Subgroups — C₂²×#452×C₆₇ #30739603

Order 17152.

5110 conjugacy classes of subgroups.

The subgroup lattice has 422 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
2	C₂³×Q₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂³×SD₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×C₂²⋊Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#452×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	—
13	C₂³×Q₈×C₆₇	C₂²	1	1	yes	characteristic
14	C₂³×Q₈×C₆₇	C₂²	1	1	yes	characteristic
15	C₂³×D₄×C₆₇	—	2	1	no	—
16	C₂³×D₄×C₆₇	C₂²	1	1	yes	characteristic
17	C₂⁴×C₄×C₆₇	—	2	1	no	—
18	C₂⁴×C₄×C₆₇	C₂²	1	1	yes	characteristic
19	C₂²×Q₁₆×C₆₇	—	2	8	no	—
20	C₂²×Q₁₆×C₆₇	C₂²	1	12	yes	complemented
21	C₂²×SD₁₆×C₆₇	—	2	8	no	—
22	C₂²×SD₁₆×C₆₇	C₂²	1	12	yes	complemented
23	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
24	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
25	C₂×C₂²⋊Q₈×C₆₇	—	2	6	no	—
26	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	12	yes	complemented
27	C₂²×C₄⋊C₄×C₆₇	—	2	1	no	—
28	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
29	C₂²×C₂²⋊C₄×C₆₇	—	2	1	no	—
30	#452×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	D₄	1	1	yes	characteristic
36	C₂×#41×C₆₇	—	2	96	no	—
37	C₂×#41×C₆₇	—	2	48	no	—
38	C₂×#41×C₆₇	—	2	48	no	—
39	C₂×#41×C₆₇	—	2	24	no	—
40	C₂×#41×C₆₇	C₂³	1	16	yes	—
41	C₂²×Q₈×C₆₇	—	2	4	no	—
42	C₂²×Q₈×C₆₇	—	2	3	no	—
43	C₂²×Q₈×C₆₇	—	2	3	no	—
44	C₂²×Q₈×C₆₇	—	2	4	no	—
45	C₂²×Q₈×C₆₇	—	2	4	no	—
46	C₂²×Q₈×C₆₇	D₄	1	8	yes	—
47	C₂²×Q₈×C₆₇	C₂³	1	6	yes	—
48	C₂²×Q₈×C₆₇	C₂³	1	6	yes	—
49	C₂²×D₄×C₆₇	—	2	16	no	—
50	C₂²×D₄×C₆₇	—	2	12	no	—
51	C₂²×D₄×C₆₇	—	2	3	no	—
52	C₂²×D₄×C₆₇	—	2	4	no	—
53	C₂²×D₄×C₆₇	D₄	1	8	yes	—
54	C₂²×D₄×C₆₇	C₂³	1	6	yes	—
55	C₂³×C₄×C₆₇	—	2	16	no	—
56	C₂³×C₄×C₆₇	—	2	1	no	—
57	C₂³×C₄×C₆₇	—	2	12	no	—
58	C₂³×C₄×C₆₇	—	2	4	no	—
59	C₂³×C₄×C₆₇	—	2	4	no	—
60	C₂³×C₄×C₆₇	—	2	1	no	—
61	C₂³×C₄×C₆₇	—	2	1	no	—
62	C₂³×C₄×C₆₇	D₄	1	6	yes	—
63	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
64	C₂³×C₄×C₆₇	C₂³	1	6	yes	—
65	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
