D₈	D₈×C₆₇	1	4	yes	complemented
351	C₂×C₈	—	2	2	no	—
352	C₂×C₈	—	2	1	no	—
353	C₂×C₈	—	2	1	no	—
354	C₂×C₈	—	2	1	no	—
355	C₂×C₈	—	4	2	no	—
356	C₂×C₈	—	4	2	no	—
357	C₂×C₈	—	8	2	no	—
358	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
359	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
360	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
361	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
362	C₄⋊C₄	—	2	2	no	—
363	C₄⋊C₄	—	2	2	no	—
364	C₄⋊C₄	—	4	2	no	—
365	C₄⋊C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
366	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
367	C₄⋊C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
368	C₂²⋊C₄	—	2	1	no	—
369	C₂²⋊C₄	—	2	2	no	—
370	C₂²⋊C₄	—	2	1	no	—
371	C₂²⋊C₄	—	4	2	no	—
372	C₂²⋊C₄	—	4	2	no	—
373	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
374	C₂³	—	2	2	no	—
375	C₂³	—	2	1	no	—
376	C₂³	—	2	1	no	—
377	C₂³	—	4	2	no	—
378	C₂³	—	8	4	no	—
379	C₂³	—	8	4	no	—
380	C₂³	—	16	2	no	—
381	Q₈	—	4	2	no	—
382	Q₈	—	4	2	no	—
383	D₄	—	2	1	no	—
384	D₄	—	2	1	no	—
385	D₄	—	2	2	no	—
386	D₄	—	2	1	no	—
387	D₄	—	4	2	no	—
388	D₄	—	4	2	no	—
389	D₄	—	4	2	no	—
390	D₄	—	4	2	no	—
391	D₄	—	8	2	no	—
392	D₄	—	8	4	no	—
393	D₄	—	8	4	no	—
394	D₄	—	8	4	no	—
395	D₄	C₂×D₈×C₆₇	1	4	yes	—
396	D₄	#135×C₆₇	1	2	yes	complemented
397	C₂×C₄	—	2	2	no	—
398	C₂×C₄	—	2	1	no	—
399	C₂×C₄	—	2	1	no	—
400	C₂×C₄	—	4	1	no	—
401	C₂×C₄	—	4	1	no	—
402	C₂×C₄	—	4	2	no	—
403	C₂×C₄	—	4	2	no	—
404	C₂×C₄	—	4	1	no	—
405	C₂×C₄	—	4	1	no	—
406	C₂×C₄	—	4	2	no	—
407	C₂×C₄	—	8	4	no	—
408	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
409	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
410	C₂×C₄	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
411	C₈	—	2	1	no	—
412	C₈	—	2	1	no	—
413	C₈	—	2	1	no	—
414	C₈	—	4	1	no	—
415	C₈	—	4	1	no	—
416	C₈	—	4	1	no	—
417	C₈	—	4	2	no	—
418	C₈	C₂×D₈×C₆₇	1	2	yes	—
419	C₂²	—	2	4	no	—
420	C₂²	—	2	2	no	—
421	C₂²	—	4	1	no	—
422	C₂²	—	4	2	no	—
423	C₂²	—	4	1	no	—
424	C₂²	—	4	2	no	—
425	C₂²	—	4	1	no	—
426	C₂²	—	4	1	no	—
427	C₂²	—	4	1	no	—
428	C₂²	—	8	2	no	—
429	C₂²	—	8	2	no	—
430	C₂²	—	8	2	no	—
431	C₂²	—	16	8	no	—
432	C₂²	D₄²×C₆₇	1	1	yes	characteristic
433	C₄	—	2	1	no	—
434	C₄	—	4	1	no	—
435	C₄	—	4	1	no	—
436	C₄	—	4	2	no	—
437	C₄	—	8	2	no	—
438	C₄	C₂²×D₈×C₆₇	1	1	yes	characteristic
439	C₄	C₂×#135×C₆₇	1	1	yes	characteristic
440	C₄	#576×C₆₇	1	1	yes	characteristic