66	C₂×Q₁₆×C₆₇	—	2	48	no	—
67	C₂×Q₁₆×C₆₇	C₂³	1	16	yes	complemented
68	C₂×SD₁₆×C₆₇	—	2	48	no	—
69	C₂×SD₁₆×C₆₇	C₂³	1	16	yes	complemented
70	C₂²×C₈×C₆₇	—	2	4	no	—
71	C₂²×C₈×C₆₇	—	2	4	no	—
72	C₂²×C₈×C₆₇	C₂³	1	6	yes	—
73	C₂²×C₈×C₆₇	C₂³	1	6	yes	—
74	C₂²⋊Q₈×C₆₇	—	2	24	no	—
75	C₂²⋊Q₈×C₆₇	C₂³	1	16	yes	complemented
76	C₂×C₄⋊C₄×C₆₇	—	2	12	no	—
77	C₂×C₄⋊C₄×C₆₇	—	2	3	no	—
78	C₂×C₄⋊C₄×C₆₇	C₂³	1	6	yes	—
79	C₂×C₂²⋊C₄×C₆₇	—	2	12	no	—
80	#103×C₆₇	C₂³	1	16	yes	complemented
81	#102×C₆₇	C₂³	1	16	yes	complemented
82	C₂²⋊C₈×C₆₇	C₂³	1	16	yes	complemented
83	C₂⁴×C₆₇	—	2	4	no	—
84	C₂⁴×C₆₇	—	2	8	no	—
85	C₂⁴×C₆₇	—	2	4	no	—
86	C₂⁴×C₆₇	—	2	4	no	—
87	C₂⁴×C₆₇	—	2	6	no	—
88	C₂⁴×C₆₇	—	4	4	no	—
89	C₂⁴×C₆₇	—	4	4	no	—
90	C₂⁴×C₆₇	C₂×D₄	1	6	yes	—
91	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
92	#41×C₆₇	—	2	192	no	—
93	#41×C₆₇	—	2	32	no	—
94	#41×C₆₇	—	2	32	no	—
95	C₂×Q₈×C₆₇	—	2	24	no	—
96	C₂×Q₈×C₆₇	—	2	12	no	—
97	C₂×Q₈×C₆₇	—	2	6	no	—
98	C₂×Q₈×C₆₇	—	2	6	no	—
99	C₂×Q₈×C₆₇	—	2	12	no	—
100	C₂×Q₈×C₆₇	—	2	12	no	—
101	C₂×Q₈×C₆₇	—	4	12	no	—
102	C₂×Q₈×C₆₇	C₂×D₄	1	24	yes	—
103	C₂×Q₈×C₆₇	C₂⁴	1	4	yes	—
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	24	no	—
108	C₂×D₄×C₆₇	—	2	6	no	—
109	C₂×D₄×C₆₇	—	2	12	no	—
110	C₂×D₄×C₆₇	C₂×D₄	1	24	yes	—
111	C₂×D₄×C₆₇	C₂⁴	1	4	yes	—
112	C₂²×C₄×C₆₇	—	2	96	no	—
113	C₂²×C₄×C₆₇	—	2	4	no	—
114	C₂²×C₄×C₆₇	—	2	8	no	—
115	C₂²×C₄×C₆₇	—	2	6	no	—
116	C₂²×C₄×C₆₇	—	2	24	no	—
117	C₂²×C₄×C₆₇	—	2	16	no	—
118	C₂²×C₄×C₆₇	—	2	12	no	—
119	C₂²×C₄×C₆₇	—	2	12	no	—
120	C₂²×C₄×C₆₇	—	2	6	no	—
121	C₂²×C₄×C₆₇	—	2	6	no	—
122	C₂²×C₄×C₆₇	—	4	4	no	—
123	C₂²×C₄×C₆₇	—	4	4	no	—
124	C₂²×C₄×C₆₇	C₂×D₄	1	6	yes	—
125	C₂²×C₄×C₆₇	C₂×D₄	1	12	yes	—
126	C₂²×C₄×C₆₇	C₂×D₄	1	3	yes	—
127	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
128	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
129	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
130	C₂²×C₄×C₆₇	C₂⁴	1	3	yes	—
131	Q₁₆×C₆₇	—	2	32	no	—
132	SD₁₆×C₆₇	—	2	32	no	—
133	C₂×C₈×C₆₇	—	2	12	no	—
134	C₂×C₈×C₆₇	—	2	12	no	—
135	C₂×C₈×C₆₇	C₂⁴	1	4	yes	—
136	C₂×C₈×C₆₇	C₂⁴	1	4	yes	—
137	C₄⋊C₄×C₆₇	—	2	16	no	—
138	C₄⋊C₄×C₆₇	—	2	6	no	—
139	C₄⋊C₄×C₆₇	C₂⁴	1	4	yes	—
140	C₂²⋊C₄×C₆₇	—	2	16	no	—
141	C₂³×C₆₇	—	2	12	no	—
142	C₂³×C₆₇	—	2	24	no	—
143	C₂³×C₆₇	—	2	12	no	—
144	C₂³×C₆₇	—	2	12	no	—
145	C₂³×C₆₇	—	2	4	no	—
146	C₂³×C₆₇	—	4	12	no	—
147	C₂³×C₆₇	—	4	12	no	—
148	C₂³×C₆₇	—	4	16	no	—
149	C₂³×C₆₇	—	4	32	no	—
150	C₂³×C₆₇	#120	1	4	yes	—
151	C₂³×C₆₇	#135	1	4	yes	—
152	C₂³×C₆₇	#137	1	4	yes	—
153	C₂³×C₆₇	C₂²×D₄	1	4	yes	—
154	C₂³×C₆₇	C₂²×D₄	1	3	yes	—
155	Q₈×C₆₇	—	2	24	no	—
156	Q₈×C₆₇	—	2	4	no	—
157	Q₈×C₆₇	—	2	4	no	—
158	Q₈×C₆₇	—	2	4	no	—
159	Q₈×C₆₇	—	4	12	no	—
160	Q₈×C₆₇	C₂²×D₄	1	8	yes	—
161	D₄×C₆₇	—	2	64	no	—
162	D₄×C₆₇	—	2	24	no	—
163	D₄×C₆₇	—	2	4	no	—
164	D₄×C₆₇	C₂²×D₄	1	8	yes	—
165	C₂×C₄×C₆₇	—	2	64	no	—
166	C₂×C₄×C₆₇	—	2	12	no	—
167	C₂×C₄×C₆₇	—	2	24	no	—
168	C₂×C₄×C₆₇	—	2	4	no	—
169	C₂×C₄×C₆₇	—	2	24	no	—
170	C₂×C₄×C₆₇	—	2	4	no	—
171	C₂×C₄×C₆₇	—	2	4	no	—
172	C₂×C₄×C₆₇	—	2	4	no	—
173	C₂×C₄×C₆₇	—	2	4	no	—
174	C₂×C₄×C₆₇	—	4	12	no	—
175	C₂×C₄×C₆₇	—	4	12	no	—
176	C₂×C₄×C₆₇	#120	1	12	yes	—
177	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
178	C₂×C₄×C₆₇	C₂²×D₄	1	3	yes	—
179	C₂×C₄×C₆₇	C₂²×D₄	1	6	yes	—
180	C₂×C₄×C₆₇	C₂²×D₄	1	6	yes	—
181	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
182	C₈×C₆₇	—	2	4	no	—
183	C₈×C₆₇	—	2	4	no	—
184	C₂²×C₆₇	—	2	4	no	—
185	C₂²×C₆₇	—	2	8	no	—
186	C₂²×C₆₇	—	2	4	no	—
187	C₂²×C₆₇	—	2	4	no	—
188	C₂²×C₆₇	—	4	24	no	—
189	C₂²×C₆₇	—	4	48	no	—
190	C₂²×C₆₇	—	4	4	no	—
191	C₂²×C₆₇	—	4	4	no	—
192	C₂²×C₆₇	#452	1	16	yes	direct product factor
193	C₂²×C₆₇	C₂×#120	1	6	yes	—
194	C₂²×C₆₇	C₂×#135	1	6	yes	—
195	C₂²×C₆₇	C₂×#137	1	6	yes	—
196	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
197	C₄×C₆₇	—	2	4	no	—
198	C₄×C₆₇	—	2	8	no	—
199	C₄×C₆₇	—	4	4	no	—
200	C₄×C₆₇	—	4	4	no	—
201	C₄×C₆₇	C₂×#120	1	6	yes	—
202	C₄×C₆₇	C₂³×D₄	1	1	yes	characteristic
203	C₄×C₆₇	C₂³×D₄	1	1	yes	characteristic
204	C₂²×#452	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
205	C₂×C₆₇	—	4	4	no	—
206	C₂×C₆₇	—	4	8	no	—
207	C₂×C₆₇	C₂×#452	1	12	yes	direct product factor