441	C₄	#576×C₆₇	1	1	yes	characteristic
442	C₂	—	4	2	no	—
443	C₂	—	4	4	no	—
444	C₂	—	8	1	no	—
445	C₂	—	16	2	no	—
446	C₂	D₄×D₈×C₆₇	1	1	yes	characteristic
447	C₂	C₆₇×#3280	1	1	yes	characteristic
448	C₂	C₆₇×#3290	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₇	D₄²
commutator, Frattini, cosocle	C₄²	C₂⁴×C₆₇
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ C₄²×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₆₇×#3294	C₂	semidirect
2. C₆₇×#3283	C₂	semidirect
3. D₄×D₈×C₆₇	C₂	semidirect
4. C₆₇×#1709	C₂	semidirect
5. C₆₇×#1681	C₂	semidirect
6. C₆₇×#1669	C₂	semidirect
7. C₆₇×#1667	C₂	semidirect
8. C₆₇×#1639	C₂	semidirect
9. C₆₇×#1625	C₂	semidirect
10. C₈×D₈×C₆₇	C₂	semidirect
11. C₈×D₈×C₆₇	C₂	semidirect
14. #547×C₆₇	C₂²	semidirect
16. #494×C₆₇	C₂²	semidirect
18. #486×C₆₇	C₂²	semidirect
19. #486×C₆₇	C₂²	semidirect
25. #461×C₆₇	C₂²	semidirect
27. #459×C₆₇	C₂²	semidirect
34. C₈×D₄×C₆₇	C₂²	semidirect
35. C₈×D₄×C₆₇	C₂²	semidirect
37. #335×C₆₇	C₂²	semidirect
38. #335×C₆₇	C₂²	semidirect
39. #332×C₆₇	C₂²	semidirect
40. #331×C₆₇	C₂²	semidirect
42. D₄⋊C₈×C₆₇	C₂²	semidirect
43. D₄⋊C₈×C₆₇	C₂²	semidirect
44. C₈²×C₆₇	C₂²	semidirect
125. D₈×C₆₇	D₈	semidirect
171. D₄×C₆₇	#135	semidirect
217. #19968	C₆₇	direct (N × H)
225. #3294	C₂×C₆₇	semidirect
226. #3283	C₂×C₆₇	semidirect
227. D₄×D₈	C₂×C₆₇	semidirect
228. #1709	C₂×C₆₇	semidirect
229. #1681	C₂×C₆₇	semidirect
230. #1669	C₂×C₆₇	semidirect
231. #1667	C₂×C₆₇	semidirect
232. #1639	C₂×C₆₇	semidirect
233. #1625	C₂×C₆₇	semidirect
234. C₈×D₈	C₂×C₆₇	semidirect
235. C₈×D₈	C₂×C₆₇	semidirect
236. C₆₇	#19968	direct (N × H)
239. #547	C₂²×C₆₇	semidirect
241. #494	C₂²×C₆₇	semidirect
243. #486	C₂²×C₆₇	semidirect
244. #486	C₂²×C₆₇	semidirect
250. #461	C₂²×C₆₇	semidirect
252. #459	C₂²×C₆₇	semidirect
259. C₈×D₄	C₂²×C₆₇	semidirect
260. C₈×D₄	C₂²×C₆₇	semidirect
262. #335	C₂²×C₆₇	semidirect
263. #335	C₂²×C₆₇	semidirect
264. #332	C₂²×C₆₇	semidirect
265. #331	C₂²×C₆₇	semidirect
267. D₄⋊C₈	C₂²×C₆₇	semidirect
268. D₄⋊C₈	C₂²×C₆₇	semidirect
269. C₈²	C₂²×C₆₇	semidirect
350. D₈	D₈×C₆₇	semidirect
396. D₄	#135×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₆₇×#3294	2
C₆₇×#3283	2
D₄×D₈×C₆₇	2
C₆₇×#1709	2
C₆₇×#1681	2
C₆₇×#1669	2
C₆₇×#1667	2
C₆₇×#1639	2
C₆₇×#1625	2
C₈×D₈×C₆₇	2
#19968	67

Subgroups — C67×#19968 #30699219

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#19968 #30699219