208	C₂×C₆₇	C₂²×#120	1	1	yes	characteristic
209	C₂×C₆₇	C₂²×#135	1	1	yes	characteristic
210	C₂×C₆₇	C₂²×#137	1	1	yes	characteristic
211	C₂³×#41	C₂×C₆₇	1	1	yes	characteristic
212	C₂³×Q₁₆	C₂×C₆₇	1	1	yes	characteristic, complemented
213	C₂³×SD₁₆	C₂×C₆₇	1	1	yes	characteristic, complemented
214	C₂²×C₂²⋊Q₈	C₂×C₆₇	1	1	yes	characteristic, complemented
215	C₂×#452	C₂×C₆₇	1	24	yes	direct product factor
216	C₂²×#103	C₂×C₆₇	1	1	yes	characteristic, complemented
217	C₂²×#102	C₂×C₆₇	1	1	yes	characteristic, complemented
218	C₂²×C₂²⋊C₈	C₂×C₆₇	1	1	yes	characteristic, complemented
219	C₆₇	C₂²×#452	1	1	yes	characteristic, direct product factor, Sylow
220	C₂²×#41	—	2	8	no	—
221	C₂²×#41	—	2	8	no	—
222	C₂²×#41	—	2	6	no	—
223	C₂²×#41	C₂²×C₆₇	1	12	yes	—
224	C₂³×Q₈	C₂²×C₆₇	1	1	yes	characteristic
225	C₂³×Q₈	C₂²×C₆₇	1	1	yes	characteristic
226	C₂³×D₄	—	2	1	no	—
227	C₂³×D₄	C₂²×C₆₇	1	1	yes	characteristic
228	C₂⁴×C₄	—	2	1	no	—
229	C₂⁴×C₄	C₂²×C₆₇	1	1	yes	characteristic
230	C₂²×Q₁₆	—	2	8	no	—
231	C₂²×Q₁₆	C₂²×C₆₇	1	12	yes	complemented
232	C₂²×SD₁₆	—	2	8	no	—
233	C₂²×SD₁₆	C₂²×C₆₇	1	12	yes	complemented
234	C₂³×C₈	C₂²×C₆₇	1	1	yes	characteristic
235	C₂³×C₈	C₂²×C₆₇	1	1	yes	characteristic
236	C₂×C₂²⋊Q₈	—	2	6	no	—
237	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	12	yes	complemented
238	C₂²×C₄⋊C₄	—	2	1	no	—
239	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
240	C₂²×C₂²⋊C₄	—	2	1	no	—
241	#452	C₂²×C₆₇	1	64	yes	direct product factor
242	C₂×#103	C₂²×C₆₇	1	12	yes	complemented
243	C₂×#102	C₂²×C₆₇	1	12	yes	complemented
244	C₂×C₂²⋊C₈	C₂²×C₆₇	1	12	yes	complemented
245	C₂⁵	—	2	1	no	—
246	C₂⁵	D₄×C₆₇	1	1	yes	characteristic
247	C₂×#41	—	2	96	no	—
248	C₂×#41	—	2	48	no	—
249	C₂×#41	—	2	48	no	—
250	C₂×#41	—	2	24	no	—
251	C₂×#41	C₂³×C₆₇	1	16	yes	—
252	C₂²×Q₈	—	2	4	no	—
253	C₂²×Q₈	—	2	3	no	—
254	C₂²×Q₈	—	2	3	no	—
255	C₂²×Q₈	—	2	4	no	—
256	C₂²×Q₈	—	2	4	no	—
257	C₂²×Q₈	D₄×C₆₇	1	8	yes	—
258	C₂²×Q₈	C₂³×C₆₇	1	6	yes	—
259	C₂²×Q₈	C₂³×C₆₇	1	6	yes	—
260	C₂²×D₄	—	2	16	no	—
261	C₂²×D₄	—	2	12	no	—
262	C₂²×D₄	—	2	3	no	—
263	C₂²×D₄	—	2	4	no	—
264	C₂²×D₄	D₄×C₆₇	1	8	yes	—
265	C₂²×D₄	C₂³×C₆₇	1	6	yes	—
266	C₂³×C₄	—	2	16	no	—
267	C₂³×C₄	—	2	1	no	—
268	C₂³×C₄	—	2	12	no	—
269	C₂³×C₄	—	2	4	no	—
270	C₂³×C₄	—	2	4	no	—
271	C₂³×C₄	—	2	1	no	—
272	C₂³×C₄	—	2	1	no	—
273	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
274	C₂³×C₄	D₄×C₆₇	1	6	yes	—
275	C₂³×C₄	C₂³×C₆₇	1	6	yes	—
276	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
277	C₂×Q₁₆	—	2	48	no	—
278	C₂×Q₁₆	C₂³×C₆₇	1	16	yes	complemented
279	C₂×SD₁₆	—	2	48	no	—
280	C₂×SD₁₆	C₂³×C₆₇	1	16	yes	complemented
281	C₂²×C₈	—	2	4	no	—
282	C₂²×C₈	—	2	4	no	—
283	C₂²×C₈	C₂³×C₆₇	1	6	yes	—
284	C₂²×C₈	C₂³×C₆₇	1	6	yes	—
285	C₂²⋊Q₈	—	2	24	no	—
286	C₂²⋊Q₈	C₂³×C₆₇	1	16	yes	complemented
287	C₂×C₄⋊C₄	—	2	12	no	—
288	C₂×C₄⋊C₄	—	2	3	no	—
289	C₂×C₄⋊C₄	C₂³×C₆₇	1	6	yes	—
290	C₂×C₂²⋊C₄	—	2	12	no	—
291	#103	C₂³×C₆₇	1	16	yes	complemented
292	#102	C₂³×C₆₇	1	16	yes	complemented
293	C₂²⋊C₈	C₂³×C₆₇	1	16	yes	complemented
294	C₂⁴	—	2	4	no	—
295	C₂⁴	—	2	8	no	—
296	C₂⁴	—	2	4	no	—
297	C₂⁴	—	2	4	no	—
298	C₂⁴	—	2	6	no	—
299	C₂⁴	—	4	4	no	—
300	C₂⁴	—	4	4	no	—
301	C₂⁴	C₂×D₄×C₆₇	1	6	yes	—
302	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
303	#41	—	2	192	no	—
304	#41	—	2	32	no	—
305	#41	—	2	32	no	—
306	C₂×Q₈	—	2	24	no	—
307	C₂×Q₈	—	2	12	no	—
308	C₂×Q₈	—	2	6	no	—
309	C₂×Q₈	—	2	6	no	—
310	C₂×Q₈	—	2	12	no	—
311	C₂×Q₈	—	2	12	no	—
312	C₂×Q₈	—	4	12	no	—
313	C₂×Q₈	C₂×D₄×C₆₇	1	24	yes	—
314	C₂×Q₈	C₂⁴×C₆₇	1	4	yes	—
315	C₂×Q₈	C₂⁴×C₆₇	1	4	yes	—
316	C₂×D₄	—	2	96	no	—
317	C₂×D₄	—	2	16	no	—
318	C₂×D₄	—	2	24	no	—
319	C₂×D₄	—	2	6	no	—
320	C₂×D₄	—	2	12	no	—
321	C₂×D₄	C₂×D₄×C₆₇	1	24	yes	—
322	C₂×D₄	C₂⁴×C₆₇	1	4	yes	—
323	C₂²×C₄	—	2	96	no	—
324	C₂²×C₄	—	2	4	no	—
325	C₂²×C₄	—	2	8	no	—
326	C₂²×C₄	—	2	6	no	—
327	C₂²×C₄	—	2	24	no	—
328	C₂²×C₄	—	2	16	no	—
329	C₂²×C₄	—	2	12	no	—
330	C₂²×C₄	—	2	12	no	—
331	C₂²×C₄	—	2	6	no	—
332	C₂²×C₄	—	2	6	no	—
333	C₂²×C₄	—	4	4	no	—
334	C₂²×C₄	—	4	4	no	—
335	C₂²×C₄	C₂×D₄×C₆₇	1	6	yes	—
336	C₂²×C₄	C₂×D₄×C₆₇	1	12	yes	—
337	C₂²×C₄	C₂×D₄×C₆₇	1	3	yes	—
338	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
339	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
340	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
341	C₂²×C₄	C₂⁴×C₆₇	1	3	yes	—
342	Q₁₆	—	2	32	no	—
343	SD₁₆	—	2	32	no	—
344	C₂×C₈	—	2	12	no	—
345	C₂×C₈	—	2	12	no	—
346	C₂×C₈	C₂⁴×C₆₇	1	4	yes	—
347	C₂×C₈	C₂⁴×C₆₇	1	4	yes	—
348	C₄⋊C₄	—	2	16	no	—
349	C₄⋊C₄	—	2	6	no	—
350	C₄⋊C₄	C₂⁴×C₆₇	1	4	yes	—
351	C₂²⋊C₄	—	2	16	no	—
352	C₂³	—	2	12	no	—
353	C₂³	—	2	24	no	—
354	C₂³	—	2	12	no	—
355	C₂³	—	2	12	no	—
356	C₂³	—	2	4	no	—
357	C₂³	—	4	16	no	—
358	C₂³	—	4	32	no	—
359	C₂³	—	4	12	no	—
360	C₂³	—	4	12	no	—
361	C₂³	#120×C₆₇	1	4	yes	—
362	C₂³	#135×C₆₇	1	4	yes	—
363	C₂³	#137×C₆₇	1	4	yes	—
364	C₂³	C₂²×D₄×C₆₇	1	4	yes	—
365	C₂³	C₂²×D₄×C₆₇	1	3	yes	—
366	Q₈	—	2	24	no	—
367	Q₈	—	2	4	no	—
368	Q₈	—	2	4	no	—
369	Q₈	—	2	4	no	—
370	Q₈	—	4	12	no	—
371	Q₈	C₂²×D₄×C₆₇	1	8	yes	—
372	D₄	—	2	64	no	—
373	D₄	—	2	24	no	—
374	D₄	—	2	4	no	—
375	D₄	C₂²×D₄×C₆₇	1	8	yes	—
376	C₂×C₄	—	2	64	no	—
377	C₂×C₄	—	2	12	no	—
378	C₂×C₄	—	2	24	no	—
379	C₂×C₄	—	2	4	no	—
380	C₂×C₄	—	2	24	no	—
381	C₂×C₄	—	2	4	no	—
382	C₂×C₄	—	2	4	no	—
383	C₂×C₄	—	2	4	no	—
384	C₂×C₄	—	2	4	no	—
385	C₂×C₄	—	4	12	no	—
386	C₂×C₄	—	4	12	no	—
387	C₂×C₄	#120×C₆₇	1	12	yes	—
388	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
389	C₂×C₄	C₂²×D₄×C₆₇	1	3	yes	—
390	C₂×C₄	C₂²×D₄×C₆₇	1	6	yes	—
391	C₂×C₄	C₂²×D₄×C₆₇	1	6	yes	—
392	C₂×C₄	C₂⁵×C₆₇	1	1	yes	characteristic
393	C₈	—	2	4	no	—
394	C₈	—	2	4	no	—
395	C₂²	—	2	4	no	—
396	C₂²	—	2	4	no	—
397	C₂²	—	2	4	no	—
398	C₂²	—	2	8	no	—
399	C₂²	—	4	4	no	—
400	C₂²	—	4	4	no	—
401	C₂²	—	4	24	no	—
402	C₂²	—	4	48	no	—
403	C₂²	#452×C₆₇	1	16	yes	direct product factor
404	C₂²	C₂×#120×C₆₇	1	6	yes	—
405	C₂²	C₂×#135×C₆₇	1	6	yes	—
406	C₂²	C₂×#137×C₆₇	1	6	yes	—
407	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
408	C₄	—	2	4	no	—
409	C₄	—	2	8	no	—
410	C₄	—	4	4	no	—
411	C₄	—	4	4	no	—
412	C₄	C₂×#120×C₆₇	1	6	yes	—
413	C₄	C₂³×D₄×C₆₇	1	1	yes	characteristic
414	C₄	C₂³×D₄×C₆₇	1	1	yes	characteristic
415	C₂	—	4	4	no	—
416	C₂	—	4	8	no	—
417	C₂	C₂×#452×C₆₇	1	12	yes	direct product factor
418	C₂	C₂²×#120×C₆₇	1	1	yes	characteristic
419	C₂	C₂²×#135×C₆₇	1	1	yes	characteristic
420	C₂	C₂²×#137×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂⁴×C₆₇	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₆₇
prime radical	C₂³×#41×C₆₇	C₂
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
2. C₂³×Q₁₆×C₆₇	C₂	semidirect
3. C₂³×SD₁₆×C₆₇	C₂	semidirect
4. C₂²×C₂²⋊Q₈×C₆₇	C₂	semidirect
5. C₂×#452×C₆₇	C₂	direct (N × H)
6. C₂²×#103×C₆₇	C₂	semidirect
7. C₂²×#102×C₆₇	C₂	semidirect
8. C₂²×C₂²⋊C₈×C₆₇	C₂	semidirect
20. C₂²×Q₁₆×C₆₇	C₂²	semidirect
22. C₂²×SD₁₆×C₆₇	C₂²	semidirect
26. C₂×C₂²⋊Q₈×C₆₇	C₂²	semidirect
30. #452×C₆₇	C₂²	direct (N × H)
31. C₂×#103×C₆₇	C₂²	semidirect
32. C₂×#102×C₆₇	C₂²	semidirect
33. C₂×C₂²⋊C₈×C₆₇	C₂²	semidirect
67. C₂×Q₁₆×C₆₇	C₂³	semidirect
69. C₂×SD₁₆×C₆₇	C₂³	semidirect
75. C₂²⋊Q₈×C₆₇	C₂³	semidirect
80. #103×C₆₇	C₂³	semidirect
81. #102×C₆₇	C₂³	semidirect
82. C₂²⋊C₈×C₆₇	C₂³	semidirect
192. C₂²×C₆₇	#452	direct (N × H)
204. C₂²×#452	C₆₇	direct (N × H)
207. C₂×C₆₇	C₂×#452	direct (N × H)
212. C₂³×Q₁₆	C₂×C₆₇	semidirect
213. C₂³×SD₁₆	C₂×C₆₇	semidirect
214. C₂²×C₂²⋊Q₈	C₂×C₆₇	semidirect
215. C₂×#452	C₂×C₆₇	direct (N × H)
216. C₂²×#103	C₂×C₆₇	semidirect
217. C₂²×#102	C₂×C₆₇	semidirect
218. C₂²×C₂²⋊C₈	C₂×C₆₇	semidirect
219. C₆₇	C₂²×#452	direct (N × H)
231. C₂²×Q₁₆	C₂²×C₆₇	semidirect
233. C₂²×SD₁₆	C₂²×C₆₇	semidirect
237. C₂×C₂²⋊Q₈	C₂²×C₆₇	semidirect
241. #452	C₂²×C₆₇	direct (N × H)
242. C₂×#103	C₂²×C₆₇	semidirect
243. C₂×#102	C₂²×C₆₇	semidirect
244. C₂×C₂²⋊C₈	C₂²×C₆₇	semidirect
278. C₂×Q₁₆	C₂³×C₆₇	semidirect
280. C₂×SD₁₆	C₂³×C₆₇	semidirect
286. C₂²⋊Q₈	C₂³×C₆₇	semidirect
291. #103	C₂³×C₆₇	semidirect
292. #102	C₂³×C₆₇	semidirect
293. C₂²⋊C₈	C₂³×C₆₇	semidirect
403. C₂²	#452×C₆₇	direct (N × H)
417. C₂	C₂×#452×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×Q₁₆×C₆₇	2
C₂³×SD₁₆×C₆₇	2
C₂²×C₂²⋊Q₈×C₆₇	2
C₂×#452×C₆₇	2
C₂²×#103×C₆₇	2
C₂²×#102×C₆₇	2
C₂²×C₂²⋊C₈×C₆₇	2
C₂²×#452	67

Subgroups — C22×#452×C67 #30739603

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#452×C₆₇ #30739603